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Davidnet/Gd.cardozo684.ipynb

Created February 10, 2017 14:32
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{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Tarea 1"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Punto 1\n",
"Escriba una fórmula que involucre $a,b$ y $\\epsilon$ para calcular el número de iteraciones necesarias para garantizar que el método de la bisección tenga un error menor a $\\epsilon$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Solución\n",
"Tenemos la siguiente relación:\n",
"$$\n",
"| r - c_n | \\leq 2^{-(n+1)}(b - a) \\\\\n",
"\\epsilon \\leq 2^{-(n+1)}(b - a) \\leq 2^{-n} (b-a)\\\\\n",
"\\frac{\\epsilon}{b-a} \\leq 2^{-n} \\\\\n",
"n \\geq \\log_2 \\frac{ b - a } {\\epsilon}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": true
},
"source": [
"# Punto 2\n",
"Sea $ f \\in C^{m+1}([a,b])$ y $r$ una raíz de multiplicidad $m > 1$. Muestre que en este caso el método de Newton tiene órden de convergencia lineal y además:\n",
"\n",
"$$\n",
"\\lim_{n \\rightarrow \\infty} \\frac{ | e_{n+1} | }{ | e_n | } = \\frac{m-1}{m} . \n",
"$$\n",
"\n",
"Mostramos ahora que\n",
"\n",
"$$\n",
"\\lim_{n \\rightarrow \\infty} \\frac{ | e_{n+1} | }{ | e_n | } = \\frac{m-1}{m} . \n",
"$$\n",
"\n",
"Observamos que $ e_{n+1} = x_{n+1} - r $\n",
"Es decir que:\n",
"$$\n",
"e_{n+1} = x_n - \\frac{(x-r)^m g(x) }{ m (x-r)^{m-1}g(x) + (x-r)^m g'(x) } - r\n",
"$$\n",
"Tomando la razon de $e_{n}$, tenemos $x=x_n$\n",
"$$\n",
"\\frac{ e_{n+1} }{e_n} = \\frac{ e_n - \\frac{(x-r)^m g(x) }{ m (x-r)^{m-1}g(x) + (x-r)^m g'(x) }}{e_n} \\\\\n",
"\\frac{ e_{n+1} }{e_n} = 1 - \\frac{(x-r)^m g(x) }{ e_n(m (x-r)^{m-1}g(x) + (x-r)^m g'(x)) } \\\\\n",
"\\frac{ e_{n+1} }{e_n} = 1 - \\frac{(x-r)g(x)}{(x-r)(mg(x) + (x-r)g'(x)} \\\\\n",
"\\frac{ e_{n+1} }{e_n} = 1 - \\frac{ g(x_n) }{mg(x_n) + (x_n -r)g'(x_n)}\n",
"$$\n",
"Tomando el límite\n",
"$$\n",
"\\lim_{n \\rightarrow \\infty} 1 - \\frac{ g(x_n) }{mg(x_n) + (x_n -r)g'(x_n)} \\\\\n",
"\\lim_{n \\rightarrow \\infty} 1 - \\frac{g(x_n)}{mg(x_n)} \\\\\n",
"\\lim_{n \\rightarrow \\infty} 1 - \\frac{1}{m} \\\\\n",
"\\frac{m-1}{m}\n",
"$$\n",
"Como el límite para $m>1$ es mas pequeño que $1$ concluimos que la convergencia es lineal.\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Punto 3\n",
"\n",
"Condiciones para que Newton-Raphson tenga convergencia cubica"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Para que $f$ tenga convergencia cubica, se necesita que $f''($"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"using Plots\n",
"using Calculus"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"Plots.GRBackend()"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"gr()"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"BisectionMethod (generic function with 1 method)"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function BisectionMethod(f::Function,a::Number,b::Number;steps::UInt64=typemax(UInt64),δ::Float64=eps(),ϵ::Float64=eps())\n",
" u = f(a)\n",
" v = f(b) \n",
" e = b - a\n",
" w = 0\n",
" c = 0 \n",
" if sign(u) == sign(v)\n",
" error(\"Not a suitable interval\")\n",
" end\n",
" for i in 1:steps\n",
" e = e/2\n",
" c = a + e\n",
" w = f(c)\n",
" if (abs(e) < δ) || (abs(w) < ϵ)\n",
" return c\n",
" end\n",
" if sign(w) != sign(u)\n",
" b = c\n",
" v = w\n",
" else\n",
" a = c\n",
" u = w\n",
" end\n",
" end\n",
" return c\n",
"end "
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"BisectionMethod! (generic function with 1 method)"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function BisectionMethod!(f::Function,a::Number,b::Number;steps::UInt64=typemax(UInt64),δ::Float64=eps(),ϵ::Float64=eps())\n",
" u = f(a)\n",
" v = f(b) \n",
" e = b - a\n",
" w = 0\n",
" c = 0\n",
" roots = Array{Float64}(0)\n",
" if sign(u) == sign(v)\n",
" error(\"Not a suitable interval\")\n",
" end\n",
" for i in 1:steps\n",
" e = e/2\n",
" c = a + e\n",
" w = f(c)\n",
" push!(roots,c)\n",
" if (abs(e) < δ) || (abs(w) < ϵ)\n",
" return roots\n",
" end\n",
" if sign(w) != sign(u)\n",
" b = c\n",
" v = w\n",
" else\n",
" a = c\n",
" u = w\n",
" end\n",
" end\n",
" return roots\n",
"end "
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"NewtonRaphsonMethod (generic function with 1 method)"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function NewtonRaphsonMethod(f::Function,fder::Function,x0::Number;steps::UInt64=typemax(UInt64),δ::Float64=eps(),ϵ::Float64=eps())\n",
" v = f(x0)\n",
" if abs(v) < ϵ\n",
" return x0\n",
" end\n",
" for i in 1:steps\n",
" x1 = x0 - v/(fder(x0))\n",
" v = f(x1)\n",
" if (abs(x1 - x0) < δ) || abs(v) < ϵ\n",
" return x1\n",
" end\n",
" x0 = x1\n",
" end\n",
" return x0\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"2.220446049250313e-16"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"eps()"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"NewtonRaphsonMethod! (generic function with 1 method)"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function NewtonRaphsonMethod!(f::Function,fder::Function,x0::Number;steps::UInt64=typemax(UInt64),δ::Float64=eps(),ϵ::Float64=eps())\n",
" v = f(x0)\n",
" roots = Array{Float64}(1)\n",
" if abs(v) < ϵ\n",
" return push!(roots,x0)\n",
" end\n",
" for i in 1:steps\n",
" x1 = x0 - v/(fder(x0))\n",
" push!(roots,x1)\n",
" v = f(x1)\n",
" if (abs(x1 - x0) < δ) || abs(v) < ϵ\n",
" return roots\n",
" end\n",
" x0 = x1\n",
" end\n",
" return roots\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"SecantMethod (generic function with 2 methods)"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function SecantMethod(f::Function, x0::Number, x1::Number, args::Tuple=();\n",
" tol::AbstractFloat=1e-12, maxiter::Integer=500)\n",
" x = 0.0\n",
" for _ in 1:maxiter\n",
" y1 = f(x1, args...)\n",
" y0 = f(x0, args...)\n",
" x = x1 - y1 * (x1-x0)/(y1-y0)\n",
" if abs(x-x1) < tol\n",
" return x\n",
" end\n",
" x0 = x1\n",
" x1 = x\n",
" end\n",
" warn(\"Max iteration exceeded\")\n",
" return x\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"SecantMethod! (generic function with 2 methods)"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function SecantMethod!(f::Function, x0::Number, x1::Number, args::Tuple=();\n",
" tol::AbstractFloat=1e-12, maxiter::Integer=500)\n",
" x = 0.0\n",
" roots = Array{Float64}(0)\n",
" for _ in 1:maxiter\n",
" y1 = f(x1, args...)\n",
" y0 = f(x0, args...)\n",
" x = x1 - y1 * (x1-x0)/(y1-y0)\n",
" push!(roots,x)\n",
" if abs(x-x1) < tol\n",
" return roots\n",
" end\n",
" x0 = x1\n",
" x1 = x\n",
" end\n",
" warn(\"Max iteration exceeded\")\n",
" return roots\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"logpoints (generic function with 1 method)"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function logpoints(x::Array{Float64,1})\n",
" points = Array{Float64}(0)\n",
" for i in 1:(size(x)[1] - 2)\n",
" a = log(abs(x[i+1]))/log(abs(x[i]))\n",
" if (a != 0) && (a != Inf)\n",
" push!(points,a)\n",
" end\n",
" end\n",
" return points\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"f(x) = 1/x - exp(x)\n",
"fprime = f'\n",
"g(x) = x - tan(x)\n",
"gprime = g'\n",
"h(x) = sin(x) + x^2*cos(x) - x^2 - x\n",
"hprime = h';"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Encontramos un cero de $f$."
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"0.5671432904097838"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"rootf = BisectionMethod(f,0,1)"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"0.0"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f(rootf)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Encontremos los valores de $\\frac{e_{n+1}}{e_n}$"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"errors = abs(BisectionMethod!(f,0,1) .- rootf);"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {
"collapsed": false,
"scrolled": true
},
"outputs": [
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"function NewtonRaphsonMethodModified(f::Function,fder::Function,x0::Number;m::UInt64=1,steps::UInt64=typemax(UInt64),δ::Float64=eps(),ϵ::Float64=eps())\n",
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"rooth = NewtonRaphsonMethodModified(h,hprime,0.2,m=UInt(3),steps=UInt(100))"
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"Vemos que la convergencia es lineal para h, aunque de orden 4 para las demas funciones, tiene mejor convergencia que el método de la secante ya que evalua en dos puntos la funcion $f$ (aunque sería una desventaja si $f$ es complicada)."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": []
}
],
"metadata": {
"anaconda-cloud": {},
"kernelspec": {
"display_name": "Julia 0.5.0",
"language": "julia",
"name": "julia-0.5"
},
"language_info": {
"file_extension": ".jl",
"mimetype": "application/julia",
"name": "julia",
"version": "0.5.0"
}
},
"nbformat": 4,
"nbformat_minor": 1
}
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