389 lines
31 KiB
Plaintext
389 lines
31 KiB
Plaintext
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{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {
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"collapsed": true
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},
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"source": [
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"# Solving Non-linear Equations\n",
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"\n",
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"The [Roots](https://github.com/JuliaMath/Roots.jl) package provides methods for solving a non-linear equation (one variable, one function). \n",
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"\n",
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"For a system of non-linear equations, use [NLsolve](https://github.com/JuliaNLSolvers/NLsolve.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, Roots, NLsolve\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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"# Defining and Plotting the Function\n",
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"\n",
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"The next few cells define a fairly simple function and then plots it. \n",
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"\n",
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"If possible, plot your function before trying to find the roots. Maybe you see something strange, perhaps there are several roots? It also helps you set the initial guesses (or brackets) for root solving."
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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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"fn1 (generic function with 1 method)"
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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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"fn1(x,c) = 2*(x - 1.1)^2 - c"
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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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{
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"data": {
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" 218.818,151.29 258.962,237.499 299.106,319.503 339.251,397.301 379.395,470.894 419.539,540.282 459.683,605.464 499.827,666.441 539.971,723.213 580.115,775.779 \n",
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||
|
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" 620.259,824.14 660.403,868.296 700.548,908.247 740.692,943.992 780.836,975.532 820.98,1002.87 861.124,1026 901.268,1044.92 941.412,1059.64 981.556,1070.15 \n",
|
||
|
|
" 1021.7,1076.46 1061.84,1078.56 1101.99,1076.46 1142.13,1070.15 1182.28,1059.64 1222.42,1044.92 1262.57,1026 1302.71,1002.87 1342.85,975.532 1383,943.992 \n",
|
||
|
|
" 1423.14,908.247 1463.29,868.296 1503.43,824.14 1543.57,775.779 1583.72,723.213 1623.86,666.441 1664.01,605.464 1704.15,540.282 1744.29,470.894 1784.44,397.301 \n",
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||
|
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" 1824.58,319.503 \n",
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||
|
|
" \"/>\n",
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||
|
|
"</svg>\n"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "display_data"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"x = -1:0.1:3\n",
|
||
|
|
"\n",
|
||
|
|
"p1 = plot( x,fn1.(x,0.5),\n",
|
||
|
|
" linecolor = :red,\n",
|
||
|
|
" linewidth = 2,\n",
|
||
|
|
" legend = nothing,\n",
|
||
|
|
" title = \"the fn1(x,0.5) function\",\n",
|
||
|
|
" xlabel = \"x\",\n",
|
||
|
|
" ylabel = \"y\" )\n",
|
||
|
|
"display(p1)"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"There seems to be two roots: around 0.6 and 1.6."
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"# Solving a Non-Linear Equation\n",
|
||
|
|
"\n",
|
||
|
|
"The Roots package wants a function with only one input. An easy way to turn ```fn1(a,0.5)``` into that form is by defining an anonymous function:\n",
|
||
|
|
"```\n",
|
||
|
|
"x->fn1(x,0.5)\n",
|
||
|
|
"```\n",
|
||
|
|
"\n",
|
||
|
|
"Then, running \n",
|
||
|
|
"```\n",
|
||
|
|
"find_zero(x->fn1(x,0.5),(x₀,x₁))\n",
|
||
|
|
"```\n",
|
||
|
|
"searches for a root in the `[x₀,x₁]` interval. Alternatively, you can also do \n",
|
||
|
|
"```\n",
|
||
|
|
"find_zero(x->fn1(x,0.5),x₂)\n",
|
||
|
|
"``` \n",
|
||
|
|
"where `x₂` is a single starting guess.\n",
|
||
|
|
"\n",
|
||
|
|
"Instead, running\n",
|
||
|
|
"```\n",
|
||
|
|
"find_zeros(x->fn1(x,0.5),x₀,x₁)\n",
|
||
|
|
"```\n",
|
||
|
|
"searches for all roots between x₀ and x₁. (Notice the *s* in `find_zeros`.)"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 5,
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"name": "stdout",
|
||
|
|
"output_type": "stream",
|
||
|
|
"text": [
|
||
|
|
"at which x is fn1(x,0.5) = 0? 0.600\n",
|
||
|
|
"at which x is fn1(x,0.5) = 0? 1.600\n",
|
||
|
|
"\n",
|
||
|
|
"\u001b[34m\u001b[1myes, there are several roots. Just look at it (in the plot)\u001b[22m\u001b[39m\n"
|
||
|
|
]
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"xRoot1 = find_zero(x->fn1(x,0.5),(-1,1)) #searches for roots in [-1,1]\n",
|
||
|
|
"printlnPs(\"at which x is fn1(x,0.5) = 0? \",xRoot1)\n",
|
||
|
|
"\n",
|
||
|
|
"xRoot2 = find_zero(x->fn1(x,0.5),2) #searches for roots around 2\n",
|
||
|
|
"printlnPs(\"at which x is fn1(x,0.5) = 0? \",xRoot2)\n",
|
||
|
|
"\n",
|
||
|
|
"printblue(\"\\nyes, there are several roots. Just look at it (in the plot)\")"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 6,
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"name": "stdout",
|
||
|
|
"output_type": "stream",
|
||
|
|
"text": [
|
||
|
|
"at which x is fn1(x,0.5) = 0? 0.600 1.600\n"
|
||
|
|
]
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"xRootsAll = find_zeros(x->fn1(x,0.5),-1,3) #find_zeros (notice the \"s\")\n",
|
||
|
|
"\n",
|
||
|
|
"printlnPs(\"at which x is fn1(x,0.5) = 0? \",xRootsAll)"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"# Solving a System of Non-Linear Equations\n",
|
||
|
|
"\n",
|
||
|
|
"The NLsolve package has many options. The cells below illustrate a very simple case (2 non-linear equations with 2 unknowns, no information about the derivatives).\n",
|
||
|
|
"\n",
|
||
|
|
"The two equations are\n",
|
||
|
|
"\n",
|
||
|
|
"$ \n",
|
||
|
|
"y-x^2-1=0\n",
|
||
|
|
"$\n",
|
||
|
|
"\n",
|
||
|
|
"$\n",
|
||
|
|
"y-x-1=0\n",
|
||
|
|
"$\n",
|
||
|
|
"\n",
|
||
|
|
"and the roots are at $(x,y)=(0,1)$ and also at $(1,2)$."
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 7,
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"text/plain": [
|
||
|
|
"fn2 (generic function with 1 method)"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"execution_count": 7,
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "execute_result"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"function fn2(p) #p is a vector with 2 elements, the output too\n",
|
||
|
|
" (x,y) = (p[1],p[2])\n",
|
||
|
|
" z = [y-x^2-1;y-x-1] #equal to [0,0]\n",
|
||
|
|
" return z\n",
|
||
|
|
"end"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 8,
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"name": "stdout",
|
||
|
|
"output_type": "stream",
|
||
|
|
"text": [
|
||
|
|
"There is a solution at -0.000 1.000\n",
|
||
|
|
"There is a another solution at 1.000 2.000\n"
|
||
|
|
]
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"Sol2a = nlsolve(fn2,[0.0,0.5])\n",
|
||
|
|
"printlnPs(\"There is a solution at \",Sol2a.zero)\n",
|
||
|
|
"\n",
|
||
|
|
"Sol2b = nlsolve(fn2,[1.0,0.0]) #try again, using another starting guess\n",
|
||
|
|
"printlnPs(\"There is a another solution at \",Sol2b.zero)"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": null,
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [],
|
||
|
|
"source": []
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"metadata": {
|
||
|
|
"@webio": {
|
||
|
|
"lastCommId": null,
|
||
|
|
"lastKernelId": null
|
||
|
|
},
|
||
|
|
"anaconda-cloud": {},
|
||
|
|
"kernelspec": {
|
||
|
|
"display_name": "Julia 1.6.0",
|
||
|
|
"language": "julia",
|
||
|
|
"name": "julia-1.6"
|
||
|
|
},
|
||
|
|
"language_info": {
|
||
|
|
"file_extension": ".jl",
|
||
|
|
"mimetype": "application/julia",
|
||
|
|
"name": "julia",
|
||
|
|
"version": "1.6.0"
|
||
|
|
}
|
||
|
|
},
|
||
|
|
"nbformat": 4,
|
||
|
|
"nbformat_minor": 1
|
||
|
|
}
|