{
"cells": [
{
"cell_type": "code",
"execution_count": 12,
"metadata": {
"scrolled": false
},
"outputs": [],
"source": [
"using Plots, ComplexPhasePortrait, ApproxFun, SingularIntegralEquations, DifferentialEquations\n",
"gr();"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# M3M6: Methods of Mathematical Physics\n",
"\n",
"$$\n",
"\\def\\dashint{{\\int\\!\\!\\!\\!\\!\\!-\\,}}\n",
"\\def\\infdashint{\\dashint_{\\!\\!\\!-\\infty}^{\\,\\infty}}\n",
"\\def\\D{\\,{\\rm d}}\n",
"\\def\\E{{\\rm e}}\n",
"\\def\\dx{\\D x}\n",
"\\def\\dt{\\D t}\n",
"\\def\\dz{\\D z}\n",
"\\def\\C{{\\mathbb C}}\n",
"\\def\\R{{\\mathbb R}}\n",
"\\def\\CC{{\\cal C}}\n",
"\\def\\HH{{\\cal H}}\n",
"\\def\\I{{\\rm i}}\n",
"\\def\\qqqquad{\\qquad\\qquad}\n",
"\\def\\qqfor{\\qquad\\hbox{for}\\qquad}\n",
"\\def\\qqwhere{\\qquad\\hbox{where}\\qquad}\n",
"\\def\\Res_#1{\\underset{#1}{\\rm Res}}\\,\n",
"\\def\\sech{{\\rm sech}\\,}\n",
"\\def\\acos{\\,{\\rm acos}\\,}\n",
"\\def\\vc#1{{\\mathbf #1}}\n",
"\\def\\ip<#1,#2>{\\left\\langle#1,#2\\right\\rangle}\n",
"\\def\\norm#1{\\left\\|#1\\right\\|}\n",
"\\def\\half{{1 \\over 2}}\n",
"$$\n",
"\n",
"Dr. Sheehan Olver\n",
"<br>\n",
"s.olver@imperial.ac.uk\n",
"\n",
"<br>\n",
"Website: https://github.com/dlfivefifty/M3M6LectureNotes\n",
"\n",
"\n",
"# Lecture 16: Solving differential equations with orthogonal polynomials\n",
"\n",
"\n",
"This lecture we do the following:\n",
"\n",
"1. Recurrence relationships for Chebyshev and ultrashperical polynomials\n",
" - Conversion\n",
" - Three-term recurrence and Jacobi operators\n",
"2. Application: solving differential equations\n",
" - First order constant coefficients differential equations\n",
" - Second order constant coefficient differential equations with boundary conditions\n",
" - Non-constant coefficients\n",
" \n",
"\n",
"\n",
"That is, we introduce recurrences related to ultraspherical polynomials. This allows us to represent general linear differential equations as almost-banded systems."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Recurrence relationships for Chebyshev and ultraspherical polynomials\n",
"\n",
"\n",
"We have discussed general properties, but now we want to discuss some classical orthogonal polynomials, beginning with Chebyshev (first kind) $T_n(x)$, which is orthogonal w.r.t.\n",
"$$1\\over \\sqrt{1-x^2}$$\n",
"and ultraspherical $C_n^{(\\lambda)}(x)$, which is orthogonal w.r.t.\n",
"$$(1-x^2)^{\\lambda - \\half}$$\n",
"for $\\lambda > 0$. Note that Chebyshev (second kind) satisfies $U_n(x) = C_n^{(1)}(x)$.\n",
"\n",
"For Chebyshev, recall we have the normalization constant (here we use a superscript $T_n(x) = k_n^{\\rm T} x^n + O(x^{n-1})$)\n",
"$$\n",
"k_0^{\\rm T} = 1, k_n^{\\rm T} = 2^{n-1}\n",
"$$\n",
"For Ultraspherical $C_n^{(\\lambda)}$, this is\n",
"$$\n",
"k_n^{(\\lambda)} = {2^n (\\lambda)_n \\over n!} = {2^n \\lambda (\\lambda+1) (\\lambda+2) \\cdots (\\lambda+n-1) \\over n!}\n",
"$$\n",
"where $(\\lambda)_n$ is the Pochhammer symbol. Note for $U_n(x) = C_n^{(1)}(x)$ this simplifies to $k_n^{\\rm U} = k_n^{(1)} = 2^n$.\n",
"\n",
"We have already found the recurrence for Chebyshev:\n",
"$$\n",
"x T_n(x) = {T_{n-1}(x) \\over 2} + {T_{n+1}(x) \\over 2}\n",
"$$\n",
"We will show that we can use this to find the recurrence for _all_ ultraspherical polynomials. But first we need some special recurrences.\n",
"\n",
"**Remark** Jacobi, Laguerre, and Hermite all have similar relationships, which will be discussed further in the problem sheet.\n",
"\n",
"### Derivatives\n",
"\n",
"It turns out that the derivative of $T_n(x)$ is precisely a multiple of $U_{n-1}(x)$, and similarly the derivative of $C_n^{(\\lambda)}$ is a multiple of $C_{n-1}^{(\\lambda+1)}$.\n",
"\n",
"**Proposition (Chebyshev derivative)** $$T_n'(x) = n U_{n-1}(x)$$\n",
"\n",
"**Proof** \n",
"We first show that $T_n'(x)$ is othogonal w.r.t. $\\sqrt{1-x^2}$ to all polynomials of degree $m < n-1$, denoted $f_m$, using integration by parts:\n",
"$$\n",
"\\ip<T_n',f_m>_{\\rm U} = \\int_{-1}^1 T_n'(x) f_m(x) \\sqrt{1-x^2} \\dx = -\\int_{-1}^1 T_n(x) (f_m'(x)(1-x^2) + xf_m) {1 \\over \\sqrt{1-x^2}} \\dx = - \\ip<T_n, f_m'(1-x^2) + x f_m >_{\\rm T} = 0\n",
"$$\n",
"since $f_m'(1-x^2) + f_m $ is degree $m-1 +2 = m+1 < n$.\n",
"\n",
"The constant works out since\n",
"$$\n",
"T_n'(x) = {\\D \\over \\dx} (2^{n-1} x^n) + O(x^{n-2}) = n 2^{n-1} x^{n-1} + O(x^{n-2})\n",
"$$\n",
"⬛️\n",
"\n",
"The exact same proof shows the following:\n",
"\n",
"**Proposition (Ultraspherical derivative)** \n",
"$${\\D \\over \\dx} C_n^{(\\lambda)}(x) = 2 \\lambda C_{n-1}^{(\\lambda+1)}(x)$$\n",
"\n",
"Like the three-term recurrence and Jacobi operators, it is useful to express this in matrix form. That is, for the derivatives of $T_n(x)$ we get\n",
"$$\n",
"{\\D \\over \\dx} \\begin{pmatrix} T_0(x) \\\\ T_1(x) \\\\ T_2(x) \\\\ \\vdots \\end{pmatrix}= \\begin{pmatrix}\n",
"0 \\cr\n",
"1 \\cr \n",
"& 2 \\cr\n",
"&& 3 \\cr\n",
"&&&\\ddots \n",
"\\end{pmatrix} \\begin{pmatrix} U_0(x) \\\\ U_1(x) \\\\ U_2(x) \\\\ \\vdots \\end{pmatrix} \n",
"$$\n",
"which let's us know that, for \n",
"$$\n",
"f(x) = (T_0(x),T_1(x),\\ldots) \\begin{pmatrix} f_0\\\\f_1\\\\\\vdots \\end{pmatrix}\n",
"$$\n",
"we have a derivative operator in coefficient space as\n",
"$$\n",
"f'(x) = (U_0(x),U_1(x),\\ldots)\\begin{pmatrix}\n",
"0 & 1 \\cr \n",
"&& 2 \\cr\n",
"&&& 3 \\cr\n",
"&&&&\\ddots \n",
"\\end{pmatrix} \\begin{pmatrix} f_0\\\\f_1\\\\\\vdots \\end{pmatrix}\n",
"$$\n",
"\n",
"_Demonstration_ Here we see that applying a matrix to a vector of coefficients successfully calculates the derivative:"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"31×32 Array{Float64,2}:\n",
" 0.0 1.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0\n",
" 0.0 0.0 2.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0\n",
" 0.0 0.0 0.0 3.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0\n",
" 0.0 0.0 0.0 0.0 4.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0\n",
" 0.0 0.0 0.0 0.0 0.0 5.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0\n",
" 0.0 0.0 0.0 0.0 0.0 0.0 6.0 … 0.0 0.0 0.0 0.0 0.0 0.0\n",
" 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0\n",
" 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0\n",
" 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0\n",
" 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0\n",
" 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0\n",
" 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0\n",
" 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0\n",
" ⋮ ⋮ ⋱ ⋮ \n",
" 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0\n",
" 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0\n",
" 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0\n",
" 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0\n",
" 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0\n",
" 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0\n",
" 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 26.0 0.0 0.0 0.0 0.0 0.0\n",
" 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 27.0 0.0 0.0 0.0 0.0\n",
" 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 28.0 0.0 0.0 0.0\n",
" 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 29.0 0.0 0.0\n",
" 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 30.0 0.0\n",
" 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 31.0"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f = Fun(x -> cos(x^2), Chebyshev()) # f is expanded in Chebyshev coefficients\n",
"n = ncoefficients(f) # This is the number of coefficients\n",
"D = zeros(n-1,n)\n",
"for k=1:n-1\n",
" D[k,k+1] = k\n",
"end\n",
"D"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Here `D*f.coefficients` gives the vector of coefficients corresponding to the derivative, but now the coefficients are in the $U_n(x)$ basis, that is, `Ultraspherical(1)`:"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-0.001999966666833569"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"fp = Fun(Ultraspherical(1), D*f.coefficients)\n",
"\n",
"fp(0.1)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Indeed, it matches the \"true\" derivative:"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-0.0019999666668335634"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f'(0.1)"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-0.0019999666668333335"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"-2*0.1*sin(0.1^2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Note that in ApproxFun.jl we can construct these operators rather nicely:"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-0.001999966666833569"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"D = Derivative()\n",
"(D*f)(0.1)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Here we see that we can write produce the ∞-dimensional version as follows:"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"ConcreteDerivative : Chebyshev() → Ultraspherical(1)\n",
" ⋅ 1.0 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅\n",
" ⋅ ⋅ 2.0 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅\n",
" ⋅ ⋅ ⋅ 3.0 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅\n",
" ⋅ ⋅ ⋅ ⋅ 4.0 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ 5.0 ⋅ ⋅ ⋅ ⋅ ⋅\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 6.0 ⋅ ⋅ ⋅ ⋅\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 7.0 ⋅ ⋅ ⋅\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 8.0 ⋅ ⋅\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 9.0 ⋅\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋱\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋱"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"D : Chebyshev() → Ultraspherical(1) "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Conversion\n",
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can convert between any two polynomial bases using a lower triangular operator, because their span's are equivalent. In the case of Chebyshev and ultraspherical polynomials, they have the added property that they are banded.\n",
"\n",
"**Proposition (Chebyshev T-to-U conversion)** \n",
"\\begin{align*}\n",
" T_0(x) &= U_0(x) \\\\\n",
" T_1(x) &= {U_1(x) \\over 2} \\\\\n",
" T_n(x) &= {U_n(x) \\over 2} - {U_{n-2}(x) \\over 2}\n",
"\\end{align*}\n",
"\n",
"**Proof** \n",
"\n",
"Before we do the proof, note that the fact that there are limited non-zero entries follows immediately: if $m < n-2$ we have\n",
"$$\n",
"\\ip<T_n, U_m>_{\\rm U} = \\ip<T_n, (1-x^2)U_m>_{\\rm T} = 0\n",
"$$\n",
"\n",
"To actually determine the entries, we use the trigonometric formulae. Recall for $x = (z + z^{-1})/2$, $z = \\E^{\\I \\theta}$, we have\n",
"\\begin{align*}\n",
"T_n(x) &= \\cos n \\theta = {z^{-n} + z^n \\over 2}\\\\\n",
"U_n(x) &= {\\sin (n+1) \\theta \\over \\sin \\theta} = {z^{n+1} - z^{-n-1} \\over z - z^{-1}} = z^{-n} + z^{2-n} + \\cdots + \\cdots + z^{n-2} + z^n \n",
"\\end{align*}\n",
"The result follows immediately.\n",
"\n",
"⬛️\n",
"\n",
"**Corollary (Ultrapherical λ-to-(λ+1) conversion)**\n",
"$$\n",
"C_n^{(\\lambda)}(x) = {\\lambda \\over n+ \\lambda} (C_n^{(\\lambda+1)}(x) - C_{n-2}^{(\\lambda+1)}(x))\n",
"$$\n",
"\n",
"**Proof** This follows from differentiating the previous result. For example:\n",
"\\begin{align*}\n",
" {\\D\\over \\dx} T_0(x) &= {\\D\\over \\dx} U_0(x) \\\\\n",
" {\\D\\over \\dx} T_1(x) &= {\\D\\over \\dx} {U_1(x) \\over 2} \\\\\n",
"{\\D\\over \\dx} T_n(x) &= {\\D\\over \\dx} \\left({U_n(x) \\over 2} - {U_{n-2} \\over 2} \\right)\n",
"\\end{align*}\n",
"becomes\n",
"\\begin{align*}\n",
" 0 &= 0\\\\\n",
" U_0(x) &= C_1^{(2)}(x) \\\\\n",
" n U_{n-1}(x) &= C_{n-1}^{(2)}(x) - C_{n-3}^{(2)}(x)\n",
"\\end{align*}\n",
"\n",
"Differentiating this repeatedly completes the proof.\n",
"\n",
"⬛️\n",
"\n",
"\n",
"Note we can write this in matrix form, for example, we have\n",
"$$\n",
"\\underbrace{\\begin{pmatrix}1 \\cr\n",
" 0 & \\half\\cr\n",
" -\\half & 0 & \\half \\cr\n",
" &\\ddots &\\ddots & \\ddots\\end{pmatrix} }_{S_0^\\top} \\begin{pmatrix} \n",
" U_0(x) \\\\ U_1(x) \\\\ U_2(x) \\\\ \\vdots \\end{pmatrix} = \\begin{pmatrix} T_0(x) \\\\ T_1(x) \\\\ T_2(x) \\\\ \\vdots \\end{pmatrix}\n",
"$$\n",
"\n",
"therefore,\n",
"$$\n",
"f(x) = (T_0(x),T_1(x),\\ldots) \\begin{pmatrix} f_0\\\\f_1\\\\\\vdots \\end{pmatrix} = (U_0(x),U_1(x),\\ldots) S_0 \\begin{pmatrix} f_0\\\\f_1\\\\\\vdots \\end{pmatrix}\n",
"$$\n",
"\n",
"Again, we can construct this nicely in ApproxFun:"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"ConcreteConversion : Chebyshev() → Ultraspherical(1)\n",
" 1.0 0.0 -0.5 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅\n",
" ⋅ 0.5 0.0 -0.5 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅\n",
" ⋅ ⋅ 0.5 0.0 -0.5 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅\n",
" ⋅ ⋅ ⋅ 0.5 0.0 -0.5 ⋅ ⋅ ⋅ ⋅ ⋅\n",
" ⋅ ⋅ ⋅ ⋅ 0.5 0.0 -0.5 ⋅ ⋅ ⋅ ⋅\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ 0.5 0.0 -0.5 ⋅ ⋅ ⋅\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 0.5 0.0 -0.5 ⋅ ⋅\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 0.5 0.0 -0.5 ⋅\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 0.5 0.0 ⋱\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 0.5 ⋱\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋱"
]
},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"S₀ = I : Chebyshev() → Ultraspherical(1)"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Fun(Ultraspherical(1),[1.13032, 0.542991, 0.133011, 0.021897, 0.00271463, 0.000269864, 2.23891e-5, 1.5937e-6, 9.93309e-8, 5.5059e-9, 2.74775e-10, 1.24699e-11, 5.19555e-13, 1.99485e-14])"
]
},
"execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f = Fun(exp, Chebyshev())\n",
"g = S₀*f"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.0"
]
},
"execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"g(0.1) - exp(0.1)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Ultraspherical Three-term recurrence \n",
"\n",
"**Theorem (three-term recurrence for Chebyshev U)** \n",
"\\begin{align*}\n",
"x U_0(x) &= {U_1(x) \\over 2} \\\\\n",
"x U_n(x) &= {U_{n-1}(x) \\over 2} + {U_{n+1}(x) \\over 2}\n",
"\\end{align*}\n",
"\n",
"**Proof**\n",
"Differentiating\n",
"\\begin{align*}\n",
" x T_0(x) &= T_1(x) \\\\\n",
"x T_n(x) &= {T_{n-1}(x) \\over 2} + {T_{n+1}(x) \\over 2}\n",
"\\end{align*}\n",
"we get\n",
"\\begin{align*}\n",
" T_0(x) &= U_0(x) \\\\\n",
" T_n(x) + n x U_{n-1}(x) &= {(n-1) U_{n-2}(x) \\over 2} + {(n+1) U_n(x) \\over 2}\n",
"\\end{align*}\n",
"substituting in the conversion $T_n(x) = (U_n(x) - U_{n-2}(x))/2$ we get\n",
"\\begin{align*}\n",
" T_0(x) &= U_0(x) \\\\\n",
" n x U_{n-1}(x) &= {(n-1) U_{n-2}(x) \\over 2} + {(n+1) U_n(x) \\over 2} - (U_n(x) - U_{n-2}(x))/2 = {n U_{n-2}(x) \\over 2} + {n U_n(x) \\over 2}\n",
"\\end{align*}\n",
"\n",
"⬛️\n",
"\n",
"Differentiating this theorem again and applying the conversion we get the following\n",
"\n",
"**Corollary (three-term recurrence for ultrashperical)** \n",
"\\begin{align*}\n",
"x C_0^{(\\lambda)}(x) &= {1 \\over 2\\lambda } C_1^{(\\lambda)}(x) \\\\\n",
" x C_n^{(\\lambda)}(x) &= {n+2\\lambda-1 \\over 2(n+\\lambda)} C_{n-1}^{(\\lambda)}(x) + {n+1 \\over 2(n+\\lambda)} C_{n+1}^{(\\lambda)}(x) \n",
"\\end{align*}\n",
"\n",
"\n",
"Here's an example of the Jacobi operator (which is the transpose of the multiplciation by $x$ operator):"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"TransposeOperator : Ultraspherical(2) → Ultraspherical(2)\n",
" 0.0 0.25 … ⋅ ⋅ ⋅ ⋅\n",
" 0.6666666666666666 0.0 ⋅ ⋅ ⋅ ⋅\n",
" ⋅ 0.625 ⋅ ⋅ ⋅ ⋅\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ ⋅\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ ⋅\n",
" ⋅ ⋅ … ⋅ ⋅ ⋅ ⋅\n",
" ⋅ ⋅ 0.4375 ⋅ ⋅ ⋅\n",
" ⋅ ⋅ 0.0 0.4444444444444444 ⋅ ⋅\n",
" ⋅ ⋅ 0.55 0.0 0.45 ⋅\n",
" ⋅ ⋅ ⋅ 0.5454545454545454 0.0 ⋱\n",
" ⋅ ⋅ … ⋅ ⋅ ⋱ ⋱"
]
},
"execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Multiplication(Fun(), Ultraspherical(2))'"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Application: solving differential equations\n",
"\n",
"The preceding results allowed us to represent \n",
"\n",
"1. Differentiation\n",
"2. Conversion\n",
"3. Multiplication\n",
"\n",
"as banded operators. We will see that we can combine these, along with \n",
"\n",
"4\\. Evaluation\n",
"\n",
"to solve ordinary differential equations.\n",
"\n",
"### First order, constant coefficient differential equations\n",
"\n",
"Consider the simplest ODE:\n",
"\\begin{align*}\n",
"u(0) &= 0 \\\\\n",
"u'(x) - u(x) &= 0 \n",
"\\end{align*}\n",
"and suppose represent $u(x)$ in its Chebyshev expansion, with to be determined coefficents. In other words, we want to calculate coefficients $u_k$ such that\n",
"$$\n",
"u(x) = \\sum_{k=0}^\\infty u_k T_k(x) = (T_0(x), T_1(x), \\ldots) \\begin{pmatrix} u_0 \\\\ u_1 \\\\ \\vdots \\end{pmatrix}\n",
"$$\n",
"In this case we know that $u(x) = \\E^x$, but we would still need other means to calculate $u_k$ (They are definitely not as simple as Taylor series coefficients).\n",
"\n",
"We can express the constraints as acting on the coefficients. For example, we have\n",
"$$\n",
"u(0) = (T_0(0), T_1(0), \\ldots) \\begin{pmatrix} u_0\\\\u_1\\\\\\vdots \\end{pmatrix} = (1,0,-1,0,1,\\ldots) \\begin{pmatrix} u_0\\\\u_1\\\\\\vdots \\end{pmatrix} \n",
"$$\n",
"We also have \n",
"$$u'(x) = (U_0(x),U_1(x),\\ldots) \\begin{pmatrix}\n",
"0 & 1 \\cr \n",
"&& 2 \\cr\n",
"&&& 3 \\cr\n",
"&&&&\\ddots \n",
"\\end{pmatrix}\\begin{pmatrix} u_0\\\\u_1\\\\\\vdots \\end{pmatrix} \n",
"$$\n",
"To represent $u'(x) - u(x)$, we need to make sure the bases are compatible. In other words, we want to express $u(x)$ in it's $U_k(x)$ basis using the conversion operator $S_0$:\n",
"$$u(x) = (U_0(x),U_1(x),\\ldots) \\begin{pmatrix}\n",
" 1 &0 & -\\half \\cr \n",
"& \\half & 0 & -\\half \\cr\n",
"&&\\ddots & \\ddots & \\ddots\n",
"\\end{pmatrix}\\begin{pmatrix} u_0\\\\u_1\\\\\\vdots \\end{pmatrix} \n",
"$$\n",
"\n",
"Which gives us, \n",
"$$\n",
"u'(x) - u(x) = (U_0(x),U_1(x),\\ldots) \\begin{pmatrix}\n",
" -1 &1 & \\half \\cr \n",
"& -\\half & 2 & \\half \\cr\n",
"&& -\\half & 3 & \\half \\cr\n",
"&&&\\ddots & \\ddots & \\ddots\n",
"\\end{pmatrix} \\begin{pmatrix} u_0\\\\u_1\\\\\\vdots \\end{pmatrix} \n",
"$$\n",
"\n",
"\n",
"Combing the differential part and the evaluation part, we arrive at an (infinite) system of equations for the coefficients $u_0,u_1,\\dots$:\n",
"$$\n",
"\\begin{pmatrix}\n",
" 1 & 0 & -1 & 0 & 1 & \\cdots \\\\\n",
" -1 &1 & \\half \\cr \n",
"& -\\half & 2 & \\half \\cr\n",
"&& -\\half & 3 & \\half \\cr\n",
"&&&\\ddots & \\ddots & \\ddots\n",
"\\end{pmatrix} \\begin{pmatrix} u_0\\\\u_1\\\\\\vdots \\end{pmatrix} = \\begin{pmatrix} 1 \\\\ 0 \\\\ 0 \\\\ \\vdots \\end{pmatrix}\n",
"$$\n",
"\n",
"How to solve this system is outside the scope of this course (though a simple approach is to truncate the infinite system to finite systems). We can however do this in ApproxFun:"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"InterlaceOperator : Chebyshev() → 2-element ArraySpace:\n",
"Space{D,Float64} where D[ConstantSpace(Point(0.0)), Ultraspherical(1)]\n",
" 1.0 0.0 -1.0 0.0 1.0 0.0 -1.0 0.0 1.0 0.0 ⋯\n",
" -1.0 1.0 0.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 ⋱\n",
" 0.0 -0.5 2.0 0.5 0.0 0.0 0.0 0.0 0.0 0.0 ⋱\n",
" 0.0 0.0 -0.5 3.0 0.5 0.0 0.0 0.0 0.0 0.0 ⋱\n",
" 0.0 0.0 0.0 -0.5 4.0 0.5 0.0 0.0 0.0 0.0 ⋱\n",
" 0.0 0.0 0.0 0.0 -0.5 5.0 0.5 0.0 0.0 0.0 ⋱\n",
" 0.0 0.0 0.0 0.0 0.0 -0.5 6.0 0.5 0.0 0.0 ⋱\n",
" 0.0 0.0 0.0 0.0 0.0 0.0 -0.5 7.0 0.5 0.0 ⋱\n",
" 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.5 8.0 0.5 ⋱\n",
" 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.5 9.0 ⋱\n",
" ⋮ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱"
]
},
"execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"B = Evaluation(0.0) : Chebyshev()\n",
"D = Derivative() : Chebyshev() → Ultraspherical(1)\n",
"S₀ = I : Chebyshev() → Ultraspherical(1)\n",
"L = [B; \n",
" D - S₀]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can solve this system as follows:"
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {},
"outputs": [
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},
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"metadata": {},
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}
],
"source": [
"u = L \\ [1; 0]\n",
"plot(u)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"It matches the \"true\" result:"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-4.440892098500626e-16"
]
},
"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"u(0.1) - exp(0.1)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Note we can incorporate right-hand sides as well, for example, to solve $u'(x) - u(x) = f(x)$, by expanding $f$ in its Chebyshev U series.\n",
"\n",
"### Second-order constanst coefficient equations\n",
"\n",
"This approach extends to second-order constant-coefficient equations by using ultraspherical polynomials. Consider\n",
"\\begin{align*}\n",
"u(-1) &= 1\\\\\n",
"u(1) &= 0\\\\\n",
"u''(x) + u'(x) + u(x) &= 0\n",
"\\end{align*}\n",
"Evaluation works as in the first-order case. To handle second-derivatives, we need $C^{(2)}$ polynomials:"
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"ConcreteDerivative : Chebyshev() → Ultraspherical(2)\n",
" ⋅ ⋅ 4.0 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅\n",
" ⋅ ⋅ ⋅ 6.0 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅\n",
" ⋅ ⋅ ⋅ ⋅ 8.0 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ 10.0 ⋅ ⋅ ⋅ ⋅ ⋅\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 12.0 ⋅ ⋅ ⋅ ⋅\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 14.0 ⋅ ⋅ ⋅\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 16.0 ⋅ ⋅\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 18.0 ⋅\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋱\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋱\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋱"
]
},
"execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"D₀ = Derivative() : Chebyshev() → Ultraspherical(1)\n",
"D₁ = Derivative() : Ultraspherical(1) → Ultraspherical(2)\n",
"D₁*D₀ # 2 zeros not printed in (1,1) and (1,2) entry"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"For the identity operator, we use two conversion operators:"
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"ConversionWrapper : Chebyshev() → Ultraspherical(2)\n",
" 1.0 0.0 -0.6666666666666666 0.0 … ⋅ ⋅ ⋅\n",
" ⋅ 0.25 0.0 -0.375 ⋅ ⋅ ⋅\n",
" ⋅ ⋅ 0.16666666666666666 0.0 ⋅ ⋅ ⋅\n",
" ⋅ ⋅ ⋅ 0.125 ⋅ ⋅ ⋅\n",
" ⋅ ⋅ ⋅ ⋅ 0.07142857142857142 ⋅ ⋅\n",
" ⋅ ⋅ ⋅ ⋅ … 0.0 0.0625 ⋅\n",
" ⋅ ⋅ ⋅ ⋅ -0.12698412698412698 0.0 ⋱\n",
" ⋅ ⋅ ⋅ ⋅ 0.0 -0.1125 ⋱\n",
" ⋅ ⋅ ⋅ ⋅ 0.05555555555555555 0.0 ⋱\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ 0.05 ⋱\n",
" ⋅ ⋅ ⋅ ⋅ … ⋅ ⋅ ⋱"
]
},
"execution_count": 28,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"S₀ = I : Chebyshev() → Ultraspherical(1)\n",
"S₁ = I : Ultraspherical(1) → Ultraspherical(2)\n",
"S₁*S₀"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"And for the first derivative, we use a derivative and then a conversion:"
]
},
{
"cell_type": "code",
"execution_count": 29,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"TimesOperator : Chebyshev() → Ultraspherical(2)\n",
" ⋅ 1.0 0.0 -1.0 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅\n",
" ⋅ ⋅ 1.0 0.0 -1.0 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅\n",
" ⋅ ⋅ ⋅ 1.0 0.0 -1.0 ⋅ ⋅ ⋅ ⋅ ⋅\n",
" ⋅ ⋅ ⋅ ⋅ 1.0 0.0 -1.0 ⋅ ⋅ ⋅ ⋅\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ 1.0 0.0 -1.0 ⋅ ⋅ ⋅\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 1.0 0.0 -1.0 ⋅ ⋅\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 1.0 0.0 -1.0 ⋅\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 1.0 0.0 ⋱\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 1.0 ⋱\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋱\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋱"
]
},
"execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"S₁*D₀ # or could have been D₁*S₀"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Putting everything together we get:"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"InterlaceOperator : Chebyshev() → 3-element ArraySpace:\n",
"Space{D,Float64} where D[ConstantSpace(Point(-1)), ConstantSpace(Point(1)), Ultraspherical(2)]\n",
" 1.0 -1.0 1.0 -1.0 … 1.0 -1.0 ⋯\n",
" 1.0 1.0 1.0 1.0 1.0 1.0 ⋱\n",
" 1.0 1.0 3.3333333333333335 -1.0 0.0 0.0 ⋱\n",
" 0.0 0.25 1.0 5.625 0.0 0.0 ⋱\n",
" 0.0 0.0 0.16666666666666666 1.0 0.0 0.0 ⋱\n",
" 0.0 0.0 0.0 0.125 … 0.0 0.0 ⋱\n",
" 0.0 0.0 0.0 0.0 0.07142857142857142 0.0 ⋱\n",
" 0.0 0.0 0.0 0.0 -1.0 0.0625 ⋱\n",
" 0.0 0.0 0.0 0.0 15.873015873015873 -1.0 ⋱\n",
" 0.0 0.0 0.0 0.0 1.0 17.8875 ⋱\n",
" ⋮ ⋱ ⋱ ⋱ … ⋱ ⋱ ⋱"
]
},
"execution_count": 30,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"B₋₁ = Evaluation(-1) : Chebyshev()\n",
"B₁ = Evaluation(1) : Chebyshev()\n",
"# u(-1) \n",
"# u(1)\n",
"# u'' + u' +u\n",
"\n",
"L = [B₋₁; \n",
" B₁; \n",
" D₁*D₀ + S₁*D₀ + S₁*S₀]"
]
},
{
"cell_type": "code",
"execution_count": 31,
"metadata": {},
"outputs": [
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},
"execution_count": 31,
"metadata": {},
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}
],
"source": [
"u = L \\ [1.0,0.0,0.0]\n",
"plot(u)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Variable coefficients\n",
"\n",
"Consider the Airy ODE \n",
"\\begin{align*}\n",
"u(-1) &= 1\\\\\n",
"u(1) &= 0\\\\\n",
"u''(x) - xu(x) &= 0\n",
"\\end{align*}\n",
"\n",
"to handle, this, we need only use the Jacobi operator to represent multiplication by $x$:"
]
},
{
"cell_type": "code",
"execution_count": 32,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"ConcreteMultiplication : Chebyshev() → Chebyshev()\n",
" 0.0 0.5 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅\n",
" 1.0 0.0 0.5 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅\n",
" ⋅ 0.5 0.0 0.5 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅\n",
" ⋅ ⋅ 0.5 0.0 0.5 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅\n",
" ⋅ ⋅ ⋅ 0.5 0.0 0.5 ⋅ ⋅ ⋅ ⋅ ⋅\n",
" ⋅ ⋅ ⋅ ⋅ 0.5 0.0 0.5 ⋅ ⋅ ⋅ ⋅\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ 0.5 0.0 0.5 ⋅ ⋅ ⋅\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 0.5 0.0 0.5 ⋅ ⋅\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 0.5 0.0 0.5 ⋅\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 0.5 0.0 ⋱\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋱ ⋱"
]
},
"execution_count": 32,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x = Fun()\n",
"Jᵗ = Multiplication(x) : Chebyshev() → Chebyshev() # transpose of the Jacobi operator"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We set op ther system as follows:"
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"InterlaceOperator : Chebyshev() → 3-element ArraySpace:\n",
"Space{D,Float64} where D[ConstantSpace(Point(-1)), ConstantSpace(Point(1)), Ultraspherical(2)]\n",
" 1.0 -1.0 1.0 -1.0 … -1.0 ⋯\n",
" 1.0 1.0 1.0 1.0 1.0 ⋱\n",
" 0.0 -0.16666666666666669 4.0 0.25 0.0 ⋱\n",
" -0.25 0.0 0.0625 6.0 0.0 ⋱\n",
" 0.0 -0.08333333333333333 0.0 0.05 0.0 ⋱\n",
" 0.0 0.0 -0.0625 0.0 … 0.0 ⋱\n",
" 0.0 0.0 0.0 -0.05 -0.03571428571428571 ⋱\n",
" 0.0 0.0 0.0 0.0 0.0 ⋱\n",
" 0.0 0.0 0.0 0.0 0.03571428571428571 ⋱\n",
" 0.0 0.0 0.0 0.0 18.0 ⋱\n",
" ⋮ ⋱ ⋱ ⋱ … ⋱ ⋱"
]
},
"execution_count": 33,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"L = [B₋₁; # u(-1)\n",
" B₁ ; # u(1)\n",
" D₁*D₀ - S₁*S₀*Jᵗ] # u'' - x*u"
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {},
"outputs": [
{
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"source": [
"u = L \\ [1.0;0.0;0.0]\n",
"plot(u; legend=false)"
]
},
{
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"metadata": {},
"source": [
"If we introduce a small parameter, that is, solve\n",
"\\begin{align*}\n",
"u(-1) &= 1\\\\\n",
"u(1) &= 0\\\\\n",
"\\epsilon u''(x) - xu(x) &= 0\n",
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]
},
{
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" 2134.91,859.228 2134.23,859.228 2133.55,859.228 2132.86,859.228 2132.18,859.228 2131.49,859.228 2130.81,859.228 2130.12,859.228 2129.43,859.228 2128.73,859.228 \n",
" 2128.04,859.228 2127.34,859.228 2126.64,859.228 2125.94,859.228 2125.24,859.228 2124.54,859.228 2123.83,859.228 2123.13,859.228 2122.42,859.228 2121.71,859.228 \n",
" 2121,859.228 2120.28,859.228 2119.57,859.228 2118.85,859.228 2118.13,859.228 2117.41,859.228 2116.69,859.228 2115.97,859.228 2115.24,859.228 2114.51,859.228 \n",
" 2113.79,859.228 2113.06,859.228 2112.32,859.228 2111.59,859.228 2110.85,859.228 2110.12,859.228 2109.38,859.228 2108.64,859.228 2107.9,859.228 2107.15,859.228 \n",
" 2106.41,859.228 2105.66,859.228 2104.91,859.228 2104.16,859.228 2103.41,859.228 2102.66,859.228 2101.9,859.228 2101.14,859.228 2100.38,859.228 2099.62,859.228 \n",
" 2098.86,859.228 2098.1,859.228 2097.33,859.228 2096.57,859.228 2095.8,859.228 2095.03,859.228 2094.25,859.228 2093.48,859.228 2092.71,859.228 2091.93,859.228 \n",
" 2091.15,859.228 2090.37,859.228 2089.59,859.228 2088.81,859.228 2088.02,859.228 2087.23,859.228 2086.45,859.228 2085.66,859.228 2084.86,859.228 2084.07,859.228 \n",
" 2083.28,859.228 2082.48,859.228 2081.68,859.228 2080.88,859.228 2080.08,859.228 2079.28,859.228 2078.47,859.228 2077.67,859.228 2076.86,859.228 2076.05,859.228 \n",
" 2075.24,859.228 2074.43,859.228 2073.61,859.228 2072.8,859.228 2071.98,859.228 2071.16,859.228 2070.34,859.228 2069.52,859.228 2068.69,859.228 2067.87,859.228 \n",
" 2067.04,859.228 2066.21,859.228 2065.38,859.228 2064.55,859.228 2063.72,859.228 2062.88,859.228 2062.04,859.228 2061.21,859.228 2060.37,859.228 2059.53,859.228 \n",
" 2058.68,859.228 2057.84,859.228 2056.99,859.228 2056.14,859.228 2055.3,859.228 2054.44,859.228 2053.59,859.228 2052.74,859.228 2051.88,859.228 2051.03,859.228 \n",
" 2050.17,859.228 2049.31,859.228 2048.45,859.228 2047.58,859.228 2046.72,859.228 2045.85,859.228 2044.98,859.228 2044.11,859.228 2043.24,859.228 2042.37,859.228 \n",
" 2041.5,859.228 2040.62,859.228 2039.74,859.228 2038.86,859.228 2037.98,859.228 2037.1,859.228 2036.22,859.228 2035.33,859.228 2034.45,859.228 2033.56,859.228 \n",
" 2032.67,859.228 2031.78,859.228 2030.89,859.228 2029.99,859.228 2029.1,859.228 2028.2,859.228 2027.3,859.228 2026.4,859.228 2025.5,859.228 2024.6,859.228 \n",
" 2023.69,859.228 2022.79,859.228 2021.88,859.228 2020.97,859.228 2020.06,859.228 2019.15,859.228 2018.23,859.228 2017.32,859.228 2016.4,859.228 2015.48,859.228 \n",
" 2014.56,859.228 2013.64,859.228 2012.72,859.228 2011.79,859.228 2010.87,859.228 2009.94,859.228 2009.01,859.228 2008.08,859.228 2007.15,859.228 2006.22,859.228 \n",
" 2005.28,859.228 2004.35,859.228 2003.41,859.228 2002.47,859.228 2001.53,859.228 2000.59,859.228 1999.64,859.228 1998.7,859.228 1997.75,859.228 1996.81,859.228 \n",
" 1995.86,859.228 1994.91,859.228 1993.95,859.228 1993,859.228 1992.05,859.228 1991.09,859.228 1990.13,859.228 1989.17,859.228 1988.21,859.228 1987.25,859.228 \n",
" 1986.28,859.228 1985.32,859.228 1984.35,859.228 1983.39,859.228 1982.42,859.228 1981.44,859.228 1980.47,859.228 1979.5,859.228 1978.52,859.228 1977.55,859.228 \n",
" 1976.57,859.228 1975.59,859.228 1974.61,859.228 1973.63,859.228 1972.64,859.228 1971.66,859.228 1970.67,859.228 1969.68,859.228 1968.69,859.228 1967.7,859.228 \n",
" 1966.71,859.228 1965.72,859.228 1964.72,859.228 1963.73,859.228 1962.73,859.228 1961.73,859.228 1960.73,859.228 1959.73,859.228 1958.72,859.228 1957.72,859.228 \n",
" 1956.71,859.228 1955.71,859.228 1954.7,859.228 1953.69,859.228 1952.68,859.228 1951.66,859.228 1950.65,859.228 1949.63,859.228 1948.62,859.228 1947.6,859.228 \n",
" 1946.58,859.228 1945.56,859.228 1944.53,859.228 1943.51,859.228 1942.49,859.228 1941.46,859.228 1940.43,859.228 1939.4,859.228 1938.37,859.228 1937.34,859.228 \n",
" 1936.31,859.228 1935.27,859.228 1934.23,859.228 1933.2,859.228 1932.16,859.228 1931.12,859.228 1930.08,859.228 1929.04,859.228 1927.99,859.228 1926.95,859.228 \n",
" 1925.9,859.228 1924.85,859.228 1923.8,859.228 1922.75,859.228 1921.7,859.228 1920.65,859.228 1919.59,859.228 1918.54,859.228 1917.48,859.228 1916.42,859.228 \n",
" 1915.36,859.228 1914.3,859.228 1913.24,859.228 1912.17,859.228 1911.11,859.228 1910.04,859.228 1908.97,859.228 1907.91,859.228 1906.84,859.228 1905.76,859.228 \n",
" 1904.69,859.228 1903.62,859.228 1902.54,859.228 1901.47,859.228 1900.39,859.228 1899.31,859.228 1898.23,859.228 1897.15,859.228 1896.06,859.228 1894.98,859.228 \n",
" 1893.89,859.228 1892.81,859.228 1891.72,859.228 1890.63,859.228 1889.54,859.228 1888.45,859.228 1887.36,859.228 1886.26,859.228 1885.17,859.228 1884.07,859.228 \n",
" 1882.97,859.228 1881.87,859.228 1880.77,859.228 1879.67,859.228 1878.57,859.228 1877.46,859.228 1876.36,859.228 1875.25,859.228 1874.14,859.228 1873.03,859.228 \n",
" 1871.92,859.228 1870.81,859.228 1869.7,859.228 1868.59,859.228 1867.47,859.228 1866.35,859.228 1865.24,859.228 1864.12,859.228 1863,859.228 1861.88,859.228 \n",
" 1860.75,859.228 1859.63,859.228 1858.51,859.228 1857.38,859.228 1856.25,859.228 1855.12,859.228 1854,859.228 1852.86,859.228 1851.73,859.228 1850.6,859.228 \n",
" 1849.47,859.228 1848.33,859.228 1847.19,859.228 1846.06,859.228 1844.92,859.228 1843.78,859.228 1842.64,859.228 1841.49,859.228 1840.35,859.228 1839.2,859.228 \n",
" 1838.06,859.228 1836.91,859.228 1835.76,859.228 1834.61,859.228 1833.46,859.228 1832.31,859.228 1831.16,859.228 1830,859.228 1828.85,859.228 1827.69,859.228 \n",
" 1826.54,859.228 1825.38,859.228 1824.22,859.228 1823.06,859.228 1821.9,859.228 1820.73,859.228 1819.57,859.228 1818.4,859.228 1817.24,859.228 1816.07,859.228 \n",
" 1814.9,859.228 1813.73,859.228 1812.56,859.228 1811.39,859.228 1810.22,859.228 1809.04,859.228 1807.87,859.228 1806.69,859.228 1805.51,859.228 1804.33,859.228 \n",
" 1803.15,859.228 1801.97,859.228 1800.79,859.228 1799.61,859.228 1798.43,859.228 1797.24,859.228 1796.05,859.228 1794.87,859.228 1793.68,859.228 1792.49,859.228 \n",
" 1791.3,859.228 1790.11,859.228 1788.92,859.228 1787.72,859.228 1786.53,859.228 1785.33,859.228 1784.13,859.228 1782.94,859.228 1781.74,859.228 1780.54,859.228 \n",
" 1779.34,859.228 1778.14,859.228 1776.93,859.228 1775.73,859.228 1774.52,859.228 1773.32,859.228 1772.11,859.228 1770.9,859.228 1769.69,859.228 1768.48,859.228 \n",
" 1767.27,859.228 1766.06,859.228 1764.85,859.228 1763.63,859.228 1762.42,859.228 1761.2,859.228 1759.98,859.228 1758.76,859.228 1757.55,859.228 1756.33,859.228 \n",
" 1755.1,859.228 1753.88,859.228 1752.66,859.228 1751.43,859.228 1750.21,859.228 1748.98,859.228 1747.76,859.228 1746.53,859.228 1745.3,859.228 1744.07,859.228 \n",
" 1742.84,859.228 1741.6,859.228 1740.37,859.228 1739.14,859.228 1737.9,859.228 1736.67,859.228 1735.43,859.228 1734.19,859.228 1732.95,859.228 1731.71,859.228 \n",
" 1730.47,859.228 1729.23,859.228 1727.99,859.228 1726.74,859.228 1725.5,859.228 1724.25,859.228 1723.01,859.228 1721.76,859.228 1720.51,859.228 1719.26,859.228 \n",
" 1718.01,859.228 1716.76,859.228 1715.51,859.228 1714.26,859.228 1713,859.228 1711.75,859.228 1710.49,859.228 1709.24,859.228 1707.98,859.228 1706.72,859.228 \n",
" 1705.46,859.228 1704.2,859.228 1702.94,859.228 1701.68,859.228 1700.41,859.228 1699.15,859.228 1697.89,859.228 1696.62,859.228 1695.35,859.228 1694.09,859.228 \n",
" 1692.82,859.228 1691.55,859.228 1690.28,859.228 1689.01,859.228 1687.74,859.228 1686.47,859.228 1685.19,859.228 1683.92,859.228 1682.64,859.228 1681.37,859.228 \n",
" 1680.09,859.228 1678.81,859.228 1677.53,859.228 1676.25,859.228 1674.97,859.228 1673.69,859.228 1672.41,859.228 1671.13,859.228 1669.84,859.228 1668.56,859.228 \n",
" 1667.27,859.228 1665.99,859.228 1664.7,859.228 1663.41,859.228 1662.13,859.228 1660.84,859.228 1659.55,859.228 1658.26,859.228 1656.96,859.228 1655.67,859.228 \n",
" 1654.38,859.228 1653.08,859.228 1651.79,859.228 1650.49,859.228 1649.2,859.228 1647.9,859.228 1646.6,859.228 1645.3,859.228 1644,859.228 1642.7,859.228 \n",
" 1641.4,859.228 1640.1,859.228 1638.79,859.228 1637.49,859.228 1636.19,859.228 1634.88,859.228 1633.58,859.228 1632.27,859.228 1630.96,859.228 1629.65,859.228 \n",
" 1628.34,859.228 1627.03,859.228 1625.72,859.228 1624.41,859.228 1623.1,859.228 1621.79,859.228 1620.47,859.228 1619.16,859.228 1617.85,859.228 1616.53,859.228 \n",
" 1615.21,859.228 1613.9,859.228 1612.58,859.228 1611.26,859.228 1609.94,859.228 1608.62,859.228 1607.3,859.228 1605.98,859.228 1604.66,859.228 1603.33,859.228 \n",
" 1602.01,859.228 1600.69,859.228 1599.36,859.228 1598.04,859.228 1596.71,859.228 1595.38,859.228 1594.05,859.228 1592.73,859.228 1591.4,859.228 1590.07,859.228 \n",
" 1588.74,859.228 1587.41,859.228 1586.07,859.228 1584.74,859.228 1583.41,859.228 1582.07,859.228 1580.74,859.228 1579.4,859.228 1578.07,859.228 1576.73,859.228 \n",
" 1575.39,859.228 1574.06,859.228 1572.72,859.228 1571.38,859.228 1570.04,859.228 1568.7,859.228 1567.36,859.228 1566.02,859.228 1564.67,859.228 1563.33,859.228 \n",
" 1561.99,859.228 1560.64,859.228 1559.3,859.228 1557.95,859.228 1556.61,859.228 1555.26,859.228 1553.91,859.228 1552.57,859.228 1551.22,859.228 1549.87,859.228 \n",
" 1548.52,859.228 1547.17,859.228 1545.82,859.228 1544.47,859.228 1543.11,859.228 1541.76,859.228 1540.41,859.228 1539.06,859.228 1537.7,859.228 1536.35,859.228 \n",
" 1534.99,859.228 1533.63,859.228 1532.28,859.228 1530.92,859.228 1529.56,859.228 1528.2,859.228 1526.84,859.228 1525.49,859.228 1524.13,859.228 1522.76,859.228 \n",
" 1521.4,859.228 1520.04,859.228 1518.68,859.228 1517.32,859.228 1515.95,859.228 1514.59,859.228 1513.23,859.228 1511.86,859.228 1510.49,859.228 1509.13,859.228 \n",
" 1507.76,859.228 1506.4,859.228 1505.03,859.228 1503.66,859.228 1502.29,859.228 1500.92,859.228 1499.55,859.228 1498.18,859.228 1496.81,859.228 1495.44,859.228 \n",
" 1494.07,859.228 1492.7,859.228 1491.32,859.228 1489.95,859.228 1488.58,859.228 1487.2,859.228 1485.83,859.228 1484.45,859.228 1483.08,859.228 1481.7,859.228 \n",
" 1480.33,859.228 1478.95,859.228 1477.57,859.228 1476.19,859.228 1474.82,859.228 1473.44,859.228 1472.06,859.228 1470.68,859.228 1469.3,859.228 1467.92,859.228 \n",
" 1466.54,859.228 1465.15,859.228 1463.77,859.228 1462.39,859.228 1461.01,859.228 1459.62,859.228 1458.24,859.228 1456.86,859.228 1455.47,859.228 1454.09,859.228 \n",
" 1452.7,859.228 1451.32,859.228 1449.93,859.228 1448.54,859.228 1447.16,859.228 1445.77,859.228 1444.38,859.228 1442.99,859.228 1441.6,859.228 1440.21,859.228 \n",
" 1438.83,859.228 1437.44,859.228 1436.05,859.228 1434.66,859.228 1433.26,859.228 1431.87,859.228 1430.48,859.228 1429.09,859.228 1427.7,859.228 1426.3,859.228 \n",
" 1424.91,859.228 1423.52,859.228 1422.12,859.228 1420.73,859.228 1419.33,859.228 1417.94,859.228 1416.54,859.228 1415.15,859.228 1413.75,859.228 1412.36,859.228 \n",
" 1410.96,859.228 1409.56,859.228 1408.17,859.228 1406.77,859.228 1405.37,859.228 1403.97,859.228 1402.57,859.228 1401.17,859.228 1399.77,859.228 1398.37,859.228 \n",
" 1396.97,859.228 1395.57,859.228 1394.17,859.228 1392.77,859.228 1391.37,859.228 1389.97,859.228 1388.57,859.228 1387.17,859.228 1385.76,859.228 1384.36,859.228 \n",
" 1382.96,859.228 1381.56,859.228 1380.15,859.228 1378.75,859.228 1377.34,859.228 1375.94,859.228 1374.54,859.228 1373.13,859.228 1371.73,859.228 1370.32,859.228 \n",
" 1368.91,859.228 1367.51,859.228 1366.1,859.228 1364.7,859.228 1363.29,859.228 1361.88,859.228 1360.48,859.228 1359.07,859.228 1357.66,859.228 1356.25,859.228 \n",
" 1354.85,859.228 1353.44,859.228 1352.03,859.228 1350.62,859.228 1349.21,859.228 1347.8,859.228 1346.39,859.228 1344.98,859.228 1343.57,859.228 1342.16,859.228 \n",
" 1340.75,859.228 1339.34,859.228 1337.93,859.228 1336.52,859.228 1335.11,859.228 1333.7,859.228 1332.29,859.228 1330.88,859.228 1329.46,859.228 1328.05,859.228 \n",
" 1326.64,859.228 1325.23,859.228 1323.81,859.228 1322.4,859.228 1320.99,859.228 1319.58,859.228 1318.16,859.228 1316.75,859.228 1315.34,859.228 1313.92,859.228 \n",
" 1312.51,859.227 1311.1,859.227 1309.68,859.227 1308.27,859.227 1306.85,859.227 1305.44,859.226 1304.02,859.225 1302.61,859.224 1301.19,859.223 1299.78,859.221 \n",
" 1298.36,859.218 1296.95,859.214 1295.53,859.208 1294.12,859.201 1292.7,859.19 1291.29,859.175 1289.87,859.154 1288.46,859.127 1287.04,859.088 1285.62,859.037 \n",
" 1284.21,858.967 1282.79,858.874 1281.37,858.749 1279.96,858.582 1278.54,858.362 1277.12,858.071 1275.71,857.689 1274.29,857.191 1272.87,856.544 1271.46,855.708 \n",
" 1270.04,854.633 1268.62,853.259 1267.21,851.513 1265.79,849.306 1264.37,846.531 1262.95,843.066 1261.54,838.764 1260.12,833.457 1258.7,826.954 1257.29,819.037 \n",
" 1255.87,809.468 1254.45,797.985 1253.03,784.311 1251.61,768.154 1250.2,749.222 1248.78,727.231 1247.36,701.919 1245.94,673.068 1244.53,640.524 1243.11,604.222 \n",
" 1241.69,564.213 1240.27,520.701 1238.85,474.065 1237.44,424.896 1236.02,374.021 1234.6,322.526 1233.18,271.765 1231.77,223.361 1230.35,179.19 1228.93,141.341 \n",
" 1227.51,112.061 1226.09,93.6718 1224.68,88.4582 1223.26,98.5397 1221.84,125.717 1220.42,171.307 1219.01,235.97 1217.59,319.543 1216.17,420.901 1214.75,537.839 \n",
" 1213.34,667.028 1211.92,804.019 1210.5,943.35 1209.08,1078.73 1207.67,1203.34 1206.25,1310.19 1204.83,1392.63 1203.41,1444.81 1202,1462.26 1200.58,1442.39 \n",
" 1199.16,1384.92 1197.75,1292.2 1196.33,1169.34 1194.91,1024.12 1193.5,866.621 1192.08,708.625 1190.66,562.774 1189.25,441.518 1187.83,355.958 1186.41,314.681 \n",
" 1185,322.704 1183.58,380.663 1182.17,484.376 1180.75,624.865 1179.33,788.914 1177.92,960.149 1176.5,1120.6 1175.09,1252.59 1173.67,1340.8 1172.26,1374.21 \n",
" 1170.84,1347.73 1169.43,1263.22 1168.01,1129.71 1166.6,962.748 1165.18,782.781 1163.77,612.775 1162.35,475.293 1160.94,389.387 1159.52,367.748 1158.11,414.548 \n",
" 1156.7,524.362 1155.28,682.437 1153.87,866.384 1152.46,1049.16 1151.04,1202.99 1149.63,1303.66 1148.22,1334.53 1146.81,1289.6 1145.39,1175 1143.98,1008.53 \n",
" 1142.57,817.22 1141.16,633.141 1139.75,488.045 1138.33,407.789 1136.92,407.464 1135.51,488.205 1134.1,636.402 1132.69,825.606 1131.28,1020.99 1129.87,1185.62 \n",
" 1128.46,1287.55 1127.05,1306.19 1125.64,1236.87 1124.23,1092.35 1122.82,900.9 1121.41,701.052 1120,533.997 1118.59,435.027 1117.18,425.825 1115.78,509.341 \n",
" 1114.37,668.508 1112.96,869.307 1111.55,1067.76 1110.14,1219.46 1108.74,1289.63 1107.33,1261.44 1105.92,1140.39 1104.52,953.833 1103.11,745.087 1101.71,563.642 \n",
" 1100.3,453.39 1098.89,441.77 1097.49,532.542 1096.08,704.158 1094.68,914.439 1093.28,1110.66 1091.87,1242.7 1090.47,1276.06 1089.06,1201.24 1087.66,1036.92 \n",
" 1086.26,825.748 1084.86,623.633 1083.45,484.991 1082.05,447.936 1080.65,523.413 1079.25,691.489 1077.85,906.125 1076.45,1107.47 1075.05,1238.48 1073.65,1261.25 \n",
" 1072.25,1168.45 1070.85,986.107 1069.45,766.822 1068.05,574.877 1066.65,467.403 1065.25,477.122 1063.85,601.924 1062.46,804.73 1061.06,1024.07 1059.66,1192.49 \n",
" 1058.26,1257.41 1056.87,1197.85 1055.47,1031.74 1054.08,811.054 1052.68,605.889 1051.29,482.301 1049.89,480.807 1048.5,602.652 1047.1,808.611 1045.71,1031.16 \n",
" 1044.32,1196.38 1042.92,1248.62 1041.53,1169.62 1040.14,985.396 1038.75,758.228 1037.36,565.908 1035.97,475.118 1034.57,518.012 1033.18,680.316 1031.79,905.525 \n",
" 1030.41,1114.18 1029.02,1231.8 1027.63,1215.72 1026.24,1071.03 1024.85,849.57 1023.46,631.788 1022.08,497.665 1020.69,497.154 1019.3,631.119 1017.92,850.193 \n",
" 1016.53,1072.51 1015.15,1214.12 1013.76,1220.84 1012.38,1089.47 1011,869.789 1009.61,646.179 1008.23,505.406 1006.85,502.775 1005.47,639.948 1004.08,863.479 \n",
" 1002.7,1085.17 1001.32,1216.7 999.943,1205.03 998.563,1054.21 997.184,824.645 995.805,609.297 994.427,496.18 993.049,532.152 991.672,702.994 990.295,938.463 \n",
" 988.918,1140.77 987.542,1225.14 986.167,1155.64 984.792,961.005 983.417,723.326 982.043,543.742 980.669,499.373 979.296,609.818 977.924,827.944 976.552,1059.49 \n",
" 975.18,1203.57 973.809,1196.79 972.438,1041.57 971.068,805.93 969.699,594.102 968.329,500.523 966.961,567.468 965.593,765.423 964.225,1005.55 962.858,1179.22 \n",
" 961.492,1207.24 960.126,1076.31 958.76,845.949 957.395,621.866 956.031,507.652 954.667,556.654 953.304,746.568 951.941,989.026 950.579,1170.37 949.217,1204.96 \n",
" 947.856,1075.97 946.495,844.158 945.135,619.702 943.776,510.002 942.417,568.07 941.058,766.413 939.7,1009.5 938.343,1179.46 936.986,1193.28 935.63,1043.77 \n",
" 934.275,803.822 932.92,591.336 931.565,511.371 930.212,603.932 928.858,823.432 927.506,1060.64 926.154,1196.78 924.802,1163.18 923.451,976.336 922.101,730.39 \n",
" 920.752,550.025 919.402,527.252 918.054,674.095 916.706,915.711 915.359,1128.1 914.012,1201.65 912.666,1097.97 911.321,870.46 909.976,637.276 908.632,520.179 \n",
" 907.289,580.771 905.946,787.62 904.603,1032.05 903.262,1184.94 901.921,1165.05 900.58,982.574 899.241,734.469 897.902,553.274 896.563,536.307 895.225,693.033 \n",
" 893.888,939.278 892.552,1142.03 891.216,1191.24 889.881,1059.83 888.546,819.115 887.213,600.526 885.88,523.969 884.547,631.806 883.215,864.936 881.884,1094.73 \n",
" 880.554,1193.82 879.224,1106.92 877.895,882.103 876.566,644.685 875.239,527.585 873.912,596.748 872.585,813.57 871.26,1055.98 869.935,1186.92 868.61,1131.97 \n",
" 867.287,922.063 865.964,676.521 864.642,535.512 863.32,579.916 862,784.546 860.68,1031.96 859.36,1179.61 858.042,1142.03 856.724,940.709 855.407,692.26 \n",
" 854.09,541.138 852.775,575.588 851.46,775.743 850.146,1024.62 848.832,1176.24 847.519,1141.29 846.208,940.117 844.896,691.334 843.586,542.128 842.276,581.127 \n",
" 840.967,785.388 839.659,1033.46 838.351,1177.35 837.045,1130.88 835.739,921.666 834.434,675.137 833.129,539.559 831.826,596.781 830.523,812.462 829.221,1056.26 \n",
" 827.919,1180.41 826.619,1109.36 825.319,886.147 824.02,646.707 822.722,537.277 821.425,624.952 820.128,856.179 818.832,1089.09 817.537,1180.38 816.243,1073.66 \n",
" 814.95,834.631 813.657,611.015 812.365,541.368 811.074,669.049 809.784,914.852 808.495,1125.81 807.206,1170.31 805.918,1020.42 804.632,769.777 803.346,575.414 \n",
" 802.06,559.395 800.776,731.928 799.492,984.489 798.21,1157.82 796.928,1142.37 795.647,947.784 794.366,697.241 793.087,549.712 791.809,599.028 790.531,813.922 \n",
" 789.254,1057.51 787.978,1174.32 786.703,1089.6 785.429,857.471 784.155,626.667 782.883,545.355 781.611,665.884 780.34,910.769 779.07,1122.22 777.801,1163.75 \n",
" 776.533,1008.43 775.266,756.687 773.999,571.609 772.734,573.297 771.469,760.73 770.206,1012.07 768.943,1163.65 767.681,1116.61 766.42,901.585 765.16,659.058 \n",
" 763.9,547.728 762.642,640.572 761.384,876.752 760.128,1101.22 758.872,1166.41 757.617,1029.37 756.364,780.359 755.111,583.581 753.859,568.98 752.608,746.244 \n",
" 751.358,998.112 750.108,1157.77 748.86,1119.33 747.613,908.294 746.366,664.842 745.121,550.881 743.876,642.286 742.633,878.168 741.39,1101.21 740.148,1162.51 \n",
" 738.908,1021.09 737.668,771.544 736.429,580.916 735.191,576.934 733.954,762.266 732.718,1012.51 731.483,1159.57 730.249,1104.56 729.016,884.523 727.784,647.648 \n",
" 726.553,553.584 725.322,665.764 724.093,908.438 722.865,1117.65 721.638,1151.96 720.412,988.194 719.186,737.253 717.962,569.182 716.739,597.92 715.516,803.928 \n",
" 714.295,1047.29 713.075,1162.67 711.856,1071.64 710.637,836.19 709.42,616.639 708.204,562.528 706.988,710.689 705.774,960.015 704.561,1140.34 703.349,1128.56 \n",
" 702.138,932.796 700.927,686.924 699.718,559.119 698.51,636.787 697.303,866.673 696.097,1091.25 694.892,1156.61 693.688,1018.01 692.485,770.622 691.283,584.271 \n",
" 690.082,586.877 688.882,776.557 687.683,1022.87 686.486,1156.46 685.289,1085.5 684.093,858.925 682.899,632.774 681.705,562.757 680.513,697.011 679.321,942.911 \n",
" 678.131,1130.9 676.942,1131.4 675.753,944.181 674.566,698.529 673.38,564.009 672.195,633.407 671.011,858.666 669.828,1084.1 668.647,1153.97 667.466,1020.1 \n",
" 666.286,775.176 665.108,588.587 663.931,589.31 662.754,776.702 661.579,1021 660.405,1153.14 659.232,1081.79 658.06,856.486 656.889,633.294 655.72,566.713 \n",
" 654.551,702.704 653.384,946.936 652.217,1130.2 651.052,1125.64 649.888,936.582 648.725,694.133 647.563,566.268 646.402,641.446 645.243,867.401 644.084,1087.47 \n",
" 642.927,1149.23 641.771,1010.04 640.616,766.53 639.462,587.462 638.309,596.804 637.157,787.891 636.007,1028.19 634.857,1151.25 633.709,1071.97 632.562,845.467 \n",
" 631.416,628.716 630.271,571.752 629.128,713.825 627.985,956.385 626.844,1131.47 625.704,1118.03 624.565,925.571 623.427,687.421 622.291,568.315 621.155,650.457 \n",
" 620.021,876.824 618.888,1090.8 617.756,1144.57 616.625,1001.18 615.496,759.905 614.368,587.451 613.24,602.757 612.114,795.018 610.99,1031.38 609.866,1148.78 \n",
" 608.744,1066.44 607.623,841.388 606.503,628.853 605.384,575.197 604.266,717.141 603.15,956.723 602.035,1128.99 600.921,1115.56 599.808,925.944 598.697,690.68 \n",
" 597.587,571.426 596.478,649.887 595.37,871.926 594.263,1085.02 593.158,1143.09 592.054,1006.56 590.951,769.259 589.849,593.803 588.749,600.161 587.65,783.698 \n",
" 586.552,1018.67 585.455,1144.46 584.359,1075.35 583.265,858.812 582.172,642.79 581.08,574.542 579.99,700.32 578.901,934.162 577.813,1116.76 576.726,1124.03 \n",
" 575.641,951.325 574.557,716.255 573.474,578.48 572.392,631.286 571.312,838.543 570.233,1059.59 569.155,1144.76 568.078,1036.7 567.003,808.791 565.929,615.199 \n",
" 564.856,586.531 563.785,741.821 562.715,976.068 561.646,1131.37 560.578,1103.37 559.512,911.241 558.447,684.378 557.383,575.147 556.321,656.589 555.26,873.752 \n",
" 554.2,1080.95 553.142,1139.56 552.085,1010.71 551.029,780.819 549.974,603.516 548.921,596.906 547.869,765.046 546.819,995.677 545.769,1135.32 544.721,1091.4 \n",
" 543.675,893.442 542.629,673.027 541.585,576.202 540.543,666.779 539.501,884.478 538.461,1085.25 537.423,1136.67 536.385,1005.17 535.349,777.731 534.315,604.159 \n",
" 533.282,598.378 532.25,763.81 531.219,991.608 530.19,1132.44 529.162,1094.37 528.135,902.682 527.11,682.854 526.086,578.284 525.064,656.81 524.043,866.923 \n",
" 523.023,1071.79 522.004,1138.45 520.987,1024.03 519.972,803.039 518.958,618.65 517.945,589.874 516.933,734.9 515.923,959.86 514.914,1119.79 513.907,1112.06 \n",
" 512.901,942.042 511.896,719.143 510.893,586.206 509.891,628.008 508.89,817.413 507.891,1033.25 506.894,1137.99 505.897,1065.3 504.902,861.803 503.909,656.926 \n",
" 502.917,580.447 501.926,680.407 500.937,893.199 499.949,1084.3 498.962,1133.39 497.977,1009.98 496.994,792.039 496.011,616.513 495.031,593.207 494.051,736.277 \n",
" 493.073,955.935 492.097,1115.02 491.121,1114.67 490.148,955.549 489.175,736.851 488.205,594.265 487.235,615.791 486.267,787.665 485.301,1003.45 484.335,1130.15 \n",
" 483.372,1090.13 482.41,908.434 481.449,696.823 480.489,584.846 479.531,640.603 478.575,829.597 477.62,1036.38 476.666,1135.23 475.714,1066.5 474.764,872.319 \n",
" 473.814,670.485 472.867,582.842 471.92,661.849 470.975,859.47 470.032,1056.68 469.09,1135.28 468.15,1048.61 467.211,848.96 466.273,655.699 465.337,583.811 \n",
" 464.403,675.608 463.47,876.142 462.538,1066.39 461.608,1134.01 460.679,1039.54 459.752,839.097 458.827,650.752 457.902,584.877 456.98,679.608 456.059,879.093 \n",
" 455.139,1066.72 454.221,1133.4 453.304,1040.82 452.389,843.169 451.475,655.082 450.563,585.061 449.652,673.024 448.743,867.873 447.835,1057.48 446.929,1133.35 \n",
" 446.024,1052.54 445.121,861.667 444.22,669.679 443.319,585.552 442.421,656.55 441.524,841.991 440.628,1036.88 439.734,1131.51 438.842,1073.11 437.951,894.96 \n",
" 437.061,697.093 436.173,589.964 435.287,632.864 434.402,801.412 433.519,1001.74 432.637,1122.98 431.756,1098.55 430.878,942.449 430,740.884 429.125,604.456 \n",
" 428.251,607.479 427.378,747.768 426.507,948.475 425.638,1100.44 424.77,1121.57 423.903,1000.97 423.039,804.183 422.175,637.286 421.314,589.661 420.453,686.309 \n",
" 419.595,875.078 418.738,1055.11 417.882,1130.85 417.029,1062.6 416.176,887.01 415.325,697.054 414.476,592.652 413.629,628.194 412.783,784.51 411.938,979.552 \n",
" 411.095,1111.67 410.254,1112.57 409.414,982.336 408.576,788.757 407.74,631.778 406.905,591.858 406.071,688.939 405.239,872.841 404.409,1049.5 403.581,1129.21 \n",
" 402.754,1071.99 401.928,907.369 401.104,718.762 400.282,601.055 399.461,612.916 398.642,747.838 397.825,938.038 397.009,1088.75 396.195,1125.49 395.382,1030.6 \n",
" 394.571,851.448 393.762,676.426 392.954,591.248 392.148,637.103 391.343,791.051 390.54,977.835 389.739,1106.87 388.939,1116.14 388.141,1001.76 387.345,819.18 \n",
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" \n",
" \"/>\n",
"</svg>\n"
]
},
"execution_count": 35,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ε = 1E-6\n",
"L = [B₋₁; \n",
" B₁ ; \n",
" ε*D₁*D₀ - S₁*S₀*Jᵗ]\n",
"\n",
"u = L \\ [1.0;0.0;0.0]\n",
"plot(u; legend=false)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Because of the banded structure, this can be solved fast:"
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" 0.466365 seconds (13.06 M allocations: 296.400 MiB, 19.59% gc time)\n",
"ncoefficients(u) = 62496\n"
]
}
],
"source": [
"ε = 1E-10\n",
"L = [B₋₁; \n",
" B₁ ; \n",
" ε*D₁*D₀ - S₁*S₀*Jᵗ]\n",
"\n",
"\n",
"@time u = L \\ [1.0;0.0;0.0]\n",
"@show ncoefficients(u);"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"To handle other variable coefficients, first consider a polynomial $p(x)$. If Multiplication by $x$ is represented by multiplying the coefficients by $J^\\top$, then multiplication by $p$ is represented by $p(J^\\top)$:"
]
},
{
"cell_type": "code",
"execution_count": 38,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"PlusOperator : Chebyshev() → Chebyshev()\n",
" -0.5 0.5 0.25 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅\n",
" 1.0 -0.25 0.5 0.25 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅\n",
" 0.5 0.5 -0.5 0.5 0.25 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅\n",
" ⋅ 0.25 0.5 -0.5 0.5 0.25 ⋅ ⋅ ⋅ ⋅ ⋅\n",
" ⋅ ⋅ 0.25 0.5 -0.5 0.5 0.25 ⋅ ⋅ ⋅ ⋅\n",
" ⋅ ⋅ ⋅ 0.25 0.5 -0.5 0.5 0.25 ⋅ ⋅ ⋅\n",
" ⋅ ⋅ ⋅ ⋅ 0.25 0.5 -0.5 0.5 0.25 ⋅ ⋅\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ 0.25 0.5 -0.5 0.5 0.25 ⋅\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 0.25 0.5 -0.5 0.5 ⋱\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 0.25 0.5 -0.5 ⋱\n",
" ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋱ ⋱ ⋱"
]
},
"execution_count": 38,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"M = -I + Jᵗ + (Jᵗ)^2 # represents -1+x+x^2"
]
},
{
"cell_type": "code",
"execution_count": 39,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" 0.023297 seconds (295.98 k allocations: 7.463 MiB)\n",
"ε * ((u')')(0.1) - (-1 + 0.1 + 0.1 ^ 2) * u(0.1) = -1.4863110742169283e-14\n"
]
},
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" 2151.72,859.091 2151.32,859.091 2150.92,859.091 2150.51,859.091 2150.11,859.091 2149.71,859.091 2149.3,859.091 2148.9,859.091 2148.49,859.091 2148.09,859.091 \n",
" 2147.68,859.091 2147.27,859.091 2146.87,859.091 2146.46,859.091 2146.05,859.091 2145.64,859.091 2145.23,859.091 2144.82,859.091 2144.4,859.091 2143.99,859.091 \n",
" 2143.58,859.091 2143.16,859.091 2142.75,859.091 2142.33,859.091 2141.91,859.091 2141.5,859.091 2141.08,859.091 2140.66,859.091 2140.24,859.091 2139.82,859.091 \n",
" 2139.4,859.091 2138.98,859.091 2138.56,859.091 2138.14,859.091 2137.71,859.091 2137.29,859.091 2136.86,859.091 2136.44,859.091 2136.01,859.091 2135.58,859.091 \n",
" 2135.16,859.091 2134.73,859.091 2134.3,859.091 2133.87,859.091 2133.44,859.091 2133.01,859.091 2132.58,859.091 2132.15,859.091 2131.71,859.091 2131.28,859.091 \n",
" 2130.84,859.091 2130.41,859.091 2129.97,859.091 2129.54,859.091 2129.1,859.091 2128.66,859.091 2128.22,859.091 2127.78,859.091 2127.34,859.091 2126.9,859.091 \n",
" 2126.46,859.091 2126.02,859.091 2125.58,859.091 2125.14,859.091 2124.69,859.091 2124.25,859.091 2123.8,859.091 2123.36,859.091 2122.91,859.091 2122.46,859.091 \n",
" 2122.01,859.091 2121.57,859.091 2121.12,859.091 2120.67,859.091 2120.22,859.091 2119.76,859.091 2119.31,859.091 2118.86,859.091 2118.41,859.091 2117.95,859.091 \n",
" 2117.5,859.091 2117.04,859.091 2116.59,859.091 2116.13,859.091 2115.67,859.091 2115.21,859.091 2114.75,859.091 2114.3,859.091 2113.84,859.091 2113.37,859.091 \n",
" 2112.91,859.091 2112.45,859.091 2111.99,859.091 2111.52,859.091 2111.06,859.091 2110.6,859.091 2110.13,859.091 2109.66,859.091 2109.2,859.091 2108.73,859.091 \n",
" 2108.26,859.091 2107.79,859.091 2107.32,859.091 2106.85,859.091 2106.38,859.091 2105.91,859.091 2105.44,859.091 2104.97,859.091 2104.49,859.091 2104.02,859.091 \n",
" 2103.54,859.091 2103.07,859.091 2102.59,859.091 2102.12,859.091 2101.64,859.091 2101.16,859.091 2100.68,859.091 2100.2,859.091 2099.72,859.091 2099.24,859.091 \n",
" 2098.76,859.091 2098.28,859.091 2097.8,859.091 2097.31,859.091 2096.83,859.091 2096.34,859.091 2095.86,859.091 2095.37,859.091 2094.89,859.091 2094.4,859.091 \n",
" 2093.91,859.091 2093.42,859.091 2092.93,859.091 2092.44,859.091 2091.95,859.091 2091.46,859.091 2090.97,859.091 2090.48,859.091 2089.98,859.091 2089.49,859.091 \n",
" 2089,859.091 2088.5,859.091 2088,859.091 2087.51,859.091 2087.01,859.091 2086.51,859.091 2086.01,859.091 2085.52,859.091 2085.02,859.091 2084.52,859.091 \n",
" 2084.01,859.091 2083.51,859.091 2083.01,859.091 2082.51,859.091 2082,859.091 2081.5,859.091 2080.99,859.091 2080.49,859.091 2079.98,859.091 2079.48,859.091 \n",
" 2078.97,859.091 2078.46,859.091 2077.95,859.091 2077.44,859.091 2076.93,859.091 2076.42,859.091 2075.91,859.091 2075.4,859.091 2074.89,859.091 2074.37,859.091 \n",
" 2073.86,859.091 2073.34,859.091 2072.83,859.091 2072.31,859.091 2071.8,859.091 2071.28,859.091 2070.76,859.091 2070.24,859.091 2069.73,859.091 2069.21,859.091 \n",
" 2068.69,859.091 2068.16,859.091 2067.64,859.091 2067.12,859.091 2066.6,859.091 2066.07,859.091 2065.55,859.091 2065.03,859.091 2064.5,859.091 2063.97,859.091 \n",
" 2063.45,859.091 2062.92,859.091 2062.39,859.091 2061.86,859.091 2061.34,859.091 2060.81,859.091 2060.28,859.091 2059.74,859.091 2059.21,859.091 2058.68,859.091 \n",
" 2058.15,859.091 2057.61,859.091 2057.08,859.091 2056.55,859.091 2056.01,859.091 2055.47,859.091 2054.94,859.091 2054.4,859.091 2053.86,859.091 2053.32,859.091 \n",
" 2052.78,859.091 2052.24,859.091 2051.7,859.091 2051.16,859.091 2050.62,859.091 2050.08,859.091 2049.54,859.091 2048.99,859.091 2048.45,859.091 2047.9,859.091 \n",
" 2047.36,859.091 2046.81,859.091 2046.27,859.091 2045.72,859.091 2045.17,859.091 2044.62,859.091 2044.07,859.091 2043.52,859.091 2042.97,859.091 2042.42,859.091 \n",
" 2041.87,859.091 2041.32,859.091 2040.77,859.091 2040.21,859.091 2039.66,859.091 2039.1,859.091 2038.55,859.091 2037.99,859.091 2037.44,859.091 2036.88,859.091 \n",
" 2036.32,859.091 2035.76,859.091 2035.2,859.091 2034.64,859.091 2034.08,859.091 2033.52,859.091 2032.96,859.091 2032.4,859.091 2031.84,859.091 2031.27,859.091 \n",
" 2030.71,859.091 2030.15,859.091 2029.58,859.091 2029.02,859.091 2028.45,859.091 2027.88,859.091 2027.32,859.091 2026.75,859.091 2026.18,859.091 2025.61,859.091 \n",
" 2025.04,859.091 2024.47,859.091 2023.9,859.091 2023.33,859.091 2022.75,859.091 2022.18,859.091 2021.61,859.091 2021.03,859.091 2020.46,859.091 2019.88,859.091 \n",
" 2019.31,859.091 2018.73,859.091 2018.16,859.091 2017.58,859.091 2017,859.091 2016.42,859.091 2015.84,859.091 2015.26,859.091 2014.68,859.091 2014.1,859.091 \n",
" 2013.52,859.091 2012.93,859.091 2012.35,859.091 2011.77,859.091 2011.18,859.091 2010.6,859.091 2010.01,859.091 2009.43,859.091 2008.84,859.091 2008.25,859.091 \n",
" 2007.67,859.091 2007.08,859.091 2006.49,859.091 2005.9,859.091 2005.31,859.091 2004.72,859.091 2004.13,859.091 2003.54,859.091 2002.94,859.091 2002.35,859.091 \n",
" 2001.76,859.091 2001.16,859.091 2000.57,859.091 1999.97,859.091 1999.38,859.091 1998.78,859.091 1998.18,859.091 1997.59,859.091 1996.99,859.091 1996.39,859.091 \n",
" 1995.79,859.091 1995.19,859.091 1994.59,859.091 1993.99,859.091 1993.39,859.091 1992.78,859.091 1992.18,859.091 1991.58,859.091 1990.97,859.091 1990.37,859.091 \n",
" 1989.76,859.091 1989.16,859.091 1988.55,859.091 1987.94,859.091 1987.34,859.091 1986.73,859.091 1986.12,859.091 1985.51,859.091 1984.9,859.091 1984.29,859.091 \n",
" 1983.68,859.091 1983.07,859.091 1982.46,859.091 1981.84,859.091 1981.23,859.091 1980.62,859.091 1980,859.091 1979.39,859.091 1978.77,859.091 1978.15,859.091 \n",
" 1977.54,859.091 1976.92,859.091 1976.3,859.091 1975.68,859.091 1975.07,859.091 1974.45,859.091 1973.83,859.091 1973.21,859.091 1972.58,859.091 1971.96,859.091 \n",
" 1971.34,859.091 1970.72,859.091 1970.09,859.091 1969.47,859.091 1968.85,859.091 1968.22,859.091 1967.6,859.091 1966.97,859.091 1966.34,859.091 1965.71,859.091 \n",
" 1965.09,859.091 1964.46,859.091 1963.83,859.091 1963.2,859.091 1962.57,859.091 1961.94,859.091 1961.31,859.091 1960.68,859.091 1960.04,859.091 1959.41,859.091 \n",
" 1958.78,859.091 1958.14,859.091 1957.51,859.091 1956.87,859.091 1956.24,859.091 1955.6,859.091 1954.97,859.091 1954.33,859.091 1953.69,859.091 1953.05,859.091 \n",
" 1952.41,859.091 1951.77,859.091 1951.13,859.091 1950.49,859.091 1949.85,859.091 1949.21,859.091 1948.57,859.091 1947.93,859.091 1947.28,859.091 1946.64,859.091 \n",
" 1945.99,859.091 1945.35,859.091 1944.7,859.091 1944.06,859.091 1943.41,859.091 1942.76,859.091 1942.12,859.091 1941.47,859.091 1940.82,859.091 1940.17,859.091 \n",
" 1939.52,859.091 1938.87,859.091 1938.22,859.091 1937.57,859.091 1936.92,859.091 1936.26,859.091 1935.61,859.091 1934.96,859.091 1934.3,859.091 1933.65,859.091 \n",
" 1932.99,859.091 1932.34,859.091 1931.68,859.091 1931.02,859.091 1930.37,859.091 1929.71,859.091 1929.05,859.091 1928.39,859.091 1927.73,859.091 1927.07,859.091 \n",
" 1926.41,859.091 1925.75,859.091 1925.09,859.091 1924.43,859.091 1923.77,859.09 1923.1,859.09 1922.44,859.09 1921.77,859.09 1921.11,859.089 1920.44,859.089 \n",
" 1919.78,859.088 1919.11,859.087 1918.45,859.086 1917.78,859.085 1917.11,859.083 1916.44,859.081 1915.77,859.079 1915.1,859.076 1914.43,859.072 1913.76,859.068 \n",
" 1913.09,859.063 1912.42,859.056 1911.75,859.048 1911.08,859.038 1910.4,859.026 1909.73,859.011 1909.06,858.993 1908.38,858.971 1907.71,858.945 1907.03,858.912 \n",
" 1906.36,858.874 1905.68,858.827 1905,858.77 1904.33,858.703 1903.65,858.621 1902.97,858.523 1902.29,858.407 1901.61,858.267 1900.93,858.101 1900.25,857.903 \n",
" 1899.57,857.668 1898.89,857.389 1898.2,857.06 1897.52,856.671 1896.84,856.212 1896.15,855.672 1895.47,855.039 1894.79,854.297 1894.1,853.43 1893.41,852.419 \n",
" 1892.73,851.242 1892.04,849.876 1891.35,848.293 1890.67,846.463 1889.98,844.353 1889.29,841.925 1888.6,839.14 1887.91,835.951 1887.22,832.311 1886.53,828.166 \n",
" 1885.84,823.459 1885.15,818.128 1884.46,812.107 1883.76,805.328 1883.07,797.717 1882.38,789.199 1881.68,779.696 1880.99,769.128 1880.29,757.415 1879.6,744.48 \n",
" 1878.9,730.245 1878.2,714.64 1877.51,697.598 1876.81,679.064 1876.11,658.993 1875.41,637.353 1874.71,614.133 1874.01,589.34 1873.31,563.005 1872.61,535.19 \n",
" 1871.91,505.987 1871.21,475.523 1870.51,443.965 1869.81,411.522 1869.1,378.448 1868.4,345.044 1867.7,311.661 1866.99,278.698 1866.29,246.605 1865.58,215.878 \n",
" 1864.88,187.058 1864.17,160.726 1863.46,137.495 1862.75,118.002 1862.05,102.898 1861.34,92.8373 1860.63,88.4582 1859.92,90.3708 1859.21,99.1374 1858.5,115.253 \n",
" 1857.79,139.124 1857.08,171.048 1856.37,211.192 1855.66,259.572 1854.94,316.035 1854.23,380.237 1853.52,451.638 1852.8,529.484 1852.09,612.806 1851.37,700.423 \n",
" 1850.66,790.947 1849.94,882.8 1849.23,974.237 1848.51,1063.38 1847.79,1148.25 1847.07,1226.83 1846.36,1297.11 1845.64,1357.15 1844.92,1405.14 1844.2,1439.48 \n",
" 1843.48,1458.85 1842.76,1462.26 1842.04,1449.14 1841.32,1419.37 1840.59,1373.34 1839.87,1311.97 1839.15,1236.7 1838.43,1149.54 1837.7,1052.98 1836.98,949.981 \n",
" 1836.25,843.853 1835.53,738.2 1834.8,636.778 1834.08,543.362 1833.35,461.6 1832.62,394.852 1831.9,346.032 1831.17,317.457 1830.44,310.712 1829.71,326.529 \n",
" 1828.98,364.708 1828.25,424.066 1827.52,502.431 1826.79,596.69 1826.06,702.879 1825.33,816.322 1824.6,931.828 1823.86,1043.91 1823.13,1147.06 1822.4,1236.01 \n",
" 1821.66,1306.04 1820.93,1353.24 1820.2,1374.76 1819.46,1369.05 1818.73,1335.94 1817.99,1276.82 1817.25,1194.52 1816.52,1093.32 1815.78,978.695 1815.04,857.087 \n",
" 1814.3,735.541 1813.56,621.314 1812.83,521.431 1812.09,442.237 1811.35,388.965 1810.61,365.351 1809.87,373.329 1809.12,412.824 1808.38,481.678 1807.64,575.704 \n",
" 1806.9,688.9 1806.15,813.781 1805.41,941.854 1804.67,1064.18 1803.92,1171.99 1803.18,1257.36 1802.43,1313.78 1801.69,1336.73 1800.94,1324.08 1800.2,1276.35 \n",
" 1799.45,1196.75 1798.7,1091.04 1797.95,967.165 1797.2,834.7 1796.46,704.129 1795.71,586.025 1794.96,490.183 1794.21,424.775 1793.46,395.61 1792.71,405.558 \n",
" 1791.96,454.205 1791.2,537.786 1790.45,649.398 1789.7,779.519 1788.95,916.771 1788.19,1048.9 1787.44,1163.84 1786.69,1250.89 1785.93,1301.68 1785.18,1311.09 \n",
" 1784.42,1277.8 1783.66,1204.57 1782.91,1098.03 1782.15,968.219 1781.4,827.643 1780.64,690.128 1779.88,569.459 1779.12,477.982 1778.36,425.297 1777.6,417.194 \n",
" 1776.84,454.935 1776.08,535 1775.32,649.33 1774.56,786.056 1773.8,930.686 1773.04,1067.61 1772.28,1181.77 1771.52,1260.37 1770.75,1294.34 1769.99,1279.48 \n",
" 1769.23,1217.04 1768.46,1113.71 1767.7,981.049 1766.93,834.185 1766.17,690.181 1765.4,566.046 1764.64,476.711 1763.87,433.175 1763.1,441.084 1762.33,499.902 \n",
" 1761.57,602.829 1760.8,737.495 1760.03,887.368 1759.26,1033.74 1758.49,1158.06 1757.72,1244.27 1756.95,1281.01 1756.18,1263.11 1755.41,1192.48 1754.64,1078.01 \n",
" 1753.87,934.495 1753.1,780.875 1752.32,637.714 1751.55,524.472 1750.78,456.817 1750,444.383 1749.23,489.306 1748.45,585.753 1747.68,720.578 1746.9,875.015 \n",
" 1746.13,1027.24 1745.35,1155.43 1744.58,1240.92 1743.8,1270.98 1743.02,1240.84 1742.24,1154.48 1741.47,1024.32 1740.69,869.47 1739.91,713.008 1739.13,578.582 \n",
" 1738.35,486.803 1737.57,452.026 1736.79,480.003 1736.01,566.821 1735.23,699.317 1734.45,856.971 1733.67,1015.01 1732.88,1148.27 1732.1,1235.25 1731.32,1261.6 \n",
" 1730.54,1222.66 1729.75,1124.36 1728.97,982.453 1728.18,820.115 1727.4,664.21 1726.61,540.861 1725.83,471.037 1725.04,466.903 1724.26,529.576 1723.47,648.737 \n",
" 1722.68,804.195 1721.89,969.219 1721.11,1115.08 1720.32,1216.07 1719.53,1254.09 1718.74,1222.01 1717.95,1125.13 1717.16,980.453 1716.37,813.817 1715.58,655.357 \n",
" 1714.79,534.061 1714,472.429 1713.21,482.196 1712.42,561.969 1711.62,697.256 1710.83,862.973 1710.04,1028 1709.24,1161.01 1708.45,1236.41 1707.66,1239.41 \n",
" 1706.86,1169.01 1706.07,1038.47 1705.27,872.895 1704.48,704.572 1703.68,566.666 1702.89,486.679 1702.09,480.868 1701.29,550.804 1700.49,682.823 1699.7,850.549 \n",
" 1698.9,1020.06 1698.1,1156.7 1697.3,1232.21 1696.5,1230.67 1695.7,1151.99 1694.9,1012.17 1694.1,840.215 1693.3,672.193 1692.5,543.717 1691.7,482.344 \n",
" 1690.9,501.551 1690.1,597.627 1689.3,750.213 1688.49,926.474 1687.69,1088.08 1686.89,1199.53 1686.08,1236.01 1685.28,1189.09 1684.48,1068.76 1683.67,901.555 \n",
" 1682.87,724.777 1682.06,578.255 1681.26,495.345 1680.45,495.241 1679.64,578.373 1678.84,726.042 1678.03,904.465 1677.22,1072.4 1676.41,1190.67 1675.61,1231.35 \n",
" 1674.8,1184.49 1673.99,1060.75 1673.18,889.134 1672.37,710.313 1671.56,567.077 1670.75,494.041 1669.94,509.18 1669.13,609.23 1668.32,770.2 1667.51,952.966 \n",
" 1666.7,1112.69 1665.88,1209.82 1665.07,1219.99 1664.26,1140.25 1663.45,990.161 1662.63,807.107 1661.82,637.132 1661.01,523.367 1660.19,495.016 1659.38,559.703 \n",
" 1658.56,701.239 1657.75,883.486 1656.93,1059.46 1656.12,1183.39 1655.3,1222.72 1654.48,1166.77 1653.67,1029.85 1652.85,847.791 1652.03,668.69 1651.21,540.271 \n",
" 1650.4,497.098 1649.58,551.125 1648.76,688.198 1647.94,871.559 1647.12,1051.57 1646.3,1179.09 1645.48,1218.96 1644.66,1159.86 1643.84,1017.7 1643.02,831.54 \n",
" 1642.2,653.013 1641.38,532.031 1640.55,502.761 1639.73,573.819 1638.91,725.525 1638.09,915.158 1637.26,1088.85 1636.44,1196.89 1635.62,1207.98 1634.79,1118.57 \n",
" 1633.97,954.093 1633.14,762.013 1632.32,598.187 1631.49,510.638 1630.67,525.367 1629.84,638.43 1629.02,816.771 1628.19,1007.7 1627.36,1154.39 1626.54,1212.81 \n",
" 1625.71,1165.08 1624.88,1025.17 1624.05,835.042 1623.22,652.176 1622.4,532.283 1621.57,512.235 1620.74,598.554 1619.91,765.07 1619.08,960.652 1618.25,1124.79 \n",
" 1617.42,1206.3 1616.59,1179.45 1615.76,1052.25 1614.93,864.316 1614.09,674.703 1613.26,543.425 1612.43,512.385 1611.6,591.839 1610.77,756.772 1609.93,954.563 \n",
" 1609.1,1121.64 1608.27,1203.92 1607.43,1174.42 1606.6,1042.35 1605.76,850.447 1604.93,661.35 1604.09,537.197 1603.26,519.142 1602.42,613.524 1601.59,789.41 \n",
" 1600.75,988.52 1599.92,1144.42 1599.08,1204.72 1598.24,1148.82 1597.4,995.173 1596.57,795.455 1595.73,617.317 1594.89,521.507 1594.05,541.04 1593.22,669.604 \n",
" 1592.38,863.349 1591.54,1055.65 1590.7,1179.95 1589.86,1192.87 1589.02,1089.54 1588.18,905.689 1587.34,705.484 1586.5,559.258 1585.66,518.752 1584.82,598.636 \n",
" 1583.97,770.95 1583.13,974.657 1582.29,1137.13 1581.45,1200.04 1580.61,1140.49 1579.76,979.543 1578.92,775.042 1578.08,600.986 1577.23,520.755 1576.39,563.901 \n",
" 1575.54,715.024 1574.7,919.017 1573.86,1100.98 1573.01,1193.7 1572.17,1162.57 1571.32,1018.78 1570.47,815.517 1569.63,628.506 1568.78,527.831 1567.94,551.566 \n",
" 1567.09,691.143 1566.24,894.181 1565.4,1083.94 1564.55,1188.29 1563.7,1167.21 1562.85,1028.39 1562,824.67 1561.16,634.163 1560.31,530.324 1559.46,553.538 \n",
" 1558.61,695.176 1557.76,900.534 1556.91,1089.76 1556.06,1188.85 1555.21,1158.71 1554.36,1010.82 1553.51,803.125 1552.66,617.526 1551.81,527.631 1550.96,569.429 \n",
" 1550.1,726.637 1549.25,936.773 1548.4,1115.81 1547.55,1191.74 1546.7,1133.71 1545.84,964.775 1544.99,753.033 1544.14,584.332 1543.28,527.464 1542.43,605.944 \n",
" 1541.58,788.025 1540.72,999.277 1539.87,1152.88 1539.01,1185.31 1538.16,1082.85 1537.3,887.601 1536.45,680.492 1535.59,547.802 1534.73,545.171 1533.88,674.045 \n",
" 1533.02,880.645 1532.17,1078.18 1531.31,1183.25 1530.45,1151.14 1529.6,995.114 1528.74,781.225 1527.88,600.526 1527.02,530.349 1526.16,601.053 1525.31,782.621 \n",
" 1524.45,997.135 1523.59,1152.06 1522.73,1180.19 1521.87,1069 1521.01,866.498 1520.15,660.759 1519.29,541.697 1518.43,561.696 1517.57,712.317 1516.71,927.567 \n",
" 1515.85,1112.61 1514.99,1185.5 1514.13,1113.65 1513.27,928.636 1512.41,712.626 1511.54,562.023 1510.68,544.409 1509.82,668.01 1508.96,877.566 1508.09,1078.8 \n",
" 1507.23,1180.76 1506.37,1137.01 1505.5,967.075 1504.64,747.994 1503.78,579.579 1502.91,538.945 1502.05,645.012 1501.19,849.406 1500.32,1058.27 1499.46,1175.24 \n",
" 1498.59,1146.02 1497.73,983.816 1496.86,763.669 1496,587.939 1495.13,538.725 1494.26,639.33 1493.4,842.939 1492.53,1054.14 1491.67,1173.53 1490.8,1144.56 \n",
" 1489.93,980.604 1489.07,759.061 1488.2,585.019 1487.33,541.422 1486.46,649.353 1485.6,857.55 1484.73,1066.49 1483.86,1175.86 1482.99,1132.83 1482.12,957.802 \n",
" 1481.25,735.077 1480.38,572.398 1479.52,548.835 1478.65,676.145 1477.78,892.61 1476.91,1092.71 1476.04,1178.49 1475.17,1107.64 1474.3,914.654 1473.43,694.301 \n",
" 1472.56,555.26 1471.68,566.454 1470.81,722.646 1469.94,946.587 1469.07,1127 1468.2,1173.85 1467.33,1063.46 1466.46,850.771 1465.58,642.25 1464.71,542.71 \n",
" 1463.84,602.501 1462.97,791.773 1462.09,1015.08 1461.22,1159.37 1460.35,1151.22 1459.48,994.48 1458.6,768.741 1457.73,589.139 1456.85,547.659 1455.98,665.837 \n",
" 1455.11,883.282 1454.23,1088.28 1453.36,1175.09 1452.48,1098.64 1451.61,898.192 1450.73,677.46 1449.86,551.111 1448.98,585.105 1448.11,762.017 1447.23,989.66 \n",
" 1446.36,1148.94 1445.48,1156.16 1444.6,1007.25 1443.73,780.33 1442.85,594.957 1441.98,549.188 1441.1,667.517 1440.22,887.474 1439.34,1092.34 1438.47,1173.02 \n",
" 1437.59,1086.23 1436.71,878.061 1435.83,659.737 1434.96,548.34 1434.08,603.92 1433.2,796.857 1432.32,1023.35 1431.44,1161.11 1430.57,1135.4 1429.69,959.893 \n",
" 1428.81,729.636 1427.93,569.819 1427.05,567.678 1426.17,724.656 1425.29,955.166 1424.41,1133.03 1423.53,1160.52 1422.65,1022.27 1421.77,794.14 1420.89,601.853 \n",
" 1420.01,551.751 1419.13,671.817 1418.25,895.746 1417.37,1099.31 1416.49,1169.18 1415.61,1066.18 1414.73,847.551 1413.85,635.379 1412.96,548.557 1412.08,636.02 \n",
" 1411.2,848.886 1410.32,1067.49 1409.44,1168.54 1408.56,1094.76 1407.67,887.604 1406.79,664.295 1405.91,551.595 1405.03,613.774 1404.14,815.694 1403.26,1042.37 \n",
" 1402.38,1164.29 1401.49,1111.51 1400.61,914.01 1399.73,684.977 1398.84,556.099 1397.96,601.778 1397.08,795.929 1396.19,1026.58 1395.31,1160.27 1394.43,1119.34 \n",
" 1393.54,927.303 1392.66,695.598 1391.77,559.135 1390.89,597.669 1390,788.951 1389.12,1021.27 1388.23,1158.53 1387.35,1120.01 1386.46,928.096 1385.58,695.563 \n",
" 1384.69,559.472 1383.81,600.34 1382.92,794.254 1382.04,1026.57 1381.15,1159.51 1380.26,1113.95 1379.38,916.743 1378.49,685.186 1377.6,557.429 1376.72,609.98 \n",
" 1375.83,811.641 1374.95,1041.74 1374.06,1162.17 1373.17,1100.39 1372.28,893.323 1371.4,665.631 1370.51,554.767 1369.62,627.935 1368.74,841.08 1367.85,1065.14 \n",
" 1366.96,1164.05 1366.07,1077.49 1365.19,857.928 1364.3,639.086 1363.41,554.623 1362.52,656.379 1361.63,882.282 1360.75,1093.95 1359.86,1161.36 1358.97,1042.87 \n",
" 1358.08,811.201 1357.19,609.081 1356.3,561.369 1355.41,697.772 1354.53,934.063 1353.64,1123.85 1352.75,1149.23 1351.86,994.211 1350.97,755.076 1350.08,580.803 \n",
" 1349.19,580.267 1348.3,754.022 1347.41,993.532 1346.52,1148.76 1345.63,1122.19 1344.74,930.292 1343.85,693.6 1342.96,561.192 1342.07,616.726 1341.18,825.354 \n",
" 1340.29,1055.31 1339.4,1160.98 1338.51,1075.21 1337.62,852.226 1336.73,633.623 1335.84,558.557 1334.95,675.043 1334.06,908.972 1333.17,1111.01 1332.27,1151.87 \n",
" 1331.38,1005.18 1330.49,764.812 1329.6,584.999 1328.71,581.424 1327.82,756.591 1326.93,997.809 1326.04,1149.49 1325.14,1113.54 1324.25,912.992 1323.36,677.552 \n",
" 1322.47,559.846 1321.58,636.424 1320.69,857.728 1319.79,1079.9 1318.9,1158.19 1318.01,1041.37 1317.12,805.514 1316.23,604.71 1315.33,570.444 1314.44,725.352 \n",
" 1313.55,967.999 1312.66,1139.08 1311.76,1126.02 1310.87,937.203 1309.98,696.916 1309.09,563.672 1308.19,625.614 1307.3,841.977 1306.41,1069.65 1305.51,1157.73 \n",
" 1304.62,1047.6 1303.73,812.24 1302.84,608.062 1301.94,571.055 1301.05,726.059 1300.16,969.827 1299.26,1139.59 1298.37,1121.74 1297.48,928.042 1296.58,688.142 \n",
" 1295.69,562.968 1294.8,636.713 1293.9,859.984 1293.01,1082.63 1292.12,1154.63 1291.22,1027.25 1290.33,786.332 1289.43,594.621 1288.54,581.896 1287.65,756.949 \n",
" 1286.75,1001.25 1285.86,1149.03 1284.96,1099.77 1284.07,886.804 1283.18,654.988 1282.28,562.313 1281.39,672.142 1280.49,909.647 1279.6,1112.56 1278.71,1141.99 \n",
" 1277.81,977.61 1276.92,731.973 1276.02,573.618 1275.13,611.429 1274.23,819.574 1273.34,1054.9 1272.45,1155.27 1271.55,1051.32 1270.66,814.651 1269.76,608.722 \n",
" 1268.87,576.021 1267.97,739.341 1267.08,985.647 1266.18,1144.12 1265.29,1104.64 1264.39,894.484 1263.5,659.721 1262.61,563.826 1261.71,673.757 1260.82,912.986 \n",
" 1259.92,1114.55 1259.03,1137.53 1258.13,965.702 1257.24,719.235 1256.34,570.822 1255.45,624.658 1254.55,843.103 1253.66,1072.76 1252.76,1152.09 1251.87,1025.12 \n",
" 1250.97,781.166 1250.08,592.201 1249.18,591.663 1248.29,780.077 1247.39,1024.35 1246.5,1151.62 1245.6,1071.66 1244.71,840.992 1243.81,623.195 1242.92,572.96 \n",
" 1242.02,726.12 1241.13,973.861 1240.23,1139.85 1239.34,1105.75 1238.44,895.73 1237.55,659.58 1236.65,565.988 1235.76,681.973 1234.86,924.687 1233.97,1120.39 \n",
" 1233.07,1128.79 1232.18,943.724 1231.28,697.921 1230.39,567.944 1229.49,647.334 1228.6,879.101 1227.7,1096.43 1226.81,1142.67 1225.91,984.346 1225.02,735.624 \n",
" 1224.12,576.132 1223.23,621.244 1222.33,838.45 1221.44,1070.58 1220.54,1149.41 1219.65,1017.69 1218.75,770.874 1217.86,588.163 1216.96,602.405 1216.07,803.362 \n",
" 1215.17,1044.86 1214.28,1151 1213.38,1044.31 1212.49,802.509 1211.59,602.046 1210.7,589.411 1209.8,773.955 1208.91,1020.77 1208.02,1149.17 1207.12,1065.01 \n",
" 1206.23,829.873 1205.33,616.206 1204.44,580.909 1203.54,750.019 1202.65,999.307 1201.75,1145.39 1200.86,1080.65 1199.96,852.677 1199.07,629.458 1198.17,575.693 \n",
" 1197.28,731.168 1196.39,981.132 1195.49,1140.84 1194.6,1092.09 1193.7,870.873 1192.81,640.953 1191.91,572.755 1191.02,716.946 1190.13,966.617 1189.23,1136.4 \n",
" 1188.34,1100.08 1187.44,884.56 1186.55,650.121 1185.66,571.304 1184.76,706.904 1183.87,955.939 1182.97,1132.7 1182.08,1105.21 1181.19,893.911 1180.29,656.62 \n",
" 1179.4,570.763 1178.51,700.649 1177.61,949.138 1176.72,1130.14 1175.82,1107.95 1174.93,899.121 1174.04,660.284 1173.14,570.758 1172.25,697.871 1171.36,946.16 \n",
" 1170.46,1128.92 1169.57,1108.59 1168.68,900.379 1167.79,661.092 1166.89,571.103 1166,698.35 1165.11,946.89 1164.21,1129.1 1163.32,1107.28 1162.43,897.855 \n",
" 1161.53,659.141 1160.64,571.772 1159.75,701.958 1158.86,951.161 1157.96,1130.54 1157.07,1104.05 1156.18,891.697 1155.29,654.627 1154.4,572.885 1153.5,708.643 \n",
" 1152.61,958.765 1151.72,1133.03 1150.83,1098.79 1149.93,882.041 1149.04,647.841 1148.15,574.686 1147.26,718.413 1146.37,969.444 1145.48,1136.21 1144.58,1091.33 \n",
" 1143.69,869.024 1142.8,639.162 1141.91,577.52 1141.02,731.311 1140.13,982.885 1139.24,1139.62 1138.35,1081.41 1137.45,852.808 1136.56,629.057 1135.67,581.813 \n",
" 1134.78,747.387 1133.89,998.706 1133,1142.74 1132.11,1068.77 1131.22,833.6 1130.33,618.078 1129.44,588.045 1128.55,766.67 1127.66,1016.44 1126.77,1144.95 \n",
" 1125.88,1053.1 1124.99,811.677 1124.1,606.864 1123.21,596.726 1122.32,789.135 1121.43,1035.54 1120.54,1145.58 1119.65,1034.17 1118.76,787.408 1117.87,596.129 \n",
" 1116.98,608.359 1116.1,814.675 1115.21,1055.35 1114.32,1143.96 1113.43,1011.79 1112.54,761.272 1111.65,586.652 1110.76,623.413 1109.88,843.067 1108.99,1075.13 \n",
" 1108.1,1139.37 1107.21,985.871 1106.32,733.869 1105.43,579.255 1104.55,642.282 1103.66,873.957 1102.77,1094.03 1101.88,1131.18 1101,956.461 1100.11,705.928 \n",
" 1099.22,574.776 1098.34,665.245 1097.45,906.831 1096.56,1111.15 1095.68,1118.79 1094.79,923.773 1093.9,678.298 1093.02,574.033 1092.13,692.433 1091.24,941.019 \n",
" 1090.36,1125.54 1089.47,1101.73 1088.59,888.202 1087.7,651.937 1086.81,577.785 1085.93,723.793 1085.04,975.688 1084.16,1136.24 1083.27,1079.7 1082.39,850.341 \n",
" 1081.5,627.876 1080.62,586.679 1079.73,759.063 1078.85,1009.86 1077.96,1142.35 1077.08,1052.58 1076.2,810.979 1075.31,607.188 1074.43,601.211 1073.54,797.752 \n",
" 1072.66,1042.44 1071.78,1143.03 1070.89,1020.48 1070.01,771.09 1069.13,590.932 1068.24,621.672 1067.36,839.138 1066.48,1072.26 1065.59,1137.62 1064.71,983.764 \n",
" 1063.83,731.805 1062.95,580.102 1062.06,648.113 1061.18,882.275 1060.3,1098.13 1059.42,1125.62 1058.54,943.07 1057.66,694.368 1056.77,575.561 1055.89,680.316 \n",
" 1055.01,926.026 1054.13,1118.9 1053.25,1106.8 1052.37,899.278 1051.49,660.083 1050.61,577.984 1049.73,717.775 1048.85,969.096 1047.97,1133.52 1047.09,1081.22 \n",
" 1046.21,853.504 1045.33,630.246 1044.45,587.804 1043.57,759.699 1042.69,1010.1 1041.81,1141.12 1040.93,1049.22 1040.05,807.056 1039.18,606.074 1038.3,605.166 \n",
" 1037.42,805.033 1036.54,1047.61 1035.66,1141.05 1034.79,1011.46 1033.91,761.375 1033.03,588.634 1032.15,629.897 1031.28,852.495 1030.4,1080.27 1029.52,1132.97 \n",
" 1028.65,968.912 1027.77,717.964 1026.89,578.77 1026.02,661.5 1025.14,900.634 1024.26,1106.83 1023.39,1116.85 1022.51,922.773 1021.64,678.313 1020.76,577.051 \n",
" 1019.89,699.158 1019.01,947.898 1018.14,1126.26 1017.26,1092.98 1016.39,874.466 1015.51,643.813 1014.64,583.725 1013.77,741.77 1012.89,992.717 1012.02,1137.76 \n",
" 1011.15,1062.03 1010.27,825.542 1009.4,615.683 1008.53,598.695 1007.65,787.998 1006.78,1033.58 1005.91,1140.89 1005.04,1024.92 1004.16,777.609 1003.29,594.896 \n",
" 1002.42,621.517 1001.55,836.338 1000.68,1069.14 999.807,1135.5 998.936,982.888 998.065,732.242 997.194,582.127 996.324,651.425 995.454,885.195 994.584,1098.24 \n",
" 993.714,1121.83 992.844,937.333 991.975,690.906 991.105,577.718 990.236,687.366 989.367,932.97 988.498,1120.03 987.63,1100.45 986.762,889.792 985.893,654.875 \n",
" 985.025,581.665 984.158,728.067 983.29,978.142 982.423,1133.94 981.555,1072.2 980.688,841.836 979.821,625.171 978.955,593.625 978.088,772.098 977.222,1019.35 \n",
" 976.356,1139.79 975.49,1038.2 974.624,794.996 973.759,602.523 972.894,612.945 972.029,817.959 971.164,1055.44 970.299,1137.68 969.435,999.736 968.57,750.681 \n",
" 967.706,587.336 966.843,638.714 965.979,864.158 965.115,1085.55 964.252,1128.06 963.389,958.187 962.526,710.114 961.664,579.694 960.801,669.818 959.939,909.285 \n",
" 959.077,1109.1 958.215,1111.63 957.354,914.97 956.493,674.281 955.631,579.373 954.77,705.016 953.91,952.083 953.049,1125.81 952.189,1089.32 951.329,871.452 \n",
" 950.469,643.898 949.61,585.876 948.75,743.007 947.891,991.494 947.032,1135.71 946.173,1062.19 945.315,828.887 944.456,619.396 943.598,598.479 942.74,782.503 \n",
" 941.883,1026.7 941.025,1139.09 940.168,1031.42 939.311,788.366 938.454,600.929 937.598,616.287 936.742,822.289 935.886,1057.13 935.03,1136.49 934.174,998.171 \n",
" 933.319,750.776 932.464,588.388 931.609,638.296 930.754,861.276 929.9,1082.49 929.045,1128.62 928.191,963.603 927.338,716.782 926.484,581.437 925.631,663.452 \n",
" 924.778,898.539 923.925,1102.71 923.073,1116.3 922.22,928.771 921.368,686.822 920.516,579.554 919.665,690.702 918.813,933.341 917.962,1117.95 917.111,1100.45 \n",
" 916.261,894.607 915.41,661.107 914.56,582.073 913.71,719.043 912.861,965.14 912.011,1128.54 911.162,1082.01 910.313,861.896 909.464,639.651 908.616,588.238 \n",
" 907.768,747.561 906.92,993.589 906.072,1134.98 905.225,1061.91 904.378,831.262 903.531,622.289 902.684,597.24 901.838,775.451 900.992,1018.52 900.146,1137.86 \n",
" 899.3,1041.01 898.455,803.164 897.609,608.718 896.765,608.263 895.92,802.033 895.076,1039.92 894.232,1137.83 893.388,1020.1 892.544,777.913 891.701,598.528 \n",
" 890.858,620.511 890.015,826.758 889.172,1057.92 888.33,1135.57 887.488,999.903 886.647,755.682 885.805,591.239 884.964,633.241 884.123,849.199 883.282,1072.72 \n",
" 882.442,1131.76 881.602,981.005 880.762,736.532 879.922,586.338 879.083,645.772 878.244,869.041 877.405,1084.61 876.567,1127.05 875.729,963.913 874.891,720.441 \n",
" 874.053,583.305 873.216,657.502 872.378,886.062 871.542,1093.92 870.705,1122.04 869.869,949.043 869.033,707.328 868.197,581.647 867.362,667.912 866.526,900.111 \n",
" 865.691,1100.95 864.857,1117.25 864.023,936.725 863.189,697.088 862.355,580.914 861.521,676.562 860.688,911.088 859.855,1106 859.023,1113.14 858.19,927.226 \n",
" 857.358,689.61 856.527,580.724 855.695,683.095 854.864,918.919 854.033,1109.33 853.202,1110.08 852.372,920.76 851.542,684.812 850.712,580.78 849.883,687.226 \n",
" 849.054,923.536 848.225,1111.1 847.397,1108.34 846.568,917.497 845.74,682.654 844.913,580.88 844.085,688.749 843.258,924.862 842.432,1111.41 841.605,1108.1 \n",
" 840.779,917.579 839.953,683.164 839.128,580.941 838.302,687.53 837.477,922.794 836.653,1110.23 835.829,1109.41 835.005,921.116 834.181,686.446 833.357,581.003 \n",
" 832.534,683.517 831.711,917.204 830.889,1107.42 830.067,1112.22 829.245,928.189 828.423,692.69 827.602,581.25 826.781,676.751 825.961,907.934 825.14,1102.72 \n",
" 824.32,1116.34 823.501,938.832 822.681,702.172 821.862,582.024 821.043,667.388 820.225,894.81 819.407,1095.71 818.589,1121.39 817.772,953.012 816.954,715.246 \n",
" 816.138,583.838 815.321,655.725 814.505,877.67 813.689,1085.85 812.873,1126.83 812.058,970.591 811.243,732.321 810.429,587.382 809.614,642.24 808.801,856.402 \n",
" 807.987,1072.47 807.174,1131.91 806.361,991.278 805.548,753.824 804.736,593.531 803.924,627.632 803.112,830.998 802.301,1054.83 801.49,1135.64 800.679,1014.57 \n",
" 799.869,780.136 799.059,603.326 798.249,612.87 797.44,801.636 796.631,1032.14 795.822,1136.78 795.014,1039.67 794.206,811.516 793.398,617.941 792.591,599.234 \n",
" 791.784,768.768 790.977,1003.63 790.171,1133.86 789.365,1065.44 788.56,847.983 787.754,638.615 786.949,588.343 786.145,733.235 785.341,968.73 784.537,1125.24 \n",
" 783.733,1090.28 782.93,889.184 782.127,666.54 781.325,582.149 780.522,696.378 779.721,927.164 778.919,1109.13 778.118,1112.14 777.317,934.227 776.517,702.695 \n",
" 775.717,582.886 774.917,660.144 774.118,879.183 773.318,1083.84 772.52,1128.48 771.721,981.524 770.924,747.621 770.126,592.945 769.329,627.151 768.532,825.777 \n",
" 767.735,1047.94 766.939,1136.35 766.143,1028.64 765.348,801.143 764.553,614.644 763.758,600.686 762.963,768.913 762.169,1000.61 761.376,1132.61 760.582,1072.2 \n",
" 759.789,862.048 758.997,649.891 758.205,584.58 757.413,711.73 756.621,942.053 755.83,1114.24 755.039,1107.94 754.249,927.775 753.459,699.724 752.669,582.917 \n",
" 751.88,658.63 751.091,873.91 750.303,1078.8 749.514,1130.89 748.727,994.163 747.939,763.749 747.152,599.527 746.365,615.192 745.579,799.68 744.793,1025.19 \n",
" 744.007,1135.85 743.222,1055.39 742.437,839.55 741.653,637.259 740.869,587.793 740.085,724.995 739.302,954.322 738.519,1118.16 737.736,1104.18 736.954,922.185 \n",
" 736.172,697.038 735.391,582.869 734.61,657.594 733.829,869.93 733.049,1074.79 732.269,1132.5 731.49,1003.94 730.711,776.845 729.932,605.755 729.153,606.83 \n",
" 728.375,779.118 727.598,1005.57 726.821,1132.7 726.044,1074.61 725.268,870.8 724.491,659.161 723.716,582.538 722.941,692.418 722.166,914.534 721.391,1099.25 \n",
" 720.617,1122.43 719.844,968.704 719.07,741.482 718.297,593.195 717.525,623.058 716.753,810.847 715.981,1030.82 715.21,1136.01 714.439,1056.41 713.668,845.323 \n",
" 712.898,643.481 712.129,585.147 711.359,708.89 710.59,932.305 709.822,1107.07 709.054,1117.46 708.286,956.825 707.519,731.622 706.752,590.714 705.985,626.974 \n",
" 705.219,816.248 704.453,1033.66 703.688,1136.14 702.923,1056.47 702.159,847.244 701.395,645.869 700.631,584.287 699.868,702.611 699.105,923.209 698.343,1101.83 \n",
" 697.581,1121.99 696.819,970.76 696.058,746.877 695.297,596.08 694.537,616.254 693.777,794.055 693.017,1013.96 692.258,1133.45 691.499,1075.36 690.741,877.507 \n",
" 689.983,667.839 689.225,581.618 688.468,674.212 687.712,885.761 686.955,1080.07 686.2,1132.38 685.444,1009.34 684.689,790.068 683.935,615.068 683.181,596.18 \n",
" 682.427,745.232 681.674,966.868 680.921,1119.72 680.168,1106.58 679.416,936.065 678.665,716.724 677.914,587.811 677.163,630.892 676.413,818.426 675.663,1031.6 \n",
" 674.913,1135.71 674.164,1065.24 673.416,864.878 672.667,660.96 671.92,581.719 671.172,676.731 670.425,886.111 669.679,1078.47 668.933,1133.42 668.187,1016.83 \n",
" 667.442,801.787 666.697,622.579 665.953,590.812 665.209,726.017 664.466,943.882 663.723,1109.01 662.98,1119.07 662.238,968.093 661.497,749.744 660.755,599.066 \n",
" 660.014,608.951 659.274,773.044 658.534,990.047 657.795,1126.41 657.056,1098.5 656.317,923.708 655.579,709.398 654.841,586.811 654.104,630.738 653.367,814.079 \n",
" 652.631,1024.79 651.895,1134.47 651.159,1076.56 650.424,886.542 649.69,679.947 648.955,582.036 648.222,651.901 647.489,847.013 646.756,1049.33 646.023,1136.8 \n",
" 645.291,1056.94 644.56,858.08 643.829,659.918 643.098,581.385 642.368,669.346 641.639,870.857 640.909,1065.24 640.181,1136.43 639.452,1042.18 638.725,838.932 \n",
" 637.997,647.769 637.27,582.245 636.544,681.02 635.818,885.269 635.092,1073.93 634.367,1135.51 633.642,1033.9 632.918,829.274 632.195,642.308 631.471,582.904 \n",
" 630.748,685.717 630.026,890.159 629.304,1076.3 628.583,1135.3 627.862,1032.92 627.141,829.172 626.421,642.96 625.702,582.599 624.983,682.933 624.264,885.444 \n",
" 623.546,1072.58 622.828,1136.06 622.111,1039.43 621.394,838.748 620.678,649.895 619.962,581.563 619.247,672.812 618.532,870.957 617.818,1062.2 617.104,1137.06 \n",
" 616.39,1052.94 615.677,858.19 614.965,664.05 614.253,581.068 613.541,656.223 612.83,846.537 612.119,1043.87 611.409,1136.47 610.699,1072.16 609.99,887.567 \n",
" 609.282,687.012 608.573,583.464 607.866,634.967 607.158,812.3 606.451,1015.78 605.745,1131.41 605.039,1094.71 604.334,926.457 603.629,720.735 602.925,592.16 \n",
" 602.221,612.069 601.517,769.117 600.814,975.965 600.112,1118 599.41,1116.82 598.708,973.375 598.007,767.006 597.307,611.436 596.607,592.071 595.907,719.267 \n",
" 595.208,923.022 594.51,1091.68 593.812,1133.15 593.114,1025.06 592.417,826.606 591.72,645.931 591.024,581.151 590.329,667.169 589.633,857.088 588.939,1047.99 \n",
" 588.245,1136.77 587.551,1075.78 586.858,898.158 586.165,699.633 585.473,586.815 584.781,619.995 584.09,781.115 583.399,983.811 582.709,1119.85 582.02,1116.89 \n",
" 581.33,976.801 580.642,774.273 579.953,616.842 579.266,587.772 578.579,702.164 577.892,899.329 577.206,1075.17 576.52,1137.2 575.835,1053.06 575.15,867.209 \n",
" 574.466,677.213 573.782,582.45 573.099,632.2 572.417,800.215 571.734,998.78 571.053,1124.63 570.372,1112.61 569.691,969.247 569.011,768.99 568.331,615.385 \n",
" 567.652,587.539 566.973,699.513 566.295,893.446 565.618,1069.63 564.941,1137.84 564.264,1063.41 563.588,884.672 562.913,693 562.238,586 561.563,617.869 \n",
" 560.889,772.126 560.216,970.348 559.543,1112.18 558.87,1126.11 558.198,1005.42 557.527,811.15 556.856,641.09 556.186,580.502 555.516,659.487 554.846,838.232 \n",
" 554.178,1027.28 553.509,1132.39 552.842,1101.45 552.174,950.139 551.508,753.775 550.841,609.674 550.176,588.94 549.511,701.52 548.846,891.649 548.182,1065.65 \n",
" 547.518,1138.13 546.855,1073.8 546.193,904.457 545.531,713.064 544.869,593.002 544.208,602.55 543.548,736.766 542.888,930.241 542.228,1089.13 541.57,1136.7 \n",
" 540.911,1050.24 540.253,871.742 539.596,687.272 538.939,585.424 538.283,614.822 537.628,761.085 536.972,954.067 536.318,1101.63 535.664,1133.65 535.01,1035.18 \n",
" 534.357,853.246 533.705,674.202 533.053,582.671 532.401,621.631 531.751,772.414 531.1,963.851 530.45,1105.99 529.801,1132.35 529.153,1030.9 528.504,849.303 \n",
" 527.857,672.391 527.21,582.45 526.563,621.031 525.917,769.933 525.272,959.986 524.627,1103.31 523.982,1133.95 523.338,1038.07 522.695,860.053 522.052,681.713 \n",
" 521.41,584.681 520.769,613.081 520.127,753.646 519.487,942.111 518.847,1092.72 518.207,1137.28 517.568,1055.87 516.93,885.603 516.292,703.501 515.655,591.571 \n",
" 515.018,599.928 514.382,724.516 513.746,909.261 513.111,1071.46 512.477,1138.82 511.843,1081.61 511.209,925.568 510.576,740.154 509.944,607.524 509.312,586.138 \n",
" 508.681,685.139 508.051,860.635 507.42,1035.33 506.791,1132.61 506.162,1110.12 505.533,977.99 504.906,794.118 504.278,638.655 503.651,579.044 503.025,640.873 \n",
" 502.4,797.068 501.775,979.929 501.15,1110.58 500.526,1132.95 499.903,1037.74 499.28,866.053 498.657,691.516 498.036,588.633 497.415,601.034 496.794,723.136 \n",
" 496.174,902.832 495.555,1063.87 494.936,1138.24 494.317,1094.81 493.699,952.178 493.082,770.55 492.466,626.186 491.85,579.396 491.234,649.453 490.619,806.834 \n",
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" 384.108,662.048 383.61,789.303 383.113,934.74 382.616,1060.19 382.121,1132.97 381.625,1134.34 381.131,1064.22 380.637,941.07 380.144,796.896 379.651,668.907 \n",
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" 374.277,847.276 373.792,711.622 373.308,612.698 372.825,574.808 372.342,607.036 371.86,701.169 371.379,833.883 370.898,972.589 370.418,1083.48 369.939,1139.73 \n",
" 369.46,1127.96 368.982,1051.29 368.504,928.462 368.028,789.136 367.552,666.723 367.076,590.353 366.601,577.993 366.127,632.299 365.654,740.148 365.181,875.892 \n",
" 364.709,1007.51 364.237,1104.2 363.767,1143.52 363.297,1116.56 362.827,1029.88 362.358,903.787 361.89,767.546 361.423,652.536 360.956,585.036 360.49,580.267 \n",
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" 355.407,664.368 354.949,590.825 354.491,576.259 354.035,623.592 353.579,722.22 353.123,850.471 352.668,980.416 352.214,1083.98 351.761,1138.97 351.308,1133.82 \n",
" 350.856,1069.89 350.405,961.1 349.954,830.772 349.504,706.624 349.055,614.851 348.606,574.624 348.158,594.147 347.711,669.06 347.264,783.486 346.819,913.464 \n",
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" 262.616,634.458 262.335,592.485 262.055,571.356 261.776,572.537 261.498,595.757 261.22,639.061 260.943,698.991 260.667,770.875 260.392,849.208 260.117,928.073 \n",
" 259.843,1001.6 259.57,1064.39 259.298,1111.91 259.026,1140.82 258.755,1149.16 258.485,1136.5 258.216,1103.93 257.947,1053.95 257.679,990.248 257.412,917.454 \n",
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},
"execution_count": 39,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ε = 1E-6\n",
"L = [B₋₁; \n",
" B₁ ; \n",
" ε*D₁*D₀ - S₁*S₀*M]\n",
"\n",
"@time u = L \\ [1.0;0.0;0.0]\n",
"\n",
"@show ε*u''(0.1) - (-1+0.1+0.1^2)*u(0.1)\n",
"plot(u)"
]
},
{
"cell_type": "markdown",
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"For other smooth functions, we first approximate in a polynomial basis, and without loss of generality we use Chebyshev T basis. For example, consider \n",
"\\begin{align*}\n",
"u(-1) &= 1\\\\\n",
"u(1) &= 0\\\\\n",
"\\epsilon u''(x) - \\E^x u(x) &= 0\n",
"\\end{align*}\n",
"where\n",
"$$\n",
"\\E^x \\approx p(x) = \\sum_{k=0}^{m-1} p_k T_k(x)\n",
"$$\n",
"Evaluating at a point $x$, recall Clenshaw's algorithm:\n",
"\\begin{align*}\n",
"\\gamma_{n-1} &= 2p_{n-1} \\\\\n",
"\\gamma_{n-2} &= 2p_{n-2} + 2x \\gamma_{n-1} \\\\\n",
"\\gamma_{n-3} &= 2 p_{n-3} + 2x \\gamma_{n-2} - \\gamma_{n-1} \\\\\n",
"& \\vdots \\\\\n",
"\\gamma_1 &= p_1 + x \\gamma_2 - \\half \\gamma_3 \\\\\n",
"p(x) = \\gamma_0 &= p_0 + x \\gamma_1 - \\half \\gamma_2\n",
"\\end{align*}\n",
"If multiplication by $x$ becomes $J^\\top$, then multiplication by $p(x)$ becomes $p(J^\\top)$, and hence we calculate:\\\n",
"\\begin{align*}\n",
"\\Gamma_{n-1} &= 2p_{n-1}I \\\\\n",
"\\Gamma_{n-2} &= 2p_{n-2}I + 2J^\\top \\Gamma_{n-1} \\\\\n",
"\\Gamma_{n-3} &= 2 p_{n-3}I + 2J^\\top \\Gamma_{n-2} - \\Gamma_{n-1} \\\\\n",
"& \\vdots \\\\\n",
"\\Gamma_1 &= p_1I + J^\\top \\Gamma_2 - \\half \\Gamma_3 \\\\\n",
"p(J^\\top) = \\Gamma_0 &= p_0 + x \\Gamma_1 - \\half \\Gamma_2\n",
"\\end{align*}\n",
"\n",
"Here is an example:"
]
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"metadata": {},
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"data": {
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"ConcreteMultiplication : Chebyshev() → Chebyshev()\n",
" 1.2660658777520084 0.5651591039924851 … 5.518385951076946e-9 ⋱\n",
" 1.1303182079849703 1.4018135475190467 9.988153506976674e-8 ⋱\n",
" 0.27149533953407656 0.587327528916817 1.59923072107804e-6 ⋱\n",
" 0.044336849848663804 0.1384847899880851 2.248866199669348e-5 ⋱\n",
" 0.0054742404420936785 0.022439888080288854 0.00027146315597690023 ⋱\n",
" 0.0005429263119139036 0.0027596088825239777 … 0.0027371202210468393 ⋱\n",
" 4.497732295427654e-5 0.00027306237418820746 0.022168424924331902 ⋱\n",
" 3.198436462511398e-6 2.2588267717379123e-5 0.13574766976703828 ⋱\n",
" 1.992124804817033e-7 1.6047366172067758e-6 0.5651591039924851 ⋱\n",
" 1.1036771902153892e-8 9.988153506976674e-8 1.2660658777520084 ⋱\n",
" ⋱ ⋱ … ⋱ ⋱"
]
},
"execution_count": 40,
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],
"source": [
"p = Fun(exp, Chebyshev()) # polynomial approximation to exp(x)\n",
"M = Multiplication(p) : Chebyshev() # constructed using Clenshaw:"
]
},
{
"cell_type": "code",
"execution_count": 41,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(13, 13)"
]
},
"execution_count": 41,
"metadata": {},
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],
"source": [
"ApproxFun.bandwidths(M) # still banded"
]
},
{
"cell_type": "code",
"execution_count": 42,
"metadata": {},
"outputs": [
{
"name": "stdout",
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" 0.039351 seconds (1.04 M allocations: 21.628 MiB, 28.97% gc time)\n",
"ε * ((u')')(0.1) + exp(0.1) * u(0.1) = 4.884981308350689e-15\n"
]
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" 2220.34,811.067 2220.1,652.269 2219.86,510.729 2219.62,407.084 2219.38,356.732 2219.14,367.475 2218.9,438.223 2218.66,558.99 2218.42,712.213 2218.18,875.243 \n",
" 2217.93,1023.65 2217.69,1134.9 2217.45,1191.8 2217.2,1185.24 2216.95,1115.72 2216.71,993.483 2216.46,837.068 2216.21,670.589 2215.96,520.049 2215.71,409.274 \n",
" 2215.46,356.099 2215.21,369.425 2214.96,447.627 2214.71,578.634 2214.46,741.691 2214.2,910.578 2213.95,1057.79 2213.7,1159.04 2213.44,1197.31 2213.18,1165.78 \n",
" 2212.93,1069.19 2212.67,923.202 2212.41,751.973 2212.15,584.224 2211.89,448.437 2211.63,367.99 2211.37,357.065 2211.11,418.049 2210.85,540.916 2210.59,704.763 \n",
" 2210.32,881.281 2210.06,1039.59 2209.8,1151.67 2209.53,1197.34 2209.26,1167.99 2209,1068.35 2208.73,915.813 2208.46,737.528 2208.19,565.631 2207.92,431.472 \n",
" 2207.65,359.859 2207.38,364.387 2207.11,444.75 2206.84,586.564 2206.57,763.829 2206.29,943.608 2206.02,1092.11 2205.74,1181.09 2205.47,1193.23 2205.19,1125.73 \n",
" 2204.92,990.944 2204.64,814.354 2204.36,629.782 2204.08,472.984 2203.8,374.7 2203.52,354.557 2203.24,417.033 2202.96,550.34 2202.68,728.507 2202.4,916.344 \n",
" 2202.11,1076.34 2201.83,1176.16 2201.54,1195.28 2201.26,1129.3 2200.97,991.085 2200.68,808.43 2200.4,618.537 2200.11,460.503 2199.82,367.247 2199.53,358.569 \n",
" 2199.24,436.82 2198.95,586.146 2198.66,775.561 2198.37,965.277 2198.07,1115.02 2197.78,1192.62 2197.49,1181.02 2197.19,1082.16 2196.9,916.819 2196.6,720.425 \n",
" 2196.3,535.516 2196.01,402.562 2195.71,351.046 2195.41,392.815 2195.11,519.202 2194.81,702.658 2194.51,902.634 2194.21,1074.47 2193.91,1179.38 2193.61,1193.29 \n",
" 2193.3,1112.51 2193,954.84 2192.69,755.814 2192.39,560.807 2192.08,414.715 2191.78,351.576 2191.47,386.523 2191.16,511.997 2190.85,699.202 2190.54,904.579 \n",
" 2190.23,1079.86 2189.92,1183.43 2189.61,1190.29 2189.3,1098.24 2188.99,928.738 2188.67,722.019 2188.36,527.689 2188.05,392.825 2187.73,350.512 2187.42,411.59 \n",
" 2187.1,561.702 2186.78,764.493 2186.47,970.279 2186.15,1128.17 2185.83,1198.69 2185.51,1163.79 2185.19,1031.59 2184.87,834.752 2184.55,622.514 2184.22,448.463 \n",
" 2183.9,356.987 2183.58,371.844 2183.25,489.817 2182.93,681.201 2182.6,897.112 2182.28,1081.88 2181.95,1187.42 2181.62,1185.83 2181.29,1076.95 2180.97,888.737 \n",
" 2180.64,670.287 2180.31,479.11 2179.98,366 2179.64,361.446 2179.31,467.238 2178.98,655.637 2178.65,876.461 2178.31,1070.35 2177.98,1184.7 2177.64,1188.07 \n",
" 2177.31,1078.94 2176.97,886.571 2176.63,663.287 2176.3,470.38 2175.96,361.279 2175.62,366.647 2175.28,485.578 2174.94,685.477 2174.6,910.802 2174.26,1098.4 \n",
" 2173.91,1195.22 2173.57,1173.41 2173.23,1038.58 2172.88,828.525 2172.54,602.851 2172.19,426.14 2171.85,349.433 2171.5,395.351 2171.15,551.218 2170.8,772.331 \n",
" 2170.46,994.617 2170.11,1153.13 2169.76,1201.1 2169.41,1123.86 2169.05,943.6 2168.7,713.166 2168.35,500.737 2168,369.68 2167.64,359.553 2167.29,473.975 \n",
" 2166.93,679.136 2166.58,913.577 2166.22,1106.49 2165.86,1199.1 2165.51,1162.74 2165.15,1007.89 2164.79,781.452 2164.43,552.694 2164.07,392.145 2163.71,349.787 \n",
" 2163.35,439.317 2162.98,633.487 2162.62,872.169 2162.26,1080.83 2161.89,1193.79 2161.53,1175.01 2161.16,1029.82 2160.8,803.669 2160.43,568.074 2160.06,398.123 \n",
" 2159.7,348.477 2159.33,435.61 2158.96,632.107 2158.59,875.05 2158.22,1086.01 2157.85,1196.36 2157.48,1169.7 2157.1,1014.14 2156.73,780.048 2156.36,543.967 \n",
" 2155.98,383.67 2155.61,352.458 2155.23,461.246 2154.86,674.498 2154.48,921.596 2154.1,1120.09 2153.72,1203.23 2153.35,1142.55 2152.97,957.896 2152.59,711.152 \n",
" 2152.21,485.721 2151.83,358.354 2151.44,372.908 2151.06,525.018 2150.68,763.109 2150.3,1005.67 2149.91,1169.06 2149.53,1196.44 2149.14,1077.73 2148.75,853.606 \n",
" 2148.37,601.772 2147.98,410.181 2147.59,346.273 2147.2,433.056 2146.81,640.561 2146.42,895.925 2146.03,1108.81 2145.64,1203.37 2145.25,1145.39 2144.86,954.97 \n",
" 2144.47,699.814 2144.07,471.349 2143.68,352.006 2143.28,385.353 2142.89,559.932 2142.49,812.904 2142.09,1052.46 2141.7,1191.06 2141.3,1177.55 2140.9,1016.28 \n",
" 2140.5,766.011 2140.1,518.748 2139.7,365.99 2139.3,364.784 2138.9,516.18 2138.5,764.33 2138.09,1016.84 2137.69,1179.07 2137.29,1189.71 2136.88,1044.14 \n",
" 2136.47,796.743 2136.07,540.812 2135.66,373.484 2135.26,358.796 2134.85,502.946 2134.44,751.402 2134.03,1009.27 2133.62,1177.44 2133.21,1190.73 2132.8,1043.42 \n",
" 2132.39,791.964 2131.97,533.619 2131.56,368.946 2131.15,362.575 2130.73,517.597 2130.32,773.817 2129.9,1030.87 2129.49,1187.43 2129.07,1181.27 2128.66,1014.21 \n",
" 2128.24,751.989 2127.82,498.621 2127.4,355.235 2126.98,379.585 2126.56,562.531 2126.14,831.131 2125.72,1077.5 2125.3,1202.06 2124.87,1153.95 2124.45,952.061 \n",
" 2124.03,677.942 2123.6,443.079 2123.18,343.588 2122.75,420.709 2122.33,643.415 2121.9,920.584 2121.47,1138.08 2121.04,1205.78 2120.61,1095.08 2120.18,851.328 \n",
" 2119.75,575.538 2119.32,382.675 2118.89,353.697 2118.46,501.329 2118.03,764.164 2117.6,1031.92 2117.16,1191.6 2116.73,1175.27 2116.29,989.246 2115.86,711.989 \n",
" 2115.42,461.271 2114.99,344.208 2114.55,411.345 2114.11,634.527 2113.67,918.38 2113.23,1140.86 2112.79,1205.71 2112.35,1084.35 2111.91,828.765 2111.47,549.548 \n",
" 2111.03,368.216 2110.59,364.237 2110.14,539.949 2109.7,818.935 2109.26,1078.99 2108.81,1205.55 2108.37,1142.35 2107.92,916.693 2107.47,628.028 2107.03,404.327 \n",
" 2106.58,345.351 2106.13,477.944 2105.68,743.478 2105.23,1023.47 2104.78,1192.32 2104.33,1173.71 2103.88,975.428 2103.43,686.44 2102.97,437.241 2102.52,340.98 \n",
" 2102.07,441.897 2101.61,694.667 2101.16,984.422 2100.7,1178.77 2100.24,1188.35 2099.79,1008.18 2099.33,720.546 2098.87,457.74 2098.41,341.259 2097.95,425.492 \n",
" 2097.49,671.995 2097.03,966.691 2096.57,1172.47 2096.11,1193.01 2095.65,1018.12 2095.19,729.164 2094.72,461.449 2094.26,340.996 2093.79,425.043 2093.33,674.521 \n",
" 2092.86,971.864 2092.4,1176.22 2091.93,1190.19 2091.46,1006.57 2090.99,712.319 2090.53,447.755 2090.06,339.656 2089.59,440.362 2089.12,702.088 2088.65,999.254 \n",
" 2088.17,1188.55 2087.7,1178.14 2087.23,972.448 2086.76,670.748 2086.28,419.469 2085.81,341.191 2085.33,474.637 2084.86,755.166 2084.38,1045.75 2083.9,1203.76 \n",
" 2083.43,1151.1 2082.95,913.088 2082.47,606.702 2081.99,383.283 2081.51,353.782 2081.03,533.407 2080.55,833.42 2080.07,1104.67 2079.59,1211.68 2079.1,1100.44 \n",
" 2078.62,826.077 2078.14,525.837 2077.65,350.63 2077.17,389.074 2076.68,622.373 2076.2,933.12 2075.71,1164.14 2075.22,1197.97 2074.73,1016.87 2074.25,712.467 \n",
" 2073.76,439.699 2073.27,338.059 2072.78,460.059 2072.29,743.747 2071.8,1043.82 2071.3,1205.88 2070.81,1145.99 2070.32,894.476 2069.82,580.943 2069.33,367.722 \n",
" 2068.84,365.789 2068.34,576.723 2067.84,891.207 2067.35,1145.4 2066.85,1206.24 2066.35,1041.36 2065.86,736.641 2065.36,451.824 2064.86,336.882 2064.36,452.872 \n",
" 2063.86,739.148 2063.36,1044.71 2062.86,1207.69 2062.35,1141.19 2061.85,879.97 2061.35,562.64 2060.84,358.341 2060.34,376.54 2059.83,608.075 2059.33,929.369 \n",
" 2058.82,1168.12 2058.32,1195.68 2057.81,996.662 2057.3,677.883 2056.79,411.346 2056.28,341.493 2055.78,506.708 2055.27,817.851 2054.75,1106.02 2054.24,1214.12 \n",
" 2053.73,1082.7 2053.22,782.999 2052.71,478.684 2052.19,336.637 2051.68,435.29 2051.17,720.971 2050.65,1036.73 2050.14,1208.38 2049.62,1140.69 2049.1,870.527 \n",
" 2048.59,547.199 2048.07,350.132 2047.55,389.247 2047.03,643.335 2046.51,971.16 2045.99,1189.72 2045.47,1176.43 2044.95,938.17 2044.43,608.152 2043.91,371.69 \n",
" 2043.38,362.156 2042.86,585.472 2042.34,916.076 2041.81,1167.23 2041.29,1196.46 2040.76,986.667 2040.23,656.472 2039.71,393.478 2039.18,347.717 2038.65,545.819 \n",
" 2038.13,875.133 2037.6,1147.47 2037.07,1206.58 2036.54,1018.03 2036.01,689.642 2035.48,410.164 2034.94,340.856 2034.41,522.225 2033.88,849.972 2033.35,1134.63 \n",
" 2032.81,1211.03 2032.28,1034.31 2031.74,706.67 2031.21,418.6 2030.67,338.206 2030.14,512.917 2029.6,841.122 2029.06,1130.93 2028.52,1212.26 2027.98,1036.87 \n",
" 2027.45,707.348 2026.91,417.45 2026.37,338.258 2025.82,517.018 2025.28,848.571 2024.74,1136.88 2024.2,1210.84 2023.66,1026.06 2023.11,691.857 2022.57,406.945 \n",
" 2022.02,341.342 2021.48,534.705 2020.93,871.971 2020.39,1151.5 2019.84,1205.57 2019.29,1001.21 2018.75,660.743 2018.2,388.829 2017.65,349.53 2017.1,567.068 \n",
" 2016.55,910.511 2016,1172.27 2015.45,1193.55 2014.9,960.924 2014.34,615.193 2013.79,366.458 2013.24,366.448 2012.68,615.693 2012.13,962.489 2011.58,1194.96 \n",
" 2011.02,1170.51 2010.46,903.605 2009.91,557.585 2009.35,344.983 2008.79,396.925 2008.24,681.948 2007.68,1024.67 2007.12,1213.46 2006.56,1131.24 2006,828.361 \n",
" 2005.44,492.208 2004.88,331.43 2004.32,446.336 2003.75,765.962 2003.19,1091.52 2002.63,1219.89 2002.07,1070.46 2001.5,736.154 2000.94,425.97 2000.37,334.457 \n",
" 1999.81,519.502 1999.24,865.342 1998.67,1154.65 1998.11,1205.02 1997.54,984.212 1996.97,631.149 1996.4,368.834 1995.83,363.517 1995.26,619.071 1994.69,973.827 \n",
" 1994.12,1202.58 1993.55,1159.58 1992.98,871.681 1992.4,521.974 1991.83,333.573 1991.26,427.191 1990.68,743.468 1990.11,1080.24 1989.53,1221.57 1988.96,1076.34 \n",
" 1988.38,737.375 1987.8,422.418 1987.23,334.428 1986.65,530.581 1986.07,884.757 1985.49,1168.3 1984.91,1197.58 1984.33,953.16 1983.75,593.055 1983.17,350.882 \n",
" 1982.59,384.285 1982.01,672.034 1981.43,1027.13 1980.85,1218.1 1980.26,1119.96 1979.68,796.317 1979.09,458.548 1978.51,327.887 1977.92,490.394 1977.34,839.915 \n",
" 1976.75,1147.08 1976.16,1209.74 1975.58,986.169 1974.99,623.27 1974.4,360.311 1973.81,371.211 1973.22,649.248 1972.63,1010.8 1972.04,1216.38 1971.45,1129.3 \n",
" 1970.86,806.889 1970.27,463.102 1969.68,326.722 1969.08,488.98 1968.49,842.175 1967.9,1150.83 1967.3,1208.57 1966.71,976.349 1966.11,609.173 1965.52,352.964 \n",
" 1964.92,379.848 1964.32,672.25 1963.73,1034 1963.13,1221.73 1962.53,1108.65 1961.93,770.522 1961.33,435.236 1960.73,329.383 1960.13,524.955 1959.53,890.031 \n",
" 1958.93,1177.45 1958.33,1192.08 1957.73,923.535 1957.12,553.892 1956.52,334.503 1955.92,415.042 1955.31,741.097 1954.71,1090.57 1954.1,1224.78 1953.5,1051.63 \n",
" 1952.89,689.097 1952.28,385.226 1951.68,348.461 1951.07,604.467 1950.46,977.852 1949.85,1212.07 1949.24,1145.69 1948.63,823.969 1948.02,468.324 1947.41,324.154 \n",
" 1946.8,491.387 1946.19,854.888 1945.58,1163.4 1944.96,1203.14 1944.35,946.123 1943.74,570.245 1943.12,336.396 1942.51,407.37 1941.89,734.258 1941.28,1089.79 \n",
" 1940.66,1226.19 1940.05,1047.95 1939.43,679.042 1938.81,377.025 1938.19,353.269 1937.58,624.833 1936.96,1001.87 1936.34,1220.09 1935.72,1126.06 1935.1,785.387 \n",
" 1934.48,437.245 1933.86,326.651 1933.23,531.865 1932.61,908.683 1931.99,1191.48 1931.37,1180.39 1930.74,882.833 1930.12,508.861 1929.49,323.133 1928.87,457.532 \n",
" 1928.24,817.216 1927.62,1147.23 1926.99,1213.1 1926.36,967.612 1925.74,584.986 1925.11,337.487 1924.48,401.701 1923.85,732.276 1923.22,1093.62 1922.59,1227.62 \n",
" 1921.96,1038.15 1921.33,660.321 1920.7,364.441 1920.07,362.687 1919.44,656.736 1918.81,1035.92 1918.17,1227.96 1917.54,1094.51 1916.91,731.141 1916.27,399.191 \n",
" 1915.64,337.926 1915,591.908 1914.37,978.259 1913.73,1218.09 1913.09,1137.83 1912.46,795.103 1911.82,437.672 1911.18,324.488 1910.54,537.95 1909.9,923.612 \n",
" 1909.26,1201.65 1908.62,1169.88 1907.98,850.97 1907.34,476.642 1906.7,319.45 1906.06,494.254 1905.42,874 1904.77,1181.77 1904.13,1192.67 1903.49,898.322 \n",
" 1902.84,513.652 1902.2,320.129 1901.55,459.774 1900.91,830.651 1900.26,1160.99 1899.62,1208.22 1898.97,937.282 1898.32,546.944 1897.68,324.212 1897.03,433.274 \n",
" 1896.38,794.213 1895.73,1141.26 1895.08,1218.38 1894.43,968.296 1893.78,575.332 1893.13,329.801 1892.48,413.518 1891.83,764.929 1891.18,1124.07 1890.52,1224.74 \n",
" 1889.87,991.951 1889.22,598.065 1888.56,335.425 1887.91,399.389 1887.25,742.793 1886.6,1110.43 1885.94,1228.56 1885.29,1008.85 1884.63,614.717 1883.97,340.009 \n",
" 1883.32,389.968 1882.66,727.665 1882,1101 1881.34,1230.76 1880.68,1019.53 1880.02,625.091 1879.36,342.842 1878.7,384.573 1878.04,719.355 1877.38,1096.17 \n",
" 1876.72,1231.97 1876.06,1024.38 1875.39,629.15 1874.73,343.539 1874.07,382.778 1873.4,717.676 1872.74,1096.05 1872.07,1232.47 1871.41,1023.67 1870.74,626.974 \n",
" 1870.08,342.004 1869.41,384.406 1868.74,722.468 1868.08,1100.56 1867.41,1232.28 1866.74,1017.5 1866.07,618.735 1865.4,338.408 1864.73,389.517 1864.06,733.602 \n",
" 1863.39,1109.41 1862.72,1231.13 1862.05,1005.83 1861.38,604.699 1860.71,333.17 1860.03,398.382 1859.36,750.963 1858.69,1122.15 1858.01,1228.52 1857.34,988.52 \n",
" 1856.66,585.238 1855.99,326.94 1855.31,411.444 1854.64,774.418 1853.96,1138.1 1853.29,1223.74 1852.61,965.363 1851.93,560.856 1851.25,320.59 1850.57,429.282 \n",
" 1849.9,803.772 1849.22,1156.41 1848.54,1215.89 1847.86,936.142 1847.18,532.223 1846.49,315.198 1845.81,452.549 1845.13,838.715 1844.45,1175.99 1843.77,1203.95 \n",
" 1843.08,900.696 1842.4,500.215 1841.72,312.029 1841.03,481.911 1840.35,878.759 1839.66,1195.55 1838.98,1186.81 1838.29,858.992 1837.61,465.948 1836.92,312.499 \n",
" 1836.23,517.97 1835.54,923.18 1834.86,1213.56 1834.17,1163.33 1833.48,811.208 1832.79,430.806 1832.1,318.134 1831.41,561.174 1830.72,970.96 1830.03,1228.32 \n",
" 1829.34,1132.46 1828.65,757.809 1827.96,396.452 1827.26,330.492 1826.57,611.724 1825.88,1020.74 1825.18,1237.94 1824.49,1093.3 1823.8,699.628 1823.1,364.819 \n",
" 1822.41,351.083 1821.71,669.476 1821.01,1070.79 1820.32,1240.48 1819.62,1045.24 1818.92,637.924 1818.23,338.074 1817.53,381.25 1816.83,733.843 1816.13,1119.02 \n",
" 1815.43,1233.98 1814.73,988.058 1814.03,574.432 1813.33,318.547 1812.63,422.045 1811.93,803.71 1811.23,1163.01 1810.53,1216.65 1809.83,922.065 1809.12,511.371 \n",
" 1808.42,308.62 1807.72,474.08 1807.01,877.377 1806.31,1200.04 1805.6,1186.96 1804.9,848.178 1804.19,451.414 1803.49,310.586 1802.78,537.389 1802.08,952.532 \n",
" 1801.37,1227.31 1800.66,1143.85 1799.95,768.026 1799.25,397.615 1798.54,326.465 1797.83,611.284 1797.12,1026.27 1796.41,1241.99 1795.7,1086.92 1794.99,683.985 \n",
" 1794.28,353.27 1793.57,357.806 1792.86,694.256 1792.15,1095.18 1791.43,1241.52 1790.72,1016.56 1790.01,599.167 1789.29,321.734 1788.58,405.47 1787.87,783.904 \n",
" 1787.15,1155.48 1786.44,1223.8 1785.72,934.093 1785.01,517.336 1784.29,306.178 1783.57,469.423 1782.86,876.941 1782.14,1203.25 1781.42,1187.45 1780.71,841.904 \n",
" 1779.99,442.756 1779.27,309.311 1778.55,548.567 1777.83,969.269 1777.11,1234.67 1776.39,1132.08 1775.67,743.412 1774.95,379.951 1774.23,333.08 1773.51,640.615 \n",
" 1772.79,1056.15 1772.06,1246.38 1771.34,1058.48 1770.62,643.014 1769.9,333.401 1769.17,378.377 1768.45,742.064 1767.72,1132.45 1767,1235.78 1766.27,968.79 \n",
" 1765.55,545.907 1764.82,307.171 1764.1,444.781 1763.37,848.26 1762.64,1193.02 1761.92,1201.45 1761.19,866.53 1760.46,457.801 1759.73,304.519 1759.01,530.38 \n",
" 1758.28,953.587 1757.55,1233.07 1756.82,1143.35 1756.09,756.559 1755.36,384.538 1754.63,327.5 1753.9,631.687 1753.16,1051.77 1752.43,1248.63 1751.7,1063.13 \n",
" 1750.97,644.869 1750.23,331.631 1749.5,376.619 1748.77,743.699 1748.03,1136.31 1747.3,1237.01 1746.57,964.169 1745.83,538.254 1745.1,303.766 1744.36,450.575 \n",
" 1743.62,860.097 1742.89,1200.98 1742.15,1197.18 1741.41,851.559 1740.68,443.852 1739.94,304.3 1739.2,546.137 1738.46,973.613 1737.73,1240.36 1736.99,1130.05 \n",
" 1736.25,731.888 1735.51,368.585 1734.77,334.82 1734.03,658.176 1733.29,1076.52 1732.55,1250.45 1731.8,1038.66 1731.06,612.846 1730.32,318.556 1729.58,394.81 \n",
" 1728.84,779.898 1728.09,1161.24 1727.35,1229.08 1726.61,928.123 1725.86,502.697 1725.12,298.44 1724.37,481.48 1723.63,903.248 1722.88,1221 1722.14,1176.34 \n",
" 1721.39,805.379 1720.65,409.625 1719.9,310.953 1719.15,589.787 1718.4,1019.48 1717.66,1250.51 1716.91,1094.76 1716.16,678.789 1715.41,341.022 1714.66,356.445 \n",
" 1713.91,712.686 1713.16,1119.88 1712.42,1246.49 1711.66,989.265 1710.91,557.506 1710.16,302.789 1709.41,432.692 1708.66,841.581 1707.91,1196.46 1707.16,1208.15 \n",
" 1706.41,866.946 1705.65,450.746 1704.9,298.719 1704.15,534.908 1703.39,966.973 1702.64,1242.78 1701.88,1137.4 1701.13,736.571 1700.37,366.982 1699.62,330.032 \n",
" 1698.86,656.011 1698.11,1079.23 1697.35,1254.56 1696.6,1038.86 1695.84,607.915 1695.08,313.167 1694.32,395.14 1693.57,787.126 1692.81,1169.44 1692.05,1230.14 \n",
" 1691.29,919.543 1690.53,490.944 1689.77,294.052 1689.01,489.651 1688.26,918.279 1687.49,1230.21 1686.73,1170.79 1685.97,788.366 1685.21,394.953 1684.45,311.681 \n",
" 1683.69,606.655 1682.93,1039.24 1682.17,1256.35 1681.4,1080.65 1680.64,655.427 1679.88,327.726 1679.11,365.133 1678.35,737.256 1677.59,1140.41 1676.82,1245.44 \n",
" 1676.06,966.483 1675.29,531.143 1674.53,294.813 1673.76,450.533 1673,871.31 1672.23,1213.68 1671.46,1198.07 1670.7,837.081 1669.93,425.339 1669.16,299.009 \n",
" 1668.4,561.344 1667.63,998.296 1666.86,1253.17 1666.09,1117.79 1665.32,702.563 1664.55,346.386 1663.78,340.08 1663.01,688.918 1662.24,1108.23 1661.47,1255.73 \n",
" 1660.7,1010.9 1659.93,573.473 1659.16,300.464 1658.39,414.784 1657.62,823.251 1656.85,1192.55 1656.08,1221.25 1655.3,885.804 1654.53,459.87 1653.76,291.033 \n",
" 1652.99,517.162 1652.21,953.853 1651.44,1244.83 1650.66,1152.61 1649.89,752.325 1649.11,370.462 1648.34,318.572 1647.56,639.088 1646.79,1070.66 1646.01,1261.29 \n",
" 1645.24,1055.41 1644.46,620.817 1643.69,311.885 1642.91,380.584 1642.13,771.005 1641.35,1164.89 1640.58,1241.1 1639.8,937.4 1639.02,501.271 1638.24,288.187 \n",
" 1637.46,471.894 1636.68,902.776 1635.91,1229.75 1635.13,1186.34 1634.35,807.811 1633.57,402.487 1632.79,300.569 1632.01,585.16 1631.22,1024.55 1630.44,1260.92 \n",
" 1629.66,1101.72 1628.88,676.482 1628.1,331.382 1627.32,347.398 1626.53,711.58 1625.75,1127.58 1624.97,1256.84 1624.19,994.09 1623.4,553.046 1622.62,292.503 \n",
" 1621.83,424.463 1621.05,841.676 1620.27,1204.87 1619.48,1218.67 1618.7,871.779 1617.91,446.147 1617.13,287.778 1616.34,525.459 1615.55,966.103 1614.77,1251.67 \n",
" 1613.98,1150.1 1613.19,743.819 1612.41,362.802 1611.62,316.513 1610.83,642.613 1610.04,1076.39 1609.26,1265.69 1608.47,1056.81 1607.68,619.199 1606.89,307.944 \n",
" 1606.1,375.622 1605.31,767.398 1604.52,1165.54 1603.73,1247.15 1602.94,945.982 1602.15,506.158 1601.36,284.193 1600.57,460.034 1599.78,891.248 1598.99,1228.48 \n",
" 1598.2,1198.52 1597.41,825.568 1596.62,411.608 1595.82,291.826 1595.03,563.242 1594.24,1006.22 1593.45,1262.29 1592.65,1124.22 1591.86,703.645 1591.07,340.723 \n",
" 1590.27,328.961 1589.48,677.919 1588.68,1105.54 1587.89,1266.21 1587.1,1030.03 1586.3,587.796 1585.51,296.709 1584.71,391.893 1583.91,796.536 1583.12,1183.99 \n",
" 1582.32,1241.51 1581.53,922.631 1580.73,484.602 1579.93,280.765 1579.14,475.548 1578.34,911.929 1577.54,1238.13 1576.74,1191.2 1575.95,808.933 1575.15,399.271 \n",
" 1574.35,292.202 1573.55,573.985 1572.75,1017.78 1571.95,1266.34 1571.15,1119.62 1570.35,695.628 1569.55,335.423 1568.75,328.711 1567.95,680.919 1567.15,1108.94 \n",
" 1566.35,1268.77 1565.55,1031.96 1564.75,588.722 1563.95,295.024 1563.15,386.704 1562.35,790.192 1561.54,1181.68 1560.74,1247.05 1559.94,933.834 1559.14,493.211 \n",
" 1558.34,278.457 1557.53,461.728 1556.73,896.175 1555.93,1233.74 1555.12,1204.03 1554.32,830.828 1553.51,412.883 1552.71,284.705 1551.91,548.867 1551.1,994.071 \n",
" 1550.3,1264.28 1549.49,1143.45 1548.69,728.145 1547.88,350.228 1547.07,311.606 1546.27,643.135 1545.46,1080.12 1544.66,1273.75 1543.85,1069.5 1543.04,630.317 \n",
" 1542.24,306.463 1541.43,356.162 1540.62,739.801 1539.81,1151.67 1539.01,1263.65 1538.2,986.565 1537.39,541.016 1536.58,281.647 1535.77,414.843 1534.97,834.648 \n",
" 1534.16,1207.21 1533.35,1236.33 1532.54,898.871 1531.73,462.963 1530.92,274.848 1530.11,483.898 1529.3,924.158 1528.49,1246.24 1527.68,1194.64 1526.87,810.277 \n",
" 1526.06,397.904 1525.25,284.364 1524.44,559.608 1523.62,1005.61 1522.81,1269.2 1522,1141.73 1521.19,724.093 1520.38,346.674 1519.56,307.944 1518.75,638.505 \n",
" 1517.94,1077.1 1517.13,1277.21 1516.31,1080.81 1515.5,642.984 1514.69,309.293 1513.87,343.013 1513.06,717.527 1512.25,1137.53 1511.43,1271.97 1510.62,1014.95 \n",
" 1509.8,568.929 1508.99,285.108 1508.17,386.871 1507.36,794.119 1506.54,1186.51 1505.73,1255.51 1504.91,946.921 1504.1,503.232 1503.28,272.947 1502.47,436.858 \n",
" 1501.65,866.28 1500.83,1224.27 1500.02,1230.05 1499.2,879.134 1498.38,446.575 1497.57,271.272 1496.75,490.49 1495.93,932.569 1495.11,1251.52 1494.3,1197.85 \n",
" 1493.48,813.561 1492.66,399.096 1491.84,278.323 1491.02,545.545 1490.2,992.065 1489.38,1269.35 1488.57,1161.07 1487.75,751.729 1486.93,360.491 1486.11,292.249 \n",
" 1485.29,600.122 1484.47,1044.32 1483.65,1279.06 1482.83,1121.72 1482.01,694.739 1481.19,330.111 1480.37,311.206 1479.55,652.661 1478.73,1089.27 1477.91,1282.11 \n",
" 1477.08,1081.6 1476.26,643.302 1475.44,307.064 1474.62,333.439 1473.8,701.937 1472.98,1127.17 1472.15,1279.99 1471.33,1042.22 1470.51,597.798 1469.69,290.313 \n",
" 1468.86,357.337 1468.04,747.039 1467.22,1158.49 1466.4,1274.16 1465.57,1004.88 1464.75,558.341 1463.92,278.749 1463.1,381.467 1462.28,787.322 1461.45,1183.86 \n",
" 1460.63,1265.98 1459.8,970.595 1458.98,524.844 1458.16,271.267 1457.33,404.585 1456.51,822.37 1455.68,1203.97 1454.86,1256.7 1454.03,940.196 1453.21,497.087 \n",
" 1452.38,266.817 1451.55,425.644 1450.73,851.935 1449.9,1219.54 1449.08,1247.4 1448.25,914.305 1447.42,474.779 1446.6,264.447 1445.77,443.78 1444.94,875.887 \n",
" 1444.12,1231.24 1443.29,1239.02 1442.46,893.4 1441.64,457.611 1440.81,263.335 1439.98,458.298 1439.15,894.168 1438.32,1239.64 1437.5,1232.31 1436.67,877.839 \n",
" 1435.84,445.308 1435.01,262.818 1434.18,468.658 1433.35,906.748 1432.53,1245.21 1431.7,1227.85 1430.87,867.892 1430.04,437.663 1429.21,262.402 1428.38,474.46 \n",
" 1427.55,913.589 1426.72,1248.27 1425.89,1226.05 1425.06,863.771 1424.23,434.575 1423.4,261.788 1422.57,475.429 1421.74,914.617 1420.91,1248.95 1420.08,1227.12 \n",
" 1419.25,865.639 1418.42,436.07 1417.59,260.877 1416.76,471.418 1415.93,909.705 1415.1,1247.2 1414.26,1231.12 1413.43,873.62 1412.6,442.314 1411.77,259.792 \n",
" 1410.94,462.411 1410.11,898.672 1409.27,1242.74 1408.44,1237.87 1407.61,887.793 1406.78,453.621 1405.94,258.889 1405.11,448.54 1404.28,881.285 1403.45,1235.12 \n",
" 1402.61,1246.98 1401.78,908.168 1400.95,470.448 1400.11,258.775 1399.28,430.114 1398.45,857.285 1397.61,1223.62 1396.78,1257.81 1395.95,934.651 1395.11,493.369 \n",
" 1394.28,260.317 1393.45,407.658 1392.61,826.425 1391.78,1207.37 1390.94,1269.45 1390.11,966.994 1389.27,523.044 1388.44,264.653 1387.61,381.967 1386.77,788.535 \n",
" 1385.94,1185.29 1385.1,1280.67 1384.27,1004.72 1383.43,560.156 1382.6,273.186 1381.76,354.16 1380.93,743.6 1380.09,1156.17 1379.25,1289.89 1378.42,1047.04 \n",
" 1377.58,605.326 1376.75,287.561 1375.91,325.744 1375.08,691.866 1374.24,1118.75 1373.4,1295.21 1372.57,1092.77 1371.73,659.001 1370.9,309.616 1370.06,298.664 \n",
" 1369.22,633.965 1368.39,1071.83 1367.55,1294.36 1366.71,1140.21 1365.88,721.305 1365.04,341.289 1364.2,275.343 1363.36,571.06 1362.53,1014.4 1361.69,1284.83 \n",
" 1360.85,1187.07 1360.02,791.86 1359.18,384.476 1358.34,258.676 1357.5,504.991 1356.67,945.864 1355.83,1263.9 1354.99,1230.39 1354.15,869.591 1353.31,440.833 \n",
" 1352.48,251.974 1351.64,438.408 1350.8,866.284 1349.96,1228.88 1349.12,1266.56 1348.29,952.509 1347.45,511.505 1346.61,258.827 1345.77,374.878 1344.93,776.641 \n",
" 1344.09,1177.36 1343.25,1291.35 1342.42,1037.52 1341.58,596.792 1340.74,282.859 1339.9,318.901 1339.06,679.136 1338.22,1107.57 1337.38,1300.1 1336.54,1120.29 \n",
" 1335.7,695.768 1334.86,327.362 1334.02,275.826 1333.18,577.459 1332.35,1018.85 1331.51,1288.1 1330.67,1195.2 1329.83,805.884 1328.99,394.792 1328.15,251.588 \n",
" 1327.31,476.969 1326.47,912.154 1325.63,1251.02 1324.79,1255.5 1323.95,922.612 1323.11,486.147 1322.27,252.245 1321.43,384.719 1320.59,790.6 1319.75,1185.68 \n",
" 1318.91,1293.66 1318.07,1039.22 1317.23,600.264 1316.39,283.258 1315.55,309.238 1314.71,659.911 1313.86,1090.85 1313.02,1302.08 1312.18,1146.79 1311.34,733.12 \n",
" 1310.5,348.542 1309.66,259.969 1308.82,528.67 1307.98,968.205 1307.14,1274.04 1306.3,1234.57 1305.46,877.285 1304.62,449.315 1303.78,246.29 1302.93,408.208 \n",
" 1302.09,823.185 1301.25,1205.07 1300.41,1290.86 1299.57,1021.7 1298.73,582.889 1297.89,276.109 1297.05,311.961 1296.21,665.641 1295.36,1094.42 1294.52,1304.43 \n",
" 1293.68,1151.98 1292.84,741.615 1292,354.081 1291.16,254.163 1290.32,509.909 1289.47,946.614 1288.63,1266.43 1287.79,1251.54 1286.95,912.283 1286.11,479.651 \n",
" 1285.27,247.797 1284.43,374.091 1283.58,772.588 1282.74,1172.83 1281.9,1303.54 1281.06,1076.36 1280.22,645.263 1279.38,301.908 1278.53,278.261 1277.69,590.09 \n",
" 1276.85,1026.78 1276.01,1293.82 1275.17,1211.41 1274.32,835.249 1273.48,418.573 1272.64,241.494 1271.8,422.8 1270.96,840.571 1270.12,1214.47 1269.27,1293.96 \n",
" 1268.43,1026 1267.59,590.116 1266.75,277.857 1265.91,297.76 1265.06,636.32 1264.22,1067.48 1263.38,1303.79 1262.54,1187.99 1261.7,797.379 1260.85,391.899 \n",
" 1260.01,240.997 1259.17,444.696 1258.33,867.613 1257.49,1229.24 1256.64,1290.06 1255.8,1010.04 1254.96,574.593 1254.12,271.6 1253.28,301.041 1252.44,643.262 \n",
" 1251.59,1072.43 1250.75,1305.76 1249.91,1189.8 1249.07,801.074 1248.23,395.227 1247.38,238.779 1246.54,434.207 1245.7,853.137 1244.86,1220.98 1244.02,1296.99 \n",
" 1243.17,1031.67 1242.33,598.643 1241.49,280.791 1240.65,285.41 1239.81,609.258 1238.96,1041.15 1238.12,1299.92 1237.28,1217.39 1236.44,847.448 1235.6,430.491 \n",
" 1234.76,237.048 1233.91,392.344 1233.07,795.544 1232.23,1185.54 1231.39,1310.17 1230.55,1089.11 1229.71,665.316 1228.86,312.291 1228.02,257.492 1227.18,535.981 \n",
" 1226.34,968.331 1225.5,1276.49 1224.66,1262.61 1223.81,935.844 1222.97,505.969 1222.13,248.485 1221.29,328.093 1220.45,693.616 1219.61,1111.23 1218.77,1314.24 \n",
" 1217.92,1173.27 1217.08,778.464 1216.24,381.477 1215.4,234.947 1214.56,431.851 1213.72,846.826 1212.88,1216.29 1212.03,1305.97 1211.19,1059.24 1210.35,632.59 \n",
" 1209.51,296.052 1208.67,262.26 1207.83,552.312 1206.99,982.943 1206.15,1282.57 1205.31,1262.57 1204.47,935.791 1203.62,507.944 1202.78,247.893 1201.94,318.729 \n",
" 1201.1,675.762 1200.26,1095.05 1199.42,1314.06 1198.58,1195.96 1197.74,814.844 1196.9,408.877 1196.06,231.338 1195.22,392.695 1194.38,792.186 1193.54,1181.06 \n",
" 1192.7,1317.59 1191.86,1117.2 1191.02,704.501 1190.18,335.545 1189.34,238.827 1188.5,473.975 1187.66,895.315 1186.82,1242.4 1185.98,1301.08 1185.14,1035.41 \n",
" 1184.3,609.338 1183.46,285.272 1182.62,262.418 1181.78,554.559 1180.94,982.088 1180.1,1282.68 1179.26,1272.25 1178.42,957.459 1177.58,531.103 1176.74,253.824 \n",
" 1175.9,294.777 1175.06,628.706 1174.22,1051.85 1173.38,1306.51 1172.54,1237.91 1171.7,888.037 1170.87,469.568 1170.03,236.414 1169.19,329.715 1168.35,692.689 \n",
" 1167.51,1105.52 1166.67,1318.66 1165.83,1203.59 1164.99,830.023 1164.16,423.332 1163.32,228.426 1162.48,362.364 1161.64,744.351 1160.8,1144.8 1159.96,1323.48 \n",
" 1159.13,1173.51 1158.29,784.976 1157.45,390.515 1156.61,225.869 1155.77,389.129 1154.94,782.612 1154.1,1171.62 1153.26,1324.56 1152.42,1150.68 1151.59,753.608 \n",
" 1150.75,369.279 1149.91,225.623 1149.07,407.533 1148.24,807.024 1147.4,1187.63 1146.56,1324.51 1145.73,1137.08 1144.89,736.19 1144.05,358.204 1143.21,225.567 \n",
" 1142.38,416.036 1141.54,817.419 1140.71,1193.96 1139.87,1324.94 1139.03,1133.86 1138.2,732.846 1137.36,356.522 1136.52,224.633 1135.69,413.891 1134.85,813.676 \n",
" 1134.02,1190.99 1133.18,1326.37 1132.34,1141.38 1131.51,743.722 1130.67,364.254 1129.84,222.831 1129,401.083 1128.17,795.616 1127.33,1178.29 1126.5,1328.26 \n",
" 1125.66,1159.29 1124.83,769.019 1123.99,382.24 1123.16,221.287 1122.32,378.365 1121.49,763.041 1120.65,1154.62 1119.82,1328.89 1118.99,1186.4 1118.15,808.885 \n",
" 1117.32,412.075 1116.48,222.275 1115.65,347.4 1114.82,715.92 1113.98,1118.08 1113.15,1325.43 1112.32,1220.52 1111.48,863.15 1110.65,455.915 1109.82,229.222 \n",
" 1108.98,310.983 1108.15,654.726 1107.32,1066.34 1106.48,1313.87 1105.65,1258.19 1104.82,930.916 1103.99,516.132 1103.15,246.637 1102.32,273.317 1101.49,580.925 \n",
" 1100.66,997.121 1099.83,1289.25 1098.99,1294.4 1098.16,1010 1097.33,594.758 1096.5,279.863 1095.67,240.234 1094.84,497.581 1094.01,908.846 1093.18,1245.98 \n",
" 1092.34,1322.47 1091.51,1096.28 1090.68,692.689 1089.85,334.573 1089.02,219.274 1088.19,410.016 1087.36,801.552 1086.53,1178.53 1085.7,1334.06 1084.87,1183.1 \n",
" 1084.04,808.671 1083.21,415.877 1082.38,219.422 1081.55,326.35 1080.72,678.009 1079.9,1082.54 1079.07,1319.74 1078.24,1260.82 1077.41,938.123 1076.58,526.992 \n",
" 1075.75,250.32 1074.92,257.725 1074.1,544.887 1073.27,956.38 1072.44,1270.03 1071.61,1316.9 1070.78,1072.04 1069.96,667.487 1069.13,320.756 1068.3,217.864 \n",
" 1067.47,413.694 1066.65,803.04 1065.82,1177.41 1064.99,1336.79 1064.17,1196.35 1063.34,831.307 1062.51,436.343 1061.69,221.612 1060.86,301.021 1060.04,632.077 \n",
" 1059.21,1038.95 1058.38,1306.03 1057.56,1292.18 1056.73,1005.05 1055.91,596.524 1055.08,282.187 1054.26,227.507 1053.43,460.951 1052.61,859.487 1051.78,1213.69 \n",
" 1050.96,1337.85 1050.13,1167.21 1049.31,791.397 1048.49,407.083 1047.66,214.923 1046.84,314.904 1046.02,654.596 1045.19,1056.96 1044.37,1312.79 1043.55,1289.44 \n",
" 1042.72,999.359 1041.9,593.12 1041.08,281.062 1040.25,224.355 1039.43,451.966 1038.61,846.182 1037.79,1203.75 1036.97,1340.74 1036.14,1187.03 1035.32,821.861 \n",
" 1034.5,432.74 1033.68,218.985 1032.86,289.736 1032.04,608.491 1031.22,1012.25 1030.4,1295.07 1029.58,1313.06 1028.76,1057.41 1027.93,658.339 1027.11,318.548 \n",
" 1026.29,210.206 1025.48,387.883 1024.66,761.367 1023.84,1141.75 1023.02,1337.09 1022.2,1249.17 1021.38,922.609 1020.56,521.898 1019.74,248.381 1018.92,239.109 \n",
" 1018.1,498.395 1017.29,896.135 1016.47,1233.33 1015.65,1341.68 1014.83,1167.45 1014.02,797.781 1013.2,416.857 1012.38,214 1011.56,289.681 1010.75,606.021 \n",
" 1009.93,1006.05 1009.12,1291.8 1008.3,1322.23 1007.48,1082.66 1006.67,691.513 1005.85,341.549 1005.04,204.8 1004.22,348.154 1003.41,700.924 1002.59,1089.99 \n",
" 1001.78,1324.91 1000.96,1291.04 1000.15,1005.28 999.332,607.3 998.518,291.081 997.704,210.351 996.89,404.026 996.077,777.755 995.263,1150.15 994.45,1340.92 \n",
" 993.637,1258.05 992.824,941.942 992.012,545.377 991.199,259.533 990.387,221.819 989.575,450.03 988.763,834.259 987.951,1190.07 987.14,1347.1 986.328,1230.52 \n",
" 985.517,896.375 984.706,504.4 983.895,241.473 983.085,232.567 982.274,481.575 981.464,869.914 980.654,1213.23 979.844,1348.98 979.035,1213.21 978.225,870.457 \n",
" 977.416,482.739 976.607,232.862 975.798,238.266 974.989,496.062 974.181,884.848 973.373,1222.1 972.565,1349.92 971.757,1208.79 970.949,865.028 970.142,479.34 \n",
" 969.334,231.551 968.527,236.82 967.72,492.347 966.914,879.128 966.107,1217.6 965.301,1351.06 964.495,1218.04 963.689,880.382 962.883,494.184 962.078,237.503 \n",
" 961.273,228.305 960.468,470.472 959.663,852.462 958.858,1198.91 958.054,1351.21 957.25,1239.96 956.446,916.371 955.642,528.367 954.838,252.843 954.035,215.025 \n",
" 953.232,431.763 952.429,804.31 951.626,1163.55 950.824,1346.85 950.021,1271.57 949.219,972.086 948.417,583.795 947.616,281.734 946.814,201.672 946.013,379.259 \n",
" 945.212,734.435 944.411,1107.77 943.611,1332.14 942.81,1307.48 942.01,1045.14 941.21,662.452 940.411,329.976 939.611,195.492 938.812,318.436 938.013,643.9 \n",
" 937.214,1027.42 936.415,1299.24 935.617,1339.44 934.819,1130.62 934.021,765.163 933.223,404.139 932.426,206.213 931.629,258.021 930.832,536.481 930.035,919.349 \n",
" 929.238,1239.07 928.442,1356.15 927.646,1219.82 926.85,889.836 926.054,510.009 925.259,245.385 924.464,210.579 923.669,420.322 922.874,783.505 922.08,1142.98 \n",
" 921.285,1343.65 920.491,1299.29 919.698,1029.35 918.904,650.185 918.111,324.704 917.318,192.327 916.525,309.423 915.732,625.556 914.94,1005.46 914.148,1287.08 \n",
" 913.356,1350.65 912.564,1169.48 911.773,820.884 910.982,452.933 910.191,221.481 909.4,224.2 908.61,459.581 907.82,827.985 907.03,1174.03 906.24,1352.2 \n",
" 905.451,1287.9 904.661,1008.5 903.872,631.366 903.084,314.439 902.295,190.031 901.507,309.747 900.719,623.323 899.931,999.979 899.144,1283.11 898.357,1355.37 \n",
" 897.57,1187.16 896.783,848.456 895.996,479.508 895.21,232.619 894.424,209.345 893.639,418.908 892.853,774.81 892.068,1130.75 891.283,1340.83 890.498,1319.32 \n",
" 889.714,1075.37 888.93,708.905 888.146,369.499 887.362,195.257 886.579,256.737 885.795,528.603 885.013,900.317 884.23,1221.25 883.448,1361.77 882.665,1265.44 \n",
" 881.884,971.472 881.102,598.522 880.321,296.632 879.54,186.864 878.759,312.905 877.978,623.918 877.198,995.281 876.418,1278.66 875.638,1361.26 874.859,1210.51 \n",
" 874.08,886.732 873.301,518.716 872.522,252.403 871.744,193.028 870.965,363.718 870.188,696.722 869.41,1060.53 868.633,1311.91 867.856,1352.25 867.079,1166.09 \n",
" 866.302,826.736 865.526,467.259 864.75,228.16 863.974,202.537 863.199,400.016 862.424,743.383 861.649,1098.98 860.874,1328.82 860.1,1344.07 859.326,1139.19 \n",
" 858.552,793.706 857.779,441.125 857.006,217.279 856.233,208.052 855.46,416.622 854.688,762.624 853.916,1113.31 853.144,1334.55 852.372,1342.09 851.601,1133.41 \n",
" 850.83,788.344 850.059,438.331 849.289,216.267 848.519,206.12 847.749,411.366 846.98,754.047 846.21,1104.58 845.441,1330.82 844.673,1347.85 843.904,1149.62 \n",
" 843.136,810.9 842.369,458.853 841.601,225.224 840.834,197.094 840.067,384.597 839.3,717.509 838.534,1071.74 837.768,1315.7 837.002,1359.09 836.237,1186.2 \n",
" 835.471,861.34 834.707,504.729 833.942,247.894 833.178,185.207 832.414,339.368 831.65,653.408 830.887,1011.95 830.123,1283.71 829.361,1369.54 828.598,1238.45 \n",
" 827.836,938.596 827.074,579.348 826.312,291.302 825.551,178.798 824.79,282.264 824.029,563.898 823.269,921.697 822.509,1226.53 821.749,1368.78 820.99,1297.59 \n",
" 820.23,1038.91 819.472,685.77 818.713,364.695 817.955,190.332 817.197,224.611 816.439,454.945 815.682,799.06 814.925,1134.56 814.168,1342.5 813.412,1349.5 \n",
" 812.655,1153.46 811.9,823.937 811.144,477.304 810.389,235.558 809.634,183.436 808.88,338.874 808.125,647.046 807.371,999.896 806.618,1274.14 805.865,1374.3 \n",
" 805.112,1265.84 804.359,986.845 803.607,634.453 802.854,330.904 802.103,181.125 801.351,236.567 800.6,477.724 799.85,821.202 799.099,1148.71 798.349,1347.85 \n",
" 797.599,1350.6 796.85,1156.38 796.1,831.949 795.352,488.221 794.603,242.325 793.855,177.706 793.107,315.965 792.36,609.839 791.612,959.552 790.865,1246.8 \n",
" 790.119,1374.77 789.373,1300.64 788.627,1049.75 787.881,706.686 787.136,386.681 786.391,196.83 785.646,200.347 784.902,395.688 784.158,717.387 783.414,1058.17 \n",
" 782.671,1304.8 781.928,1375.66 781.186,1247.63 780.443,963.417 779.701,617.142 778.96,323.041 778.218,177.78 777.478,228.79 776.737,458.957 775.997,792.561 \n",
" 775.257,1120.33 774.517,1335.29 773.778,1367.6 773.039,1207.12 772.3,906.345 771.562,563.11 770.824,288.64 770.086,171.532 769.349,249.274 768.612,496.43 \n",
" 767.876,833.145 767.14,1151.09 766.404,1348.33 765.668,1361.96 764.933,1187.99 764.198,882.313 763.464,542.584 762.729,276.942 761.996,169.604 761.262,254.287 \n",
" 760.529,503.799 759.796,838.953 759.064,1153.83 758.332,1349.27 757.6,1364.07 756.869,1193.92 756.138,892.556 755.407,554.542 754.677,285.565 753.947,169.387 \n",
" 753.217,241.879 752.488,480.118 751.759,809.82 751.03,1128.63 750.302,1337.94 749.574,1373.32 748.847,1224.22 748.12,936.924 747.393,600.001 746.666,316.932 \n",
" 745.94,174.315 745.215,215.473 744.489,427.462 743.764,745.317 743.04,1072.1 742.315,1308.53 741.591,1383.08 740.868,1273.53 740.145,1013.36 739.422,681.399 \n",
" 738.699,377.797 737.977,193.801 737.255,184.403 736.534,352.057 735.813,646.224 735.092,978.763 734.372,1250.42 733.652,1380.45 732.932,1330.5 732.213,1115.77 \n",
" 731.494,800.208 730.776,477.355 730.058,242.532 729.34,164.761 728.623,266.589 727.906,517.727 727.189,844.22 726.473,1150.34 725.757,1346.67 725.041,1376.17 \n",
" 724.326,1230.61 723.612,952.678 722.897,623.326 722.183,338.086 721.469,179.367 720.756,192.727 720.043,373.944 719.331,670.511 718.619,996.988 717.907,1259.69 \n",
" 717.195,1383.54 716.484,1333.46 715.774,1124.11 715.063,815.446 714.354,495.448 713.644,254.957 712.935,161.934 712.226,242.368 711.518,473.17 710.81,788.961 \n",
" 710.102,1100.71 709.395,1320.85 708.688,1387.88 707.982,1283.36 707.276,1036.88 706.57,717.495 705.865,414.289 705.16,211.5 704.455,165.172 703.751,287.789 \n",
" 703.047,545.076 702.344,865.725 701.641,1161.25 700.938,1350.44 700.236,1381.58 699.534,1246.49 698.833,982.506 698.132,662 697.431,372.473 696.731,192.643 \n",
" 696.031,171.113 695.332,313.366 694.633,580.528 693.934,900.132 693.236,1185.86 692.538,1360.85 691.84,1378.33 691.143,1233.96 690.446,966.78 689.75,648.482 \n",
" 689.054,364.121 688.359,189.366 687.663,170.432 686.969,311.989 686.274,576.168 685.58,892.823 684.887,1178.24 684.194,1357.28 683.501,1383.1 682.809,1249.26 \n",
" 682.117,991.216 681.425,676.755 680.734,388.129 680.043,200.513 679.353,162.492 678.663,283.595 677.974,532.056 677.285,843.284 676.596,1136.74 675.908,1336.79 \n",
" 675.22,1392.18 674.532,1289.01 673.845,1054.11 673.159,747.9 672.472,448.797 671.786,233.06 671.101,155.411 670.416,235.257 669.731,451.977 669.047,750.382 \n",
" 668.363,1054.88 667.68,1288.64 666.997,1392.99 666.314,1342 665.632,1148.84 664.951,862.202 664.269,553.943 663.588,301.01 662.908,166.247 662.228,182.865 \n",
" 661.548,346.381 660.869,615.956 660.19,924.715 659.512,1196.4 658.834,1364.2 658.156,1387.14 657.479,1259.92 656.802,1014.07 656.126,709.941 655.45,421.872 \n",
" 654.774,219.954 654.099,153.049 653.425,237.058 652.751,451.277 652.077,743.624 651.403,1043.4 650.73,1278.42 650.058,1392.37 649.386,1358.22 648.714,1184.5 \n",
" 648.043,913.06 647.372,608.871 646.702,344.411 646.032,182.41 645.362,161.028 644.693,284.98 644.024,524.629 643.356,823.19 642.688,1110.26 642.021,1318.46 \n",
" 641.354,1399.16 640.687,1333.81 640.021,1138.03 639.356,857.765 638.69,558.432 638.026,309.57 637.361,168.722 636.697,168.195 636.034,307.758 635.371,554.908 \n",
" 634.708,852.544 634.046,1132.23 633.384,1329.95 632.723,1400.7 632.062,1328.67 631.401,1130.61 630.741,851.864 630.082,555.863 629.423,309.668 628.764,168.784 \n",
" 628.106,164.717 627.448,298.035 626.79,538.44 626.134,831.759 625.477,1112.22 624.821,1317.23 624.165,1401.27 623.51,1345.97 622.855,1163.96 622.201,895.89 \n",
" 621.547,601.268 620.894,345.186 620.241,183.954 619.588,152.771 618.936,258.144 618.285,476.648 617.634,760.236 616.983,1046.89 616.333,1274.21 615.683,1392.95 \n",
" 615.033,1377.67 614.384,1232.01 613.736,987.777 613.088,697.852 612.44,424.718 611.793,226.961 611.146,146.75 610.5,200.924 609.854,377.572 609.209,638.756 \n",
" 608.564,928.748 607.92,1185.97 607.276,1356.06 606.632,1403.32 605.989,1318.12 605.347,1118.72 604.705,847.303 604.063,560.926 603.422,319.514 602.781,173.328 \n",
" 602.141,152.56 601.501,261.193 600.861,476.347 600.222,753.2 599.584,1034.4 598.946,1261.95 598.308,1389.19 597.671,1390.27 597.035,1265.29 596.398,1040.16 \n",
" 595.763,761.085 595.127,485.044 594.493,268.127 593.858,154.167 593.224,165.949 592.591,300.753 591.958,531.073 591.326,810.321 590.694,1082.31 590.062,1292.58 \n",
" 589.431,1399.26 588.8,1381.36 588.17,1242.78 587.541,1011.38 586.911,733.286 586.283,463.654 585.654,255.687 585.027,150.188 584.399,167.617 583.772,304.214 \n",
" 583.146,532.906 582.52,808.758 581.895,1077.86 581.27,1287.87 580.645,1398.18 580.021,1387.69 579.398,1258.78 578.774,1036.64 578.152,764.295 577.53,494.214 \n",
" 576.908,278.166 576.287,157.331 575.666,154.51 575.046,269.909 574.426,481.258 573.807,748.194 573.188,1020.03 572.57,1245.42 571.952,1381.99 571.335,1404.32 \n",
" 570.718,1308.53 570.101,1112.91 569.485,854.299 568.87,581.095 568.255,344.185 567.641,187.456 567.027,139.718 566.413,209.488 565.8,383.556 565.188,629.577 \n",
" 564.575,902.151 563.964,1151.24 563.353,1331.34 562.742,1409.79 562.132,1372.59 561.522,1226.84 560.913,999.275 560.305,731.314 559.697,471.452 559.089,266.48 \n",
" 558.482,153.087 557.875,151.345 557.269,261.241 556.663,462.821 556.058,719.898 555.453,986.596 554.849,1215.57 554.245,1366.39 553.642,1412.65 553.039,1346.51 \n",
" 552.437,1179.91 551.835,942.406 551.234,675.781 550.633,426.706 550.032,238.549 549.433,143.859 548.833,158.792 548.234,280.437 547.636,487.456 547.038,743.906 \n",
" 546.441,1005.52 545.844,1227.39 545.248,1371.62 544.652,1413.82 544.057,1347.08 543.462,1183.09 542.867,949.974 542.273,687.423 541.68,439.888 541.087,249.058 \n",
" 540.495,146.863 539.903,150.187 539.312,258.154 538.721,452.407 538.13,700.3 537.541,960.437 536.951,1189.61 536.362,1349.96 535.774,1415.22 535.186,1374.91 \n",
" 534.599,1235.98 534.012,1021.54 533.426,766.928 532.84,513.847 532.255,303.542 531.67,170.076 531.086,134.865 530.502,203.303 529.919,364.019 529.336,590.833 \n",
" 528.754,847.082 528.172,1091.58 527.591,1285.23 527.01,1397.29 526.43,1410.16 525.85,1322.1 525.271,1147.39 524.692,913.938 524.114,658.765 523.536,422.117 \n",
" 522.959,241.106 522.383,143.926 521.806,145.506 521.231,245.289 520.656,427.436 520.081,663.408 519.507,916.48 518.934,1147.49 518.361,1320.89 517.788,1410.2 \n",
" 517.216,1401.97 516.645,1297.77 516.074,1113.83 515.503,878.435 514.933,627.525 514.364,399.206 513.795,227.952 513.227,139.43 512.659,146.705 512.092,248.375 \n",
" 511.525,428.899 510.959,661.029 510.393,909.972 509.828,1138.63 509.263,1313.12 508.699,1407.77 508.135,1408.86 507.572,1316.54 507.009,1144.68 506.447,918.747 \n",
" 505.886,671.992 505.325,440.503 504.764,257.946 504.204,150.685 503.645,134.02 503.086,210.062 502.527,367.538 501.969,583.519 501.412,826.808 500.855,1062.47 \n",
" 500.299,1256.87 499.743,1382.44 499.188,1421.56 498.633,1368.96 498.079,1232.39 497.526,1031.4 496.972,794.512 496.42,555.108 495.868,346.74 495.316,198.422 \n",
" 494.765,130.633 494.215,152.529 493.665,260.772 493.116,440.09 492.567,665.489 492.018,905.775 491.471,1127.89 490.923,1301.45 490.377,1402.91 489.831,1418.66 \n",
" 489.285,1346.87 488.74,1197.57 488.195,991.222 487.651,755.89 487.108,523.363 486.565,324.867 486.023,186.885 485.481,127.652 484.939,154.796 484.399,264.41 \n",
" 483.858,441.668 483.319,662.879 482.779,898.692 482.241,1118.01 481.703,1292.08 481.165,1398.25 480.628,1422.9 480.092,1363.08 479.556,1226.88 479.021,1032.21 \n",
" 478.486,804.476 477.951,573.165 477.418,368.064 476.885,215.411 476.352,134.568 475.82,135.609 475.288,218.112 474.757,371.308 474.227,575.529 473.697,804.778 \n",
" 473.168,1030.06 472.639,1223.05 472.111,1359.66 471.583,1422.99 471.056,1405.4 470.529,1309.37 470.003,1147.09 469.478,938.916 468.953,710.709 468.428,490.66 \n",
" 467.904,305.779 467.381,178.599 466.858,124.458 466.336,149.694 465.815,250.953 465.293,415.678 464.773,623.712 464.253,849.799 463.734,1066.67 463.215,1248.32 \n",
" 462.696,1373.14 462.179,1426.44 461.661,1402.12 461.145,1303.36 460.629,1142.13 460.113,937.706 459.598,714.333 459.084,498.329 458.57,314.982 458.057,185.604 \n",
" 457.544,125.078 457.032,140.187 456.52,228.899 456.009,380.682 455.499,577.795 454.989,797.395 454.479,1014.21 453.97,1203.43 453.462,1343.58 452.954,1418.89 \n",
" 452.447,1421.06 451.941,1350.1 451.435,1214.32 450.929,1029.23 450.424,815.808 449.92,598.053 449.416,400.298 448.913,244.491 448.411,147.776 447.909,120.643 \n",
" 447.407,165.823 446.906,278.065 446.406,444.783 445.906,647.505 445.407,863.946 444.908,1070.47 444.41,1244.68 443.913,1367.81 443.416,1426.76 442.919,1415.43 \n",
" 442.424,1335.3 441.928,1195.23 441.434,1010.41 440.94,800.713 440.446,588.528 439.953,396.348 439.461,244.408 438.969,148.564 438.478,118.671 437.987,157.604 \n",
" 437.497,261.026 437.007,417.905 436.518,611.728 436.03,822.262 435.542,1027.68 435.055,1206.84 434.568,1341.43 434.082,1417.83 433.597,1428.47 433.112,1372.53 \n",
" 432.628,1255.94 432.144,1090.77 431.661,893.897 431.178,685.28 430.696,485.923 430.215,315.76 429.734,191.671 429.253,125.831 428.774,124.546 428.295,187.682 \n",
" 427.816,308.746 427.338,475.585 426.861,671.632 426.384,877.578 425.908,1073.28 425.432,1239.73 424.957,1360.89 424.482,1425.22 424.009,1426.75 423.535,1365.58 \n",
" 423.063,1247.84 422.59,1085.03 422.119,892.849 421.648,689.7 421.177,494.889 420.708,326.799 420.238,201.164 419.77,129.617 419.302,118.638 418.834,169.003 \n",
" 418.367,275.765 417.901,428.773 417.435,613.665 416.97,813.237 416.506,1009.05 416.042,1183.16 415.578,1319.7 415.116,1406.4 414.653,1435.55 414.192,1404.78 \n",
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]
},
"execution_count": 42,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ε = 1E-6\n",
"L = [B₋₁; \n",
" B₁ ; \n",
" ε*D₁*D₀ + S₁*S₀*M]\n",
"\n",
"@time u = L \\ [1.0;0.0;0.0]\n",
"\n",
"@show ε*u''(0.1) + exp(0.1)*u(0.1)\n",
"plot(u)"
]
}
],
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