702 lines
105 KiB
Plaintext
702 lines
105 KiB
Plaintext
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"cells": [
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Numerical Integration and Differentiation\n",
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"\n",
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"This notebook illustrates how to perform numerical integration and differentiation. \n",
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"\n",
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"There are several packages for doing this. Here, the focus is on [QuadGK](https://github.com/JuliaMath/QuadGK.jl) and [ForwardDiff](https://github.com/JuliaDiff/ForwardDiff.jl)."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Load Packages and Extra Functions"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"printyellow (generic function with 1 method)"
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]
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},
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"execution_count": 1,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"using Printf, QuadGK, ForwardDiff\n",
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"\n",
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"include(\"jlFiles/printmat.jl\")"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {},
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"outputs": [],
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"source": [
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"using Plots\n",
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"\n",
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"#pyplot(size=(600,400))\n",
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"gr(size=(480,320))\n",
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"default(fmt = :svg)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Numerical Integration\n",
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"\n",
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"As a simple illustration, the next cells plot and integrate the $N(0,\\sigma)$ pdf."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## The Pdf of the N() Distribution \n",
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"\n",
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"Notice: the function takes $\\mu$ and $\\sigma$ (not $\\sigma^2$) as inputs, similar to the Distributions package. "
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"ϕNS (generic function with 3 methods)"
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]
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},
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"execution_count": 3,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"function ϕNS(x,μ=0,σ=1) #pdf of N(μ,σ), defaults to N(0,1)\n",
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" z = (x - μ)/σ\n",
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" pdf = exp(-0.5*z^2)/(sqrt(2*pi)*σ)\n",
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" return pdf\n",
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"end"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"metadata": {},
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"outputs": [
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|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
"metadata": {},
|
|||
|
|
"output_type": "display_data"
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"x = -3:0.1:3\n",
|
|||
|
|
"xb = x[x.<=1.645]\n",
|
|||
|
|
"\n",
|
|||
|
|
"p1 = plot( x,ϕNS.(x),\n",
|
|||
|
|
" linecolor = :red,\n",
|
|||
|
|
" linewidth = 2,\n",
|
|||
|
|
" ylims = (0,0.42),\n",
|
|||
|
|
" xlims = (-3,3),\n",
|
|||
|
|
" legend = nothing,\n",
|
|||
|
|
" title = \"pdf of N(0,1)\",\n",
|
|||
|
|
" xlabel = \"x\",\n",
|
|||
|
|
" ylabel = \"\",\n",
|
|||
|
|
" annotation = (1.75,0.25,text(\"the area covers\\nup to x=1.645\",8)) )\n",
|
|||
|
|
"plot!(xb,ϕNS.(xb),fillcolor=:red,linewidth=2,legend=nothing,fill=(0,:red))\n",
|
|||
|
|
"display(p1)"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {},
|
|||
|
|
"source": [
|
|||
|
|
"## Calculating Prob(x<=1.645)\n",
|
|||
|
|
"\n",
|
|||
|
|
"The next cell calculates (by numeric integration)\n",
|
|||
|
|
"\n",
|
|||
|
|
"$\\int_{-\\infty}^{1.645}\\phi(x,0,\\sigma)dx$,\n",
|
|||
|
|
"\n",
|
|||
|
|
"where $\\phi(x,\\mu,\\sigma)$ is the pdf of an $N(\\mu,\\sigma)$ variable.\n",
|
|||
|
|
"\n",
|
|||
|
|
"\n",
|
|||
|
|
"The input to `quadgk` should be a function with only one argument. We do that by creating an anonymous function\n",
|
|||
|
|
"```\n",
|
|||
|
|
"x->fn(x,a)\n",
|
|||
|
|
"```\n",
|
|||
|
|
"assuming that `a` has a value already."
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 5,
|
|||
|
|
"metadata": {},
|
|||
|
|
"outputs": [
|
|||
|
|
{
|
|||
|
|
"name": "stdout",
|
|||
|
|
"output_type": "stream",
|
|||
|
|
"text": [
|
|||
|
|
"\n",
|
|||
|
|
"Pr(x<=1.64) according to N(0,1): 0.950\n",
|
|||
|
|
"\n",
|
|||
|
|
"Pr(x<=1.64) according to N(0,2): 0.795\n"
|
|||
|
|
]
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"cdf1, = quadgk(x->ϕNS(x),-Inf,1.645) #N(0,1)\n",
|
|||
|
|
"printlnPs(\"\\nPr(x<=1.64) according to N(0,1):\", cdf1)\n",
|
|||
|
|
"\n",
|
|||
|
|
"cdf2, = quadgk(x->ϕNS(x,0,2),-Inf,1.645) #N(0,σ=2)\n",
|
|||
|
|
"printlnPs(\"\\nPr(x<=1.64) according to N(0,2):\", cdf2)"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {
|
|||
|
|
"collapsed": true
|
|||
|
|
},
|
|||
|
|
"source": [
|
|||
|
|
"# Numerical Derivatives\n",
|
|||
|
|
"\n",
|
|||
|
|
"Numerical derivatives can be calculated by a crude finite difference (see `NumDer()` below) or by the much more sophisticated routines in the `ForwardDiff` package.\n",
|
|||
|
|
"\n",
|
|||
|
|
"To do this calculation for many `x` values, we loop (either a `for` loop or list comprehension). Dot syntax, `ForwardDiff.derivative.()`, seems to work, but is not documented."
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 6,
|
|||
|
|
"metadata": {},
|
|||
|
|
"outputs": [
|
|||
|
|
{
|
|||
|
|
"data": {
|
|||
|
|
"text/plain": [
|
|||
|
|
"NumDer (generic function with 1 method)"
|
|||
|
|
]
|
|||
|
|
},
|
|||
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|
"execution_count": 6,
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|||
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|
"metadata": {},
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"output_type": "execute_result"
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|
}
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|
],
|
|||
|
|
"source": [
|
|||
|
|
"function NumDer(fun,b0,h) #crude function for a centered numerical derivative\n",
|
|||
|
|
" bminus = b0 .- h\n",
|
|||
|
|
" bplus = b0 .+ h\n",
|
|||
|
|
" hh = bplus - bminus\n",
|
|||
|
|
" fplus = fun(bplus)\n",
|
|||
|
|
" fminus = fun(bminus)\n",
|
|||
|
|
" D = (fplus-fminus)/hh\n",
|
|||
|
|
" return D\n",
|
|||
|
|
"end"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
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|
"cell_type": "code",
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|
"execution_count": 7,
|
|||
|
|
"metadata": {},
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|
"outputs": [
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|
|
{
|
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|
|
"data": {
|
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|
"text/plain": [
|
|||
|
|
"fn1 (generic function with 1 method)"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
"execution_count": 7,
|
|||
|
|
"metadata": {},
|
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|
"output_type": "execute_result"
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"function fn1(x,a) #a simple function, to be differentiated\n",
|
|||
|
|
" return (x - 1.1)^2 - a\n",
|
|||
|
|
"end"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 8,
|
|||
|
|
"metadata": {},
|
|||
|
|
"outputs": [
|
|||
|
|
{
|
|||
|
|
"name": "stdout",
|
|||
|
|
"output_type": "stream",
|
|||
|
|
"text": [
|
|||
|
|
"The derivative at x=2 is (from two different methods): \n",
|
|||
|
|
" 1.800 1.800\n",
|
|||
|
|
"\n"
|
|||
|
|
]
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"x0 = 2\n",
|
|||
|
|
"\n",
|
|||
|
|
"dydx_A = NumDer(x->fn1(x,0.5),x0,0.01) #differentiate fn1(x,0.5) at x = x0\n",
|
|||
|
|
"dydx_B = ForwardDiff.derivative(x->fn1(x,0.5),x0)\n",
|
|||
|
|
"\n",
|
|||
|
|
"println(\"The derivative at x=$x0 is (from two different methods): \")\n",
|
|||
|
|
"printmat([dydx_A dydx_B])"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 9,
|
|||
|
|
"metadata": {},
|
|||
|
|
"outputs": [
|
|||
|
|
{
|
|||
|
|
"name": "stdout",
|
|||
|
|
"output_type": "stream",
|
|||
|
|
"text": [
|
|||
|
|
"now lets plot this\n"
|
|||
|
|
]
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"x = -3:6/99:6 #calculate the derivative at many points\n",
|
|||
|
|
"\n",
|
|||
|
|
"dydx_A = [NumDer(x->fn1(x,0.5),x[i],0.01) for i=1:length(x)] #list comprehension as a quick loop\n",
|
|||
|
|
"\n",
|
|||
|
|
"dydx_B = [ForwardDiff.derivative(x->fn1(x,0.5),x[i]) for i=1:length(x)]\n",
|
|||
|
|
"\n",
|
|||
|
|
"println(\"now lets plot this\")"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 10,
|
|||
|
|
"metadata": {},
|
|||
|
|
"outputs": [
|
|||
|
|
{
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|
"data": {
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|
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|
" \"/>\n",
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|||
|
|
" 137.886,541.337 158.705,541.337 \n",
|
|||
|
|
" \"/>\n",
|
|||
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|
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|
|||
|
|
" 137.886,393.748 158.705,393.748 \n",
|
|||
|
|
" \"/>\n",
|
|||
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|
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|
|||
|
|
" 137.886,246.159 158.705,246.159 \n",
|
|||
|
|
" \"/>\n",
|
|||
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" 499.486,337.676 615.144,337.676 \n",
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" \"/>\n",
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|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
"metadata": {},
|
|||
|
|
"output_type": "display_data"
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"p1 = plot( [x x x],[fn1.(x,0.5) dydx_A dydx_B],\n",
|
|||
|
|
" line = [:solid :dot :dash],\n",
|
|||
|
|
" linecolor = [:black :red :blue],\n",
|
|||
|
|
" label = [\"fn1()\" \"crude derivative\" \"from ForwardDiff\"],\n",
|
|||
|
|
" legend = :top,\n",
|
|||
|
|
" title = \"fn1() and its derivative\",\n",
|
|||
|
|
" xlabel = \"x\",\n",
|
|||
|
|
" ylabel = \"\" )\n",
|
|||
|
|
"display(p1)"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {
|
|||
|
|
"collapsed": true
|
|||
|
|
},
|
|||
|
|
"source": [
|
|||
|
|
"## Comments on Using the ForwardDiff Package (extra)\n",
|
|||
|
|
"\n",
|
|||
|
|
"The `ForwardDiff` package applies an interesting approach to calculate derivatives, using a special number type (\"dual numbers\"). This means that your code must be able to handle such numbers. In most cases, that is not a problem, but you may have to watch out if you create arrays to store (intermediate?) results inside the function. See the examples below"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 11,
|
|||
|
|
"metadata": {},
|
|||
|
|
"outputs": [
|
|||
|
|
{
|
|||
|
|
"data": {
|
|||
|
|
"text/plain": [
|
|||
|
|
"fnDoesWork (generic function with 1 method)"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
"execution_count": 11,
|
|||
|
|
"metadata": {},
|
|||
|
|
"output_type": "execute_result"
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"function fnDoesNotWork(b,a)\n",
|
|||
|
|
" z = zeros(length(b)) #will not work with ForwardDiff, since\n",
|
|||
|
|
" for i = 1:length(z) #z cannot store dual numbers\n",
|
|||
|
|
" z[i] = b[i]*i\n",
|
|||
|
|
" end\n",
|
|||
|
|
" return sum(z) + a\n",
|
|||
|
|
"end\n",
|
|||
|
|
"\n",
|
|||
|
|
"function fnDoesWork(b,a)\n",
|
|||
|
|
" z = zeros(eltype(b),length(b)) #will work with ForwardDiff, since\n",
|
|||
|
|
" for i = 1:length(z) #when b is a dual number, so is z \n",
|
|||
|
|
" z[i] = b[i]*i #could also start with z = similar(b)\n",
|
|||
|
|
" end\n",
|
|||
|
|
" return sum(z) + a\n",
|
|||
|
|
"end"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 12,
|
|||
|
|
"metadata": {},
|
|||
|
|
"outputs": [
|
|||
|
|
{
|
|||
|
|
"data": {
|
|||
|
|
"text/plain": [
|
|||
|
|
"2-element Vector{Float64}:\n",
|
|||
|
|
" 1.0\n",
|
|||
|
|
" 2.0"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
"execution_count": 12,
|
|||
|
|
"metadata": {},
|
|||
|
|
"output_type": "execute_result"
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"b0 = [1.5,2]\n",
|
|||
|
|
"\n",
|
|||
|
|
"#ForwardDiff.gradient(b->fnDoesNotWork(b,1),b0) #uncomment to get an error\n",
|
|||
|
|
"ForwardDiff.gradient(b->fnDoesWork(b,1),b0)"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": null,
|
|||
|
|
"metadata": {},
|
|||
|
|
"outputs": [],
|
|||
|
|
"source": []
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"metadata": {
|
|||
|
|
"@webio": {
|
|||
|
|
"lastCommId": null,
|
|||
|
|
"lastKernelId": null
|
|||
|
|
},
|
|||
|
|
"anaconda-cloud": {},
|
|||
|
|
"kernelspec": {
|
|||
|
|
"display_name": "Julia 1.6.1",
|
|||
|
|
"language": "julia",
|
|||
|
|
"name": "julia-1.6"
|
|||
|
|
},
|
|||
|
|
"language_info": {
|
|||
|
|
"file_extension": ".jl",
|
|||
|
|
"mimetype": "application/julia",
|
|||
|
|
"name": "julia",
|
|||
|
|
"version": "1.6.1"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"nbformat": 4,
|
|||
|
|
"nbformat_minor": 1
|
|||
|
|
}
|