From 504ab18cde656c449a21c8f317bdfc3aaeb8addb Mon Sep 17 00:00:00 2001 From: "github-actions[bot]" Date: Wed, 1 Jul 2020 06:35:56 +0000 Subject: [PATCH 1/8] Rebuild ode_extras/01-ModelingToolkit.jmd --- html/ode_extras/01-ModelingToolkit.html | 596 +++++++++--------- markdown/ode_extras/01-ModelingToolkit.md | 556 ++++++++-------- .../figures/01-ModelingToolkit_1_1.png | Bin 47469 -> 135924 bytes notebook/ode_extras/01-ModelingToolkit.ipynb | 8 +- pdf/ode_extras/01-ModelingToolkit.pdf | Bin 165311 -> 170344 bytes tutorials/ode_extras/Project.toml | 4 +- 6 files changed, 565 insertions(+), 599 deletions(-) diff --git a/html/ode_extras/01-ModelingToolkit.html b/html/ode_extras/01-ModelingToolkit.html index f024aa5a..5bb5bf61 100644 --- a/html/ode_extras/01-ModelingToolkit.html +++ b/html/ode_extras/01-ModelingToolkit.html @@ -650,22 +650,19 @@ div.title {text-align: center;} + + +
+
+
+
+

ModelingToolkit.jl, An IR and Compiler for Scientific Models

+
Chris Rackauckas
+ +
- - - -
-
-
- -
-

ModelingToolkit.jl, An IR and Compiler for Scientific Models

-
Chris Rackauckas
- -
- -

A lot of people are building modeling languages for their specific domains. However, while the syntax my vary greatly between these domain-specific languages (DSLs), the internals of modeling frameworks are surprisingly similar: building differential equations, calculating Jacobians, etc.

+

A lot of people are building modeling languages for their specific domains. However, while the syntax my vary greatly between these domain-specific languages (DSLs), the internals of modeling frameworks are surprisingly similar: building differential equations, calculating Jacobians, etc.

ModelingToolkit.jl is metamodeling systemitized

After building our third modeling interface, we realized that this problem can be better approached by having a reusable internal structure which DSLs can target. This internal is ModelingToolkit.jl: an Intermediate Representation (IR) with a well-defined interface for defining system transformations and compiling to Julia functions for use in numerical libraries. Now a DSL can easily be written by simply defining the translation to ModelingToolkit.jl's primatives and querying for the mathematical quantities one needs.

Basic usage: defining differential equation systems, with performance!

@@ -701,7 +698,7 @@

Basic usage: defining differential equation systems, with performance! - +

ModelingToolkit is a compiler for mathematical systems

At its core, ModelingToolkit is a compiler. It's IR is its type system, and its output are Julia functions (it's a compiler for Julia code to Julia code, written in Julia).

@@ -714,41 +711,34 @@

ModelingToolkit is a compiler for mathematical systems

-:((u, p, t)->begin
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:61 =#
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:62 =#
-          if u isa Array
-              #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:63 =#
-              return #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:55 =# @inbounds(begin
-                          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:55 =#
-                          let (x, y, z, σ, ρ, β) = (u[1], u[2], u[3], p[1], p[2], p[3])
-                              [σ * (y - x), x * (ρ - z) - y, x * y - β * z]
+:((var"##MTKArg#262", var"##MTKArg#263", var"##MTKArg#264")->begin
+          if var"##MTKArg#262" isa Array || !(typeof(var"##MTKArg#262") <: StaticArray) && false
+              return @inbounds(begin
+                          let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#262"[1], var"##MTKArg#262"[2], var"##MTKArg#262"[3], var"##MTK
+Arg#263"[1], var"##MTKArg#263"[2], var"##MTKArg#263"[3], var"##MTKArg#264")
+                              [(getproperty(Base, :*))(σ, (getproperty(Base, :-))(y, x)), (getproperty(Base, :-))((getproperty(Bas
+e, :*))(x, (getproperty(Base, :-))(ρ, z)), y), (getproperty(Base, :-))((getproperty(Base, :*))(x, y), (getproperty(Base, :*))(β, z
+))]
                           end
                       end)
           else
-              #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:65 =#
-              X = #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:54 =# @inbounds(begin
-                          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:54 =#
-                          let (x, y, z, σ, ρ, β) = (u[1], u[2], u[3], p[1], p[2], p[3])
-                              (σ * (y - x), x * (ρ - z) - y, x * y - β * z)
+              X = @inbounds(begin
+                          let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#262"[1], var"##MTKArg#262"[2], var"##MTKArg#262"[3], var"##MTK
+Arg#263"[1], var"##MTKArg#263"[2], var"##MTKArg#263"[3], var"##MTKArg#264")
+                              ((getproperty(Base, :*))(σ, (getproperty(Base, :-))(y, x)), (getproperty(Base, :-))((getproperty(Bas
+e, :*))(x, (getproperty(Base, :-))(ρ, z)), y), (getproperty(Base, :-))((getproperty(Base, :*))(x, y), (getproperty(Base, :*))(β, z
+)))
                           end
                       end)
+              construct = if var"##MTKArg#262" isa ModelingToolkit.StaticArrays.StaticArray
+                      (getproperty(ModelingToolkit.StaticArrays, :similar_type))(typeof(var"##MTKArg#262"), eltype(X))
+                  else
+                      x->begin
+                              convert(typeof(var"##MTKArg#262"), x)
+                          end
+                  end
+              return construct(X)
           end
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:67 =#
-          T = promote_type(map(typeof, X)...)
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:68 =#
-          map(T, X)
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:69 =#
-          construct = if u isa ModelingToolkit.StaticArrays.StaticArray
-                  ModelingToolkit.StaticArrays.similar_type(typeof(u), eltype(X))
-              else
-                  x->begin
-                          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:69 =#
-                          convert(typeof(u), x)
-                      end
-              end
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:70 =#
-          construct(X)
       end)
@@ -762,21 +752,17 @@

ModelingToolkit is a compiler for mathematical systems

-:((var"##MTIIPVar#1252", u, p, t)->begin
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:75 =#
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:76 =#
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:76 =# @inbounds begin
-                  #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:77 =#
-                  #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:56 =# @inbounds begin
-                          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:56 =#
-                          let (x, y, z, σ, ρ, β) = (u[1], u[2], u[3], p[1], p[2], p[3])
-                              var"##MTIIPVar#1252"[1] = σ * (y - x)
-                              var"##MTIIPVar#1252"[2] = x * (ρ - z) - y
-                              var"##MTIIPVar#1252"[3] = x * y - β * z
-                          end
-                      end
+:((var"##MTIIPVar#272", var"##MTKArg#268", var"##MTKArg#269", var"##MTKArg#270")->begin
+          @inbounds begin
+                  let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#268"[1], var"##MTKArg#268"[2], var"##MTKArg#268"[3], var"##MTKArg#269"
+[1], var"##MTKArg#269"[2], var"##MTKArg#269"[3], var"##MTKArg#270")
+                      var"##MTIIPVar#272"[1] = (getproperty(Base, :*))(σ, (getproperty(Base, :-))(y, x))
+                      var"##MTIIPVar#272"[2] = (getproperty(Base, :-))((getproperty(Base, :*))(x, (getproperty(Base, :-))(ρ, z)), 
+y)
+                      var"##MTIIPVar#272"[3] = (getproperty(Base, :-))((getproperty(Base, :*))(x, y), (getproperty(Base, :*))(β, z
+))
+                  end
               end
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:79 =#
           nothing
       end)
@@ -791,10 +777,10 @@

ModelingToolkit is a compiler for mathematical systems

-3×3 Array{Expression,2}:
-   σ * -1             σ  Constant(0)
- ρ - z(t)  Constant(-1)    x(t) * -1
-     y(t)          x(t)          -1β
+3×3 Array{ModelingToolkit.Expression,2}:
+           -1σ             σ  Constant(0)
+ -1 * z(t) + ρ  Constant(-1)    -1 * x(t)
+          y(t)          x(t)          -1β
@@ -807,62 +793,48 @@

ModelingToolkit is a compiler for mathematical systems

-(:((u, p, t)->begin
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:61 =#
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:62 =#
-          if u isa Array
-              #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:63 =#
-              return #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:55 =# @inbounds(begin
-                          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:55 =#
-                          let (x, y, z, σ, ρ, β) = (u[1], u[2], u[3], p[1], p[2], p[3])
-                              [σ * -1, ρ - z, y, σ, -1, x, 0, x * -1, -1β]
+(:((var"##MTKArg#274", var"##MTKArg#275", var"##MTKArg#276")->begin
+          if var"##MTKArg#274" isa Array || !(typeof(var"##MTKArg#274") <: StaticArray) && false
+              return @inbounds(begin
+                          let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#274"[1], var"##MTKArg#274"[2], var"##MTKArg#274"[3], var"##MTK
+Arg#275"[1], var"##MTKArg#275"[2], var"##MTKArg#275"[3], var"##MTKArg#276")
+                              [(getproperty(Base, :*))(-1, σ) σ 0; (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ) -1 (
+getproperty(Base, :*))(-1, x); y x (getproperty(Base, :*))(-1, β)]
                           end
                       end)
           else
-              #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:65 =#
-              X = #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:54 =# @inbounds(begin
-                          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:54 =#
-                          let (x, y, z, σ, ρ, β) = (u[1], u[2], u[3], p[1], p[2], p[3])
-                              (σ * -1, ρ - z, y, σ, -1, x, 0, x * -1, -1β)
+              X = @inbounds(begin
+                          let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#274"[1], var"##MTKArg#274"[2], var"##MTKArg#274"[3], var"##MTK
+Arg#275"[1], var"##MTKArg#275"[2], var"##MTKArg#275"[3], var"##MTKArg#276")
+                              ((getproperty(Base, :*))(-1, σ), (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ), y, σ, -
+1, x, 0, (getproperty(Base, :*))(-1, x), (getproperty(Base, :*))(-1, β))
                           end
                       end)
-          end
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:67 =#
-          T = promote_type(map(typeof, X)...)
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:68 =#
-          map(T, X)
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:69 =#
-          construct = if u isa ModelingToolkit.StaticArrays.StaticArray
-                  ModelingToolkit.StaticArrays.similar_type(typeof(u), eltype(X))
-              else
-                  x->begin
-                          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:69 =#
-                          convert(typeof(u), x)
-                      end
-              end
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:70 =#
-          construct(X)
-      end), :((var"##MTIIPVar#1254", u, p, t)->begin
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:75 =#
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:76 =#
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:76 =# @inbounds begin
-                  #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:77 =#
-                  #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:56 =# @inbounds begin
-                          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:56 =#
-                          let (x, y, z, σ, ρ, β) = (u[1], u[2], u[3], p[1], p[2], p[3])
-                              var"##MTIIPVar#1254"[1] = σ * -1
-                              var"##MTIIPVar#1254"[2] = ρ - z
-                              var"##MTIIPVar#1254"[3] = y
-                              var"##MTIIPVar#1254"[4] = σ
-                              var"##MTIIPVar#1254"[5] = -1
-                              var"##MTIIPVar#1254"[6] = x
-                              var"##MTIIPVar#1254"[7] = 0
-                              var"##MTIIPVar#1254"[8] = x * -1
-                              var"##MTIIPVar#1254"[9] = -1β
+              construct = if var"##MTKArg#274" isa ModelingToolkit.StaticArrays.StaticArray
+                      ModelingToolkit.StaticArrays.SMatrix{3, 3}
+                  else
+                      x->begin
+                              out = similar(typeof(var"##MTKArg#274"), 3, 3)
+                              out .= x
                           end
-                      end
+                  end
+              return construct(X)
+          end
+      end), :((var"##MTIIPVar#278", var"##MTKArg#274", var"##MTKArg#275", var"##MTKArg#276")->begin
+          @inbounds begin
+                  let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#274"[1], var"##MTKArg#274"[2], var"##MTKArg#274"[3], var"##MTKArg#275"
+[1], var"##MTKArg#275"[2], var"##MTKArg#275"[3], var"##MTKArg#276")
+                      var"##MTIIPVar#278"[1] = (getproperty(Base, :*))(-1, σ)
+                      var"##MTIIPVar#278"[2] = (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ)
+                      var"##MTIIPVar#278"[3] = y
+                      var"##MTIIPVar#278"[4] = σ
+                      var"##MTIIPVar#278"[5] = -1
+                      var"##MTIIPVar#278"[6] = x
+                      var"##MTIIPVar#278"[7] = 0
+                      var"##MTIIPVar#278"[8] = (getproperty(Base, :*))(-1, x)
+                      var"##MTIIPVar#278"[9] = (getproperty(Base, :*))(-1, β)
+                  end
               end
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:79 =#
           nothing
       end))
@@ -877,76 +849,88 @@

ModelingToolkit is a compiler for mathematical systems

-(:((u, p, gam, t)->begin
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:61 =#
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:62 =#
-          if u isa Array
-              #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:63 =#
-              return #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:55 =# @inbounds(begin
-                          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:55 =#
-                          let (x, y, z, σ, ρ, β) = (u[1], u[2], u[3], p[1], p[2], p[3])
-                              [σ * -1 * gam + -1, gam * (ρ - z) * inv(σ * -1 * gam + -1), gam * y * inv(σ * -1 * gam + -1), gam * 
-σ, (gam * -1 + -1) - gam * (ρ - z) * inv(σ * -1 * gam + -1) * gam * σ, (gam * x - gam * y * inv(σ * -1 * gam + -1) * gam * σ) * in
-v((gam * -1 + -1) - gam * (ρ - z) * inv(σ * -1 * gam + -1) * gam * σ), 0, x * -1 * gam - 0, ((-1 * β * gam + -1) - 0) - (gam * x -
- gam * y * inv(σ * -1 * gam + -1) * gam * σ) * inv((gam * -1 + -1) - gam * (ρ - z) * inv(σ * -1 * gam + -1) * gam * σ) * (x * -1 *
- gam - 0)]
+(:((var"##MTKArg#280", var"##MTKArg#281", var"##MTKArg#282", var"##MTKArg#283")->begin
+          if var"##MTKArg#280" isa Array || !(typeof(var"##MTKArg#280") <: StaticArray) && false
+              return @inbounds(begin
+                          let (x, y, z, σ, ρ, β, __MTKWgamma, t) = (var"##MTKArg#280"[1], var"##MTKArg#280"[2], var"##MTKArg#280"[
+3], var"##MTKArg#281"[1], var"##MTKArg#281"[2], var"##MTKArg#281"[3], var"##MTKArg#282", var"##MTKArg#283")
+                              [(getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ)) (getproperty(Base, :*))(__
+MTKWgamma, σ) 0; (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWg
+amma, σ))), __MTKWgamma, (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ)) (getproperty(Base, :*))(-1, (getproperty(Base
+, :+))(1, (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ
+))), __MTKWgamma ^ 2, σ, (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ)), __MTKWgamma)) (getproperty(Base, :*))(-1, x,
+ __MTKWgamma); (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgam
+ma, σ))), y, __MTKWgamma) (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :*))(-1, (getproperty(Base, :+))(1,
+ (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ))), __MT
+KWgamma ^ 2, σ, (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ)), __MTKWgamma))), (getproperty(Base, :+))((getproperty(
+Base, :*))(x, __MTKWgamma), (getproperty(Base, :*))(-1, (getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :
+*))(-1, __MTKWgamma, σ))), y, __MTKWgamma ^ 2, σ))) (getproperty(Base, :+))((getproperty(Base, :*))(-1, (getproperty(Base, :+))(1,
+ (getproperty(Base, :*))(-1, (getproperty(Base, :inv))((getproperty(Base, :*))(-1, (getproperty(Base, :+))(1, (getproperty(Base, :
+*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ))), __MTKWgamma ^ 2, σ, (getp
+roperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ)), __MTKWgamma))), x, __MTKWgamma, (getproperty(Base, :+))((getproperty(Base,
+ :*))(x, __MTKWgamma), (getproperty(Base, :*))(-1, (getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-
+1, __MTKWgamma, σ))), y, __MTKWgamma ^ 2, σ))))), (getproperty(Base, :*))(-1, __MTKWgamma, β))]
                           end
                       end)
           else
-              #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:65 =#
-              X = #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:54 =# @inbounds(begin
-                          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:54 =#
-                          let (x, y, z, σ, ρ, β) = (u[1], u[2], u[3], p[1], p[2], p[3])
-                              (σ * -1 * gam + -1, gam * (ρ - z) * inv(σ * -1 * gam + -1), gam * y * inv(σ * -1 * gam + -1), gam * 
-σ, (gam * -1 + -1) - gam * (ρ - z) * inv(σ * -1 * gam + -1) * gam * σ, (gam * x - gam * y * inv(σ * -1 * gam + -1) * gam * σ) * in
-v((gam * -1 + -1) - gam * (ρ - z) * inv(σ * -1 * gam + -1) * gam * σ), 0, x * -1 * gam - 0, ((-1 * β * gam + -1) - 0) - (gam * x -
- gam * y * inv(σ * -1 * gam + -1) * gam * σ) * inv((gam * -1 + -1) - gam * (ρ - z) * inv(σ * -1 * gam + -1) * gam * σ) * (x * -1 *
- gam - 0))
+              X = @inbounds(begin
+                          let (x, y, z, σ, ρ, β, __MTKWgamma, t) = (var"##MTKArg#280"[1], var"##MTKArg#280"[2], var"##MTKArg#280"[
+3], var"##MTKArg#281"[1], var"##MTKArg#281"[2], var"##MTKArg#281"[3], var"##MTKArg#282", var"##MTKArg#283")
+                              ((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ)), (getproperty(Base, :*))((
+getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ))), __MTKWgamma, (getproperty(Base
+, :+))((getproperty(Base, :*))(-1, z), ρ)), (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getprop
+erty(Base, :*))(-1, __MTKWgamma, σ))), y, __MTKWgamma), (getproperty(Base, :*))(__MTKWgamma, σ), (getproperty(Base, :*))(-1, (getp
+roperty(Base, :+))(1, (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __
+MTKWgamma, σ))), __MTKWgamma ^ 2, σ, (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ)), __MTKWgamma)), (getproperty(Base
+, :*))((getproperty(Base, :inv))((getproperty(Base, :*))(-1, (getproperty(Base, :+))(1, (getproperty(Base, :*))((getproperty(Base,
+ :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ))), __MTKWgamma ^ 2, σ, (getproperty(Base, :+))((ge
+tproperty(Base, :*))(-1, z), ρ)), __MTKWgamma))), (getproperty(Base, :+))((getproperty(Base, :*))(x, __MTKWgamma), (getproperty(Ba
+se, :*))(-1, (getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ))), y, __MTKWgamma ^
+ 2, σ))), 0, (getproperty(Base, :*))(-1, x, __MTKWgamma), (getproperty(Base, :+))((getproperty(Base, :*))(-1, (getproperty(Base, :
++))(1, (getproperty(Base, :*))(-1, (getproperty(Base, :inv))((getproperty(Base, :*))(-1, (getproperty(Base, :+))(1, (getproperty(B
+ase, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ))), __MTKWgamma ^ 2, σ,
+ (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ)), __MTKWgamma))), x, __MTKWgamma, (getproperty(Base, :+))((getproperty
+(Base, :*))(x, __MTKWgamma), (getproperty(Base, :*))(-1, (getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, 
+:*))(-1, __MTKWgamma, σ))), y, __MTKWgamma ^ 2, σ))))), (getproperty(Base, :*))(-1, __MTKWgamma, β)))
                           end
                       end)
+              construct = (x->begin
+                          A = SMatrix{(3, 3)...}(x)
+                          (getproperty(StaticArrays, :LU))(LowerTriangular(SMatrix{(3, 3)...}(UnitLowerTriangular(A))), UpperTrian
+gular(A), SVector(ntuple((n->begin
+                                              n
+                                          end), max((3, 3)...))))
+                      end)
+              return construct(X)
           end
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:67 =#
-          T = promote_type(map(typeof, X)...)
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:68 =#
-          map(T, X)
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:69 =#
-          construct = (x->begin
-                      #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\systems\diffeqs\diffeqsystem.jl:202 =#
-                      #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\systems\diffeqs\diffeqsystem.jl:203 =#
-                      A = SMatrix{(3, 3)...}(x)
-                      #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\systems\diffeqs\diffeqsystem.jl:204 =#
-                      StaticArrays.LU(LowerTriangular(SMatrix{(3, 3)...}(UnitLowerTriangular(A))), UpperTriangular(A), SVector(ntu
-ple((n->begin
-                                          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\systems\diffeqs\diffeqsystem
-.jl:204 =#
-                                          n
-                                      end), max((3, 3)...))))
-                  end)
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:70 =#
-          construct(X)
-      end), :((var"##MTIIPVar#1256", u, p, gam, t)->begin
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:75 =#
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:76 =#
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:76 =# @inbounds begin
-                  #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:77 =#
-                  #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:56 =# @inbounds begin
-                          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:56 =#
-                          let (x, y, z, σ, ρ, β) = (u[1], u[2], u[3], p[1], p[2], p[3])
-                              var"##MTIIPVar#1256"[1] = σ * -1 * gam + -1
-                              var"##MTIIPVar#1256"[2] = gam * (ρ - z) * inv(σ * -1 * gam + -1)
-                              var"##MTIIPVar#1256"[3] = gam * y * inv(σ * -1 * gam + -1)
-                              var"##MTIIPVar#1256"[4] = gam * σ
-                              var"##MTIIPVar#1256"[5] = (gam * -1 + -1) - gam * (ρ - z) * inv(σ * -1 * gam + -1) * gam * σ
-                              var"##MTIIPVar#1256"[6] = (gam * x - gam * y * inv(σ * -1 * gam + -1) * gam * σ) * inv((gam * -1 + -
-1) - gam * (ρ - z) * inv(σ * -1 * gam + -1) * gam * σ)
-                              var"##MTIIPVar#1256"[7] = 0
-                              var"##MTIIPVar#1256"[8] = x * -1 * gam - 0
-                              var"##MTIIPVar#1256"[9] = ((-1 * β * gam + -1) - 0) - (gam * x - gam * y * inv(σ * -1 * gam + -1) * 
-gam * σ) * inv((gam * -1 + -1) - gam * (ρ - z) * inv(σ * -1 * gam + -1) * gam * σ) * (x * -1 * gam - 0)
-                          end
-                      end
+      end), :((var"##MTIIPVar#285", var"##MTKArg#280", var"##MTKArg#281", var"##MTKArg#282", var"##MTKArg#283")->begin
+          @inbounds begin
+                  let (x, y, z, σ, ρ, β, __MTKWgamma, t) = (var"##MTKArg#280"[1], var"##MTKArg#280"[2], var"##MTKArg#280"[3], var"
+##MTKArg#281"[1], var"##MTKArg#281"[2], var"##MTKArg#281"[3], var"##MTKArg#282", var"##MTKArg#283")
+                      var"##MTIIPVar#285"[1] = (getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ))
+                      var"##MTIIPVar#285"[2] = (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getp
+roperty(Base, :*))(-1, __MTKWgamma, σ))), __MTKWgamma, (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ))
+                      var"##MTIIPVar#285"[3] = (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getp
+roperty(Base, :*))(-1, __MTKWgamma, σ))), y, __MTKWgamma)
+                      var"##MTIIPVar#285"[4] = (getproperty(Base, :*))(__MTKWgamma, σ)
+                      var"##MTIIPVar#285"[5] = (getproperty(Base, :*))(-1, (getproperty(Base, :+))(1, (getproperty(Base, :*))((get
+property(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ))), __MTKWgamma ^ 2, σ, (getproperty(
+Base, :+))((getproperty(Base, :*))(-1, z), ρ)), __MTKWgamma))
+                      var"##MTIIPVar#285"[6] = (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :*))(-1, (getp
+roperty(Base, :+))(1, (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __
+MTKWgamma, σ))), __MTKWgamma ^ 2, σ, (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ)), __MTKWgamma))), (getproperty(Bas
+e, :+))((getproperty(Base, :*))(x, __MTKWgamma), (getproperty(Base, :*))(-1, (getproperty(Base, :inv))((getproperty(Base, :+))(-1,
+ (getproperty(Base, :*))(-1, __MTKWgamma, σ))), y, __MTKWgamma ^ 2, σ)))
+                      var"##MTIIPVar#285"[7] = 0
+                      var"##MTIIPVar#285"[8] = (getproperty(Base, :*))(-1, x, __MTKWgamma)
+                      var"##MTIIPVar#285"[9] = (getproperty(Base, :+))((getproperty(Base, :*))(-1, (getproperty(Base, :+))(1, (get
+property(Base, :*))(-1, (getproperty(Base, :inv))((getproperty(Base, :*))(-1, (getproperty(Base, :+))(1, (getproperty(Base, :*))((
+getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ))), __MTKWgamma ^ 2, σ, (getproper
+ty(Base, :+))((getproperty(Base, :*))(-1, z), ρ)), __MTKWgamma))), x, __MTKWgamma, (getproperty(Base, :+))((getproperty(Base, :*))
+(x, __MTKWgamma), (getproperty(Base, :*))(-1, (getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __
+MTKWgamma, σ))), y, __MTKWgamma ^ 2, σ))))), (getproperty(Base, :*))(-1, __MTKWgamma, β))
+                  end
               end
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:79 =#
           nothing
       end))
@@ -970,56 +954,39 @@

Solving Nonlinear systems

-(:((u, p)->begin
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:61 =#
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:62 =#
-          if u isa Array
-              #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:63 =#
-              return #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:55 =# @inbounds(begin
-                          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:55 =#
-                          let (x, y, z, σ, ρ, β) = (u[1], u[2], u[3], p[1], p[2], p[3])
+(:((var"##MTKArg#295", var"##MTKArg#296")->begin
+          if var"##MTKArg#295" isa Array || !(typeof(var"##MTKArg#295") <: StaticArray) && false
+              return @inbounds(begin
+                          let (x, y, z, σ, ρ, β) = (var"##MTKArg#295"[1], var"##MTKArg#295"[2], var"##MTKArg#295"[3], var"##MTKArg
+#296"[1], var"##MTKArg#296"[2], var"##MTKArg#296"[3])
                               [(*)(σ, (-)(y, x)), (-)((*)(x, (-)(ρ, z)), y), (-)((*)(x, y), (*)(β, z))]
                           end
                       end)
           else
-              #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:65 =#
-              X = #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:54 =# @inbounds(begin
-                          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:54 =#
-                          let (x, y, z, σ, ρ, β) = (u[1], u[2], u[3], p[1], p[2], p[3])
+              X = @inbounds(begin
+                          let (x, y, z, σ, ρ, β) = (var"##MTKArg#295"[1], var"##MTKArg#295"[2], var"##MTKArg#295"[3], var"##MTKArg
+#296"[1], var"##MTKArg#296"[2], var"##MTKArg#296"[3])
                               ((*)(σ, (-)(y, x)), (-)((*)(x, (-)(ρ, z)), y), (-)((*)(x, y), (*)(β, z)))
                           end
                       end)
-          end
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:67 =#
-          T = promote_type(map(typeof, X)...)
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:68 =#
-          map(T, X)
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:69 =#
-          construct = if u isa ModelingToolkit.StaticArrays.StaticArray
-                  ModelingToolkit.StaticArrays.similar_type(typeof(u), eltype(X))
-              else
-                  x->begin
-                          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:69 =#
-                          convert(typeof(u), x)
-                      end
-              end
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:70 =#
-          construct(X)
-      end), :((var"##MTIIPVar#1260", u, p)->begin
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:75 =#
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:76 =#
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:76 =# @inbounds begin
-                  #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:77 =#
-                  #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:56 =# @inbounds begin
-                          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:56 =#
-                          let (x, y, z, σ, ρ, β) = (u[1], u[2], u[3], p[1], p[2], p[3])
-                              var"##MTIIPVar#1260"[1] = (*)(σ, (-)(y, x))
-                              var"##MTIIPVar#1260"[2] = (-)((*)(x, (-)(ρ, z)), y)
-                              var"##MTIIPVar#1260"[3] = (-)((*)(x, y), (*)(β, z))
+              construct = if var"##MTKArg#295" isa ModelingToolkit.StaticArrays.StaticArray
+                      (getproperty(ModelingToolkit.StaticArrays, :similar_type))(typeof(var"##MTKArg#295"), eltype(X))
+                  else
+                      x->begin
+                              convert(typeof(var"##MTKArg#295"), x)
                           end
-                      end
+                  end
+              return construct(X)
+          end
+      end), :((var"##MTIIPVar#298", var"##MTKArg#295", var"##MTKArg#296")->begin
+          @inbounds begin
+                  let (x, y, z, σ, ρ, β) = (var"##MTKArg#295"[1], var"##MTKArg#295"[2], var"##MTKArg#295"[3], var"##MTKArg#296"[1]
+, var"##MTKArg#296"[2], var"##MTKArg#296"[3])
+                      var"##MTIIPVar#298"[1] = (*)(σ, (-)(y, x))
+                      var"##MTIIPVar#298"[2] = (-)((*)(x, (-)(ρ, z)), y)
+                      var"##MTIIPVar#298"[3] = (-)((*)(x, y), (*)(β, z))
+                  end
               end
-          #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:79 =#
           nothing
       end))
@@ -1042,7 +1009,7 @@

Solving Nonlinear systems

Results of Nonlinear Solver Algorithm * Algorithm: Trust-region with dogleg and autoscaling * Starting Point: [1.0, 1.0, 1.0] - * Zero: [2.2228042242798023e-10, 2.222804224296743e-10, -9.990339458476605e-11] + * Zero: [2.2228042243306243e-10, 2.2228042243645056e-10, -9.990339599422887e-11] * Inf-norm of residuals: 0.000000 * Iterations: 3 * Convergence: true @@ -1069,10 +1036,13 @@

Library of transformations on mathematical systems

-ODESystem(ModelingToolkit.DiffEq[ModelingToolkit.DiffEq(u_tt, 1, ((2 * u_tt(t) + u_t(t)) + x_t(t)) + 1), ModelingToolkit.DiffEq(x_
-t, 1, x_t(t) + 2), ModelingToolkit.DiffEq(u_t, 1, u_tt(t)), ModelingToolkit.DiffEq(u, 1, u_t(t)), ModelingToolkit.DiffEq(x, 1, x_t
-(t))], t, Variable[u, x, u_tt, u_t, x_t], Variable[], Base.RefValue{Array{Expression,2}}(Array{Expression}(undef,0,0)), Base.RefVa
-lue{Array{Expression,2}}(Array{Expression}(undef,0,0)), Base.RefValue{Array{Expression,2}}(Array{Expression}(undef,0,0)))
+ModelingToolkit.ODESystem(ModelingToolkit.Equation[ModelingToolkit.Equation(derivative(uˍtt(t), t), ((2 * uˍtt(t) + uˍt(t)) + xˍt(
+t)) + 1), ModelingToolkit.Equation(derivative(xˍt(t), t), xˍt(t) + 2), ModelingToolkit.Equation(derivative(uˍt(t), t), uˍtt(t)), M
+odelingToolkit.Equation(derivative(u(t), t), uˍt(t)), ModelingToolkit.Equation(derivative(x(t), t), xˍt(t))], t, ModelingToolkit.V
+ariable[uˍtt, xˍt, uˍt, u, x], ModelingToolkit.Variable[], Base.RefValue{Array{ModelingToolkit.Expression,1}}(ModelingToolkit.Expr
+ession[]), Base.RefValue{Array{ModelingToolkit.Expression,2}}(Array{ModelingToolkit.Expression}(undef,0,0)), Base.RefValue{Array{M
+odelingToolkit.Expression,2}}(Array{ModelingToolkit.Expression}(undef,0,0)), Base.RefValue{Array{ModelingToolkit.Expression,2}}(Ar
+ray{ModelingToolkit.Expression}(undef,0,0)), Symbol("##ODESystem#301"), ModelingToolkit.ODESystem[])
@@ -1083,12 +1053,12 @@

Library of transformations on mathematical systems

-5-element Array{ModelingToolkit.DiffEq,1}:
- ModelingToolkit.DiffEq(u_tt, 1, ((2 * u_tt(t) + u_t(t)) + x_t(t)) + 1)
- ModelingToolkit.DiffEq(x_t, 1, x_t(t) + 2)                            
- ModelingToolkit.DiffEq(u_t, 1, u_tt(t))                               
- ModelingToolkit.DiffEq(u, 1, u_t(t))                                  
- ModelingToolkit.DiffEq(x, 1, x_t(t))
+5-element Array{ModelingToolkit.Equation,1}:
+ ModelingToolkit.Equation(derivative(uˍtt(t), t), ((2 * uˍtt(t) + uˍt(t)) + xˍt(t)) + 1)
+ ModelingToolkit.Equation(derivative(xˍt(t), t), xˍt(t) + 2)
+ ModelingToolkit.Equation(derivative(uˍt(t), t), uˍtt(t))
+ ModelingToolkit.Equation(derivative(u(t), t), uˍt(t))
+ ModelingToolkit.Equation(derivative(x(t), t), xˍt(t))
@@ -1111,7 +1081,7 @@

Linear Algebra... for free?

-3×3 Array{Operation,2}:
+3×3 Array{ModelingToolkit.Operation,2}:
         derivative(σ * (y(t) - x(t)), x(t))  …         derivative(σ * (y(t) - x(t)), z(t))
  derivative(x(t) * (ρ - z(t)) - y(t), x(t))     derivative(x(t) * (ρ - z(t)) - y(t), z(t))
    derivative(x(t) * y(t) - β * z(t), x(t))       derivative(x(t) * y(t) - β * z(t), z(t))
@@ -1127,10 +1097,10 @@ 
Linear Algebra... for free?

 
 
 
-3×3 Array{Expression,2}:
-   σ * -1             σ  Constant(0)
- ρ - z(t)  Constant(-1)    x(t) * -1
-     y(t)          x(t)          -1β
+3×3 Array{ModelingToolkit.Expression,2}:
+           -1σ             σ  Constant(0)
+ -1 * z(t) + ρ  Constant(-1)    -1 * x(t)
+          y(t)          x(t)          -1β
 

 
 
@@ -1144,20 +1114,20 @@ 
Linear Algebra... for free?

 
 
 
-LU{Expression,Array{Expression,2}}
+LinearAlgebra.LU{ModelingToolkit.Expression,Array{ModelingToolkit.Expression,2}}
 L factor:
-3×3 Array{Expression,2}:
-              Constant(1)  …  Constant(0)
- (ρ - z(t)) * inv(σ * -1)     identity(0)
-       y(t) * inv(σ * -1)     Constant(1)
+3×3 Array{ModelingToolkit.Expression,2}:
+                Constant(1)  …  Constant(0)
+ (-1 * z(t) + ρ) * inv(-1σ)     Constant(0)
+            y(t) * inv(-1σ)     Constant(1)
 U factor:
-3×3 Array{Expression,2}:
-      σ * -1  …                                                                                                                   
-                                    Constant(0)
- identity(0)                                                                                                                      
-     x(t) * -1 - ((ρ - z(t)) * inv(σ * -1)) * 0
- identity(0)     (-1β - (y(t) * inv(σ * -1)) * 0) - ((x(t) - (y(t) * inv(σ * -1)) * σ) * inv(-1 - ((ρ - z(t)) * inv(σ * -1)) * σ))
- * (x(t) * -1 - ((ρ - z(t)) * inv(σ * -1)) * 0)
+3×3 Array{ModelingToolkit.Expression,2}:
+         -1σ  …                                                                                                                   
+                                  Constant(0)
+ Constant(0)                                                                                                                      
+ -1 * x(t) - ((-1 * z(t) + ρ) * inv(-1σ)) * 0
+ Constant(0)     (-1β - (y(t) * inv(-1σ)) * 0) - ((x(t) - (y(t) * inv(-1σ)) * σ) * inv(-1 - ((-1 * z(t) + ρ) * inv(-1σ)) * σ)) * (
+-1 * x(t) - ((-1 * z(t) + ρ) * inv(-1σ)) * 0)
 

 
 
@@ -1168,10 +1138,10 @@ 
Linear Algebra... for free?

 
 
 
-3×3 Array{Expression,2}:
-              Constant(1)  …  Constant(0)
- (ρ - z(t)) * inv(σ * -1)     identity(0)
-       y(t) * inv(σ * -1)     Constant(1)
+3×3 Array{ModelingToolkit.Expression,2}:
+                Constant(1)  …  Constant(0)
+ (-1 * z(t) + ρ) * inv(-1σ)     Constant(0)
+            y(t) * inv(-1σ)     Constant(1)
 

 
 
@@ -1184,46 +1154,46 @@ 
Linear Algebra... for free?

 
 
 
-3×3 Array{Expression,2}:
- (σ * -1) \ ((true - 0 * (((-1β - (y(t) * inv(σ * -1)) * 0) - ((x(t) - (y(t) * inv(σ * -1)) * σ) * inv(-1 - ((ρ - z(t)) * inv(σ * 
--1)) * σ)) * (x(t) * -1 - ((ρ - z(t)) * inv(σ * -1)) * 0)) \ ((0 - (y(t) * inv(σ * -1)) * true) - ((x(t) - (y(t) * inv(σ * -1)) * 
-σ) * inv(-1 - ((ρ - z(t)) * inv(σ * -1)) * σ)) * (0 - ((ρ - z(t)) * inv(σ * -1)) * true)))) - σ * ((-1 - ((ρ - z(t)) * inv(σ * -1)
-) * σ) \ ((0 - ((ρ - z(t)) * inv(σ * -1)) * true) - (x(t) * -1 - ((ρ - z(t)) * inv(σ * -1)) * 0) * (((-1β - (y(t) * inv(σ * -1)) *
- 0) - ((x(t) - (y(t) * inv(σ * -1)) * σ) * inv(-1 - ((ρ - z(t)) * inv(σ * -1)) * σ)) * (x(t) * -1 - ((ρ - z(t)) * inv(σ * -1)) * 0
-)) \ ((0 - (y(t) * inv(σ * -1)) * true) - ((x(t) - (y(t) * inv(σ * -1)) * σ) * inv(-1 - ((ρ - z(t)) * inv(σ * -1)) * σ)) * (0 - ((
-ρ - z(t)) * inv(σ * -1)) * true))))))  …  (σ * -1) \ ((0 - 0 * (((-1β - (y(t) * inv(σ * -1)) * 0) - ((x(t) - (y(t) * inv(σ * -1)) 
-* σ) * inv(-1 - ((ρ - z(t)) * inv(σ * -1)) * σ)) * (x(t) * -1 - ((ρ - z(t)) * inv(σ * -1)) * 0)) \ ((true - (y(t) * inv(σ * -1)) *
- 0) - ((x(t) - (y(t) * inv(σ * -1)) * σ) * inv(-1 - ((ρ - z(t)) * inv(σ * -1)) * σ)) * (0 - ((ρ - z(t)) * inv(σ * -1)) * 0)))) - σ
- * ((-1 - ((ρ - z(t)) * inv(σ * -1)) * σ) \ ((0 - ((ρ - z(t)) * inv(σ * -1)) * 0) - (x(t) * -1 - ((ρ - z(t)) * inv(σ * -1)) * 0) *
- (((-1β - (y(t) * inv(σ * -1)) * 0) - ((x(t) - (y(t) * inv(σ * -1)) * σ) * inv(-1 - ((ρ - z(t)) * inv(σ * -1)) * σ)) * (x(t) * -1 
-- ((ρ - z(t)) * inv(σ * -1)) * 0)) \ ((true - (y(t) * inv(σ * -1)) * 0) - ((x(t) - (y(t) * inv(σ * -1)) * σ) * inv(-1 - ((ρ - z(t)
-) * inv(σ * -1)) * σ)) * (0 - ((ρ - z(t)) * inv(σ * -1)) * 0))))))
+3×3 Array{ModelingToolkit.Expression,2}:
+ (-1σ) \ ((true - 0 * (((-1β - (y(t) * inv(-1σ)) * 0) - ((x(t) - (y(t) * inv(-1σ)) * σ) * inv(-1 - ((-1 * z(t) + ρ) * inv(-1σ)) * 
+σ)) * (-1 * x(t) - ((-1 * z(t) + ρ) * inv(-1σ)) * 0)) \ ((0 - (y(t) * inv(-1σ)) * true) - ((x(t) - (y(t) * inv(-1σ)) * σ) * inv(-1
+ - ((-1 * z(t) + ρ) * inv(-1σ)) * σ)) * (0 - ((-1 * z(t) + ρ) * inv(-1σ)) * true)))) - σ * ((-1 - ((-1 * z(t) + ρ) * inv(-1σ)) * σ
+) \ ((0 - ((-1 * z(t) + ρ) * inv(-1σ)) * true) - (-1 * x(t) - ((-1 * z(t) + ρ) * inv(-1σ)) * 0) * (((-1β - (y(t) * inv(-1σ)) * 0) 
+- ((x(t) - (y(t) * inv(-1σ)) * σ) * inv(-1 - ((-1 * z(t) + ρ) * inv(-1σ)) * σ)) * (-1 * x(t) - ((-1 * z(t) + ρ) * inv(-1σ)) * 0)) 
+\ ((0 - (y(t) * inv(-1σ)) * true) - ((x(t) - (y(t) * inv(-1σ)) * σ) * inv(-1 - ((-1 * z(t) + ρ) * inv(-1σ)) * σ)) * (0 - ((-1 * z(
+t) + ρ) * inv(-1σ)) * true))))))  …  (-1σ) \ ((0 - 0 * (((-1β - (y(t) * inv(-1σ)) * 0) - ((x(t) - (y(t) * inv(-1σ)) * σ) * inv(-1 
+- ((-1 * z(t) + ρ) * inv(-1σ)) * σ)) * (-1 * x(t) - ((-1 * z(t) + ρ) * inv(-1σ)) * 0)) \ ((true - (y(t) * inv(-1σ)) * 0) - ((x(t) 
+- (y(t) * inv(-1σ)) * σ) * inv(-1 - ((-1 * z(t) + ρ) * inv(-1σ)) * σ)) * (0 - ((-1 * z(t) + ρ) * inv(-1σ)) * 0)))) - σ * ((-1 - ((
+-1 * z(t) + ρ) * inv(-1σ)) * σ) \ ((0 - ((-1 * z(t) + ρ) * inv(-1σ)) * 0) - (-1 * x(t) - ((-1 * z(t) + ρ) * inv(-1σ)) * 0) * (((-1
+β - (y(t) * inv(-1σ)) * 0) - ((x(t) - (y(t) * inv(-1σ)) * σ) * inv(-1 - ((-1 * z(t) + ρ) * inv(-1σ)) * σ)) * (-1 * x(t) - ((-1 * z
+(t) + ρ) * inv(-1σ)) * 0)) \ ((true - (y(t) * inv(-1σ)) * 0) - ((x(t) - (y(t) * inv(-1σ)) * σ) * inv(-1 - ((-1 * z(t) + ρ) * inv(-
+1σ)) * σ)) * (0 - ((-1 * z(t) + ρ) * inv(-1σ)) * 0))))))
                                                                                                                                   
                                                                                                                                   
-                                                                                                     (-1 - ((ρ - z(t)) * inv(σ * -
-1)) * σ) \ ((0 - ((ρ - z(t)) * inv(σ * -1)) * true) - (x(t) * -1 - ((ρ - z(t)) * inv(σ * -1)) * 0) * (((-1β - (y(t) * inv(σ * -1))
- * 0) - ((x(t) - (y(t) * inv(σ * -1)) * σ) * inv(-1 - ((ρ - z(t)) * inv(σ * -1)) * σ)) * (x(t) * -1 - ((ρ - z(t)) * inv(σ * -1)) *
- 0)) \ ((0 - (y(t) * inv(σ * -1)) * true) - ((x(t) - (y(t) * inv(σ * -1)) * σ) * inv(-1 - ((ρ - z(t)) * inv(σ * -1)) * σ)) * (0 - 
-((ρ - z(t)) * inv(σ * -1)) * true))))                                                                                             
+                                                                                              (-1 - ((-1 * z(t) + ρ) * inv(-1σ)) *
+ σ) \ ((0 - ((-1 * z(t) + ρ) * inv(-1σ)) * true) - (-1 * x(t) - ((-1 * z(t) + ρ) * inv(-1σ)) * 0) * (((-1β - (y(t) * inv(-1σ)) * 0
+) - ((x(t) - (y(t) * inv(-1σ)) * σ) * inv(-1 - ((-1 * z(t) + ρ) * inv(-1σ)) * σ)) * (-1 * x(t) - ((-1 * z(t) + ρ) * inv(-1σ)) * 0)
+) \ ((0 - (y(t) * inv(-1σ)) * true) - ((x(t) - (y(t) * inv(-1σ)) * σ) * inv(-1 - ((-1 * z(t) + ρ) * inv(-1σ)) * σ)) * (0 - ((-1 * 
+z(t) + ρ) * inv(-1σ)) * true))))                                                                                                  
                                                                                                                                   
+                                                                                                                            (-1 - 
+((-1 * z(t) + ρ) * inv(-1σ)) * σ) \ ((0 - ((-1 * z(t) + ρ) * inv(-1σ)) * 0) - (-1 * x(t) - ((-1 * z(t) + ρ) * inv(-1σ)) * 0) * (((
+-1β - (y(t) * inv(-1σ)) * 0) - ((x(t) - (y(t) * inv(-1σ)) * σ) * inv(-1 - ((-1 * z(t) + ρ) * inv(-1σ)) * σ)) * (-1 * x(t) - ((-1 *
+ z(t) + ρ) * inv(-1σ)) * 0)) \ ((true - (y(t) * inv(-1σ)) * 0) - ((x(t) - (y(t) * inv(-1σ)) * σ) * inv(-1 - ((-1 * z(t) + ρ) * inv
+(-1σ)) * σ)) * (0 - ((-1 * z(t) + ρ) * inv(-1σ)) * 0))))
                                                                                                                                   
-      (-1 - ((ρ - z(t)) * inv(σ * -1)) * σ) \ ((0 - ((ρ - z(t)) * inv(σ * -1)) * 0) - (x(t) * -1 - ((ρ - z(t)) * inv(σ * -1)) * 0)
- * (((-1β - (y(t) * inv(σ * -1)) * 0) - ((x(t) - (y(t) * inv(σ * -1)) * σ) * inv(-1 - ((ρ - z(t)) * inv(σ * -1)) * σ)) * (x(t) * -
-1 - ((ρ - z(t)) * inv(σ * -1)) * 0)) \ ((true - (y(t) * inv(σ * -1)) * 0) - ((x(t) - (y(t) * inv(σ * -1)) * σ) * inv(-1 - ((ρ - z(
-t)) * inv(σ * -1)) * σ)) * (0 - ((ρ - z(t)) * inv(σ * -1)) * 0))))
                                                                                                                                   
                                                                                                                                   
+                                                                                                       ((-1β - (y(t) * inv(-1σ)) *
+ 0) - ((x(t) - (y(t) * inv(-1σ)) * σ) * inv(-1 - ((-1 * z(t) + ρ) * inv(-1σ)) * σ)) * (-1 * x(t) - ((-1 * z(t) + ρ) * inv(-1σ)) * 
+0)) \ ((0 - (y(t) * inv(-1σ)) * true) - ((x(t) - (y(t) * inv(-1σ)) * σ) * inv(-1 - ((-1 * z(t) + ρ) * inv(-1σ)) * σ)) * (0 - ((-1 
+* z(t) + ρ) * inv(-1σ)) * true))                                                                                                  
                                                                                                                                   
-                                                                                                        ((-1β - (y(t) * inv(σ * -1
-)) * 0) - ((x(t) - (y(t) * inv(σ * -1)) * σ) * inv(-1 - ((ρ - z(t)) * inv(σ * -1)) * σ)) * (x(t) * -1 - ((ρ - z(t)) * inv(σ * -1))
- * 0)) \ ((0 - (y(t) * inv(σ * -1)) * true) - ((x(t) - (y(t) * inv(σ * -1)) * σ) * inv(-1 - ((ρ - z(t)) * inv(σ * -1)) * σ)) * (0 
-- ((ρ - z(t)) * inv(σ * -1)) * true))                                                                                             
                                                                                                                                   
                                                                                                                                   
-                                                                                                                                  
-      ((-1β - (y(t) * inv(σ * -1)) * 0) - ((x(t) - (y(t) * inv(σ * -1)) * σ) * inv(-1 - ((ρ - z(t)) * inv(σ * -1)) * σ)) * (x(t) *
- -1 - ((ρ - z(t)) * inv(σ * -1)) * 0)) \ ((true - (y(t) * inv(σ * -1)) * 0) - ((x(t) - (y(t) * inv(σ * -1)) * σ) * inv(-1 - ((ρ - 
-z(t)) * inv(σ * -1)) * σ)) * (0 - ((ρ - z(t)) * inv(σ * -1)) * 0))
+((-1β - (y(t) * inv(-1σ)) * 0) - ((x(t) - (y(t) * inv(-1σ)) * σ) * inv(-1 - ((-1 * z(t) + ρ) * inv(-1σ)) * σ)) * (-1 * x(t) - ((-1
+ * z(t) + ρ) * inv(-1σ)) * 0)) \ ((true - (y(t) * inv(-1σ)) * 0) - ((x(t) - (y(t) * inv(-1σ)) * σ) * inv(-1 - ((-1 * z(t) + ρ) * i
+nv(-1σ)) * σ)) * (0 - ((-1 * z(t) + ρ) * inv(-1σ)) * 0))
 

 
 
@@ -1260,7 +1230,7 @@ 
Automatically convert numerical codes to symbolic

 
 
 
-3-element Array{Operation,1}:
+3-element Array{ModelingToolkit.Operation,1}:
         σ * (y(t) - x(t))
  x(t) * (ρ - z(t)) - y(t)
    x(t) * y(t) - β * z(t)
@@ -1279,10 +1249,10 @@ 
Automatically convert numerical codes to symbolic

 
 
 
-3×3 Array{Expression,2}:
-   σ * -1             σ  Constant(0)
- ρ - z(t)  Constant(-1)    x(t) * -1
-     y(t)          x(t)          -1β
+3×3 Array{ModelingToolkit.Expression,2}:
+           -1σ             σ  Constant(0)
+ -1 * z(t) + ρ  Constant(-1)    -1 * x(t)
+          y(t)          x(t)          -1β
 

 
 
@@ -1304,14 +1274,14 @@ 
Automated Sparsity Detection

 
 
 
-3×3 SparseMatrixCSC{Expression,Int64} with 8 stored entries:
-  [1, 1]  =  σ * -1
-  [2, 1]  =  ρ - z(t)
+3×3 SparseArrays.SparseMatrixCSC{ModelingToolkit.Expression,Int64} with 8 stored entries:
+  [1, 1]  =  -1σ
+  [2, 1]  =  -1 * z(t) + ρ
   [3, 1]  =  y(t)
   [1, 2]  =  σ
   [2, 2]  =  Constant(-1)
   [3, 2]  =  x(t)
-  [2, 3]  =  x(t) * -1
+  [2, 3]  =  -1 * x(t)
   [3, 3]  =  -1β
 

 
@@ -1330,10 +1300,10 @@ 
Dependent Variables, Functions, Chain Rule

 
 
 
-3-element Array{Equation,1}:
- Equation(derivative(x(t), t), σ(t - 1) * (y(t) - x(t)))       
- Equation(derivative(y(t), t), x(t) * (σ(t ^ 2) - z(t)) - y(t))
- Equation(derivative(z(t), t), x(t) * y(t) - β * z(t))
+3-element Array{ModelingToolkit.Equation,1}:
+ ModelingToolkit.Equation(derivative(x(t), t), σ(t - 1) * (y(t) - x(t)))
+ ModelingToolkit.Equation(derivative(y(t), t), x(t) * (σ(t ^ 2) - z(t)) - y(t))
+ ModelingToolkit.Equation(derivative(z(t), t), x(t) * y(t) - β * z(t))
 

 
 
@@ -1348,7 +1318,8 @@ 
Dependent Variables, Functions, Chain Rule

 
 
 
-(σ(t ^ 2) - z(t)) * derivative(x(t), t) + x(t) * (derivative(σ(t ^ 2), t) + -1 * derivative(z(t), t)) + -1 * derivative(y(t), t)
+derivative(x(t), t) * (σ(t ^ 2) + -1 * z(t)) + -1 * derivative(y(t), t) + x(t) * (derivative(σ(t ^ 2), t) + -1 * derivative(z(t), 
+t))
 

 
 
@@ -1420,36 +1391,43 @@ 
Hackability: Extend directly from the language

Computer Information:

-
Julia Version 1.3.0
-Commit 46ce4d7933 (2019-11-26 06:09 UTC)
+
Julia Version 1.4.2
+Commit 44fa15b150* (2020-05-23 18:35 UTC)
 Platform Info:
-  OS: Windows (x86_64-w64-mingw32)
-  CPU: Intel(R) Core(TM) i7-8550U CPU @ 1.80GHz
+  OS: Linux (x86_64-pc-linux-gnu)
+  CPU: Intel(R) Core(TM) i7-9700K CPU @ 3.60GHz
   WORD_SIZE: 64
   LIBM: libopenlibm
-  LLVM: libLLVM-6.0.1 (ORCJIT, skylake)
+  LLVM: libLLVM-8.0.1 (ORCJIT, skylake)
 Environment:
-  JULIA_EDITOR = "C:\Users\accou\AppData\Local\atom\app-1.42.0\atom.exe"  -a
+  JULIA_DEPOT_PATH = /builds/JuliaGPU/DiffEqTutorials.jl/.julia
+  JULIA_CUDA_MEMORY_LIMIT = 536870912
+  JULIA_PROJECT = @.
   JULIA_NUM_THREADS = 4
 

 

 
Package Information:

 

-
Status `~\.julia\dev\DiffEqTutorials\Project.toml`

+
Status `/builds/JuliaGPU/DiffEqTutorials.jl/tutorials/ode_extras/Project.toml`
+[961ee093-0014-501f-94e3-6117800e7a78] ModelingToolkit 3.11.0
+[2774e3e8-f4cf-5e23-947b-6d7e65073b56] NLsolve 4.4.0
+[1dea7af3-3e70-54e6-95c3-0bf5283fa5ed] OrdinaryDiffEq 5.41.0
+[91a5bcdd-55d7-5caf-9e0b-520d859cae80] Plots 1.5.0
+[37e2e46d-f89d-539d-b4ee-838fcccc9c8e] LinearAlgebra
+[2f01184e-e22b-5df5-ae63-d93ebab69eaf] SparseArrays

 

 
 
-
-          

-          

-          Published from 01-ModelingToolkit.jmd using
-          Weave.jl
-           on 2019-12-29.
-          

-
-
+        

+        

+          

+            Published from 01-ModelingToolkit.jmd
+            using Weave.jl v0.10.2 on 2020-07-01.
+
- +
+ + diff --git a/markdown/ode_extras/01-ModelingToolkit.md b/markdown/ode_extras/01-ModelingToolkit.md index 53a4833c..ac5788bf 100644 --- a/markdown/ode_extras/01-ModelingToolkit.md +++ b/markdown/ode_extras/01-ModelingToolkit.md @@ -1,3 +1,9 @@ +--- +author: "Chris Rackauckas" +title: "ModelingToolkit.jl, An IR and Compiler for Scientific Models" +--- + + A lot of people are building modeling languages for their specific domains. However, while the syntax my vary greatly between these domain-specific languages (DSLs), the internals of modeling frameworks are surprisingly similar: building differential equations, calculating Jacobians, etc. #### ModelingToolkit.jl is metamodeling systemitized @@ -54,41 +60,34 @@ generate_function(de)[1] ```` -:((u, p, t)->begin - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:61 =# - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:62 =# - if u isa Array - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:63 =# - return #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:55 =# @inbounds(begin - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:55 =# - let (x, y, z, σ, ρ, β) = (u[1], u[2], u[3], p[1], p[2], p[3]) - [σ * (y - x), x * (ρ - z) - y, x * y - β * z] +:((var"##MTKArg#360", var"##MTKArg#361", var"##MTKArg#362")->begin + if var"##MTKArg#360" isa Array || !(typeof(var"##MTKArg#360") <: StaticArray) && false + return @inbounds(begin + let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#360"[1], var"##MTKArg#360"[2], var"##MTKArg#360"[3], var"##MTK +Arg#361"[1], var"##MTKArg#361"[2], var"##MTKArg#361"[3], var"##MTKArg#362") + [(getproperty(Base, :*))(σ, (getproperty(Base, :-))(y, x)), (getproperty(Base, :-))((getproperty(Bas +e, :*))(x, (getproperty(Base, :-))(ρ, z)), y), (getproperty(Base, :-))((getproperty(Base, :*))(x, y), (getproperty(Base, :*))(β, z +))] end end) else - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:65 =# - X = #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:54 =# @inbounds(begin - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:54 =# - let (x, y, z, σ, ρ, β) = (u[1], u[2], u[3], p[1], p[2], p[3]) - (σ * (y - x), x * (ρ - z) - y, x * y - β * z) + X = @inbounds(begin + let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#360"[1], var"##MTKArg#360"[2], var"##MTKArg#360"[3], var"##MTK +Arg#361"[1], var"##MTKArg#361"[2], var"##MTKArg#361"[3], var"##MTKArg#362") + ((getproperty(Base, :*))(σ, (getproperty(Base, :-))(y, x)), (getproperty(Base, :-))((getproperty(Bas +e, :*))(x, (getproperty(Base, :-))(ρ, z)), y), (getproperty(Base, :-))((getproperty(Base, :*))(x, y), (getproperty(Base, :*))(β, z +))) end end) + construct = if var"##MTKArg#360" isa ModelingToolkit.StaticArrays.StaticArray + (getproperty(ModelingToolkit.StaticArrays, :similar_type))(typeof(var"##MTKArg#360"), eltype(X)) + else + x->begin + convert(typeof(var"##MTKArg#360"), x) + end + end + return construct(X) end - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:67 =# - T = promote_type(map(typeof, X)...) - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:68 =# - map(T, X) - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:69 =# - construct = if u isa ModelingToolkit.StaticArrays.StaticArray - ModelingToolkit.StaticArrays.similar_type(typeof(u), eltype(X)) - else - x->begin - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:69 =# - convert(typeof(u), x) - end - end - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:70 =# - construct(X) end) ```` @@ -104,21 +103,17 @@ generate_function(de)[2] ```` -:((var"##MTIIPVar#1282", u, p, t)->begin - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:75 =# - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:76 =# - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:76 =# @inbounds begin - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:77 =# - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:56 =# @inbounds begin - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:56 =# - let (x, y, z, σ, ρ, β) = (u[1], u[2], u[3], p[1], p[2], p[3]) - var"##MTIIPVar#1282"[1] = σ * (y - x) - var"##MTIIPVar#1282"[2] = x * (ρ - z) - y - var"##MTIIPVar#1282"[3] = x * y - β * z - end - end +:((var"##MTIIPVar#370", var"##MTKArg#366", var"##MTKArg#367", var"##MTKArg#368")->begin + @inbounds begin + let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#366"[1], var"##MTKArg#366"[2], var"##MTKArg#366"[3], var"##MTKArg#367" +[1], var"##MTKArg#367"[2], var"##MTKArg#367"[3], var"##MTKArg#368") + var"##MTIIPVar#370"[1] = (getproperty(Base, :*))(σ, (getproperty(Base, :-))(y, x)) + var"##MTIIPVar#370"[2] = (getproperty(Base, :-))((getproperty(Base, :*))(x, (getproperty(Base, :-))(ρ, z)), +y) + var"##MTIIPVar#370"[3] = (getproperty(Base, :-))((getproperty(Base, :*))(x, y), (getproperty(Base, :*))(β, z +)) + end end - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:79 =# nothing end) ```` @@ -135,10 +130,10 @@ jac = calculate_jacobian(de) ```` -3×3 Array{Expression,2}: - σ * -1 σ Constant(0) - ρ - z(t) Constant(-1) x(t) * -1 - y(t) x(t) -1β +3×3 Array{ModelingToolkit.Expression,2}: + -1σ σ Constant(0) + -1 * z(t) + ρ Constant(-1) -1 * x(t) + y(t) x(t) -1β ```` @@ -153,62 +148,48 @@ jac_expr = generate_jacobian(de) ```` -(:((u, p, t)->begin - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:61 =# - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:62 =# - if u isa Array - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:63 =# - return #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:55 =# @inbounds(begin - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:55 =# - let (x, y, z, σ, ρ, β) = (u[1], u[2], u[3], p[1], p[2], p[3]) - [σ * -1, ρ - z, y, σ, -1, x, 0, x * -1, -1β] +(:((var"##MTKArg#372", var"##MTKArg#373", var"##MTKArg#374")->begin + if var"##MTKArg#372" isa Array || !(typeof(var"##MTKArg#372") <: StaticArray) && false + return @inbounds(begin + let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#372"[1], var"##MTKArg#372"[2], var"##MTKArg#372"[3], var"##MTK +Arg#373"[1], var"##MTKArg#373"[2], var"##MTKArg#373"[3], var"##MTKArg#374") + [(getproperty(Base, :*))(-1, σ) σ 0; (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ) -1 ( +getproperty(Base, :*))(-1, x); y x (getproperty(Base, :*))(-1, β)] end end) else - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:65 =# - X = #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:54 =# @inbounds(begin - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:54 =# - let (x, y, z, σ, ρ, β) = (u[1], u[2], u[3], p[1], p[2], p[3]) - (σ * -1, ρ - z, y, σ, -1, x, 0, x * -1, -1β) + X = @inbounds(begin + let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#372"[1], var"##MTKArg#372"[2], var"##MTKArg#372"[3], var"##MTK +Arg#373"[1], var"##MTKArg#373"[2], var"##MTKArg#373"[3], var"##MTKArg#374") + ((getproperty(Base, :*))(-1, σ), (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ), y, σ, - +1, x, 0, (getproperty(Base, :*))(-1, x), (getproperty(Base, :*))(-1, β)) end end) - end - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:67 =# - T = promote_type(map(typeof, X)...) - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:68 =# - map(T, X) - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:69 =# - construct = if u isa ModelingToolkit.StaticArrays.StaticArray - ModelingToolkit.StaticArrays.similar_type(typeof(u), eltype(X)) - else - x->begin - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:69 =# - convert(typeof(u), x) - end - end - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:70 =# - construct(X) - end), :((var"##MTIIPVar#1284", u, p, t)->begin - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:75 =# - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:76 =# - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:76 =# @inbounds begin - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:77 =# - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:56 =# @inbounds begin - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:56 =# - let (x, y, z, σ, ρ, β) = (u[1], u[2], u[3], p[1], p[2], p[3]) - var"##MTIIPVar#1284"[1] = σ * -1 - var"##MTIIPVar#1284"[2] = ρ - z - var"##MTIIPVar#1284"[3] = y - var"##MTIIPVar#1284"[4] = σ - var"##MTIIPVar#1284"[5] = -1 - var"##MTIIPVar#1284"[6] = x - var"##MTIIPVar#1284"[7] = 0 - var"##MTIIPVar#1284"[8] = x * -1 - var"##MTIIPVar#1284"[9] = -1β + construct = if var"##MTKArg#372" isa ModelingToolkit.StaticArrays.StaticArray + ModelingToolkit.StaticArrays.SMatrix{3, 3} + else + x->begin + out = similar(typeof(var"##MTKArg#372"), 3, 3) + out .= x end - end + end + return construct(X) + end + end), :((var"##MTIIPVar#376", var"##MTKArg#372", var"##MTKArg#373", var"##MTKArg#374")->begin + @inbounds begin + let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#372"[1], var"##MTKArg#372"[2], var"##MTKArg#372"[3], var"##MTKArg#373" +[1], var"##MTKArg#373"[2], var"##MTKArg#373"[3], var"##MTKArg#374") + var"##MTIIPVar#376"[1] = (getproperty(Base, :*))(-1, σ) + var"##MTIIPVar#376"[2] = (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ) + var"##MTIIPVar#376"[3] = y + var"##MTIIPVar#376"[4] = σ + var"##MTIIPVar#376"[5] = -1 + var"##MTIIPVar#376"[6] = x + var"##MTIIPVar#376"[7] = 0 + var"##MTIIPVar#376"[8] = (getproperty(Base, :*))(-1, x) + var"##MTIIPVar#376"[9] = (getproperty(Base, :*))(-1, β) + end end - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:79 =# nothing end)) ```` @@ -225,76 +206,88 @@ ModelingToolkit.generate_factorized_W(de)[1] ```` -(:((u, p, gam, t)->begin - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:61 =# - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:62 =# - if u isa Array - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:63 =# - return #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:55 =# @inbounds(begin - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:55 =# - let (x, y, z, σ, ρ, β) = (u[1], u[2], u[3], p[1], p[2], p[3]) - [σ * -1 * gam + -1, gam * (ρ - z) * inv(σ * -1 * gam + -1), gam * y * inv(σ * -1 * gam + -1), gam * -σ, (gam * -1 + -1) - gam * (ρ - z) * inv(σ * -1 * gam + -1) * gam * σ, (gam * x - gam * y * inv(σ * -1 * gam + -1) * gam * σ) * in -v((gam * -1 + -1) - gam * (ρ - z) * inv(σ * -1 * gam + -1) * gam * σ), 0, x * -1 * gam - 0, ((-1 * β * gam + -1) - 0) - (gam * x - - gam * y * inv(σ * -1 * gam + -1) * gam * σ) * inv((gam * -1 + -1) - gam * (ρ - z) * inv(σ * -1 * gam + -1) * gam * σ) * (x * -1 * - gam - 0)] +(:((var"##MTKArg#378", var"##MTKArg#379", var"##MTKArg#380", var"##MTKArg#381")->begin + if var"##MTKArg#378" isa Array || !(typeof(var"##MTKArg#378") <: StaticArray) && false + return @inbounds(begin + let (x, y, z, σ, ρ, β, __MTKWgamma, t) = (var"##MTKArg#378"[1], var"##MTKArg#378"[2], var"##MTKArg#378"[ +3], var"##MTKArg#379"[1], var"##MTKArg#379"[2], var"##MTKArg#379"[3], var"##MTKArg#380", var"##MTKArg#381") + [(getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ)) (getproperty(Base, :*))(__ +MTKWgamma, σ) 0; (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWg +amma, σ))), __MTKWgamma, (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ)) (getproperty(Base, :*))(-1, (getproperty(Base +, :+))(1, (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ +))), __MTKWgamma ^ 2, σ, (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ)), __MTKWgamma)) (getproperty(Base, :*))(-1, x, + __MTKWgamma); (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgam +ma, σ))), y, __MTKWgamma) (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :*))(-1, (getproperty(Base, :+))(1, + (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ))), __MT +KWgamma ^ 2, σ, (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ)), __MTKWgamma))), (getproperty(Base, :+))((getproperty( +Base, :*))(x, __MTKWgamma), (getproperty(Base, :*))(-1, (getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, : +*))(-1, __MTKWgamma, σ))), y, __MTKWgamma ^ 2, σ))) (getproperty(Base, :+))((getproperty(Base, :*))(-1, (getproperty(Base, :+))(1, + (getproperty(Base, :*))(-1, (getproperty(Base, :inv))((getproperty(Base, :*))(-1, (getproperty(Base, :+))(1, (getproperty(Base, : +*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ))), __MTKWgamma ^ 2, σ, (getp +roperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ)), __MTKWgamma))), x, __MTKWgamma, (getproperty(Base, :+))((getproperty(Base, + :*))(x, __MTKWgamma), (getproperty(Base, :*))(-1, (getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(- +1, __MTKWgamma, σ))), y, __MTKWgamma ^ 2, σ))))), (getproperty(Base, :*))(-1, __MTKWgamma, β))] end end) else - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:65 =# - X = #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:54 =# @inbounds(begin - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:54 =# - let (x, y, z, σ, ρ, β) = (u[1], u[2], u[3], p[1], p[2], p[3]) - (σ * -1 * gam + -1, gam * (ρ - z) * inv(σ * -1 * gam + -1), gam * y * inv(σ * -1 * gam + -1), gam * -σ, (gam * -1 + -1) - gam * (ρ - z) * inv(σ * -1 * gam + -1) * gam * σ, (gam * x - gam * y * inv(σ * -1 * gam + -1) * gam * σ) * in -v((gam * -1 + -1) - gam * (ρ - z) * inv(σ * -1 * gam + -1) * gam * σ), 0, x * -1 * gam - 0, ((-1 * β * gam + -1) - 0) - (gam * x - - gam * y * inv(σ * -1 * gam + -1) * gam * σ) * inv((gam * -1 + -1) - gam * (ρ - z) * inv(σ * -1 * gam + -1) * gam * σ) * (x * -1 * - gam - 0)) + X = @inbounds(begin + let (x, y, z, σ, ρ, β, __MTKWgamma, t) = (var"##MTKArg#378"[1], var"##MTKArg#378"[2], var"##MTKArg#378"[ +3], var"##MTKArg#379"[1], var"##MTKArg#379"[2], var"##MTKArg#379"[3], var"##MTKArg#380", var"##MTKArg#381") + ((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ)), (getproperty(Base, :*))(( +getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ))), __MTKWgamma, (getproperty(Base +, :+))((getproperty(Base, :*))(-1, z), ρ)), (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getprop +erty(Base, :*))(-1, __MTKWgamma, σ))), y, __MTKWgamma), (getproperty(Base, :*))(__MTKWgamma, σ), (getproperty(Base, :*))(-1, (getp +roperty(Base, :+))(1, (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __ +MTKWgamma, σ))), __MTKWgamma ^ 2, σ, (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ)), __MTKWgamma)), (getproperty(Base +, :*))((getproperty(Base, :inv))((getproperty(Base, :*))(-1, (getproperty(Base, :+))(1, (getproperty(Base, :*))((getproperty(Base, + :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ))), __MTKWgamma ^ 2, σ, (getproperty(Base, :+))((ge +tproperty(Base, :*))(-1, z), ρ)), __MTKWgamma))), (getproperty(Base, :+))((getproperty(Base, :*))(x, __MTKWgamma), (getproperty(Ba +se, :*))(-1, (getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ))), y, __MTKWgamma ^ + 2, σ))), 0, (getproperty(Base, :*))(-1, x, __MTKWgamma), (getproperty(Base, :+))((getproperty(Base, :*))(-1, (getproperty(Base, : ++))(1, (getproperty(Base, :*))(-1, (getproperty(Base, :inv))((getproperty(Base, :*))(-1, (getproperty(Base, :+))(1, (getproperty(B +ase, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ))), __MTKWgamma ^ 2, σ, + (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ)), __MTKWgamma))), x, __MTKWgamma, (getproperty(Base, :+))((getproperty +(Base, :*))(x, __MTKWgamma), (getproperty(Base, :*))(-1, (getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, +:*))(-1, __MTKWgamma, σ))), y, __MTKWgamma ^ 2, σ))))), (getproperty(Base, :*))(-1, __MTKWgamma, β))) end end) + construct = (x->begin + A = SMatrix{(3, 3)...}(x) + (getproperty(StaticArrays, :LU))(LowerTriangular(SMatrix{(3, 3)...}(UnitLowerTriangular(A))), UpperTrian +gular(A), SVector(ntuple((n->begin + n + end), max((3, 3)...)))) + end) + return construct(X) end - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:67 =# - T = promote_type(map(typeof, X)...) - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:68 =# - map(T, X) - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:69 =# - construct = (x->begin - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\systems\diffeqs\diffeqsystem.jl:202 =# - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\systems\diffeqs\diffeqsystem.jl:203 =# - A = SMatrix{(3, 3)...}(x) - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\systems\diffeqs\diffeqsystem.jl:204 =# - StaticArrays.LU(LowerTriangular(SMatrix{(3, 3)...}(UnitLowerTriangular(A))), UpperTriangular(A), SVector(ntu -ple((n->begin - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\systems\diffeqs\diffeqsystem -.jl:204 =# - n - end), max((3, 3)...)))) - end) - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:70 =# - construct(X) - end), :((var"##MTIIPVar#1286", u, p, gam, t)->begin - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:75 =# - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:76 =# - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:76 =# @inbounds begin - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:77 =# - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:56 =# @inbounds begin - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:56 =# - let (x, y, z, σ, ρ, β) = (u[1], u[2], u[3], p[1], p[2], p[3]) - var"##MTIIPVar#1286"[1] = σ * -1 * gam + -1 - var"##MTIIPVar#1286"[2] = gam * (ρ - z) * inv(σ * -1 * gam + -1) - var"##MTIIPVar#1286"[3] = gam * y * inv(σ * -1 * gam + -1) - var"##MTIIPVar#1286"[4] = gam * σ - var"##MTIIPVar#1286"[5] = (gam * -1 + -1) - gam * (ρ - z) * inv(σ * -1 * gam + -1) * gam * σ - var"##MTIIPVar#1286"[6] = (gam * x - gam * y * inv(σ * -1 * gam + -1) * gam * σ) * inv((gam * -1 + - -1) - gam * (ρ - z) * inv(σ * -1 * gam + -1) * gam * σ) - var"##MTIIPVar#1286"[7] = 0 - var"##MTIIPVar#1286"[8] = x * -1 * gam - 0 - var"##MTIIPVar#1286"[9] = ((-1 * β * gam + -1) - 0) - (gam * x - gam * y * inv(σ * -1 * gam + -1) * -gam * σ) * inv((gam * -1 + -1) - gam * (ρ - z) * inv(σ * -1 * gam + -1) * gam * σ) * (x * -1 * gam - 0) - end - end + end), :((var"##MTIIPVar#383", var"##MTKArg#378", var"##MTKArg#379", var"##MTKArg#380", var"##MTKArg#381")->begin + @inbounds begin + let (x, y, z, σ, ρ, β, __MTKWgamma, t) = (var"##MTKArg#378"[1], var"##MTKArg#378"[2], var"##MTKArg#378"[3], var" +##MTKArg#379"[1], var"##MTKArg#379"[2], var"##MTKArg#379"[3], var"##MTKArg#380", var"##MTKArg#381") + var"##MTIIPVar#383"[1] = (getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ)) + var"##MTIIPVar#383"[2] = (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getp +roperty(Base, :*))(-1, __MTKWgamma, σ))), __MTKWgamma, (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ)) + var"##MTIIPVar#383"[3] = (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getp +roperty(Base, :*))(-1, __MTKWgamma, σ))), y, __MTKWgamma) + var"##MTIIPVar#383"[4] = (getproperty(Base, :*))(__MTKWgamma, σ) + var"##MTIIPVar#383"[5] = (getproperty(Base, :*))(-1, (getproperty(Base, :+))(1, (getproperty(Base, :*))((get +property(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ))), __MTKWgamma ^ 2, σ, (getproperty( +Base, :+))((getproperty(Base, :*))(-1, z), ρ)), __MTKWgamma)) + var"##MTIIPVar#383"[6] = (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :*))(-1, (getp +roperty(Base, :+))(1, (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __ +MTKWgamma, σ))), __MTKWgamma ^ 2, σ, (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ)), __MTKWgamma))), (getproperty(Bas +e, :+))((getproperty(Base, :*))(x, __MTKWgamma), (getproperty(Base, :*))(-1, (getproperty(Base, :inv))((getproperty(Base, :+))(-1, + (getproperty(Base, :*))(-1, __MTKWgamma, σ))), y, __MTKWgamma ^ 2, σ))) + var"##MTIIPVar#383"[7] = 0 + var"##MTIIPVar#383"[8] = (getproperty(Base, :*))(-1, x, __MTKWgamma) + var"##MTIIPVar#383"[9] = (getproperty(Base, :+))((getproperty(Base, :*))(-1, (getproperty(Base, :+))(1, (get +property(Base, :*))(-1, (getproperty(Base, :inv))((getproperty(Base, :*))(-1, (getproperty(Base, :+))(1, (getproperty(Base, :*))(( +getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ))), __MTKWgamma ^ 2, σ, (getproper +ty(Base, :+))((getproperty(Base, :*))(-1, z), ρ)), __MTKWgamma))), x, __MTKWgamma, (getproperty(Base, :+))((getproperty(Base, :*)) +(x, __MTKWgamma), (getproperty(Base, :*))(-1, (getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __ +MTKWgamma, σ))), y, __MTKWgamma ^ 2, σ))))), (getproperty(Base, :*))(-1, __MTKWgamma, β)) + end end - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:79 =# nothing end)) ```` @@ -321,56 +314,39 @@ nlsys_func = generate_function(ns) ```` -(:((u, p)->begin - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:61 =# - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:62 =# - if u isa Array - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:63 =# - return #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:55 =# @inbounds(begin - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:55 =# - let (x, y, z, σ, ρ, β) = (u[1], u[2], u[3], p[1], p[2], p[3]) +(:((var"##MTKArg#393", var"##MTKArg#394")->begin + if var"##MTKArg#393" isa Array || !(typeof(var"##MTKArg#393") <: StaticArray) && false + return @inbounds(begin + let (x, y, z, σ, ρ, β) = (var"##MTKArg#393"[1], var"##MTKArg#393"[2], var"##MTKArg#393"[3], var"##MTKArg +#394"[1], var"##MTKArg#394"[2], var"##MTKArg#394"[3]) [(*)(σ, (-)(y, x)), (-)((*)(x, (-)(ρ, z)), y), (-)((*)(x, y), (*)(β, z))] end end) else - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:65 =# - X = #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:54 =# @inbounds(begin - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:54 =# - let (x, y, z, σ, ρ, β) = (u[1], u[2], u[3], p[1], p[2], p[3]) + X = @inbounds(begin + let (x, y, z, σ, ρ, β) = (var"##MTKArg#393"[1], var"##MTKArg#393"[2], var"##MTKArg#393"[3], var"##MTKArg +#394"[1], var"##MTKArg#394"[2], var"##MTKArg#394"[3]) ((*)(σ, (-)(y, x)), (-)((*)(x, (-)(ρ, z)), y), (-)((*)(x, y), (*)(β, z))) end end) - end - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:67 =# - T = promote_type(map(typeof, X)...) - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:68 =# - map(T, X) - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:69 =# - construct = if u isa ModelingToolkit.StaticArrays.StaticArray - ModelingToolkit.StaticArrays.similar_type(typeof(u), eltype(X)) - else - x->begin - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:69 =# - convert(typeof(u), x) - end - end - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:70 =# - construct(X) - end), :((var"##MTIIPVar#1290", u, p)->begin - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:75 =# - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:76 =# - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:76 =# @inbounds begin - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:77 =# - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:56 =# @inbounds begin - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:56 =# - let (x, y, z, σ, ρ, β) = (u[1], u[2], u[3], p[1], p[2], p[3]) - var"##MTIIPVar#1290"[1] = (*)(σ, (-)(y, x)) - var"##MTIIPVar#1290"[2] = (-)((*)(x, (-)(ρ, z)), y) - var"##MTIIPVar#1290"[3] = (-)((*)(x, y), (*)(β, z)) + construct = if var"##MTKArg#393" isa ModelingToolkit.StaticArrays.StaticArray + (getproperty(ModelingToolkit.StaticArrays, :similar_type))(typeof(var"##MTKArg#393"), eltype(X)) + else + x->begin + convert(typeof(var"##MTKArg#393"), x) end - end + end + return construct(X) + end + end), :((var"##MTIIPVar#396", var"##MTKArg#393", var"##MTKArg#394")->begin + @inbounds begin + let (x, y, z, σ, ρ, β) = (var"##MTKArg#393"[1], var"##MTKArg#393"[2], var"##MTKArg#393"[3], var"##MTKArg#394"[1] +, var"##MTKArg#394"[2], var"##MTKArg#394"[3]) + var"##MTIIPVar#396"[1] = (*)(σ, (-)(y, x)) + var"##MTIIPVar#396"[2] = (-)((*)(x, (-)(ρ, z)), y) + var"##MTIIPVar#396"[3] = (-)((*)(x, y), (*)(β, z)) + end end - #= C:\Users\accou\.julia\packages\ModelingToolkit\xi418\src\utils.jl:79 =# nothing end)) ```` @@ -395,7 +371,7 @@ nlsolve(f2,ones(3)) Results of Nonlinear Solver Algorithm * Algorithm: Trust-region with dogleg and autoscaling * Starting Point: [1.0, 1.0, 1.0] - * Zero: [2.2228042242798023e-10, 2.222804224296743e-10, -9.990339458476605e-11] + * Zero: [2.2228042243306243e-10, 2.2228042243645056e-10, -9.990339599422887e-11] * Inf-norm of residuals: 0.000000 * Iterations: 3 * Convergence: true @@ -425,10 +401,13 @@ de1 = ode_order_lowering(de) ```` -ODESystem(ModelingToolkit.DiffEq[ModelingToolkit.DiffEq(u_tt, 1, ((2 * u_tt(t) + u_t(t)) + x_t(t)) + 1), ModelingToolkit.DiffEq(x_ -t, 1, x_t(t) + 2), ModelingToolkit.DiffEq(u_t, 1, u_tt(t)), ModelingToolkit.DiffEq(u, 1, u_t(t)), ModelingToolkit.DiffEq(x, 1, x_t -(t))], t, Variable[u, x, u_tt, u_t, x_t], Variable[], Base.RefValue{Array{Expression,2}}(Array{Expression}(undef,0,0)), Base.RefVa -lue{Array{Expression,2}}(Array{Expression}(undef,0,0)), Base.RefValue{Array{Expression,2}}(Array{Expression}(undef,0,0))) +ModelingToolkit.ODESystem(ModelingToolkit.Equation[ModelingToolkit.Equation(derivative(uˍtt(t), t), ((2 * uˍtt(t) + uˍt(t)) + xˍt( +t)) + 1), ModelingToolkit.Equation(derivative(xˍt(t), t), xˍt(t) + 2), ModelingToolkit.Equation(derivative(uˍt(t), t), uˍtt(t)), M +odelingToolkit.Equation(derivative(u(t), t), uˍt(t)), ModelingToolkit.Equation(derivative(x(t), t), xˍt(t))], t, ModelingToolkit.V +ariable[uˍtt, xˍt, uˍt, u, x], ModelingToolkit.Variable[], Base.RefValue{Array{ModelingToolkit.Expression,1}}(ModelingToolkit.Expr +ession[]), Base.RefValue{Array{ModelingToolkit.Expression,2}}(Array{ModelingToolkit.Expression}(undef,0,0)), Base.RefValue{Array{M +odelingToolkit.Expression,2}}(Array{ModelingToolkit.Expression}(undef,0,0)), Base.RefValue{Array{ModelingToolkit.Expression,2}}(Ar +ray{ModelingToolkit.Expression}(undef,0,0)), Symbol("##ODESystem#399"), ModelingToolkit.ODESystem[]) ```` @@ -439,12 +418,12 @@ de1.eqs ```` -5-element Array{ModelingToolkit.DiffEq,1}: - ModelingToolkit.DiffEq(u_tt, 1, ((2 * u_tt(t) + u_t(t)) + x_t(t)) + 1) - ModelingToolkit.DiffEq(x_t, 1, x_t(t) + 2) - ModelingToolkit.DiffEq(u_t, 1, u_tt(t)) - ModelingToolkit.DiffEq(u, 1, u_t(t)) - ModelingToolkit.DiffEq(x, 1, x_t(t)) +5-element Array{ModelingToolkit.Equation,1}: + ModelingToolkit.Equation(derivative(uˍtt(t), t), ((2 * uˍtt(t) + uˍt(t)) + xˍt(t)) + 1) + ModelingToolkit.Equation(derivative(xˍt(t), t), xˍt(t) + 2) + ModelingToolkit.Equation(derivative(uˍt(t), t), uˍtt(t)) + ModelingToolkit.Equation(derivative(u(t), t), uˍt(t)) + ModelingToolkit.Equation(derivative(x(t), t), xˍt(t)) ```` @@ -471,7 +450,7 @@ J = [Dx(eqs[1].rhs) Dy(eqs[1].rhs) Dz(eqs[1].rhs) ```` -3×3 Array{Operation,2}: +3×3 Array{ModelingToolkit.Operation,2}: derivative(σ * (y(t) - x(t)), x(t)) … derivative(σ * (y(t) - x(t)), z(t)) derivative(x(t) * (ρ - z(t)) - y(t), x(t)) derivative(x(t) * (ρ - z(t)) - y(t), z(t)) derivative(x(t) * y(t) - β * z(t), x(t)) derivative(x(t) * y(t) - β * z(t), z(t)) @@ -489,10 +468,10 @@ J = expand_derivatives.(J) ```` -3×3 Array{Expression,2}: - σ * -1 σ Constant(0) - ρ - z(t) Constant(-1) x(t) * -1 - y(t) x(t) -1β +3×3 Array{ModelingToolkit.Expression,2}: + -1σ σ Constant(0) + -1 * z(t) + ρ Constant(-1) -1 * x(t) + y(t) x(t) -1β ```` @@ -508,20 +487,20 @@ luJ = lu(J,Val(false)) ```` -LU{Expression,Array{Expression,2}} +LinearAlgebra.LU{ModelingToolkit.Expression,Array{ModelingToolkit.Expression,2}} L factor: -3×3 Array{Expression,2}: - Constant(1) … Constant(0) - (ρ - z(t)) * inv(σ * -1) identity(0) - y(t) * inv(σ * -1) Constant(1) +3×3 Array{ModelingToolkit.Expression,2}: + Constant(1) … Constant(0) + (-1 * z(t) + ρ) * inv(-1σ) Constant(0) + y(t) * inv(-1σ) Constant(1) U factor: -3×3 Array{Expression,2}: - σ * -1 … - Constant(0) - identity(0) - x(t) * -1 - ((ρ - z(t)) * inv(σ * -1)) * 0 - identity(0) (-1β - (y(t) * inv(σ * -1)) * 0) - ((x(t) - (y(t) * inv(σ * -1)) * σ) * inv(-1 - ((ρ - z(t)) * inv(σ * -1)) * σ)) - * (x(t) * -1 - ((ρ - z(t)) * inv(σ * -1)) * 0) +3×3 Array{ModelingToolkit.Expression,2}: + -1σ … + Constant(0) + Constant(0) + -1 * x(t) - ((-1 * z(t) + ρ) * inv(-1σ)) * 0 + Constant(0) (-1β - (y(t) * inv(-1σ)) * 0) - ((x(t) - (y(t) * inv(-1σ)) * σ) * inv(-1 - ((-1 * z(t) + ρ) * inv(-1σ)) * σ)) * ( +-1 * x(t) - ((-1 * z(t) + ρ) * inv(-1σ)) * 0) ```` @@ -532,10 +511,10 @@ luJ.L ```` -3×3 Array{Expression,2}: - Constant(1) … Constant(0) - (ρ - z(t)) * inv(σ * -1) identity(0) - y(t) * inv(σ * -1) Constant(1) +3×3 Array{ModelingToolkit.Expression,2}: + Constant(1) … Constant(0) + (-1 * z(t) + ρ) * inv(-1σ) Constant(0) + y(t) * inv(-1σ) Constant(1) ```` @@ -550,46 +529,46 @@ invJ = inv(luJ) ```` -3×3 Array{Expression,2}: - (σ * -1) \ ((true - 0 * (((-1β - (y(t) * inv(σ * -1)) * 0) - ((x(t) - (y(t) * inv(σ * -1)) * σ) * inv(-1 - ((ρ - z(t)) * inv(σ * --1)) * σ)) * (x(t) * -1 - ((ρ - z(t)) * inv(σ * -1)) * 0)) \ ((0 - (y(t) * inv(σ * -1)) * true) - ((x(t) - (y(t) * inv(σ * -1)) * -σ) * inv(-1 - ((ρ - z(t)) * inv(σ * -1)) * σ)) * (0 - ((ρ - z(t)) * inv(σ * -1)) * true)))) - σ * ((-1 - ((ρ - z(t)) * inv(σ * -1) -) * σ) \ ((0 - ((ρ - z(t)) * inv(σ * -1)) * true) - (x(t) * -1 - ((ρ - z(t)) * inv(σ * -1)) * 0) * (((-1β - (y(t) * inv(σ * -1)) * - 0) - ((x(t) - (y(t) * inv(σ * -1)) * σ) * inv(-1 - ((ρ - z(t)) * inv(σ * -1)) * σ)) * (x(t) * -1 - ((ρ - z(t)) * inv(σ * -1)) * 0 -)) \ ((0 - (y(t) * inv(σ * -1)) * true) - ((x(t) - (y(t) * inv(σ * -1)) * σ) * inv(-1 - ((ρ - z(t)) * inv(σ * -1)) * σ)) * (0 - (( -ρ - z(t)) * inv(σ * -1)) * true)))))) … (σ * -1) \ ((0 - 0 * (((-1β - (y(t) * inv(σ * -1)) * 0) - ((x(t) - (y(t) * inv(σ * -1)) -* σ) * inv(-1 - ((ρ - z(t)) * inv(σ * -1)) * σ)) * (x(t) * -1 - ((ρ - z(t)) * inv(σ * -1)) * 0)) \ ((true - (y(t) * inv(σ * -1)) * - 0) - ((x(t) - (y(t) * inv(σ * -1)) * σ) * inv(-1 - ((ρ - z(t)) * inv(σ * -1)) * σ)) * (0 - ((ρ - z(t)) * inv(σ * -1)) * 0)))) - σ - * ((-1 - ((ρ - z(t)) * inv(σ * -1)) * σ) \ ((0 - ((ρ - z(t)) * inv(σ * -1)) * 0) - (x(t) * -1 - ((ρ - z(t)) * inv(σ * -1)) * 0) * - (((-1β - (y(t) * inv(σ * -1)) * 0) - ((x(t) - (y(t) * inv(σ * -1)) * σ) * inv(-1 - ((ρ - z(t)) * inv(σ * -1)) * σ)) * (x(t) * -1 -- ((ρ - z(t)) * inv(σ * -1)) * 0)) \ ((true - (y(t) * inv(σ * -1)) * 0) - ((x(t) - (y(t) * inv(σ * -1)) * σ) * inv(-1 - ((ρ - z(t) -) * inv(σ * -1)) * σ)) * (0 - ((ρ - z(t)) * inv(σ * -1)) * 0)))))) - +3×3 Array{ModelingToolkit.Expression,2}: + (-1σ) \ ((true - 0 * (((-1β - (y(t) * inv(-1σ)) * 0) - ((x(t) - (y(t) * inv(-1σ)) * σ) * inv(-1 - ((-1 * z(t) + ρ) * inv(-1σ)) * +σ)) * (-1 * x(t) - ((-1 * z(t) + ρ) * inv(-1σ)) * 0)) \ ((0 - (y(t) * inv(-1σ)) * true) - ((x(t) - (y(t) * inv(-1σ)) * σ) * inv(-1 + - ((-1 * z(t) + ρ) * inv(-1σ)) * σ)) * (0 - ((-1 * z(t) + ρ) * inv(-1σ)) * true)))) - σ * ((-1 - ((-1 * z(t) + ρ) * inv(-1σ)) * σ +) \ ((0 - ((-1 * z(t) + ρ) * inv(-1σ)) * true) - (-1 * x(t) - ((-1 * z(t) + ρ) * inv(-1σ)) * 0) * (((-1β - (y(t) * inv(-1σ)) * 0) +- ((x(t) - (y(t) * inv(-1σ)) * σ) * inv(-1 - ((-1 * z(t) + ρ) * inv(-1σ)) * σ)) * (-1 * x(t) - ((-1 * z(t) + ρ) * inv(-1σ)) * 0)) +\ ((0 - (y(t) * inv(-1σ)) * true) - ((x(t) - (y(t) * inv(-1σ)) * σ) * inv(-1 - ((-1 * z(t) + ρ) * inv(-1σ)) * σ)) * (0 - ((-1 * z( +t) + ρ) * inv(-1σ)) * true)))))) … (-1σ) \ ((0 - 0 * (((-1β - (y(t) * inv(-1σ)) * 0) - ((x(t) - (y(t) * inv(-1σ)) * σ) * inv(-1 +- ((-1 * z(t) + ρ) * inv(-1σ)) * σ)) * (-1 * x(t) - ((-1 * z(t) + ρ) * inv(-1σ)) * 0)) \ ((true - (y(t) * inv(-1σ)) * 0) - ((x(t) +- (y(t) * inv(-1σ)) * σ) * inv(-1 - ((-1 * z(t) + ρ) * inv(-1σ)) * σ)) * (0 - ((-1 * z(t) + ρ) * inv(-1σ)) * 0)))) - σ * ((-1 - (( +-1 * z(t) + ρ) * inv(-1σ)) * σ) \ ((0 - ((-1 * z(t) + ρ) * inv(-1σ)) * 0) - (-1 * x(t) - ((-1 * z(t) + ρ) * inv(-1σ)) * 0) * (((-1 +β - (y(t) * inv(-1σ)) * 0) - ((x(t) - (y(t) * inv(-1σ)) * σ) * inv(-1 - ((-1 * z(t) + ρ) * inv(-1σ)) * σ)) * (-1 * x(t) - ((-1 * z +(t) + ρ) * inv(-1σ)) * 0)) \ ((true - (y(t) * inv(-1σ)) * 0) - ((x(t) - (y(t) * inv(-1σ)) * σ) * inv(-1 - ((-1 * z(t) + ρ) * inv(- +1σ)) * σ)) * (0 - ((-1 * z(t) + ρ) * inv(-1σ)) * 0)))))) - (-1 - ((ρ - z(t)) * inv(σ * - -1)) * σ) \ ((0 - ((ρ - z(t)) * inv(σ * -1)) * true) - (x(t) * -1 - ((ρ - z(t)) * inv(σ * -1)) * 0) * (((-1β - (y(t) * inv(σ * -1)) - * 0) - ((x(t) - (y(t) * inv(σ * -1)) * σ) * inv(-1 - ((ρ - z(t)) * inv(σ * -1)) * σ)) * (x(t) * -1 - ((ρ - z(t)) * inv(σ * -1)) * - 0)) \ ((0 - (y(t) * inv(σ * -1)) * true) - ((x(t) - (y(t) * inv(σ * -1)) * σ) * inv(-1 - ((ρ - z(t)) * inv(σ * -1)) * σ)) * (0 - -((ρ - z(t)) * inv(σ * -1)) * true)))) + (-1 - ((-1 * z(t) + ρ) * inv(-1σ)) * + σ) \ ((0 - ((-1 * z(t) + ρ) * inv(-1σ)) * true) - (-1 * x(t) - ((-1 * z(t) + ρ) * inv(-1σ)) * 0) * (((-1β - (y(t) * inv(-1σ)) * 0 +) - ((x(t) - (y(t) * inv(-1σ)) * σ) * inv(-1 - ((-1 * z(t) + ρ) * inv(-1σ)) * σ)) * (-1 * x(t) - ((-1 * z(t) + ρ) * inv(-1σ)) * 0) +) \ ((0 - (y(t) * inv(-1σ)) * true) - ((x(t) - (y(t) * inv(-1σ)) * σ) * inv(-1 - ((-1 * z(t) + ρ) * inv(-1σ)) * σ)) * (0 - ((-1 * +z(t) + ρ) * inv(-1σ)) * true)))) - (-1 - ((ρ - z(t)) * inv(σ * -1)) * σ) \ ((0 - ((ρ - z(t)) * inv(σ * -1)) * 0) - (x(t) * -1 - ((ρ - z(t)) * inv(σ * -1)) * 0) - * (((-1β - (y(t) * inv(σ * -1)) * 0) - ((x(t) - (y(t) * inv(σ * -1)) * σ) * inv(-1 - ((ρ - z(t)) * inv(σ * -1)) * σ)) * (x(t) * - -1 - ((ρ - z(t)) * inv(σ * -1)) * 0)) \ ((true - (y(t) * inv(σ * -1)) * 0) - ((x(t) - (y(t) * inv(σ * -1)) * σ) * inv(-1 - ((ρ - z( -t)) * inv(σ * -1)) * σ)) * (0 - ((ρ - z(t)) * inv(σ * -1)) * 0)))) + (-1 - +((-1 * z(t) + ρ) * inv(-1σ)) * σ) \ ((0 - ((-1 * z(t) + ρ) * inv(-1σ)) * 0) - (-1 * x(t) - ((-1 * z(t) + ρ) * inv(-1σ)) * 0) * ((( +-1β - (y(t) * inv(-1σ)) * 0) - ((x(t) - (y(t) * inv(-1σ)) * σ) * inv(-1 - ((-1 * z(t) + ρ) * inv(-1σ)) * σ)) * (-1 * x(t) - ((-1 * + z(t) + ρ) * inv(-1σ)) * 0)) \ ((true - (y(t) * inv(-1σ)) * 0) - ((x(t) - (y(t) * inv(-1σ)) * σ) * inv(-1 - ((-1 * z(t) + ρ) * inv +(-1σ)) * σ)) * (0 - ((-1 * z(t) + ρ) * inv(-1σ)) * 0)))) - ((-1β - (y(t) * inv(σ * -1 -)) * 0) - ((x(t) - (y(t) * inv(σ * -1)) * σ) * inv(-1 - ((ρ - z(t)) * inv(σ * -1)) * σ)) * (x(t) * -1 - ((ρ - z(t)) * inv(σ * -1)) - * 0)) \ ((0 - (y(t) * inv(σ * -1)) * true) - ((x(t) - (y(t) * inv(σ * -1)) * σ) * inv(-1 - ((ρ - z(t)) * inv(σ * -1)) * σ)) * (0 -- ((ρ - z(t)) * inv(σ * -1)) * true)) + ((-1β - (y(t) * inv(-1σ)) * + 0) - ((x(t) - (y(t) * inv(-1σ)) * σ) * inv(-1 - ((-1 * z(t) + ρ) * inv(-1σ)) * σ)) * (-1 * x(t) - ((-1 * z(t) + ρ) * inv(-1σ)) * +0)) \ ((0 - (y(t) * inv(-1σ)) * true) - ((x(t) - (y(t) * inv(-1σ)) * σ) * inv(-1 - ((-1 * z(t) + ρ) * inv(-1σ)) * σ)) * (0 - ((-1 +* z(t) + ρ) * inv(-1σ)) * true)) - ((-1β - (y(t) * inv(σ * -1)) * 0) - ((x(t) - (y(t) * inv(σ * -1)) * σ) * inv(-1 - ((ρ - z(t)) * inv(σ * -1)) * σ)) * (x(t) * - -1 - ((ρ - z(t)) * inv(σ * -1)) * 0)) \ ((true - (y(t) * inv(σ * -1)) * 0) - ((x(t) - (y(t) * inv(σ * -1)) * σ) * inv(-1 - ((ρ - -z(t)) * inv(σ * -1)) * σ)) * (0 - ((ρ - z(t)) * inv(σ * -1)) * 0)) +((-1β - (y(t) * inv(-1σ)) * 0) - ((x(t) - (y(t) * inv(-1σ)) * σ) * inv(-1 - ((-1 * z(t) + ρ) * inv(-1σ)) * σ)) * (-1 * x(t) - ((-1 + * z(t) + ρ) * inv(-1σ)) * 0)) \ ((true - (y(t) * inv(-1σ)) * 0) - ((x(t) - (y(t) * inv(-1σ)) * σ) * inv(-1 - ((-1 * z(t) + ρ) * i +nv(-1σ)) * σ)) * (0 - ((-1 * z(t) + ρ) * inv(-1σ)) * 0)) ```` @@ -633,7 +612,7 @@ du ```` -3-element Array{Operation,1}: +3-element Array{ModelingToolkit.Operation,1}: σ * (y(t) - x(t)) x(t) * (ρ - z(t)) - y(t) x(t) * y(t) - β * z(t) @@ -654,10 +633,10 @@ J = expand_derivatives.(J) ```` -3×3 Array{Expression,2}: - σ * -1 σ Constant(0) - ρ - z(t) Constant(-1) x(t) * -1 - y(t) x(t) -1β +3×3 Array{ModelingToolkit.Expression,2}: + -1σ σ Constant(0) + -1 * z(t) + ρ Constant(-1) -1 * x(t) + y(t) x(t) -1β ```` @@ -682,14 +661,14 @@ sJ = SparseMatrixCSC(J) ```` -3×3 SparseMatrixCSC{Expression,Int64} with 8 stored entries: - [1, 1] = σ * -1 - [2, 1] = ρ - z(t) +3×3 SparseArrays.SparseMatrixCSC{ModelingToolkit.Expression,Int64} with 8 stored entries: + [1, 1] = -1σ + [2, 1] = -1 * z(t) + ρ [3, 1] = y(t) [1, 2] = σ [2, 2] = Constant(-1) [3, 2] = x(t) - [2, 3] = x(t) * -1 + [2, 3] = -1 * x(t) [3, 3] = -1β ```` @@ -712,10 +691,10 @@ eqs = [D(x) ~ σ(t-1)*(y-x), ```` -3-element Array{Equation,1}: - Equation(derivative(x(t), t), σ(t - 1) * (y(t) - x(t))) - Equation(derivative(y(t), t), x(t) * (σ(t ^ 2) - z(t)) - y(t)) - Equation(derivative(z(t), t), x(t) * y(t) - β * z(t)) +3-element Array{ModelingToolkit.Equation,1}: + ModelingToolkit.Equation(derivative(x(t), t), σ(t - 1) * (y(t) - x(t))) + ModelingToolkit.Equation(derivative(y(t), t), x(t) * (σ(t ^ 2) - z(t)) - y(t)) + ModelingToolkit.Equation(derivative(z(t), t), x(t) * y(t) - β * z(t)) ```` @@ -732,7 +711,8 @@ expand_derivatives(Dₜ(x*(σ(t^2)-z)-y)) ```` -(σ(t ^ 2) - z(t)) * derivative(x(t), t) + x(t) * (derivative(σ(t ^ 2), t) + -1 * derivative(z(t), t)) + -1 * derivative(y(t), t) +derivative(x(t), t) * (σ(t ^ 2) + -1 * z(t)) + -1 * derivative(y(t), t) + x(t) * (derivative(σ(t ^ 2), t) + -1 * derivative(z(t), +t)) ```` @@ -817,16 +797,18 @@ DiffEqTutorials.weave_file("ode_extras","01-ModelingToolkit.jmd") Computer Information: ``` -Julia Version 1.3.0 -Commit 46ce4d7933 (2019-11-26 06:09 UTC) +Julia Version 1.4.2 +Commit 44fa15b150* (2020-05-23 18:35 UTC) Platform Info: - OS: Windows (x86_64-w64-mingw32) - CPU: Intel(R) Core(TM) i7-8550U CPU @ 1.80GHz + OS: Linux (x86_64-pc-linux-gnu) + CPU: Intel(R) Core(TM) i7-9700K CPU @ 3.60GHz WORD_SIZE: 64 LIBM: libopenlibm - LLVM: libLLVM-6.0.1 (ORCJIT, skylake) + LLVM: libLLVM-8.0.1 (ORCJIT, skylake) Environment: - JULIA_EDITOR = "C:\Users\accou\AppData\Local\atom\app-1.42.0\atom.exe" -a + JULIA_DEPOT_PATH = /builds/JuliaGPU/DiffEqTutorials.jl/.julia + JULIA_CUDA_MEMORY_LIMIT = 536870912 + JULIA_PROJECT = @. JULIA_NUM_THREADS = 4 ``` @@ -834,5 +816,11 @@ Environment: Package Information: ``` -Status `~\.julia\dev\DiffEqTutorials\Project.toml` +Status `/builds/JuliaGPU/DiffEqTutorials.jl/tutorials/ode_extras/Project.toml` +[961ee093-0014-501f-94e3-6117800e7a78] ModelingToolkit 3.11.0 +[2774e3e8-f4cf-5e23-947b-6d7e65073b56] NLsolve 4.4.0 +[1dea7af3-3e70-54e6-95c3-0bf5283fa5ed] OrdinaryDiffEq 5.41.0 +[91a5bcdd-55d7-5caf-9e0b-520d859cae80] Plots 1.5.0 +[37e2e46d-f89d-539d-b4ee-838fcccc9c8e] LinearAlgebra +[2f01184e-e22b-5df5-ae63-d93ebab69eaf] SparseArrays ``` diff --git a/markdown/ode_extras/figures/01-ModelingToolkit_1_1.png b/markdown/ode_extras/figures/01-ModelingToolkit_1_1.png index be8c011a594dd0199935c93690c690d88e8809bd..48c9c79c526519f77993f6f65cdfdd7e66ade3db 100644 GIT binary patch literal 135924 zcmbq)WmFwevn3WZxVw9B_u%dc?(QDk-QC>@PH>0d?(XjH&UC&vZ)SeXnm@Bxi_7h9 z`kp>jwQJX|x?u`(;_$FIupl5H@RAZDN+2L$-$6h?eWAgCBO^~z62J?jfwZ^?$mhSG 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However, while the syntax my vary greatly between these domain-specific languages (DSLs), the internals of modeling frameworks are surprisingly similar: building differential equations, calculating Jacobians, etc.\n\n#### ModelingToolkit.jl is metamodeling systemitized\n\nAfter building our third modeling interface, we realized that this problem can be better approached by having a reusable internal structure which DSLs can target. This internal is ModelingToolkit.jl: an Intermediate Representation (IR) with a well-defined interface for defining system transformations and compiling to Julia functions for use in numerical libraries. Now a DSL can easily be written by simply defining the translation to ModelingToolkit.jl's primatives and querying for the mathematical quantities one needs.\n\n### Basic usage: defining differential equation systems, with performance!\n\nLet's explore the IR itself. ModelingToolkit.jl is friendly to use, and can used as a symbolic DSL in its own right. Let's define and solve the Lorenz differential equation system using ModelingToolkit to generate the functions:" + "A lot of people are building modeling languages for their specific domains. However, while the syntax my vary greatly between these domain-specific languages (DSLs), the internals of modeling frameworks are surprisingly similar: building differential equations, calculating Jacobians, etc.\n\n#### ModelingToolkit.jl is metamodeling systemitized\n\nAfter building our third modeling interface, we realized that this problem can be better approached by having a reusable internal structure which DSLs can target. This internal is ModelingToolkit.jl: an Intermediate Representation (IR) with a well-defined interface for defining system transformations and compiling to Julia functions for use in numerical libraries. Now a DSL can easily be written by simply defining the translation to ModelingToolkit.jl's primatives and querying for the mathematical quantities one needs.\n\n### Basic usage: defining differential equation systems, with performance!\n\nLet's explore the IR itself. ModelingToolkit.jl is friendly to use, and can used as a symbolic DSL in its own right. Let's define and solve the Lorenz differential equation system using ModelingToolkit to generate the functions:" ], "metadata": {} }, @@ -393,11 +393,11 @@ "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", - "version": "1.3.0" + "version": "1.4.2" }, "kernelspec": { - "name": "julia-1.3", - "display_name": "Julia 1.3.0", + "name": "julia-1.4", + "display_name": "Julia 1.4.2", "language": "julia" } }, diff --git a/pdf/ode_extras/01-ModelingToolkit.pdf b/pdf/ode_extras/01-ModelingToolkit.pdf index 00d9fdb218de5fbfe6cf58a5b0eddc5d7c5de998..aa9d2a892b0540dc71357899f3cac48cfe97bfda 100644 GIT binary patch delta 165998 zcmZs?cT|%>`z@+e1rZUDPQU^pAWf=}*g+IfP!SPAM|zhQ@~S9R2}ML|5EVp_AiX9^ z3kVXWg&rXE7D5P+e*C`QJ$K!E*7Mz zWh-wzyCqwG9}gO*=)_^p(D9w)u;W_lB#u2i#tu%iP=6#qs`IPSZ zy#TW~q*`7`J$=)p`cK$(%!%>9Ut=e)^<9bVX};avJf6h=Q>*~!^)yRd)9Fd7ntgQ1 zG62Vkp<#>CPZSPqfBI%NfGgh0P-!W)!7z&@++Tfn{HRL~+<6)aR9)&TzQUucssHxd z!yB=$$2{uGwYo3K{&~8;Uj5i;$0z%vz?G(jo`=q{E-Dt_7cY3BiEAzgF6gH1IVGj; 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zrB@QN9&7EB?c`Z9)4;i*iCSk_8N4MH6*U&@?QCz&7wUvg0xHgUrpFXQtFr33pU&G` z=xic1(U0=pZsEQBQ!&8D7ql)Gfkw8j8TH`qtR6K;M~=1B66O4jtSDVU9$jh#pE>;HF$K8vcQ2N;Qsi!;SO+^$`|$I2$2&(JL=oTy+Emu1dAf1Z06%x4>aA$Z3QV_o0}&BEaj zakJf$LR_9iit0YHt#Q0U#?sQP@1cDN#$A-Da_rbQ768fp+4RVX;%75_x}5ifV(ypV zfyXn&WbSFvij;F8W$we$^96!<$WlzHlXsEy!oXV5pPG?0_XzbO+8hLI2O-VKuG Date: Wed, 1 Jul 2020 06:45:45 +0000 Subject: [PATCH 2/8] Rebuild ode_extras/02-feagin.jmd --- html/ode_extras/02-feagin.html | 273 ++++++++++++---------------- markdown/ode_extras/02-feagin.md | 211 +++++++++++++++++++++ notebook/ode_extras/02-feagin.ipynb | 8 +- pdf/ode_extras/02-feagin.pdf | Bin 47297 -> 42660 bytes 4 files changed, 327 insertions(+), 165 deletions(-) create mode 100644 markdown/ode_extras/02-feagin.md diff --git a/html/ode_extras/02-feagin.html b/html/ode_extras/02-feagin.html index c3f24326..81ef749a 100644 --- a/html/ode_extras/02-feagin.html +++ b/html/ode_extras/02-feagin.html @@ -650,39 +650,72 @@ div.title {text-align: center;} + + +
+
+
+
+

Feagin's Order 10, 12, and 14 Methods

+
Chris Rackauckas
+ +
- - - -
-
-
- -
-

Feagin's Order 10, 12, and 14 Methods

-
Chris Rackauckas
- -
- -

DifferentialEquations.jl includes Feagin's explicit Runge-Kutta methods of orders 10/8, 12/10, and 14/12. These methods have such high order that it's pretty much required that one uses numbers with more precision than Float64. As a prerequisite reference on how to use arbitrary number systems (including higher precision) in the numerical solvers, please see the Solving Equations in With Chosen Number Types notebook.

+

DifferentialEquations.jl includes Feagin's explicit Runge-Kutta methods of orders 10/8, 12/10, and 14/12. These methods have such high order that it's pretty much required that one uses numbers with more precision than Float64. As a prerequisite reference on how to use arbitrary number systems (including higher precision) in the numerical solvers, please see the Solving Equations in With Chosen Number Types notebook.

Investigation of the Method's Error

We can use Feagin's order 16 method as follows. Let's use a two-dimensional linear ODE. Like in the Solving Equations in With Chosen Number Types notebook, we change the initial condition to BigFloats to tell the solver to use BigFloat types.

-using DifferentialEquations
-const linear_bigα = big(1.01)
+using DifferentialEquations
+
+ + +
+ERROR: ArgumentError: Package DifferentialEquations not found in current path:
+- Run `import Pkg; Pkg.add("DifferentialEquations")` to install the DifferentialEquations package.
+
+
+ + + +
+const linear_bigα = big(1.01)
 f(u,p,t) = (linear_bigα*u)
 
 # Add analytical solution so that errors are checked
 f_analytic(u0,p,t) = u0*exp(linear_bigα*t)
-ff = ODEFunction(f,analytic=f_analytic)
-prob = ODEProblem(ff,big(0.5),(0.0,1.0))
-sol = solve(prob,Feagin14(),dt=1//16,adaptive=false);
+ff = ODEFunction(f,analytic=f_analytic)
+
+ERROR: UndefVarError: ODEFunction not defined
+
+ + + +
+prob = ODEProblem(ff,big(0.5),(0.0,1.0))
+
+ + +
+ERROR: UndefVarError: ODEProblem not defined
+
+ + + +
+sol = solve(prob,Feagin14(),dt=1//16,adaptive=false);
+
+ + +
+ERROR: UndefVarError: Feagin14 not defined
+
+ 
@@ -690,8 +723,8 @@ 
Investigation of the Method's Error
-
-Dict(:l∞=>2.19751e-23,:final=>2.19751e-23,:l2=>1.0615e-23)
+
+ERROR: UndefVarError: sol not defined
 

 
 
@@ -712,14 +745,23 @@ 
Investigation of the Method's Error

 
 
 
-sol =solve(prob,Feagin14());
-println(sol.errors); print("The length was $(length(sol))")
+sol =solve(prob,Feagin14());
 

 
 
-
-Dict(:l∞=>1.54574e-09,:final=>1.54574e-09,:l2=>8.92507e-10)
-The length was 3
+
+ERROR: UndefVarError: Feagin14 not defined
+

+
+
+
+
+println(sol.errors); print("The length was $(length(sol))")
+

+
+
+
+ERROR: UndefVarError: sol not defined
 

 
 
@@ -729,91 +771,26 @@ 
Convergence Test

 
 
 
-using DiffEqDevTools
-dts = 1.0 ./ 2.0 .^(10:-1:4)
+using DiffEqDevTools
+

+
+
+
+ERROR: ArgumentError: Package DiffEqDevTools not found in current path:
+- Run `import Pkg; Pkg.add("DiffEqDevTools")` to install the DiffEqDevTools package.
+
+

+
+
+
+
+dts = 1.0 ./ 2.0 .^(10:-1:4)
 sim = test_convergence(dts,prob,Feagin14())
 

 
 
-
-DiffEqDevTools.ConvergenceSimulation{DiffEqBase.ODESolution{BigFloat,1,Arra
-y{BigFloat,1},Array{BigFloat,1},Dict{Symbol,BigFloat},Array{Float64,1},Arra
-y{Array{BigFloat,1},1},DiffEqBase.ODEProblem{BigFloat,Tuple{Float64,Float64
-},false,Nothing,DiffEqBase.ODEFunction{false,typeof(Main.WeaveSandBox22.f),
-LinearAlgebra.UniformScaling{Bool},typeof(Main.WeaveSandBox22.f_analytic),N
-othing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing},Nothing,Dif
-fEqBase.StandardODEProblem},OrdinaryDiffEq.Feagin14,OrdinaryDiffEq.Interpol
-ationData{DiffEqBase.ODEFunction{false,typeof(Main.WeaveSandBox22.f),Linear
-Algebra.UniformScaling{Bool},typeof(Main.WeaveSandBox22.f_analytic),Nothing
-,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing},Array{BigFloat,1}
-,Array{Float64,1},Array{Array{BigFloat,1},1},OrdinaryDiffEq.Feagin14Constan
-tCache{BigFloat,Float64}},DiffEqBase.DEStats}}(DiffEqBase.ODESolution{BigFl
-oat,1,Array{BigFloat,1},Array{BigFloat,1},Dict{Symbol,BigFloat},Array{Float
-64,1},Array{Array{BigFloat,1},1},DiffEqBase.ODEProblem{BigFloat,Tuple{Float
-64,Float64},false,Nothing,DiffEqBase.ODEFunction{false,typeof(Main.WeaveSan
-dBox22.f),LinearAlgebra.UniformScaling{Bool},typeof(Main.WeaveSandBox22.f_a
-nalytic),Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing},N
-othing,DiffEqBase.StandardODEProblem},OrdinaryDiffEq.Feagin14,OrdinaryDiffE
-q.InterpolationData{DiffEqBase.ODEFunction{false,typeof(Main.WeaveSandBox22
-.f),LinearAlgebra.UniformScaling{Bool},typeof(Main.WeaveSandBox22.f_analyti
-c),Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing},Array{B
-igFloat,1},Array{Float64,1},Array{Array{BigFloat,1},1},OrdinaryDiffEq.Feagi
-n14ConstantCache{BigFloat,Float64}},DiffEqBase.DEStats}[retcode: Success
-Interpolation: 3rd order Hermite
-t: [0.0, 0.000976563, 0.00195313, 0.00292969, 0.00390625, 0.00488281, 0.005
-85938, 0.00683594, 0.0078125, 0.00878906  …  0.991211, 0.992188, 0.993164, 
-0.994141, 0.995117, 0.996094, 0.99707, 0.998047, 0.999023, 1.0]
-u: BigFloat[0.50, 0.500493, 0.500987, 0.501482, 0.501977, 0.502472, 0.50296
-8, 0.503464, 0.503961, 0.504458  …  1.36067, 1.36201, 1.36335, 1.3647, 1.36
-605, 1.3674, 1.36874, 1.3701, 1.37145, 1.3728], retcode: Success
-Interpolation: 3rd order Hermite
-t: [0.0, 0.00195313, 0.00390625, 0.00585938, 0.0078125, 0.00976563, 0.01171
-88, 0.0136719, 0.015625, 0.0175781  …  0.982422, 0.984375, 0.986328, 0.9882
-81, 0.990234, 0.992188, 0.994141, 0.996094, 0.998047, 1.0]
-u: BigFloat[0.50, 0.500987, 0.501977, 0.502968, 0.503961, 0.504956, 0.50595
-3, 0.506952, 0.507953, 0.508956  …  1.34864, 1.35131, 1.35397, 1.35665, 1.3
-5933, 1.36201, 1.3647, 1.3674, 1.3701, 1.3728], retcode: Success
-Interpolation: 3rd order Hermite
-t: [0.0, 0.00390625, 0.0078125, 0.0117188, 0.015625, 0.0195313, 0.0234375, 
-0.0273438, 0.03125, 0.0351563  …  0.964844, 0.96875, 0.972656, 0.976563, 0.
-980469, 0.984375, 0.988281, 0.992188, 0.996094, 1.0]
-u: BigFloat[0.50, 0.501977, 0.503961, 0.505953, 0.507953, 0.509961, 0.51197
-7, 0.514001, 0.516033, 0.518073  …  1.32491, 1.33015, 1.33541, 1.34069, 1.3
-4599, 1.35131, 1.35665, 1.36201, 1.3674, 1.3728], retcode: Success
-Interpolation: 3rd order Hermite
-t: [0.0, 0.0078125, 0.015625, 0.0234375, 0.03125, 0.0390625, 0.046875, 0.05
-46875, 0.0625, 0.0703125  …  0.929688, 0.9375, 0.945313, 0.953125, 0.960938
-, 0.96875, 0.976563, 0.984375, 0.992188, 1.0]
-u: BigFloat[0.50, 0.503961, 0.507953, 0.511977, 0.516033, 0.520121, 0.52424
-1, 0.528394, 0.53258, 0.536799  …  1.27869, 1.28882, 1.29903, 1.30932, 1.31
-969, 1.33015, 1.34069, 1.35131, 1.36201, 1.3728], retcode: Success
-Interpolation: 3rd order Hermite
-t: [0.0, 0.015625, 0.03125, 0.046875, 0.0625, 0.078125, 0.09375, 0.109375, 
-0.125, 0.140625  …  0.859375, 0.875, 0.890625, 0.90625, 0.921875, 0.9375, 0
-.953125, 0.96875, 0.984375, 1.0]
-u: BigFloat[0.50, 0.507953, 0.516033, 0.524241, 0.53258, 0.541051, 0.549658
-, 0.558401, 0.567283, 0.576306  …  1.19103, 1.20998, 1.22923, 1.24878, 1.26
-864, 1.28882, 1.30932, 1.33015, 1.35131, 1.3728], retcode: Success
-Interpolation: 3rd order Hermite
-t: [0.0, 0.03125, 0.0625, 0.09375, 0.125, 0.15625, 0.1875, 0.21875, 0.25, 0
-.28125  …  0.71875, 0.75, 0.78125, 0.8125, 0.84375, 0.875, 0.90625, 0.9375,
- 0.96875, 1.0]
-u: BigFloat[0.50, 0.516033, 0.53258, 0.549658, 0.567283, 0.585473, 0.604247
-, 0.623623, 0.64362, 0.664258  …  1.03333, 1.06647, 1.10067, 1.13596, 1.172
-39, 1.20998, 1.24878, 1.28882, 1.33015, 1.3728], retcode: Success
-Interpolation: 3rd order Hermite
-t: [0.0, 0.0625, 0.125, 0.1875, 0.25, 0.3125, 0.375, 0.4375, 0.5, 0.5625, 0
-.625, 0.6875, 0.75, 0.8125, 0.875, 0.9375, 1.0]
-u: BigFloat[0.50, 0.53258, 0.567283, 0.604247, 0.64362, 0.685558, 0.730229,
- 0.777811, 0.828493, 0.882477, 0.93998, 1.00123, 1.06647, 1.13596, 1.20998,
- 1.28882, 1.3728]], Dict{Any,Any}(:l∞=>BigFloat[3.35435e-49, 5.07978e-45, 6
-.96505e-41, 6.99856e-37, 2.7616e-33, 4.96506e-28, 2.19751e-23],:final=>BigF
-loat[3.35435e-49, 5.07978e-45, 6.96505e-41, 6.99856e-37, 2.7616e-33, 4.9650
-6e-28, 2.19751e-23],:l2=>BigFloat[1.55766e-49, 2.36041e-45, 3.24061e-41, 3.
-26457e-37, 1.29478e-33, 2.35149e-28, 1.0615e-23]), 7, Dict(:dts=>[0.0009765
-63, 0.00195313, 0.00390625, 0.0078125, 0.015625, 0.03125, 0.0625]), Dict{An
-y,Any}(:l∞=>14.2933,:final=>14.2933,:l2=>14.3028), [0.000976563, 0.00195313
-, 0.00390625, 0.0078125, 0.015625, 0.03125, 0.0625])
+
+ERROR: UndefVarError: Feagin14 not defined
 

 
 
@@ -827,7 +804,10 @@ 
Convergence Test

 

 
 
-
+
+ERROR: UndefVarError: sim not defined
+

+
 
 
This is a clear trend indicating that the convergence is truly Order 14, which is the estimated slope.

 
@@ -841,72 +821,43 @@ 
Convergence Test

Computer Information:

-
Julia Version 1.1.1
-Commit 55e36cc308 (2019-05-16 04:10 UTC)
+
Julia Version 1.4.2
+Commit 44fa15b150* (2020-05-23 18:35 UTC)
 Platform Info:
   OS: Linux (x86_64-pc-linux-gnu)
-  CPU: Intel(R) Core(TM) i7-3770 CPU @ 3.40GHz
+  CPU: Intel(R) Core(TM) i7-9700K CPU @ 3.60GHz
   WORD_SIZE: 64
   LIBM: libopenlibm
-  LLVM: libLLVM-6.0.1 (ORCJIT, ivybridge)
+  LLVM: libLLVM-8.0.1 (ORCJIT, skylake)
+Environment:
+  JULIA_DEPOT_PATH = /builds/JuliaGPU/DiffEqTutorials.jl/.julia
+  JULIA_CUDA_MEMORY_LIMIT = 536870912
+  JULIA_PROJECT = @.
+  JULIA_NUM_THREADS = 4
 

 

 
Package Information:

 

-
Status `~/.julia/environments/v1.1/Project.toml`
-[7e558dbc-694d-5a72-987c-6f4ebed21442] ArbNumerics 0.5.4
-[6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf] BenchmarkTools 0.4.2
-[be33ccc6-a3ff-5ff2-a52e-74243cff1e17] CUDAnative 2.2.0
-[3a865a2d-5b23-5a0f-bc46-62713ec82fae] CuArrays 1.0.2
-[55939f99-70c6-5e9b-8bb0-5071ed7d61fd] DecFP 0.4.8
-[abce61dc-4473-55a0-ba07-351d65e31d42] Decimals 0.4.0
-[ebbdde9d-f333-5424-9be2-dbf1e9acfb5e] DiffEqBayes 1.1.0
-[eb300fae-53e8-50a0-950c-e21f52c2b7e0] DiffEqBiological 3.8.2
-[459566f4-90b8-5000-8ac3-15dfb0a30def] DiffEqCallbacks 2.5.2
-[f3b72e0c-5b89-59e1-b016-84e28bfd966d] DiffEqDevTools 2.9.0
-[1130ab10-4a5a-5621-a13d-e4788d82bd4c] DiffEqParamEstim 1.6.0
-[055956cb-9e8b-5191-98cc-73ae4a59e68a] DiffEqPhysics 3.1.0
-[6d1b261a-3be8-11e9-3f2f-0b112a9a8436] DiffEqTutorials 0.1.0
-[0c46a032-eb83-5123-abaf-570d42b7fbaa] DifferentialEquations 6.4.0
-[31c24e10-a181-5473-b8eb-7969acd0382f] Distributions 0.20.0
-[497a8b3b-efae-58df-a0af-a86822472b78] DoubleFloats 0.9.1
-[f6369f11-7733-5829-9624-2563aa707210] ForwardDiff 0.10.3
-[c91e804a-d5a3-530f-b6f0-dfbca275c004] Gadfly 1.0.1
-[7073ff75-c697-5162-941a-fcdaad2a7d2a] IJulia 1.18.1
-[4138dd39-2aa7-5051-a626-17a0bb65d9c8] JLD 0.9.1
-[23fbe1c1-3f47-55db-b15f-69d7ec21a316] Latexify 0.8.2
-[eff96d63-e80a-5855-80a2-b1b0885c5ab7] Measurements 2.0.0
-[961ee093-0014-501f-94e3-6117800e7a78] ModelingToolkit 0.2.0
-[76087f3c-5699-56af-9a33-bf431cd00edd] NLopt 0.5.1
-[2774e3e8-f4cf-5e23-947b-6d7e65073b56] NLsolve 4.0.0
-[429524aa-4258-5aef-a3af-852621145aeb] Optim 0.18.1
-[1dea7af3-3e70-54e6-95c3-0bf5283fa5ed] OrdinaryDiffEq 5.8.1
-[65888b18-ceab-5e60-b2b9-181511a3b968] ParameterizedFunctions 4.1.1
-[91a5bcdd-55d7-5caf-9e0b-520d859cae80] Plots 0.25.1
-[d330b81b-6aea-500a-939a-2ce795aea3ee] PyPlot 2.8.1
-[731186ca-8d62-57ce-b412-fbd966d074cd] RecursiveArrayTools 0.20.0
-[90137ffa-7385-5640-81b9-e52037218182] StaticArrays 0.11.0
-[f3b207a7-027a-5e70-b257-86293d7955fd] StatsPlots 0.11.0
-[c3572dad-4567-51f8-b174-8c6c989267f4] Sundials 3.6.1
-[1986cc42-f94f-5a68-af5c-568840ba703d] Unitful 0.15.0
-[44d3d7a6-8a23-5bf8-98c5-b353f8df5ec9] Weave 0.9.0
-[b77e0a4c-d291-57a0-90e8-8db25a27a240] InteractiveUtils
+
Status `/builds/JuliaGPU/DiffEqTutorials.jl/tutorials/ode_extras/Project.toml`
+[961ee093-0014-501f-94e3-6117800e7a78] ModelingToolkit 3.11.0
+[2774e3e8-f4cf-5e23-947b-6d7e65073b56] NLsolve 4.4.0
+[1dea7af3-3e70-54e6-95c3-0bf5283fa5ed] OrdinaryDiffEq 5.41.0
+[91a5bcdd-55d7-5caf-9e0b-520d859cae80] Plots 1.5.0
 [37e2e46d-f89d-539d-b4ee-838fcccc9c8e] LinearAlgebra
-[44cfe95a-1eb2-52ea-b672-e2afdf69b78f] Pkg

+[2f01184e-e22b-5df5-ae63-d93ebab69eaf] SparseArrays

 

 
 
-
-          

-          

-          Published from 02-feagin.jmd using
-          Weave.jl
-           on 2019-06-27.
-          

-
-
+        

+        

+          

+            Published from 02-feagin.jmd
+            using Weave.jl v0.10.2 on 2020-07-01.
+
- +
+ + diff --git a/markdown/ode_extras/02-feagin.md b/markdown/ode_extras/02-feagin.md new file mode 100644 index 00000000..ee515a3c --- /dev/null +++ b/markdown/ode_extras/02-feagin.md @@ -0,0 +1,211 @@ +--- +author: "Chris Rackauckas" +title: "Feagin's Order 10, 12, and 14 Methods" +--- + + +DifferentialEquations.jl includes Feagin's explicit Runge-Kutta methods of orders 10/8, 12/10, and 14/12. These methods have such high order that it's pretty much required that one uses numbers with more precision than Float64. As a prerequisite reference on how to use arbitrary number systems (including higher precision) in the numerical solvers, please see the Solving Equations in With Chosen Number Types notebook. + +## Investigation of the Method's Error + +We can use Feagin's order 16 method as follows. Let's use a two-dimensional linear ODE. Like in the Solving Equations in With Chosen Number Types notebook, we change the initial condition to BigFloats to tell the solver to use BigFloat types. + +````julia +using DifferentialEquations +```` + + +```` +Error: ArgumentError: Package DifferentialEquations not found in current pa +th: +- Run `import Pkg; Pkg.add("DifferentialEquations")` to install the Differe +ntialEquations package. +```` + + + +````julia +const linear_bigα = big(1.01) +f(u,p,t) = (linear_bigα*u) + +# Add analytical solution so that errors are checked +f_analytic(u0,p,t) = u0*exp(linear_bigα*t) +ff = ODEFunction(f,analytic=f_analytic) +```` + + +```` +Error: UndefVarError: ODEFunction not defined +```` + + + +````julia +prob = ODEProblem(ff,big(0.5),(0.0,1.0)) +```` + + +```` +Error: UndefVarError: ODEProblem not defined +```` + + + +````julia +sol = solve(prob,Feagin14(),dt=1//16,adaptive=false); +```` + + +```` +Error: UndefVarError: Feagin14 not defined +```` + + + +````julia +println(sol.errors) +```` + + +```` +Error: UndefVarError: sol not defined +```` + + + + + +Compare that to machine $\epsilon$ for Float64: + +````julia +eps(Float64) +```` + + +```` +2.220446049250313e-16 +```` + + + + + +The error for Feagin's method when the stepsize is 1/16 is 8 orders of magnitude below machine $\epsilon$! However, that is dependent on the stepsize. If we instead use adaptive timestepping with the default tolerances, we get + +````julia +sol =solve(prob,Feagin14()); +```` + + +```` +Error: UndefVarError: Feagin14 not defined +```` + + + +````julia +println(sol.errors); print("The length was $(length(sol))") +```` + + +```` +Error: UndefVarError: sol not defined +```` + + + + + +Notice that when the stepsize is much higher, the error goes up quickly as well. These super high order methods are best when used to gain really accurate approximations (using still modest timesteps). Some examples of where such precision is necessary is astrodynamics where the many-body problem is highly chaotic and thus sensitive to small errors. + +## Convergence Test + +The Order 14 method is awesome, but we need to make sure it's really that awesome. The following convergence test is used in the package tests in order to make sure the implementation is correct. Note that all methods have such tests in place. + +````julia +using DiffEqDevTools +```` + + +```` +Error: ArgumentError: Package DiffEqDevTools not found in current path: +- Run `import Pkg; Pkg.add("DiffEqDevTools")` to install the DiffEqDevTools + package. +```` + + + +````julia +dts = 1.0 ./ 2.0 .^(10:-1:4) +sim = test_convergence(dts,prob,Feagin14()) +```` + + +```` +Error: UndefVarError: Feagin14 not defined +```` + + + + + +For a view of what's going on, let's plot the simulation results. + +````julia +using Plots +gr() +plot(sim) +```` + + +```` +Error: UndefVarError: sim not defined +```` + + + + + +This is a clear trend indicating that the convergence is truly Order 14, which +is the estimated slope. + + +## Appendix + + This tutorial is part of the DiffEqTutorials.jl repository, found at: + +To locally run this tutorial, do the following commands: +``` +using DiffEqTutorials +DiffEqTutorials.weave_file("ode_extras","02-feagin.jmd") +``` + +Computer Information: +``` +Julia Version 1.4.2 +Commit 44fa15b150* (2020-05-23 18:35 UTC) +Platform Info: + OS: Linux (x86_64-pc-linux-gnu) + CPU: Intel(R) Core(TM) i7-9700K CPU @ 3.60GHz + WORD_SIZE: 64 + LIBM: libopenlibm + LLVM: libLLVM-8.0.1 (ORCJIT, skylake) +Environment: + JULIA_DEPOT_PATH = /builds/JuliaGPU/DiffEqTutorials.jl/.julia + JULIA_CUDA_MEMORY_LIMIT = 536870912 + JULIA_PROJECT = @. + JULIA_NUM_THREADS = 4 + +``` + +Package Information: + +``` +Status `/builds/JuliaGPU/DiffEqTutorials.jl/tutorials/ode_extras/Project.toml` +[961ee093-0014-501f-94e3-6117800e7a78] ModelingToolkit 3.11.0 +[2774e3e8-f4cf-5e23-947b-6d7e65073b56] NLsolve 4.4.0 +[1dea7af3-3e70-54e6-95c3-0bf5283fa5ed] OrdinaryDiffEq 5.41.0 +[91a5bcdd-55d7-5caf-9e0b-520d859cae80] Plots 1.5.0 +[37e2e46d-f89d-539d-b4ee-838fcccc9c8e] LinearAlgebra +[2f01184e-e22b-5df5-ae63-d93ebab69eaf] SparseArrays +``` diff --git a/notebook/ode_extras/02-feagin.ipynb b/notebook/ode_extras/02-feagin.ipynb index d6bd08f6..450285f2 100644 --- a/notebook/ode_extras/02-feagin.ipynb +++ b/notebook/ode_extras/02-feagin.ipynb @@ -3,7 +3,7 @@ { "cell_type": "markdown", "source": [ - "# Feagin's Order 10, 12, and 14 Methods\n### Chris Rackauckas\n\nDifferentialEquations.jl includes Feagin's explicit Runge-Kutta methods of orders 10/8, 12/10, and 14/12. These methods have such high order that it's pretty much required that one uses numbers with more precision than Float64. As a prerequisite reference on how to use arbitrary number systems (including higher precision) in the numerical solvers, please see the Solving Equations in With Chosen Number Types notebook.\n\n## Investigation of the Method's Error\n\nWe can use Feagin's order 16 method as follows. Let's use a two-dimensional linear ODE. Like in the Solving Equations in With Chosen Number Types notebook, we change the initial condition to BigFloats to tell the solver to use BigFloat types." + "DifferentialEquations.jl includes Feagin's explicit Runge-Kutta methods of orders 10/8, 12/10, and 14/12. These methods have such high order that it's pretty much required that one uses numbers with more precision than Float64. As a prerequisite reference on how to use arbitrary number systems (including higher precision) in the numerical solvers, please see the Solving Equations in With Chosen Number Types notebook.\n\n## Investigation of the Method's Error\n\nWe can use Feagin's order 16 method as follows. Let's use a two-dimensional linear ODE. Like in the Solving Equations in With Chosen Number Types notebook, we change the initial condition to BigFloats to tell the solver to use BigFloat types." ], "metadata": {} }, @@ -103,11 +103,11 @@ "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", - "version": "1.1.1" + "version": "1.4.2" }, "kernelspec": { - "name": "julia-1.1", - "display_name": "Julia 1.1.1", + "name": "julia-1.4", + "display_name": "Julia 1.4.2", "language": "julia" } }, diff --git a/pdf/ode_extras/02-feagin.pdf b/pdf/ode_extras/02-feagin.pdf index f5c31ff7efd83615d9a7f45a6f2f633a5a6af535..7fb9a3eb6cfdf055a0bf90d62bee87582b610c35 100644 GIT binary patch delta 40875 zcmZs?W00Uhx20RQZFSk~vTfToy6m@Xn_aeTbamP2vTfV*orxPa&Y77%8Tl_GBlB5% z?Y%Sq+=3NPfuT|;iAgZ9Fmk|A9Nb*Kz_D;8qEY~CKUo9>;9Ojt&5iBhJlBSF?*{mo2=eBct$>aw=c*$rUwS{x5JJ2oeJWyMnb=d zq@DShCh6XeI(62u#^xx_I;H?aPid6{L&v^k`FfnPYSGc3xUq08+6DyFmnIx=y^R6Y z4nIqAKn0jQjn#J^bjW!Ru04+$X5(+8D5(K?gC>OzqcgDSI||06!*;lE8^hMlh#7&= 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Wed, 1 Jul 2020 06:58:30 +0000 Subject: [PATCH 3/8] Rebuild ode_extras/03-ode_minmax.jmd --- html/ode_extras/03-ode_minmax.html | 431 +++++++++------- markdown/ode_extras/03-ode_minmax.md | 467 ++++++++++++++++++ .../ode_extras/figures/03-ode_minmax_3_1.png | Bin 0 -> 74837 bytes .../ode_extras/figures/03-ode_minmax_4_1.png | Bin 0 -> 132097 bytes notebook/ode_extras/03-ode_minmax.ipynb | 8 +- pdf/ode_extras/03-ode_minmax.pdf | Bin 281928 -> 182027 bytes 6 files changed, 728 insertions(+), 178 deletions(-) create mode 100644 markdown/ode_extras/03-ode_minmax.md create mode 100644 markdown/ode_extras/figures/03-ode_minmax_3_1.png create mode 100644 markdown/ode_extras/figures/03-ode_minmax_4_1.png diff --git a/html/ode_extras/03-ode_minmax.html b/html/ode_extras/03-ode_minmax.html index 41248685..575ce783 100644 --- a/html/ode_extras/03-ode_minmax.html +++ b/html/ode_extras/03-ode_minmax.html @@ -650,22 +650,19 @@ div.title {text-align: center;} + + +
+
+
+
+

Finding Maxima and Minima of DiffEq Solutions

+
Chris Rackauckas
+ +
- - - -
-
-
- -
-

Finding Maxima and Minima of DiffEq Solutions

-
Chris Rackauckas
- -
- -

Setup

+

Setup

In this tutorial we will show how to use Optim.jl to find the maxima and minima of solutions. Let's take a look at the double pendulum:

@@ -710,47 +707,65 @@

Setup

retcode: Success Interpolation: specialized 4th order "free" interpolation t: 193-element Array{Float64,1}: - 0.0 + 0.0 0.08332584852065579 - 0.24175280271811872 - 0.4389536500504315 - 0.6797322542488147 - 0.964763376337819 - 1.3179449556841032 - 1.7031210236280163 - 2.0678477932001846 - 2.471782525434673 - ⋮ - 95.84571836675003 - 96.35777612654726 - 96.9291238553263 - 97.4467872981331 - 97.9624744296349 - 98.51182496995675 - 99.06081878698582 - 99.58283477685029 - 100.0 + 0.24175300587841853 + 0.4389533535703127 + 0.6797301355043014 + 0.9647629621490508 + 1.3179425637594349 + 1.7031226016307728 + 2.0678503967116617 + 2.4717899847517866 + ⋮ + 95.8457309586563 + 96.3577910122243 + 96.92913461915474 + 97.44679415429573 + 97.96248479179103 + 98.51183391850897 + 99.0608253308051 + 99.58284388126884 + 100.0 u: 193-element Array{Array{Float64,1},1}: - [0.01, 0.01, 0.01, 0.01] - [0.00917069, 0.006669, 0.0124205, 0.00826641] - [0.00767328, 0.000374625, 0.0164426, 0.00463683] - [0.00612597, -0.00730546, 0.0199674, -0.000336506] - [0.0049661, -0.0163086, 0.0214407, -0.00670509] - [0.00479557, -0.0262381, 0.0188243, -0.0139134] - [0.00605469, -0.0371246, 0.0100556, -0.0210382] - [0.00790078, -0.046676, -0.00267353, -0.025183] - [0.00827652, -0.0527843, -0.0127315, -0.0252581] - [0.00552358, -0.0552525, -0.0168439, -0.021899] - ⋮ - [-0.0148868, 0.0423324, 0.0136282, 0.0180291] - [-0.00819054, 0.0544225, 0.00944831, 0.0177401] - [0.00412448, 0.0567489, -0.00515392, 0.017597] - [0.0130796, 0.0480772, -0.0137706, 0.0182866] - [0.0153161, 0.0316313, -0.00895722, 0.0171185] - [0.0111156, 0.00992938, 0.0072972, 0.0103535] - [0.00571392, -0.0117872, 0.020508, -0.00231029] - [0.00421143, -0.0299109, 0.0187506, -0.0156505] - [0.00574124, -0.0416539, 0.00741327, -0.023349] + [0.01, 0.01, 0.01, 0.01] + [0.00917068738040534, 0.0066690004553842845, 0.012420525490765832, 0.00826 +6408515192912] + [0.007673275265972518, 0.00037461737897660346, 0.016442590227730366, 0.004 +636827483318282] + [0.0061259744192393014, -0.007305450189721184, 0.01996737108423187, -0.000 +33649798308967233] + [0.0049661106627111465, -0.01630851653373806, 0.021440659476204688, -0.006 +705037098400459] + [0.004795568331019467, -0.026238103489235838, 0.018824325208837592, -0.013 +913364556753717] + [0.006054679825355352, -0.037124551879080515, 0.010055702788069582, -0.021 +038127478647355] + [0.007900784412908595, -0.04667606960847389, -0.002673581831574413, -0.025 +183036272033735] + [0.008276510489473131, -0.05278433365633968, -0.012731546444725274, -0.025 +25804037623959] + [0.005523496816741225, -0.055252504144926044, -0.01684388188262178, -0.021 +898963191274146] + ⋮ + [-0.01488675115478854, 0.042332758272484804, 0.013628283258009553, 0.01802 +9082291418783] + [-0.008190258536392677, 0.0544226798044097, 0.009448013826703633, 0.017740 +06800908162] + [0.004124711787696242, 0.056748788205059394, -0.005154187391921515, 0.0175 +9698310394298] + [0.013079718118471311, 0.04807704307739497, -0.013770661225089886, 0.01828 +6648610391705] + [0.01531604024144831, 0.0316309595575519, -0.008956991644883512, 0.0171184 +0404984504] + [0.011115490017374378, 0.009929018220630217, 0.0072974814212210725, 0.0103 +53371812537737] + [0.005713878919291268, -0.011787427051187304, 0.02050806401368939, -0.0023 +104589058526802] + [0.0042114397261269225, -0.029911199361470082, 0.01875044642290467, -0.015 +65071229490751] + [0.005741239607321662, -0.04165385985159511, 0.007413270184092719, -0.0233 +48978525280305] @@ -763,7 +778,7 @@

Setup

- +

while it has the well-known phase-space plot:

@@ -773,7 +788,7 @@

Setup

- +

Local Optimization

Let's fine out what some of the local maxima and minima are. Optim.jl can be used to minimize functions, and the solution type has a continuous interpolation which can be used. Let's look for the local optima for the 4th variable around t=20. Thus our optimization function is:

@@ -793,21 +808,25 @@

Local Optimization

-using Optim
-opt = optimize(f,18.0,22.0)
+using Optim
-
-Results of Optimization Algorithm
- * Algorithm: Brent's Method
- * Search Interval: [18.000000, 22.000000]
- * Minimizer: 1.863213e+01
- * Minimum: -2.793164e-02
- * Iterations: 11
- * Convergence: max(|x - x_upper|, |x - x_lower|) <= 2*(1.5e-08*|x|+2.2e-16
-): true
- * Objective Function Calls: 12
+
+ERROR: ArgumentError: Package Optim not found in current path:
+- Run `import Pkg; Pkg.add("Optim")` to install the Optim package.
+
+

+
+
+
+
+opt = optimize(f,18.0,22.0)
+

+
+
+
+ERROR: UndefVarError: optimize not defined
 

 
 
@@ -819,8 +838,8 @@ 
Local Optimization
-
-18.632126799595834
+
+ERROR: UndefVarError: opt not defined
 

 
 
@@ -830,8 +849,8 @@ 
Local Optimization
-
--0.027931635264246277
+
+ERROR: UndefVarError: opt not defined
 

 
 
@@ -844,16 +863,8 @@ 
Local Optimization
-
-Results of Optimization Algorithm
- * Algorithm: Brent's Method
- * Search Interval: [0.000000, 22.000000]
- * Minimizer: 1.399975e+01
- * Minimum: -2.269411e-02
- * Iterations: 13
- * Convergence: max(|x - x_upper|, |x - x_lower|) <= 2*(1.5e-08*|x|+2.2e-16
-): true
- * Objective Function Calls: 14
+
+ERROR: UndefVarError: optimize not defined
 

 
 
@@ -862,12 +873,25 @@ 
Local Optimization

 
 
 plot(sol, vars=(0,4), plotdensity=10000)
-scatter!([opt.minimizer],[opt.minimum],label="Local Min")
-scatter!([opt2.minimizer],[-opt2.minimum],label="Local Max")
+scatter!([opt.minimizer],[opt.minimum],label="Local Min")
+

+
+
+
+ERROR: UndefVarError: opt not defined
+

+
+
+
+
+scatter!([opt2.minimizer],[-opt2.minimum],label="Local Max")
 

 
 
-
+
+ERROR: UndefVarError: opt2 not defined
+

+
 
 
Brent's method will locally minimize over the full interval. If we instead want a local maxima nearest to a point, we can use BFGS(). In this case, we need to optimize a vector [t], and thus dereference it to a number using first(t).

 
@@ -878,24 +902,8 @@ 
Local Optimization
-
-Results of Optimization Algorithm
- * Algorithm: BFGS
- * Starting Point: [20.0]
- * Minimizer: [23.297607288716723]
- * Minimum: -2.588588e-02
- * Iterations: 4
- * Convergence: true
-   * |x - x'| ≤ 0.0e+00: false 
-     |x - x'| = 1.11e-04 
-   * |f(x) - f(x')| ≤ 0.0e+00 |f(x)|: false
-     |f(x) - f(x')| = -6.49e-09 |f(x)|
-   * |g(x)| ≤ 1.0e-08: true 
-     |g(x)| = 8.41e-12 
-   * Stopped by an increasing objective: false
-   * Reached Maximum Number of Iterations: false
- * Objective Calls: 16
- * Gradient Calls: 16
+
+ERROR: UndefVarError: BFGS not defined
 

 
 
@@ -904,9 +912,20 @@ 
Global Optimization

 
 
 
-import NLopt, ForwardDiff
+import NLopt, ForwardDiff
+

 
-count = 0 # keep track of # function evaluations
+
+
+ERROR: ArgumentError: Package NLopt not found in current path:
+- Run `import Pkg; Pkg.add("NLopt")` to install the NLopt package.
+
+

+
+
+
+
+count = 0 # keep track of # function evaluations
 
 function g(t::Vector, grad::Vector)
   if length(grad) > 0
@@ -915,43 +934,136 @@ 
Global Optimization

   end
   sol(first(t),idxs=4)
 end
-opt = NLopt.Opt(:GN_ORIG_DIRECT_L, 1)
-NLopt.lower_bounds!(opt, [0.0])
-NLopt.upper_bounds!(opt, [40.0])
-NLopt.xtol_rel!(opt,1e-8)
-NLopt.min_objective!(opt, g)
-(minf,minx,ret) = NLopt.optimize(opt,[20.0])
-println(minf," ",minx," ",ret)
+opt = NLopt.Opt(:GN_ORIG_DIRECT_L, 1)
 

 
 
-
--0.027931635264246215 [18.6321] XTOL_REACHED
+
+ERROR: UndefVarError: NLopt not defined
 

 
 
 
 
-NLopt.max_objective!(opt, g)
-(maxf,maxx,ret) = NLopt.optimize(opt,[20.0])
-println(maxf," ",maxx," ",ret)
+NLopt.lower_bounds!(opt, [0.0])
 

 
 
-
-0.027968571933041936 [6.5537] XTOL_REACHED
+
+ERROR: UndefVarError: NLopt not defined
+

+
+
+
+
+NLopt.upper_bounds!(opt, [40.0])
+

+
+
+
+ERROR: UndefVarError: NLopt not defined
+

+
+
+
+
+NLopt.xtol_rel!(opt,1e-8)
+

+
+
+
+ERROR: UndefVarError: NLopt not defined
+

+
+
+
+
+NLopt.min_objective!(opt, g)
+

+
+
+
+ERROR: UndefVarError: NLopt not defined
+

+
+
+
+
+(minf,minx,ret) = NLopt.optimize(opt,[20.0])
+

+
+
+
+ERROR: UndefVarError: NLopt not defined
+

+
+
+
+
+println(minf," ",minx," ",ret)
+

+
+
+
+ERROR: UndefVarError: minf not defined
+

+
+
+
+
+NLopt.max_objective!(opt, g)
+

+
+
+
+ERROR: UndefVarError: NLopt not defined
+

+
+
+
+
+(maxf,maxx,ret) = NLopt.optimize(opt,[20.0])
+

+
+
+
+ERROR: UndefVarError: NLopt not defined
+

+
+
+
+
+println(maxf," ",maxx," ",ret)
+

+
+
+
+ERROR: UndefVarError: maxf not defined
 

 
 
 
 
 plot(sol, vars=(0,4), plotdensity=10000)
-scatter!([minx],[minf],label="Global Min")
-scatter!([maxx],[maxf],label="Global Max")
+scatter!([minx],[minf],label="Global Min")
 

 
 
-
+
+ERROR: UndefVarError: minx not defined
+

+
+
+
+
+scatter!([maxx],[maxf],label="Global Max")
+

+
+
+
+ERROR: UndefVarError: maxx not defined
+

+
 
 
 
Appendix

@@ -963,72 +1075,43 @@ 
Global Optimization

 

 
Computer Information:

 

-
Julia Version 1.1.1
-Commit 55e36cc308 (2019-05-16 04:10 UTC)
+
Julia Version 1.4.2
+Commit 44fa15b150* (2020-05-23 18:35 UTC)
 Platform Info:
   OS: Linux (x86_64-pc-linux-gnu)
-  CPU: Intel(R) Core(TM) i7-3770 CPU @ 3.40GHz
+  CPU: Intel(R) Core(TM) i7-9700K CPU @ 3.60GHz
   WORD_SIZE: 64
   LIBM: libopenlibm
-  LLVM: libLLVM-6.0.1 (ORCJIT, ivybridge)
+  LLVM: libLLVM-8.0.1 (ORCJIT, skylake)
+Environment:
+  JULIA_DEPOT_PATH = /builds/JuliaGPU/DiffEqTutorials.jl/.julia
+  JULIA_CUDA_MEMORY_LIMIT = 536870912
+  JULIA_PROJECT = @.
+  JULIA_NUM_THREADS = 4
 

 

 
Package Information:

 

-
Status `~/.julia/environments/v1.1/Project.toml`
-[7e558dbc-694d-5a72-987c-6f4ebed21442] ArbNumerics 0.5.4
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-[abce61dc-4473-55a0-ba07-351d65e31d42] Decimals 0.4.0
-[ebbdde9d-f333-5424-9be2-dbf1e9acfb5e] DiffEqBayes 1.1.0
-[eb300fae-53e8-50a0-950c-e21f52c2b7e0] DiffEqBiological 3.8.2
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-[055956cb-9e8b-5191-98cc-73ae4a59e68a] DiffEqPhysics 3.1.0
-[6d1b261a-3be8-11e9-3f2f-0b112a9a8436] DiffEqTutorials 0.1.0
-[0c46a032-eb83-5123-abaf-570d42b7fbaa] DifferentialEquations 6.4.0
-[31c24e10-a181-5473-b8eb-7969acd0382f] Distributions 0.20.0
-[497a8b3b-efae-58df-a0af-a86822472b78] DoubleFloats 0.9.1
-[f6369f11-7733-5829-9624-2563aa707210] ForwardDiff 0.10.3
-[c91e804a-d5a3-530f-b6f0-dfbca275c004] Gadfly 1.0.1
-[7073ff75-c697-5162-941a-fcdaad2a7d2a] IJulia 1.18.1
-[4138dd39-2aa7-5051-a626-17a0bb65d9c8] JLD 0.9.1
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-[1dea7af3-3e70-54e6-95c3-0bf5283fa5ed] OrdinaryDiffEq 5.8.1
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-[b77e0a4c-d291-57a0-90e8-8db25a27a240] InteractiveUtils
+
Status `/builds/JuliaGPU/DiffEqTutorials.jl/tutorials/ode_extras/Project.toml`
+[961ee093-0014-501f-94e3-6117800e7a78] ModelingToolkit 3.11.0
+[2774e3e8-f4cf-5e23-947b-6d7e65073b56] NLsolve 4.4.0
+[1dea7af3-3e70-54e6-95c3-0bf5283fa5ed] OrdinaryDiffEq 5.41.0
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 [37e2e46d-f89d-539d-b4ee-838fcccc9c8e] LinearAlgebra
-[44cfe95a-1eb2-52ea-b672-e2afdf69b78f] Pkg

+[2f01184e-e22b-5df5-ae63-d93ebab69eaf] SparseArrays

 

 
 
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-          

-          

-          Published from 03-ode_minmax.jmd using
-          Weave.jl
-           on 2019-06-27.
-          

-
-
+        

+        

+          

+            Published from 03-ode_minmax.jmd
+            using Weave.jl v0.10.2 on 2020-07-01.
+
- +
+ + diff --git a/markdown/ode_extras/03-ode_minmax.md b/markdown/ode_extras/03-ode_minmax.md new file mode 100644 index 00000000..c2fa2702 --- /dev/null +++ b/markdown/ode_extras/03-ode_minmax.md @@ -0,0 +1,467 @@ +--- +author: "Chris Rackauckas" +title: "Finding Maxima and Minima of DiffEq Solutions" +--- + + +### Setup + +In this tutorial we will show how to use Optim.jl to find the maxima and minima of solutions. Let's take a look at the double pendulum: + +````julia +#Constants and setup +using OrdinaryDiffEq +initial = [0.01, 0.01, 0.01, 0.01] +tspan = (0.,100.) + +#Define the problem +function double_pendulum_hamiltonian(udot,u,p,t) + α = u[1] + lα = u[2] + β = u[3] + lβ = u[4] + udot .= + [2(lα-(1+cos(β))lβ)/(3-cos(2β)), + -2sin(α) - sin(α+β), + 2(-(1+cos(β))lα + (3+2cos(β))lβ)/(3-cos(2β)), + -sin(α+β) - 2sin(β)*(((lα-lβ)lβ)/(3-cos(2β))) + 2sin(2β)*((lα^2 - 2(1+cos(β))lα*lβ + (3+2cos(β))lβ^2)/(3-cos(2β))^2)] +end + +#Pass to solvers +poincare = ODEProblem(double_pendulum_hamiltonian, initial, tspan) +```` + + +```` +ODEProblem with uType Array{Float64,1} and tType Float64. In-place: true +timespan: (0.0, 100.0) +u0: [0.01, 0.01, 0.01, 0.01] +```` + + + +````julia +sol = solve(poincare, Tsit5()) +```` + + +```` +retcode: Success +Interpolation: specialized 4th order "free" interpolation +t: 193-element Array{Float64,1}: + 0.0 + 0.08332584852065579 + 0.24175300587841853 + 0.4389533535703127 + 0.6797301355043014 + 0.9647629621490508 + 1.3179425637594349 + 1.7031226016307728 + 2.0678503967116617 + 2.4717899847517866 + ⋮ + 95.8457309586563 + 96.3577910122243 + 96.92913461915474 + 97.44679415429573 + 97.96248479179103 + 98.51183391850897 + 99.0608253308051 + 99.58284388126884 + 100.0 +u: 193-element Array{Array{Float64,1},1}: + [0.01, 0.01, 0.01, 0.01] + [0.00917068738040534, 0.0066690004553842845, 0.012420525490765832, 0.00826 +6408515192912] + [0.007673275265972518, 0.00037461737897660346, 0.016442590227730366, 0.004 +636827483318282] + [0.0061259744192393014, -0.007305450189721184, 0.01996737108423187, -0.000 +33649798308967233] + [0.0049661106627111465, -0.01630851653373806, 0.021440659476204688, -0.006 +705037098400459] + [0.004795568331019467, -0.026238103489235838, 0.018824325208837592, -0.013 +913364556753717] + [0.006054679825355352, -0.037124551879080515, 0.010055702788069582, -0.021 +038127478647355] + [0.007900784412908595, -0.04667606960847389, -0.002673581831574413, -0.025 +183036272033735] + [0.008276510489473131, -0.05278433365633968, -0.012731546444725274, -0.025 +25804037623959] + [0.005523496816741225, -0.055252504144926044, -0.01684388188262178, -0.021 +898963191274146] + ⋮ + [-0.01488675115478854, 0.042332758272484804, 0.013628283258009553, 0.01802 +9082291418783] + [-0.008190258536392677, 0.0544226798044097, 0.009448013826703633, 0.017740 +06800908162] + [0.004124711787696242, 0.056748788205059394, -0.005154187391921515, 0.0175 +9698310394298] + [0.013079718118471311, 0.04807704307739497, -0.013770661225089886, 0.01828 +6648610391705] + [0.01531604024144831, 0.0316309595575519, -0.008956991644883512, 0.0171184 +0404984504] + [0.011115490017374378, 0.009929018220630217, 0.0072974814212210725, 0.0103 +53371812537737] + [0.005713878919291268, -0.011787427051187304, 0.02050806401368939, -0.0023 +104589058526802] + [0.0042114397261269225, -0.029911199361470082, 0.01875044642290467, -0.015 +65071229490751] + [0.005741239607321662, -0.04165385985159511, 0.007413270184092719, -0.0233 +48978525280305] +```` + + + + + +In time, the solution looks like: + +````julia +using Plots; gr() +plot(sol, vars=[(0,3),(0,4)], leg=false, plotdensity=10000) +```` + + +![](figures/03-ode_minmax_3_1.png) + + + +while it has the well-known phase-space plot: + +````julia +plot(sol, vars=(3,4), leg=false) +```` + + +![](figures/03-ode_minmax_4_1.png) + + + +### Local Optimization + +Let's fine out what some of the local maxima and minima are. Optim.jl can be used to minimize functions, and the solution type has a continuous interpolation which can be used. Let's look for the local optima for the 4th variable around `t=20`. Thus our optimization function is: + +````julia +f = (t) -> sol(t,idxs=4) +```` + + +```` +#1 (generic function with 1 method) +```` + + + + + +`first(t)` is the same as `t[1]` which transforms the array of size 1 into a number. `idxs=4` is the same as `sol(first(t))[4]` but does the calculation without a temporary array and thus is faster. To find a local minima, we can simply call Optim on this function. Let's find a local minimum: + +````julia +using Optim +```` + + +```` +Error: ArgumentError: Package Optim not found in current path: +- Run `import Pkg; Pkg.add("Optim")` to install the Optim package. +```` + + + +````julia +opt = optimize(f,18.0,22.0) +```` + + +```` +Error: UndefVarError: optimize not defined +```` + + + + + +From this printout we see that the minimum is at `t=18.63` and the value is `-2.79e-2`. We can get these in code-form via: + +````julia +println(opt.minimizer) +```` + + +```` +Error: UndefVarError: opt not defined +```` + + + +````julia +println(opt.minimum) +```` + + +```` +Error: UndefVarError: opt not defined +```` + + + + + +To get the maximum, we just minimize the negative of the function: + +````julia +f = (t) -> -sol(first(t),idxs=4) +opt2 = optimize(f,0.0,22.0) +```` + + +```` +Error: UndefVarError: optimize not defined +```` + + + + + +Let's add the maxima and minima to the plots: + +````julia +plot(sol, vars=(0,4), plotdensity=10000) +scatter!([opt.minimizer],[opt.minimum],label="Local Min") +```` + + +```` +Error: UndefVarError: opt not defined +```` + + + +````julia +scatter!([opt2.minimizer],[-opt2.minimum],label="Local Max") +```` + + +```` +Error: UndefVarError: opt2 not defined +```` + + + + + +Brent's method will locally minimize over the full interval. If we instead want a local maxima nearest to a point, we can use `BFGS()`. In this case, we need to optimize a vector `[t]`, and thus dereference it to a number using `first(t)`. + +````julia +f = (t) -> -sol(first(t),idxs=4) +opt = optimize(f,[20.0],BFGS()) +```` + + +```` +Error: UndefVarError: BFGS not defined +```` + + + + + +### Global Optimization + +If we instead want to find global maxima and minima, we need to look somewhere else. For this there are many choices. A pure Julia option is BlackBoxOptim.jl, but I will use NLopt.jl. Following the NLopt.jl tutorial but replacing their function with out own: + +````julia +import NLopt, ForwardDiff +```` + + +```` +Error: ArgumentError: Package NLopt not found in current path: +- Run `import Pkg; Pkg.add("NLopt")` to install the NLopt package. +```` + + + +````julia + +count = 0 # keep track of # function evaluations + +function g(t::Vector, grad::Vector) + if length(grad) > 0 + #use ForwardDiff for the gradients + grad[1] = ForwardDiff.derivative((t)->sol(first(t),idxs=4),t) + end + sol(first(t),idxs=4) +end +opt = NLopt.Opt(:GN_ORIG_DIRECT_L, 1) +```` + + +```` +Error: UndefVarError: NLopt not defined +```` + + + +````julia +NLopt.lower_bounds!(opt, [0.0]) +```` + + +```` +Error: UndefVarError: NLopt not defined +```` + + + +````julia +NLopt.upper_bounds!(opt, [40.0]) +```` + + +```` +Error: UndefVarError: NLopt not defined +```` + + + +````julia +NLopt.xtol_rel!(opt,1e-8) +```` + + +```` +Error: UndefVarError: NLopt not defined +```` + + + +````julia +NLopt.min_objective!(opt, g) +```` + + +```` +Error: UndefVarError: NLopt not defined +```` + + + +````julia +(minf,minx,ret) = NLopt.optimize(opt,[20.0]) +```` + + +```` +Error: UndefVarError: NLopt not defined +```` + + + +````julia +println(minf," ",minx," ",ret) +```` + + +```` +Error: UndefVarError: minf not defined +```` + + + +````julia +NLopt.max_objective!(opt, g) +```` + + +```` +Error: UndefVarError: NLopt not defined +```` + + + +````julia +(maxf,maxx,ret) = NLopt.optimize(opt,[20.0]) +```` + + +```` +Error: UndefVarError: NLopt not defined +```` + + + +````julia +println(maxf," ",maxx," ",ret) +```` + + +```` +Error: UndefVarError: maxf not defined +```` + + + +````julia +plot(sol, vars=(0,4), plotdensity=10000) +scatter!([minx],[minf],label="Global Min") +```` + + +```` +Error: UndefVarError: minx not defined +```` + + + +````julia +scatter!([maxx],[maxf],label="Global Max") +```` + + +```` +Error: UndefVarError: maxx not defined +```` + + + + +## Appendix + + This tutorial is part of the DiffEqTutorials.jl repository, found at: + +To locally run this tutorial, do the following commands: +``` +using DiffEqTutorials +DiffEqTutorials.weave_file("ode_extras","03-ode_minmax.jmd") +``` + +Computer Information: +``` +Julia Version 1.4.2 +Commit 44fa15b150* (2020-05-23 18:35 UTC) +Platform Info: + OS: Linux (x86_64-pc-linux-gnu) + CPU: Intel(R) Core(TM) i7-9700K CPU @ 3.60GHz + WORD_SIZE: 64 + LIBM: libopenlibm + LLVM: libLLVM-8.0.1 (ORCJIT, skylake) +Environment: + JULIA_DEPOT_PATH = /builds/JuliaGPU/DiffEqTutorials.jl/.julia + JULIA_CUDA_MEMORY_LIMIT = 536870912 + JULIA_PROJECT = @. + JULIA_NUM_THREADS = 4 + +``` + +Package Information: + +``` +Status `/builds/JuliaGPU/DiffEqTutorials.jl/tutorials/ode_extras/Project.toml` +[961ee093-0014-501f-94e3-6117800e7a78] ModelingToolkit 3.11.0 +[2774e3e8-f4cf-5e23-947b-6d7e65073b56] NLsolve 4.4.0 +[1dea7af3-3e70-54e6-95c3-0bf5283fa5ed] OrdinaryDiffEq 5.41.0 +[91a5bcdd-55d7-5caf-9e0b-520d859cae80] Plots 1.5.0 +[37e2e46d-f89d-539d-b4ee-838fcccc9c8e] LinearAlgebra +[2f01184e-e22b-5df5-ae63-d93ebab69eaf] SparseArrays +``` diff --git a/markdown/ode_extras/figures/03-ode_minmax_3_1.png b/markdown/ode_extras/figures/03-ode_minmax_3_1.png new file mode 100644 index 0000000000000000000000000000000000000000..f8e5eaea21fa2779fd4e083a51314824fb5c3144 GIT binary patch literal 74837 zcmb5VWmH>F^e&tPch}-pptwVEcW-eBP$C{@*ItqY08jxGq$RXHvyR)n>TPwrDDO}FcD1%DKd$jBVo_6*aQuwy8XyiK zh!$|}Kk!T^$YeRL5V$>*Y0dY!;j-_y$H&8+)~iwWQCtSHdu-}rw|x5$m>7xbC20dn%?nTgE+KX0SKce)j3*wMA<7-(Jqr*MRA>CqIZLs>=vxKbgyj z(@!Af4Z{EY=Ioc?;0OQDjg^G7cQt~$Y36Z)_VObb3$UiJ_WX3ah_-S6`|>+n7dXEE zdf@EGh-?OC!W!CCr z^Xz@S0CKF^B0-;v$pN9*%Z+j!$HXN6-+*@q;{G27^naOWF=*3DO$7DJsxNG&&8EBP zX>KpU?JhT^)n;+&A(xe%m2dq4mHtopKRXfxcmH^Cm?Op{yofzkKIZyumsXp@8ojPG z^?qE<6S?crFc~1@ju;DgioSijDuGQtB6yJtyxTzJl&?Hf^>0D&&?eoy5p3jBdv7jg zZsd1R%Kjy85!5iUN}KzM!Y85h-_MSeXHIj(!sBeaU>vKw~}T2 zQ_qlTJN04V!{K!EaIopf##CpRp1J<3Or5xn(RsfR>T-(vzH@pd6gd9rGr_#_c=_yT~Y; 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Chris Rackauckas\n\n### Setup\n\nIn this tutorial we will show how to use Optim.jl to find the maxima and minima of solutions. Let's take a look at the double pendulum:" + "### Setup\n\nIn this tutorial we will show how to use Optim.jl to find the maxima and minima of solutions. Let's take a look at the double pendulum:" ], "metadata": {} }, @@ -185,11 +185,11 @@ "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", - "version": "1.1.1" + "version": "1.4.2" }, "kernelspec": { - "name": "julia-1.1", - "display_name": "Julia 1.1.1", + "name": "julia-1.4", + "display_name": "Julia 1.4.2", "language": "julia" } }, diff --git a/pdf/ode_extras/03-ode_minmax.pdf b/pdf/ode_extras/03-ode_minmax.pdf index 3afb4537259af103eb68f0cb3a8be0f5ace9807b..6374cb35261ff5a0cbc5b27a7847f304764227ab 100644 GIT binary patch literal 182027 zcma&MWmH_v)-4<$xJw{71Pksi9U!>7I|O(4;KAK3SmW;Q?(XicjW>Ke=RN11JHC5= z-2J1vc2(^)O6FQ~)*4MA|5KEnnSmXFV(;qw5rLVLgo(t~&;o&vk5Saz%E`omQPj%7 z$>gVrk*%=_qqK>QsgoHA3nx3ffB=G{lY@zYHGAABO7kasO{77$C9j`!i)hgW|WTLmA1(l=p?oO})C$ zEs(Z~CE1ERW2SmfuH2)hGfVf+LMCk+-*eAL7~SL+x$yFiBZ{@lB^PrGJ454;k4M}h ze0v|>Umtm8&?$q2@}RSFdq~WJ;rw9^a3N=bsnb8npz)q^yQ}DkC4Ah#C`lBJ$DFsj zMiu-?kJME8sk&?kPBg4PGJ~s%CRBI&M44YW1$y@Ase6KM2#3M@VwzO^)D%M6iP6XX 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zF1xf;+otNIxS70lGtEAMrA`g$=pp3_29{G3EBR-z7;g(^59zz3lUQR0lIB;EhAnd~ zM*4p6jkl87zZxs`-yc^#G&T}XWz*+aq);->mxUoWg3uH95T+qYwuF-NES~zVMm|v* zH2ePWr8jljf0O-y{`BhgNBX0sYGw{pb#QQ{0zYSwsz4cga|f#b$h)7n61oB=U|tXx zw;7i)HwOnVx3Q@SJIIuq!~8wuSr^lo^I5IwfBr)ssAlD51^|P2fttEtDhL(3{=Z1C z*m=4B{OZCv)}$z=f*Y^`xuTSK*P&0?u3(ioKqtb> Date: Wed, 1 Jul 2020 08:35:25 -0400 Subject: [PATCH 4/8] Update Project.toml --- tutorials/ode_extras/Project.toml | 1 + 1 file changed, 1 insertion(+) diff --git a/tutorials/ode_extras/Project.toml b/tutorials/ode_extras/Project.toml index f1ead91d..7be581b8 100644 --- a/tutorials/ode_extras/Project.toml +++ b/tutorials/ode_extras/Project.toml @@ -1,4 +1,5 @@ [deps] +DiffEqDevTools = "f3b72e0c-5b89-59e1-b016-84e28bfd966d" LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" ModelingToolkit = "961ee093-0014-501f-94e3-6117800e7a78" NLsolve = "2774e3e8-f4cf-5e23-947b-6d7e65073b56" From 27702297d302ffae59eb731bbd69c13738a057a3 Mon Sep 17 00:00:00 2001 From: Christopher Rackauckas Date: Wed, 1 Jul 2020 08:36:18 -0400 Subject: [PATCH 5/8] Update Project.toml --- tutorials/ode_extras/Project.toml | 2 ++ 1 file changed, 2 insertions(+) diff --git a/tutorials/ode_extras/Project.toml b/tutorials/ode_extras/Project.toml index 7be581b8..75b7cbef 100644 --- a/tutorials/ode_extras/Project.toml +++ b/tutorials/ode_extras/Project.toml @@ -6,6 +6,8 @@ NLsolve = "2774e3e8-f4cf-5e23-947b-6d7e65073b56" OrdinaryDiffEq = "1dea7af3-3e70-54e6-95c3-0bf5283fa5ed" Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf" +Optim = "429524aa-4258-5aef-a3af-852621145aeb" +NLopt = "76087f3c-5699-56af-9a33-bf431cd00edd" [compat] ModelingToolkit = "3.10" From 83eee4ad422f66f39c8f5c55a2b1dcf9c37dab12 Mon Sep 17 00:00:00 2001 From: Christopher Rackauckas Date: Wed, 1 Jul 2020 08:36:21 -0400 Subject: [PATCH 6/8] Update Project.toml From 640a8dab8f999842897fc648ccc4a742d47b1c59 Mon Sep 17 00:00:00 2001 From: "github-actions[bot]" Date: Wed, 1 Jul 2020 12:54:33 +0000 Subject: [PATCH 7/8] Rebuild ode_extras/01-ModelingToolkit.jmd --- html/ode_extras/01-ModelingToolkit.html | 145 +++++++++++----------- markdown/ode_extras/01-ModelingToolkit.md | 143 ++++++++++----------- pdf/ode_extras/01-ModelingToolkit.pdf | Bin 170344 -> 170520 bytes tutorials/ode_extras/Project.toml | 4 +- 4 files changed, 149 insertions(+), 143 deletions(-) diff --git a/html/ode_extras/01-ModelingToolkit.html b/html/ode_extras/01-ModelingToolkit.html index 5bb5bf61..8cacdb18 100644 --- a/html/ode_extras/01-ModelingToolkit.html +++ b/html/ode_extras/01-ModelingToolkit.html @@ -698,7 +698,7 @@

Basic usage: defining differential equation systems, with performance! - +

ModelingToolkit is a compiler for mathematical systems

At its core, ModelingToolkit is a compiler. It's IR is its type system, and its output are Julia functions (it's a compiler for Julia code to Julia code, written in Julia).

@@ -711,11 +711,11 @@

ModelingToolkit is a compiler for mathematical systems

-:((var"##MTKArg#262", var"##MTKArg#263", var"##MTKArg#264")->begin
-          if var"##MTKArg#262" isa Array || !(typeof(var"##MTKArg#262") <: StaticArray) && false
+:((var"##MTKArg#263", var"##MTKArg#264", var"##MTKArg#265")->begin
+          if var"##MTKArg#263" isa Array || !(typeof(var"##MTKArg#263") <: StaticArray) && false
               return @inbounds(begin
-                          let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#262"[1], var"##MTKArg#262"[2], var"##MTKArg#262"[3], var"##MTK
-Arg#263"[1], var"##MTKArg#263"[2], var"##MTKArg#263"[3], var"##MTKArg#264")
+                          let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#263"[1], var"##MTKArg#263"[2], var"##MTKArg#263"[3], var"##MTK
+Arg#264"[1], var"##MTKArg#264"[2], var"##MTKArg#264"[3], var"##MTKArg#265")
                               [(getproperty(Base, :*))(σ, (getproperty(Base, :-))(y, x)), (getproperty(Base, :-))((getproperty(Bas
 e, :*))(x, (getproperty(Base, :-))(ρ, z)), y), (getproperty(Base, :-))((getproperty(Base, :*))(x, y), (getproperty(Base, :*))(β, z
 ))]
@@ -723,18 +723,18 @@ 
ModelingToolkit is a compiler for mathematical systems

                       end)
           else
               X = @inbounds(begin
-                          let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#262"[1], var"##MTKArg#262"[2], var"##MTKArg#262"[3], var"##MTK
-Arg#263"[1], var"##MTKArg#263"[2], var"##MTKArg#263"[3], var"##MTKArg#264")
+                          let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#263"[1], var"##MTKArg#263"[2], var"##MTKArg#263"[3], var"##MTK
+Arg#264"[1], var"##MTKArg#264"[2], var"##MTKArg#264"[3], var"##MTKArg#265")
                               ((getproperty(Base, :*))(σ, (getproperty(Base, :-))(y, x)), (getproperty(Base, :-))((getproperty(Bas
 e, :*))(x, (getproperty(Base, :-))(ρ, z)), y), (getproperty(Base, :-))((getproperty(Base, :*))(x, y), (getproperty(Base, :*))(β, z
 )))
                           end
                       end)
-              construct = if var"##MTKArg#262" isa ModelingToolkit.StaticArrays.StaticArray
-                      (getproperty(ModelingToolkit.StaticArrays, :similar_type))(typeof(var"##MTKArg#262"), eltype(X))
+              construct = if var"##MTKArg#263" isa ModelingToolkit.StaticArrays.StaticArray
+                      (getproperty(ModelingToolkit.StaticArrays, :similar_type))(typeof(var"##MTKArg#263"), eltype(X))
                   else
                       x->begin
-                              convert(typeof(var"##MTKArg#262"), x)
+                              convert(typeof(var"##MTKArg#263"), x)
                           end
                   end
               return construct(X)
@@ -752,14 +752,14 @@ 
ModelingToolkit is a compiler for mathematical systems

 
 
 
-:((var"##MTIIPVar#272", var"##MTKArg#268", var"##MTKArg#269", var"##MTKArg#270")->begin
+:((var"##MTIIPVar#273", var"##MTKArg#269", var"##MTKArg#270", var"##MTKArg#271")->begin
           @inbounds begin
-                  let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#268"[1], var"##MTKArg#268"[2], var"##MTKArg#268"[3], var"##MTKArg#269"
-[1], var"##MTKArg#269"[2], var"##MTKArg#269"[3], var"##MTKArg#270")
-                      var"##MTIIPVar#272"[1] = (getproperty(Base, :*))(σ, (getproperty(Base, :-))(y, x))
-                      var"##MTIIPVar#272"[2] = (getproperty(Base, :-))((getproperty(Base, :*))(x, (getproperty(Base, :-))(ρ, z)), 
+                  let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#269"[1], var"##MTKArg#269"[2], var"##MTKArg#269"[3], var"##MTKArg#270"
+[1], var"##MTKArg#270"[2], var"##MTKArg#270"[3], var"##MTKArg#271")
+                      var"##MTIIPVar#273"[1] = (getproperty(Base, :*))(σ, (getproperty(Base, :-))(y, x))
+                      var"##MTIIPVar#273"[2] = (getproperty(Base, :-))((getproperty(Base, :*))(x, (getproperty(Base, :-))(ρ, z)), 
 y)
-                      var"##MTIIPVar#272"[3] = (getproperty(Base, :-))((getproperty(Base, :*))(x, y), (getproperty(Base, :*))(β, z
+                      var"##MTIIPVar#273"[3] = (getproperty(Base, :-))((getproperty(Base, :*))(x, y), (getproperty(Base, :*))(β, z
 ))
                   end
               end
@@ -793,46 +793,46 @@ 
ModelingToolkit is a compiler for mathematical systems

 
 
 
-(:((var"##MTKArg#274", var"##MTKArg#275", var"##MTKArg#276")->begin
-          if var"##MTKArg#274" isa Array || !(typeof(var"##MTKArg#274") <: StaticArray) && false
+(:((var"##MTKArg#275", var"##MTKArg#276", var"##MTKArg#277")->begin
+          if var"##MTKArg#275" isa Array || !(typeof(var"##MTKArg#275") <: StaticArray) && false
               return @inbounds(begin
-                          let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#274"[1], var"##MTKArg#274"[2], var"##MTKArg#274"[3], var"##MTK
-Arg#275"[1], var"##MTKArg#275"[2], var"##MTKArg#275"[3], var"##MTKArg#276")
+                          let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#275"[1], var"##MTKArg#275"[2], var"##MTKArg#275"[3], var"##MTK
+Arg#276"[1], var"##MTKArg#276"[2], var"##MTKArg#276"[3], var"##MTKArg#277")
                               [(getproperty(Base, :*))(-1, σ) σ 0; (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ) -1 (
 getproperty(Base, :*))(-1, x); y x (getproperty(Base, :*))(-1, β)]
                           end
                       end)
           else
               X = @inbounds(begin
-                          let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#274"[1], var"##MTKArg#274"[2], var"##MTKArg#274"[3], var"##MTK
-Arg#275"[1], var"##MTKArg#275"[2], var"##MTKArg#275"[3], var"##MTKArg#276")
+                          let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#275"[1], var"##MTKArg#275"[2], var"##MTKArg#275"[3], var"##MTK
+Arg#276"[1], var"##MTKArg#276"[2], var"##MTKArg#276"[3], var"##MTKArg#277")
                               ((getproperty(Base, :*))(-1, σ), (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ), y, σ, -
 1, x, 0, (getproperty(Base, :*))(-1, x), (getproperty(Base, :*))(-1, β))
                           end
                       end)
-              construct = if var"##MTKArg#274" isa ModelingToolkit.StaticArrays.StaticArray
+              construct = if var"##MTKArg#275" isa ModelingToolkit.StaticArrays.StaticArray
                       ModelingToolkit.StaticArrays.SMatrix{3, 3}
                   else
                       x->begin
-                              out = similar(typeof(var"##MTKArg#274"), 3, 3)
+                              out = similar(typeof(var"##MTKArg#275"), 3, 3)
                               out .= x
                           end
                   end
               return construct(X)
           end
-      end), :((var"##MTIIPVar#278", var"##MTKArg#274", var"##MTKArg#275", var"##MTKArg#276")->begin
+      end), :((var"##MTIIPVar#279", var"##MTKArg#275", var"##MTKArg#276", var"##MTKArg#277")->begin
           @inbounds begin
-                  let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#274"[1], var"##MTKArg#274"[2], var"##MTKArg#274"[3], var"##MTKArg#275"
-[1], var"##MTKArg#275"[2], var"##MTKArg#275"[3], var"##MTKArg#276")
-                      var"##MTIIPVar#278"[1] = (getproperty(Base, :*))(-1, σ)
-                      var"##MTIIPVar#278"[2] = (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ)
-                      var"##MTIIPVar#278"[3] = y
-                      var"##MTIIPVar#278"[4] = σ
-                      var"##MTIIPVar#278"[5] = -1
-                      var"##MTIIPVar#278"[6] = x
-                      var"##MTIIPVar#278"[7] = 0
-                      var"##MTIIPVar#278"[8] = (getproperty(Base, :*))(-1, x)
-                      var"##MTIIPVar#278"[9] = (getproperty(Base, :*))(-1, β)
+                  let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#275"[1], var"##MTKArg#275"[2], var"##MTKArg#275"[3], var"##MTKArg#276"
+[1], var"##MTKArg#276"[2], var"##MTKArg#276"[3], var"##MTKArg#277")
+                      var"##MTIIPVar#279"[1] = (getproperty(Base, :*))(-1, σ)
+                      var"##MTIIPVar#279"[2] = (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ)
+                      var"##MTIIPVar#279"[3] = y
+                      var"##MTIIPVar#279"[4] = σ
+                      var"##MTIIPVar#279"[5] = -1
+                      var"##MTIIPVar#279"[6] = x
+                      var"##MTIIPVar#279"[7] = 0
+                      var"##MTIIPVar#279"[8] = (getproperty(Base, :*))(-1, x)
+                      var"##MTIIPVar#279"[9] = (getproperty(Base, :*))(-1, β)
                   end
               end
           nothing
@@ -849,11 +849,11 @@ 
ModelingToolkit is a compiler for mathematical systems

 
 
 
-(:((var"##MTKArg#280", var"##MTKArg#281", var"##MTKArg#282", var"##MTKArg#283")->begin
-          if var"##MTKArg#280" isa Array || !(typeof(var"##MTKArg#280") <: StaticArray) && false
+(:((var"##MTKArg#281", var"##MTKArg#282", var"##MTKArg#283", var"##MTKArg#284")->begin
+          if var"##MTKArg#281" isa Array || !(typeof(var"##MTKArg#281") <: StaticArray) && false
               return @inbounds(begin
-                          let (x, y, z, σ, ρ, β, __MTKWgamma, t) = (var"##MTKArg#280"[1], var"##MTKArg#280"[2], var"##MTKArg#280"[
-3], var"##MTKArg#281"[1], var"##MTKArg#281"[2], var"##MTKArg#281"[3], var"##MTKArg#282", var"##MTKArg#283")
+                          let (x, y, z, σ, ρ, β, __MTKWgamma, t) = (var"##MTKArg#281"[1], var"##MTKArg#281"[2], var"##MTKArg#281"[
+3], var"##MTKArg#282"[1], var"##MTKArg#282"[2], var"##MTKArg#282"[3], var"##MTKArg#283", var"##MTKArg#284")
                               [(getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ)) (getproperty(Base, :*))(__
 MTKWgamma, σ) 0; (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWg
 amma, σ))), __MTKWgamma, (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ)) (getproperty(Base, :*))(-1, (getproperty(Base
@@ -874,8 +874,8 @@ 
ModelingToolkit is a compiler for mathematical systems

                       end)
           else
               X = @inbounds(begin
-                          let (x, y, z, σ, ρ, β, __MTKWgamma, t) = (var"##MTKArg#280"[1], var"##MTKArg#280"[2], var"##MTKArg#280"[
-3], var"##MTKArg#281"[1], var"##MTKArg#281"[2], var"##MTKArg#281"[3], var"##MTKArg#282", var"##MTKArg#283")
+                          let (x, y, z, σ, ρ, β, __MTKWgamma, t) = (var"##MTKArg#281"[1], var"##MTKArg#281"[2], var"##MTKArg#281"[
+3], var"##MTKArg#282"[1], var"##MTKArg#282"[2], var"##MTKArg#282"[3], var"##MTKArg#283", var"##MTKArg#284")
                               ((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ)), (getproperty(Base, :*))((
 getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ))), __MTKWgamma, (getproperty(Base
 , :+))((getproperty(Base, :*))(-1, z), ρ)), (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getprop
@@ -903,27 +903,27 @@ 
ModelingToolkit is a compiler for mathematical systems

                       end)
               return construct(X)
           end
-      end), :((var"##MTIIPVar#285", var"##MTKArg#280", var"##MTKArg#281", var"##MTKArg#282", var"##MTKArg#283")->begin
+      end), :((var"##MTIIPVar#286", var"##MTKArg#281", var"##MTKArg#282", var"##MTKArg#283", var"##MTKArg#284")->begin
           @inbounds begin
-                  let (x, y, z, σ, ρ, β, __MTKWgamma, t) = (var"##MTKArg#280"[1], var"##MTKArg#280"[2], var"##MTKArg#280"[3], var"
-##MTKArg#281"[1], var"##MTKArg#281"[2], var"##MTKArg#281"[3], var"##MTKArg#282", var"##MTKArg#283")
-                      var"##MTIIPVar#285"[1] = (getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ))
-                      var"##MTIIPVar#285"[2] = (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getp
+                  let (x, y, z, σ, ρ, β, __MTKWgamma, t) = (var"##MTKArg#281"[1], var"##MTKArg#281"[2], var"##MTKArg#281"[3], var"
+##MTKArg#282"[1], var"##MTKArg#282"[2], var"##MTKArg#282"[3], var"##MTKArg#283", var"##MTKArg#284")
+                      var"##MTIIPVar#286"[1] = (getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ))
+                      var"##MTIIPVar#286"[2] = (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getp
 roperty(Base, :*))(-1, __MTKWgamma, σ))), __MTKWgamma, (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ))
-                      var"##MTIIPVar#285"[3] = (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getp
+                      var"##MTIIPVar#286"[3] = (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getp
 roperty(Base, :*))(-1, __MTKWgamma, σ))), y, __MTKWgamma)
-                      var"##MTIIPVar#285"[4] = (getproperty(Base, :*))(__MTKWgamma, σ)
-                      var"##MTIIPVar#285"[5] = (getproperty(Base, :*))(-1, (getproperty(Base, :+))(1, (getproperty(Base, :*))((get
+                      var"##MTIIPVar#286"[4] = (getproperty(Base, :*))(__MTKWgamma, σ)
+                      var"##MTIIPVar#286"[5] = (getproperty(Base, :*))(-1, (getproperty(Base, :+))(1, (getproperty(Base, :*))((get
 property(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ))), __MTKWgamma ^ 2, σ, (getproperty(
 Base, :+))((getproperty(Base, :*))(-1, z), ρ)), __MTKWgamma))
-                      var"##MTIIPVar#285"[6] = (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :*))(-1, (getp
+                      var"##MTIIPVar#286"[6] = (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :*))(-1, (getp
 roperty(Base, :+))(1, (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __
 MTKWgamma, σ))), __MTKWgamma ^ 2, σ, (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ)), __MTKWgamma))), (getproperty(Bas
 e, :+))((getproperty(Base, :*))(x, __MTKWgamma), (getproperty(Base, :*))(-1, (getproperty(Base, :inv))((getproperty(Base, :+))(-1,
  (getproperty(Base, :*))(-1, __MTKWgamma, σ))), y, __MTKWgamma ^ 2, σ)))
-                      var"##MTIIPVar#285"[7] = 0
-                      var"##MTIIPVar#285"[8] = (getproperty(Base, :*))(-1, x, __MTKWgamma)
-                      var"##MTIIPVar#285"[9] = (getproperty(Base, :+))((getproperty(Base, :*))(-1, (getproperty(Base, :+))(1, (get
+                      var"##MTIIPVar#286"[7] = 0
+                      var"##MTIIPVar#286"[8] = (getproperty(Base, :*))(-1, x, __MTKWgamma)
+                      var"##MTIIPVar#286"[9] = (getproperty(Base, :+))((getproperty(Base, :*))(-1, (getproperty(Base, :+))(1, (get
 property(Base, :*))(-1, (getproperty(Base, :inv))((getproperty(Base, :*))(-1, (getproperty(Base, :+))(1, (getproperty(Base, :*))((
 getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ))), __MTKWgamma ^ 2, σ, (getproper
 ty(Base, :+))((getproperty(Base, :*))(-1, z), ρ)), __MTKWgamma))), x, __MTKWgamma, (getproperty(Base, :+))((getproperty(Base, :*))
@@ -954,37 +954,37 @@ 
Solving Nonlinear systems

 
 
 
-(:((var"##MTKArg#295", var"##MTKArg#296")->begin
-          if var"##MTKArg#295" isa Array || !(typeof(var"##MTKArg#295") <: StaticArray) && false
+(:((var"##MTKArg#296", var"##MTKArg#297")->begin
+          if var"##MTKArg#296" isa Array || !(typeof(var"##MTKArg#296") <: StaticArray) && false
               return @inbounds(begin
-                          let (x, y, z, σ, ρ, β) = (var"##MTKArg#295"[1], var"##MTKArg#295"[2], var"##MTKArg#295"[3], var"##MTKArg
-#296"[1], var"##MTKArg#296"[2], var"##MTKArg#296"[3])
+                          let (x, y, z, σ, ρ, β) = (var"##MTKArg#296"[1], var"##MTKArg#296"[2], var"##MTKArg#296"[3], var"##MTKArg
+#297"[1], var"##MTKArg#297"[2], var"##MTKArg#297"[3])
                               [(*)(σ, (-)(y, x)), (-)((*)(x, (-)(ρ, z)), y), (-)((*)(x, y), (*)(β, z))]
                           end
                       end)
           else
               X = @inbounds(begin
-                          let (x, y, z, σ, ρ, β) = (var"##MTKArg#295"[1], var"##MTKArg#295"[2], var"##MTKArg#295"[3], var"##MTKArg
-#296"[1], var"##MTKArg#296"[2], var"##MTKArg#296"[3])
+                          let (x, y, z, σ, ρ, β) = (var"##MTKArg#296"[1], var"##MTKArg#296"[2], var"##MTKArg#296"[3], var"##MTKArg
+#297"[1], var"##MTKArg#297"[2], var"##MTKArg#297"[3])
                               ((*)(σ, (-)(y, x)), (-)((*)(x, (-)(ρ, z)), y), (-)((*)(x, y), (*)(β, z)))
                           end
                       end)
-              construct = if var"##MTKArg#295" isa ModelingToolkit.StaticArrays.StaticArray
-                      (getproperty(ModelingToolkit.StaticArrays, :similar_type))(typeof(var"##MTKArg#295"), eltype(X))
+              construct = if var"##MTKArg#296" isa ModelingToolkit.StaticArrays.StaticArray
+                      (getproperty(ModelingToolkit.StaticArrays, :similar_type))(typeof(var"##MTKArg#296"), eltype(X))
                   else
                       x->begin
-                              convert(typeof(var"##MTKArg#295"), x)
+                              convert(typeof(var"##MTKArg#296"), x)
                           end
                   end
               return construct(X)
           end
-      end), :((var"##MTIIPVar#298", var"##MTKArg#295", var"##MTKArg#296")->begin
+      end), :((var"##MTIIPVar#299", var"##MTKArg#296", var"##MTKArg#297")->begin
           @inbounds begin
-                  let (x, y, z, σ, ρ, β) = (var"##MTKArg#295"[1], var"##MTKArg#295"[2], var"##MTKArg#295"[3], var"##MTKArg#296"[1]
-, var"##MTKArg#296"[2], var"##MTKArg#296"[3])
-                      var"##MTIIPVar#298"[1] = (*)(σ, (-)(y, x))
-                      var"##MTIIPVar#298"[2] = (-)((*)(x, (-)(ρ, z)), y)
-                      var"##MTIIPVar#298"[3] = (-)((*)(x, y), (*)(β, z))
+                  let (x, y, z, σ, ρ, β) = (var"##MTKArg#296"[1], var"##MTKArg#296"[2], var"##MTKArg#296"[3], var"##MTKArg#297"[1]
+, var"##MTKArg#297"[2], var"##MTKArg#297"[3])
+                      var"##MTIIPVar#299"[1] = (*)(σ, (-)(y, x))
+                      var"##MTIIPVar#299"[2] = (-)((*)(x, (-)(ρ, z)), y)
+                      var"##MTIIPVar#299"[3] = (-)((*)(x, y), (*)(β, z))
                   end
               end
           nothing
@@ -1042,7 +1042,7 @@ 
Library of transformations on mathematical systems

 ariable[uˍtt, xˍt, uˍt, u, x], ModelingToolkit.Variable[], Base.RefValue{Array{ModelingToolkit.Expression,1}}(ModelingToolkit.Expr
 ession[]), Base.RefValue{Array{ModelingToolkit.Expression,2}}(Array{ModelingToolkit.Expression}(undef,0,0)), Base.RefValue{Array{M
 odelingToolkit.Expression,2}}(Array{ModelingToolkit.Expression}(undef,0,0)), Base.RefValue{Array{ModelingToolkit.Expression,2}}(Ar
-ray{ModelingToolkit.Expression}(undef,0,0)), Symbol("##ODESystem#301"), ModelingToolkit.ODESystem[])
+ray{ModelingToolkit.Expression}(undef,0,0)), Symbol("##ODESystem#302"), ModelingToolkit.ODESystem[])
 

 
 
@@ -1409,10 +1409,13 @@ 
Hackability: Extend directly from the language

 
Package Information:

 

 
Status `/builds/JuliaGPU/DiffEqTutorials.jl/tutorials/ode_extras/Project.toml`
+[f3b72e0c-5b89-59e1-b016-84e28bfd966d] DiffEqDevTools 2.22.0
 [961ee093-0014-501f-94e3-6117800e7a78] ModelingToolkit 3.11.0
+[76087f3c-5699-56af-9a33-bf431cd00edd] NLopt 0.6.0
 [2774e3e8-f4cf-5e23-947b-6d7e65073b56] NLsolve 4.4.0
+[429524aa-4258-5aef-a3af-852621145aeb] Optim 0.22.0
 [1dea7af3-3e70-54e6-95c3-0bf5283fa5ed] OrdinaryDiffEq 5.41.0
-[91a5bcdd-55d7-5caf-9e0b-520d859cae80] Plots 1.5.0
+[91a5bcdd-55d7-5caf-9e0b-520d859cae80] Plots 1.5.1
 [37e2e46d-f89d-539d-b4ee-838fcccc9c8e] LinearAlgebra
 [2f01184e-e22b-5df5-ae63-d93ebab69eaf] SparseArrays

 

diff --git a/markdown/ode_extras/01-ModelingToolkit.md b/markdown/ode_extras/01-ModelingToolkit.md
index ac5788bf..791d4186 100644
--- a/markdown/ode_extras/01-ModelingToolkit.md
+++ b/markdown/ode_extras/01-ModelingToolkit.md
@@ -60,11 +60,11 @@ generate_function(de)[1]
 
 
 ````
-:((var"##MTKArg#360", var"##MTKArg#361", var"##MTKArg#362")->begin
-          if var"##MTKArg#360" isa Array || !(typeof(var"##MTKArg#360") <: StaticArray) && false
+:((var"##MTKArg#361", var"##MTKArg#362", var"##MTKArg#363")->begin
+          if var"##MTKArg#361" isa Array || !(typeof(var"##MTKArg#361") <: StaticArray) && false
               return @inbounds(begin
-                          let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#360"[1], var"##MTKArg#360"[2], var"##MTKArg#360"[3], var"##MTK
-Arg#361"[1], var"##MTKArg#361"[2], var"##MTKArg#361"[3], var"##MTKArg#362")
+                          let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#361"[1], var"##MTKArg#361"[2], var"##MTKArg#361"[3], var"##MTK
+Arg#362"[1], var"##MTKArg#362"[2], var"##MTKArg#362"[3], var"##MTKArg#363")
                               [(getproperty(Base, :*))(σ, (getproperty(Base, :-))(y, x)), (getproperty(Base, :-))((getproperty(Bas
 e, :*))(x, (getproperty(Base, :-))(ρ, z)), y), (getproperty(Base, :-))((getproperty(Base, :*))(x, y), (getproperty(Base, :*))(β, z
 ))]
@@ -72,18 +72,18 @@ e, :*))(x, (getproperty(Base, :-))(ρ, z)), y), (getproperty(Base, :-))((getprop
                       end)
           else
               X = @inbounds(begin
-                          let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#360"[1], var"##MTKArg#360"[2], var"##MTKArg#360"[3], var"##MTK
-Arg#361"[1], var"##MTKArg#361"[2], var"##MTKArg#361"[3], var"##MTKArg#362")
+                          let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#361"[1], var"##MTKArg#361"[2], var"##MTKArg#361"[3], var"##MTK
+Arg#362"[1], var"##MTKArg#362"[2], var"##MTKArg#362"[3], var"##MTKArg#363")
                               ((getproperty(Base, :*))(σ, (getproperty(Base, :-))(y, x)), (getproperty(Base, :-))((getproperty(Bas
 e, :*))(x, (getproperty(Base, :-))(ρ, z)), y), (getproperty(Base, :-))((getproperty(Base, :*))(x, y), (getproperty(Base, :*))(β, z
 )))
                           end
                       end)
-              construct = if var"##MTKArg#360" isa ModelingToolkit.StaticArrays.StaticArray
-                      (getproperty(ModelingToolkit.StaticArrays, :similar_type))(typeof(var"##MTKArg#360"), eltype(X))
+              construct = if var"##MTKArg#361" isa ModelingToolkit.StaticArrays.StaticArray
+                      (getproperty(ModelingToolkit.StaticArrays, :similar_type))(typeof(var"##MTKArg#361"), eltype(X))
                   else
                       x->begin
-                              convert(typeof(var"##MTKArg#360"), x)
+                              convert(typeof(var"##MTKArg#361"), x)
                           end
                   end
               return construct(X)
@@ -103,14 +103,14 @@ generate_function(de)[2]
 
 
 ````
-:((var"##MTIIPVar#370", var"##MTKArg#366", var"##MTKArg#367", var"##MTKArg#368")->begin
+:((var"##MTIIPVar#371", var"##MTKArg#367", var"##MTKArg#368", var"##MTKArg#369")->begin
           @inbounds begin
-                  let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#366"[1], var"##MTKArg#366"[2], var"##MTKArg#366"[3], var"##MTKArg#367"
-[1], var"##MTKArg#367"[2], var"##MTKArg#367"[3], var"##MTKArg#368")
-                      var"##MTIIPVar#370"[1] = (getproperty(Base, :*))(σ, (getproperty(Base, :-))(y, x))
-                      var"##MTIIPVar#370"[2] = (getproperty(Base, :-))((getproperty(Base, :*))(x, (getproperty(Base, :-))(ρ, z)), 
+                  let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#367"[1], var"##MTKArg#367"[2], var"##MTKArg#367"[3], var"##MTKArg#368"
+[1], var"##MTKArg#368"[2], var"##MTKArg#368"[3], var"##MTKArg#369")
+                      var"##MTIIPVar#371"[1] = (getproperty(Base, :*))(σ, (getproperty(Base, :-))(y, x))
+                      var"##MTIIPVar#371"[2] = (getproperty(Base, :-))((getproperty(Base, :*))(x, (getproperty(Base, :-))(ρ, z)), 
 y)
-                      var"##MTIIPVar#370"[3] = (getproperty(Base, :-))((getproperty(Base, :*))(x, y), (getproperty(Base, :*))(β, z
+                      var"##MTIIPVar#371"[3] = (getproperty(Base, :-))((getproperty(Base, :*))(x, y), (getproperty(Base, :*))(β, z
 ))
                   end
               end
@@ -148,46 +148,46 @@ jac_expr = generate_jacobian(de)
 
 
 ````
-(:((var"##MTKArg#372", var"##MTKArg#373", var"##MTKArg#374")->begin
-          if var"##MTKArg#372" isa Array || !(typeof(var"##MTKArg#372") <: StaticArray) && false
+(:((var"##MTKArg#373", var"##MTKArg#374", var"##MTKArg#375")->begin
+          if var"##MTKArg#373" isa Array || !(typeof(var"##MTKArg#373") <: StaticArray) && false
               return @inbounds(begin
-                          let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#372"[1], var"##MTKArg#372"[2], var"##MTKArg#372"[3], var"##MTK
-Arg#373"[1], var"##MTKArg#373"[2], var"##MTKArg#373"[3], var"##MTKArg#374")
+                          let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#373"[1], var"##MTKArg#373"[2], var"##MTKArg#373"[3], var"##MTK
+Arg#374"[1], var"##MTKArg#374"[2], var"##MTKArg#374"[3], var"##MTKArg#375")
                               [(getproperty(Base, :*))(-1, σ) σ 0; (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ) -1 (
 getproperty(Base, :*))(-1, x); y x (getproperty(Base, :*))(-1, β)]
                           end
                       end)
           else
               X = @inbounds(begin
-                          let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#372"[1], var"##MTKArg#372"[2], var"##MTKArg#372"[3], var"##MTK
-Arg#373"[1], var"##MTKArg#373"[2], var"##MTKArg#373"[3], var"##MTKArg#374")
+                          let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#373"[1], var"##MTKArg#373"[2], var"##MTKArg#373"[3], var"##MTK
+Arg#374"[1], var"##MTKArg#374"[2], var"##MTKArg#374"[3], var"##MTKArg#375")
                               ((getproperty(Base, :*))(-1, σ), (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ), y, σ, -
 1, x, 0, (getproperty(Base, :*))(-1, x), (getproperty(Base, :*))(-1, β))
                           end
                       end)
-              construct = if var"##MTKArg#372" isa ModelingToolkit.StaticArrays.StaticArray
+              construct = if var"##MTKArg#373" isa ModelingToolkit.StaticArrays.StaticArray
                       ModelingToolkit.StaticArrays.SMatrix{3, 3}
                   else
                       x->begin
-                              out = similar(typeof(var"##MTKArg#372"), 3, 3)
+                              out = similar(typeof(var"##MTKArg#373"), 3, 3)
                               out .= x
                           end
                   end
               return construct(X)
           end
-      end), :((var"##MTIIPVar#376", var"##MTKArg#372", var"##MTKArg#373", var"##MTKArg#374")->begin
+      end), :((var"##MTIIPVar#377", var"##MTKArg#373", var"##MTKArg#374", var"##MTKArg#375")->begin
           @inbounds begin
-                  let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#372"[1], var"##MTKArg#372"[2], var"##MTKArg#372"[3], var"##MTKArg#373"
-[1], var"##MTKArg#373"[2], var"##MTKArg#373"[3], var"##MTKArg#374")
-                      var"##MTIIPVar#376"[1] = (getproperty(Base, :*))(-1, σ)
-                      var"##MTIIPVar#376"[2] = (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ)
-                      var"##MTIIPVar#376"[3] = y
-                      var"##MTIIPVar#376"[4] = σ
-                      var"##MTIIPVar#376"[5] = -1
-                      var"##MTIIPVar#376"[6] = x
-                      var"##MTIIPVar#376"[7] = 0
-                      var"##MTIIPVar#376"[8] = (getproperty(Base, :*))(-1, x)
-                      var"##MTIIPVar#376"[9] = (getproperty(Base, :*))(-1, β)
+                  let (x, y, z, σ, ρ, β, t) = (var"##MTKArg#373"[1], var"##MTKArg#373"[2], var"##MTKArg#373"[3], var"##MTKArg#374"
+[1], var"##MTKArg#374"[2], var"##MTKArg#374"[3], var"##MTKArg#375")
+                      var"##MTIIPVar#377"[1] = (getproperty(Base, :*))(-1, σ)
+                      var"##MTIIPVar#377"[2] = (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ)
+                      var"##MTIIPVar#377"[3] = y
+                      var"##MTIIPVar#377"[4] = σ
+                      var"##MTIIPVar#377"[5] = -1
+                      var"##MTIIPVar#377"[6] = x
+                      var"##MTIIPVar#377"[7] = 0
+                      var"##MTIIPVar#377"[8] = (getproperty(Base, :*))(-1, x)
+                      var"##MTIIPVar#377"[9] = (getproperty(Base, :*))(-1, β)
                   end
               end
           nothing
@@ -206,11 +206,11 @@ ModelingToolkit.generate_factorized_W(de)[1]
 
 
 ````
-(:((var"##MTKArg#378", var"##MTKArg#379", var"##MTKArg#380", var"##MTKArg#381")->begin
-          if var"##MTKArg#378" isa Array || !(typeof(var"##MTKArg#378") <: StaticArray) && false
+(:((var"##MTKArg#379", var"##MTKArg#380", var"##MTKArg#381", var"##MTKArg#382")->begin
+          if var"##MTKArg#379" isa Array || !(typeof(var"##MTKArg#379") <: StaticArray) && false
               return @inbounds(begin
-                          let (x, y, z, σ, ρ, β, __MTKWgamma, t) = (var"##MTKArg#378"[1], var"##MTKArg#378"[2], var"##MTKArg#378"[
-3], var"##MTKArg#379"[1], var"##MTKArg#379"[2], var"##MTKArg#379"[3], var"##MTKArg#380", var"##MTKArg#381")
+                          let (x, y, z, σ, ρ, β, __MTKWgamma, t) = (var"##MTKArg#379"[1], var"##MTKArg#379"[2], var"##MTKArg#379"[
+3], var"##MTKArg#380"[1], var"##MTKArg#380"[2], var"##MTKArg#380"[3], var"##MTKArg#381", var"##MTKArg#382")
                               [(getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ)) (getproperty(Base, :*))(__
 MTKWgamma, σ) 0; (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWg
 amma, σ))), __MTKWgamma, (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ)) (getproperty(Base, :*))(-1, (getproperty(Base
@@ -231,8 +231,8 @@ roperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ)), __MTKWgamma))), x, __MT
                       end)
           else
               X = @inbounds(begin
-                          let (x, y, z, σ, ρ, β, __MTKWgamma, t) = (var"##MTKArg#378"[1], var"##MTKArg#378"[2], var"##MTKArg#378"[
-3], var"##MTKArg#379"[1], var"##MTKArg#379"[2], var"##MTKArg#379"[3], var"##MTKArg#380", var"##MTKArg#381")
+                          let (x, y, z, σ, ρ, β, __MTKWgamma, t) = (var"##MTKArg#379"[1], var"##MTKArg#379"[2], var"##MTKArg#379"[
+3], var"##MTKArg#380"[1], var"##MTKArg#380"[2], var"##MTKArg#380"[3], var"##MTKArg#381", var"##MTKArg#382")
                               ((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ)), (getproperty(Base, :*))((
 getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ))), __MTKWgamma, (getproperty(Base
 , :+))((getproperty(Base, :*))(-1, z), ρ)), (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getprop
@@ -260,27 +260,27 @@ gular(A), SVector(ntuple((n->begin
                       end)
               return construct(X)
           end
-      end), :((var"##MTIIPVar#383", var"##MTKArg#378", var"##MTKArg#379", var"##MTKArg#380", var"##MTKArg#381")->begin
+      end), :((var"##MTIIPVar#384", var"##MTKArg#379", var"##MTKArg#380", var"##MTKArg#381", var"##MTKArg#382")->begin
           @inbounds begin
-                  let (x, y, z, σ, ρ, β, __MTKWgamma, t) = (var"##MTKArg#378"[1], var"##MTKArg#378"[2], var"##MTKArg#378"[3], var"
-##MTKArg#379"[1], var"##MTKArg#379"[2], var"##MTKArg#379"[3], var"##MTKArg#380", var"##MTKArg#381")
-                      var"##MTIIPVar#383"[1] = (getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ))
-                      var"##MTIIPVar#383"[2] = (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getp
+                  let (x, y, z, σ, ρ, β, __MTKWgamma, t) = (var"##MTKArg#379"[1], var"##MTKArg#379"[2], var"##MTKArg#379"[3], var"
+##MTKArg#380"[1], var"##MTKArg#380"[2], var"##MTKArg#380"[3], var"##MTKArg#381", var"##MTKArg#382")
+                      var"##MTIIPVar#384"[1] = (getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ))
+                      var"##MTIIPVar#384"[2] = (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getp
 roperty(Base, :*))(-1, __MTKWgamma, σ))), __MTKWgamma, (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ))
-                      var"##MTIIPVar#383"[3] = (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getp
+                      var"##MTIIPVar#384"[3] = (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getp
 roperty(Base, :*))(-1, __MTKWgamma, σ))), y, __MTKWgamma)
-                      var"##MTIIPVar#383"[4] = (getproperty(Base, :*))(__MTKWgamma, σ)
-                      var"##MTIIPVar#383"[5] = (getproperty(Base, :*))(-1, (getproperty(Base, :+))(1, (getproperty(Base, :*))((get
+                      var"##MTIIPVar#384"[4] = (getproperty(Base, :*))(__MTKWgamma, σ)
+                      var"##MTIIPVar#384"[5] = (getproperty(Base, :*))(-1, (getproperty(Base, :+))(1, (getproperty(Base, :*))((get
 property(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ))), __MTKWgamma ^ 2, σ, (getproperty(
 Base, :+))((getproperty(Base, :*))(-1, z), ρ)), __MTKWgamma))
-                      var"##MTIIPVar#383"[6] = (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :*))(-1, (getp
+                      var"##MTIIPVar#384"[6] = (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :*))(-1, (getp
 roperty(Base, :+))(1, (getproperty(Base, :*))((getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __
 MTKWgamma, σ))), __MTKWgamma ^ 2, σ, (getproperty(Base, :+))((getproperty(Base, :*))(-1, z), ρ)), __MTKWgamma))), (getproperty(Bas
 e, :+))((getproperty(Base, :*))(x, __MTKWgamma), (getproperty(Base, :*))(-1, (getproperty(Base, :inv))((getproperty(Base, :+))(-1,
  (getproperty(Base, :*))(-1, __MTKWgamma, σ))), y, __MTKWgamma ^ 2, σ)))
-                      var"##MTIIPVar#383"[7] = 0
-                      var"##MTIIPVar#383"[8] = (getproperty(Base, :*))(-1, x, __MTKWgamma)
-                      var"##MTIIPVar#383"[9] = (getproperty(Base, :+))((getproperty(Base, :*))(-1, (getproperty(Base, :+))(1, (get
+                      var"##MTIIPVar#384"[7] = 0
+                      var"##MTIIPVar#384"[8] = (getproperty(Base, :*))(-1, x, __MTKWgamma)
+                      var"##MTIIPVar#384"[9] = (getproperty(Base, :+))((getproperty(Base, :*))(-1, (getproperty(Base, :+))(1, (get
 property(Base, :*))(-1, (getproperty(Base, :inv))((getproperty(Base, :*))(-1, (getproperty(Base, :+))(1, (getproperty(Base, :*))((
 getproperty(Base, :inv))((getproperty(Base, :+))(-1, (getproperty(Base, :*))(-1, __MTKWgamma, σ))), __MTKWgamma ^ 2, σ, (getproper
 ty(Base, :+))((getproperty(Base, :*))(-1, z), ρ)), __MTKWgamma))), x, __MTKWgamma, (getproperty(Base, :+))((getproperty(Base, :*))
@@ -314,37 +314,37 @@ nlsys_func = generate_function(ns)
 
 
 ````
-(:((var"##MTKArg#393", var"##MTKArg#394")->begin
-          if var"##MTKArg#393" isa Array || !(typeof(var"##MTKArg#393") <: StaticArray) && false
+(:((var"##MTKArg#394", var"##MTKArg#395")->begin
+          if var"##MTKArg#394" isa Array || !(typeof(var"##MTKArg#394") <: StaticArray) && false
               return @inbounds(begin
-                          let (x, y, z, σ, ρ, β) = (var"##MTKArg#393"[1], var"##MTKArg#393"[2], var"##MTKArg#393"[3], var"##MTKArg
-#394"[1], var"##MTKArg#394"[2], var"##MTKArg#394"[3])
+                          let (x, y, z, σ, ρ, β) = (var"##MTKArg#394"[1], var"##MTKArg#394"[2], var"##MTKArg#394"[3], var"##MTKArg
+#395"[1], var"##MTKArg#395"[2], var"##MTKArg#395"[3])
                               [(*)(σ, (-)(y, x)), (-)((*)(x, (-)(ρ, z)), y), (-)((*)(x, y), (*)(β, z))]
                           end
                       end)
           else
               X = @inbounds(begin
-                          let (x, y, z, σ, ρ, β) = (var"##MTKArg#393"[1], var"##MTKArg#393"[2], var"##MTKArg#393"[3], var"##MTKArg
-#394"[1], var"##MTKArg#394"[2], var"##MTKArg#394"[3])
+                          let (x, y, z, σ, ρ, β) = (var"##MTKArg#394"[1], var"##MTKArg#394"[2], var"##MTKArg#394"[3], var"##MTKArg
+#395"[1], var"##MTKArg#395"[2], var"##MTKArg#395"[3])
                               ((*)(σ, (-)(y, x)), (-)((*)(x, (-)(ρ, z)), y), (-)((*)(x, y), (*)(β, z)))
                           end
                       end)
-              construct = if var"##MTKArg#393" isa ModelingToolkit.StaticArrays.StaticArray
-                      (getproperty(ModelingToolkit.StaticArrays, :similar_type))(typeof(var"##MTKArg#393"), eltype(X))
+              construct = if var"##MTKArg#394" isa ModelingToolkit.StaticArrays.StaticArray
+                      (getproperty(ModelingToolkit.StaticArrays, :similar_type))(typeof(var"##MTKArg#394"), eltype(X))
                   else
                       x->begin
-                              convert(typeof(var"##MTKArg#393"), x)
+                              convert(typeof(var"##MTKArg#394"), x)
                           end
                   end
               return construct(X)
           end
-      end), :((var"##MTIIPVar#396", var"##MTKArg#393", var"##MTKArg#394")->begin
+      end), :((var"##MTIIPVar#397", var"##MTKArg#394", var"##MTKArg#395")->begin
           @inbounds begin
-                  let (x, y, z, σ, ρ, β) = (var"##MTKArg#393"[1], var"##MTKArg#393"[2], var"##MTKArg#393"[3], var"##MTKArg#394"[1]
-, var"##MTKArg#394"[2], var"##MTKArg#394"[3])
-                      var"##MTIIPVar#396"[1] = (*)(σ, (-)(y, x))
-                      var"##MTIIPVar#396"[2] = (-)((*)(x, (-)(ρ, z)), y)
-                      var"##MTIIPVar#396"[3] = (-)((*)(x, y), (*)(β, z))
+                  let (x, y, z, σ, ρ, β) = (var"##MTKArg#394"[1], var"##MTKArg#394"[2], var"##MTKArg#394"[3], var"##MTKArg#395"[1]
+, var"##MTKArg#395"[2], var"##MTKArg#395"[3])
+                      var"##MTIIPVar#397"[1] = (*)(σ, (-)(y, x))
+                      var"##MTIIPVar#397"[2] = (-)((*)(x, (-)(ρ, z)), y)
+                      var"##MTIIPVar#397"[3] = (-)((*)(x, y), (*)(β, z))
                   end
               end
           nothing
@@ -407,7 +407,7 @@ odelingToolkit.Equation(derivative(u(t), t), uˍt(t)), ModelingToolkit.Equation(
 ariable[uˍtt, xˍt, uˍt, u, x], ModelingToolkit.Variable[], Base.RefValue{Array{ModelingToolkit.Expression,1}}(ModelingToolkit.Expr
 ession[]), Base.RefValue{Array{ModelingToolkit.Expression,2}}(Array{ModelingToolkit.Expression}(undef,0,0)), Base.RefValue{Array{M
 odelingToolkit.Expression,2}}(Array{ModelingToolkit.Expression}(undef,0,0)), Base.RefValue{Array{ModelingToolkit.Expression,2}}(Ar
-ray{ModelingToolkit.Expression}(undef,0,0)), Symbol("##ODESystem#399"), ModelingToolkit.ODESystem[])
+ray{ModelingToolkit.Expression}(undef,0,0)), Symbol("##ODESystem#400"), ModelingToolkit.ODESystem[])
 ````
 
 
@@ -817,10 +817,13 @@ Package Information:
 
 ```
 Status `/builds/JuliaGPU/DiffEqTutorials.jl/tutorials/ode_extras/Project.toml`
+[f3b72e0c-5b89-59e1-b016-84e28bfd966d] DiffEqDevTools 2.22.0
 [961ee093-0014-501f-94e3-6117800e7a78] ModelingToolkit 3.11.0
+[76087f3c-5699-56af-9a33-bf431cd00edd] NLopt 0.6.0
 [2774e3e8-f4cf-5e23-947b-6d7e65073b56] NLsolve 4.4.0
+[429524aa-4258-5aef-a3af-852621145aeb] Optim 0.22.0
 [1dea7af3-3e70-54e6-95c3-0bf5283fa5ed] OrdinaryDiffEq 5.41.0
-[91a5bcdd-55d7-5caf-9e0b-520d859cae80] Plots 1.5.0
+[91a5bcdd-55d7-5caf-9e0b-520d859cae80] Plots 1.5.1
 [37e2e46d-f89d-539d-b4ee-838fcccc9c8e] LinearAlgebra
 [2f01184e-e22b-5df5-ae63-d93ebab69eaf] SparseArrays
 ```
diff --git a/pdf/ode_extras/01-ModelingToolkit.pdf b/pdf/ode_extras/01-ModelingToolkit.pdf
index aa9d2a892b0540dc71357899f3cac48cfe97bfda..1f4bf8aedabf8add7c1f49cc728bd14236bcea1e 100644
GIT binary patch
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Date: Wed, 1 Jul 2020 13:17:52 +0000
Subject: [PATCH 8/8] Rebuild ode_extras/02-feagin.jmd

---
 html/ode_extras/02-feagin.html   |  21 ++++++---------------
 markdown/ode_extras/02-feagin.md |  17 ++++-------------
 pdf/ode_extras/02-feagin.pdf     | Bin 42660 -> 42681 bytes
 3 files changed, 10 insertions(+), 28 deletions(-)

diff --git a/html/ode_extras/02-feagin.html b/html/ode_extras/02-feagin.html
index 81ef749a..65aeb3f2 100644
--- a/html/ode_extras/02-feagin.html
+++ b/html/ode_extras/02-feagin.html
@@ -771,20 +771,8 @@ 
Convergence Test

 
 
 
-using DiffEqDevTools
-

-
-
-
-ERROR: ArgumentError: Package DiffEqDevTools not found in current path:
-- Run `import Pkg; Pkg.add("DiffEqDevTools")` to install the DiffEqDevTools package.
-
-

-
-
-
-
-dts = 1.0 ./ 2.0 .^(10:-1:4)
+using DiffEqDevTools
+dts = 1.0 ./ 2.0 .^(10:-1:4)
 sim = test_convergence(dts,prob,Feagin14())
 

 
@@ -839,10 +827,13 @@ 
Convergence Test

 
Package Information:

 

 
Status `/builds/JuliaGPU/DiffEqTutorials.jl/tutorials/ode_extras/Project.toml`
+[f3b72e0c-5b89-59e1-b016-84e28bfd966d] DiffEqDevTools 2.22.0
 [961ee093-0014-501f-94e3-6117800e7a78] ModelingToolkit 3.11.0
+[76087f3c-5699-56af-9a33-bf431cd00edd] NLopt 0.6.0
 [2774e3e8-f4cf-5e23-947b-6d7e65073b56] NLsolve 4.4.0
+[429524aa-4258-5aef-a3af-852621145aeb] Optim 0.22.0
 [1dea7af3-3e70-54e6-95c3-0bf5283fa5ed] OrdinaryDiffEq 5.41.0
-[91a5bcdd-55d7-5caf-9e0b-520d859cae80] Plots 1.5.0
+[91a5bcdd-55d7-5caf-9e0b-520d859cae80] Plots 1.5.1
 [37e2e46d-f89d-539d-b4ee-838fcccc9c8e] LinearAlgebra
 [2f01184e-e22b-5df5-ae63-d93ebab69eaf] SparseArrays

 

diff --git a/markdown/ode_extras/02-feagin.md b/markdown/ode_extras/02-feagin.md
index ee515a3c..8cbae22e 100644
--- a/markdown/ode_extras/02-feagin.md
+++ b/markdown/ode_extras/02-feagin.md
@@ -124,18 +124,6 @@ The Order 14 method is awesome, but we need to make sure it's really that awesom
 
 ````julia
 using DiffEqDevTools
-````
-
-
-````
-Error: ArgumentError: Package DiffEqDevTools not found in current path:
-- Run `import Pkg; Pkg.add("DiffEqDevTools")` to install the DiffEqDevTools
- package.
-````
-
-
-
-````julia
 dts = 1.0 ./ 2.0 .^(10:-1:4)
 sim = test_convergence(dts,prob,Feagin14())
 ````
@@ -202,10 +190,13 @@ Package Information:
 
 ```
 Status `/builds/JuliaGPU/DiffEqTutorials.jl/tutorials/ode_extras/Project.toml`
+[f3b72e0c-5b89-59e1-b016-84e28bfd966d] DiffEqDevTools 2.22.0
 [961ee093-0014-501f-94e3-6117800e7a78] ModelingToolkit 3.11.0
+[76087f3c-5699-56af-9a33-bf431cd00edd] NLopt 0.6.0
 [2774e3e8-f4cf-5e23-947b-6d7e65073b56] NLsolve 4.4.0
+[429524aa-4258-5aef-a3af-852621145aeb] Optim 0.22.0
 [1dea7af3-3e70-54e6-95c3-0bf5283fa5ed] OrdinaryDiffEq 5.41.0
-[91a5bcdd-55d7-5caf-9e0b-520d859cae80] Plots 1.5.0
+[91a5bcdd-55d7-5caf-9e0b-520d859cae80] Plots 1.5.1
 [37e2e46d-f89d-539d-b4ee-838fcccc9c8e] LinearAlgebra
 [2f01184e-e22b-5df5-ae63-d93ebab69eaf] SparseArrays
 ```
diff --git a/pdf/ode_extras/02-feagin.pdf b/pdf/ode_extras/02-feagin.pdf
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