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"ellipse_rule (generic function with 1 method)"
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"using Plots, ComplexPhasePortrait, Interact, ApproxFun, Statistics, SpecialFunctions, LinearAlgebra\n",
"gr();\n",
"\n",
"periodic_rule(n) = 2π/n*(0:(n-1)), 2π/n*ones(n)\n",
"\n",
"function circle_rule(n, r) \n",
" θ = periodic_rule(n)[1]\n",
" r*exp.(im*θ), 2π*im*r/n*exp.(im*θ)\n",
"end\n",
"\n",
"function ellipse_rule(n, a, b) \n",
" θ = periodic_rule(n)[1]\n",
" a*cos.(θ) + b*im*sin.(θ), 2π/n*(-a*sin.(θ) + im*b*cos.(θ))\n",
"end"
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"$$\n",
"\\def\\I{{\\rm i}}\n",
"\\def\\E{{\\rm e}}\n",
"\\def\\D{{\\rm d}}\n",
"$$\n",
"\n",
"# M3M6: Methods of Mathematical Physics\n",
"\n",
"Dr. Sheehan Olver\n",
"<br>\n",
"s.olver@imperial.ac.uk\n",
"\n",
"\n",
"# Lecture 4: Trapezium rule, Fourier series and Laurent series\n",
"\n",
"\n",
"This lecture we cover\n",
"\n",
"1. Periodic and complex Trapezium rule \n",
" - Application: Calculating matrix exponentials\n",
" - Gershorwin's theorem\n",
"1. Laurent series and Fourier series\n",
" - Application: Decay of Fourier coefficients"
]
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"### Periodic and complex Trapezium rule\n",
"\n",
"Quadrature rules are pairs of _nodes_ $x_0,\\ldots,x_{n-1}$ and weights $w_0,\\ldots,w_{n-1}$ to approximate integrals\n",
"$$\n",
"\\int_a^b f(x) dx \\approx \\sum_{j=0}^{n-1} w_j f(x_j)\n",
"$$\n",
"In this lecture we construct quadrature rules on complex contours $\\gamma$ to approximate contour integrals.\n",
"\n",
"\n",
"The trapezium rule gives an easy approximation to integrals. On $[0,2\\pi)$ for periodic $f(\\theta)$, we have a simplified form:\n",
"\n",
"**Definition (Periodic trapezium rule)** The _periodic trapezium rule_ is the approximation\n",
"\n",
"$$\\int_0^{2 \\pi} f(\\theta) d \\theta \\approx {2\\pi \\over n} \\sum_{j=0}^{n-1} f(\\theta_k)$$\n",
"\n",
"for $\\theta_j = {2 \\pi j \\over n}$.\n",
"\n",
"The periodic trapezium rule is amazingly accurate for smooth, periodic functions:"
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"f = θ -> 1/(2 + cos(θ))\n",
"\n",
"errs = [((x, w) = periodic_rule(n); abs(sum(w.*f.(x)) - sum(Fun(f, 0 .. 2π)))) for n = 1:45];\n",
"plot(errs.+eps(); yscale=:log10, title=\"exponential convergence of n-point trapezium rule\", legend=false, xlabel=\"n\")"
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"The accuracy in integration is remarkable as the trapezoidal interpolant does not accurately approximate $f$: we can see clear differences between the functions here:"
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"source": [
"n=20\n",
"(x, w) = periodic_rule(n)\n",
"plot(Fun(f, 0 .. 2π); label = \"integrand\")\n",
"plot!(x, f.(x); label = \"trapezium approximation\")"
]
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"### Complex Trapezium rule\n",
"\n",
"We can use the map that defines a contour to construct an approximation to integrals over closed contour $\\gamma$:\n",
"\n",
"**Definition (Complex trapezium rule)** The _complex trapezium rule_ on a contour $\\gamma$ (mapped from $[0,2\\pi)$) is the approximation\n",
"\n",
"$$\\oint_\\gamma f(z) dz \\approx \\sum_{j=0}^{n-1} w_j f(z_j)$$\n",
"\n",
"for\n",
"\n",
"$$z_j = \\gamma(\\theta_j) \\qquad\\hbox{and}\\qquad w_j = {2\\pi \\over n} \\gamma'(\\theta_j)$$\n",
"\n",
"\n",
"*Example (Circle trapezium rule)* On a circle $\\{r e^{i \\theta} : 0 \\leq \\theta < 2 \\pi\\}$, we have \n",
"\n",
"$$\\oint_\\gamma f(z) dz \\approx \\sum_{j=0}^{n-1} w_j f(z_j)$$\n",
"\n",
"for $z_j = r e^{i \\theta_j}$ and $w_j = {2 \\pi i r \\over n} e^{i \\theta_j}$.\n",
"\n",
"Here we plot the quadrature points:"
]
},
{
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"source": [
"ζ, w = circle_rule(20, 1.0)\n",
"scatter(ζ; title=\"quadrature points\", legend=false, ratio=1.0)"
]
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{
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"metadata": {},
"source": [
"The Circle trapezium rule is surprisingly accurate for analytic functions:"
]
},
{
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"data": {
"text/plain": [
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"source": [
"ζ, w = circle_rule(20, 1.0)\n",
"\n",
"f = z -> cos(z)\n",
"\n",
"z = 0.1+0.2im\n",
"\n",
"sum(f.(ζ)./(ζ .- z).*w)/(2π*im) - f(z)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Application: Numerical differentiation\n",
"\n",
"Calculating high-order derivatives using limits is numerically unstable. Here is a demonstration using finite-difference: making $h$ small does not increase the accuracy after a certain point:"
]
},
{
"cell_type": "code",
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"f = z -> gamma(z)\n",
"fp = z -> gamma(z)polygamma(0,z) # exact derivative \n",
"x = 1.2\n",
"fp_fd = [(h=2.0^(-n); (f(x+h)-f(x))/h) for n = 1:50]\n",
"plot(abs.(fp_fd .- fp(x)); yscale=:log10, legend=false, title = \"error of finite-difference with h=2^(-n)\", xlabel=\"n\")"
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"But the previous formula tells us that we can reduce a derivative to a contour integral. The example above shows that it's still numerically unstable, but we can deform the integration contour, to make it stable! "
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"trap_fp = [((ζ, w) = circle_rule(n, 0.5); \n",
" ζ .+= x; # circle around x\n",
" sum(f.(ζ)./(ζ .- x).^2 .*w)/(2π*im)) for n=1:50]\n",
"\n",
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" title=\"Error of trapezium rule applied to Cauchy integral formula\", xlabel=\"n\", legend=false)"
]
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"We can take things further and use this to calculate the higher order derivatives, with some care taken for choosing the radius:"
]
},
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"source": [
"k=100\n",
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]
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{
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"metadata": {},
"source": [
"[Bournemann 2011](https://www-m3.ma.tum.de/foswiki/pub/M3/Allgemeines/FolkmarBornemannPublications/FoCM_Stability_Cauchy_Integrals.pdf) investigates this further and optimizes the radius."
]
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"source": [
"The exponential convergence of the complex trapezium rule is a consequence of $f(\\gamma(t))$ being 2π-periodic, as we will see later in a few lectures:"
]
},
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"source": [
"θ = periodic_rule(100)[1]\n",
"plot(θ, real(f.(0.6exp.(im*θ) .+ x)./(0.5exp.(im*θ))))"
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"*Example (Ellipse trapezium rule)* On an ellipse $\\{a \\cos \\theta + b i \\sin \\theta : 0 \\leq \\theta < 2 \\pi\\}$ we have \n",
"$$\\oint_\\gamma f(z) dz \\approx \\sum_{j=0}^{n-1} w_j f(z_j)$$\n",
"for $z_j = a \\cos \\theta_j + b i \\sin \\theta_j$ and $w_j = {2 \\pi \\over n} (-a \\sin \\theta_j + i b \\cos \\theta_j)$.\n",
"\n",
"We can use the ellipse trapezium rule in place of the circle trapzium rule and still achieve accurate results. This gives us flexibility in avoiding singularities. Consider\n",
"$$\n",
"f(z) = 1/(25z^2 + 1)\n",
"$$\n",
"which has poles at $\\pm \\I/5$. Using an ellipse allows us to design a contour that avoids these singularities:"
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" 496.692,551.055 494.896,551.892 493.108,552.731 491.328,553.573 489.556,554.416 487.791,555.26 486.035,556.107 484.286,556.956 482.545,557.806 480.813,558.658 \n",
" 479.088,559.512 477.371,560.368 475.662,561.226 473.961,562.085 472.269,562.947 470.584,563.81 468.907,564.675 467.239,565.541 465.578,566.41 463.926,567.28 \n",
" 462.282,568.152 460.646,569.025 459.018,569.901 457.398,570.778 455.786,571.656 454.183,572.537 452.588,573.419 451.001,574.303 449.423,575.188 447.852,576.076 \n",
" 446.29,576.964 444.736,577.855 443.191,578.747 441.654,579.641 440.125,580.536 438.605,581.433 437.093,582.332 435.589,583.232 434.094,584.134 432.607,585.037 \n",
" 431.129,585.942 429.659,586.849 428.198,587.757 426.745,588.666 425.3,589.578 423.865,590.49 422.437,591.404 421.018,592.32 419.608,593.237 418.206,594.156 \n",
" 416.813,595.076 415.429,595.998 414.053,596.921 412.686,597.845 411.327,598.771 409.977,599.699 408.636,600.627 407.303,601.558 405.979,602.489 404.664,603.422 \n",
" 403.357,604.357 402.059,605.293 400.77,606.23 399.49,607.168 398.218,608.108 396.955,609.049 395.701,609.992 394.456,610.936 393.22,611.881 391.992,612.828 \n",
" 390.774,613.775 389.564,614.724 388.363,615.675 387.171,616.626 385.987,617.579 384.813,618.533 383.648,619.489 382.491,620.445 381.344,621.403 380.205,622.362 \n",
" 379.075,623.322 377.955,624.284 376.843,625.246 375.74,626.21 374.647,627.175 373.562,628.141 372.486,629.108 371.42,630.077 370.362,631.046 369.313,632.017 \n",
" 368.274,632.989 367.244,633.961 366.222,634.935 365.21,635.91 364.207,636.886 363.213,637.863 362.228,638.842 361.252,639.821 360.286,640.801 359.328,641.782 \n",
" 358.38,642.764 357.441,643.748 356.511,644.732 355.59,645.717 354.679,646.703 353.776,647.69 352.883,648.678 351.999,649.667 351.125,650.657 350.259,651.648 \n",
" 349.403,652.64 348.556,653.633 347.719,654.626 346.89,655.621 346.071,656.616 345.262,657.612 344.461,658.609 343.67,659.607 342.888,660.606 342.116,661.605 \n",
" 341.353,662.606 340.599,663.607 339.854,664.609 339.119,665.611 338.394,666.615 337.677,667.619 336.97,668.624 336.273,669.63 335.584,670.636 334.905,671.644 \n",
" 334.236,672.652 333.576,673.66 332.925,674.67 332.284,675.68 331.653,676.69 331.03,677.702 330.417,678.714 329.814,679.726 329.22,680.739 328.636,681.753 \n",
" 328.061,682.768 327.495,683.783 326.939,684.799 326.392,685.815 325.855,686.832 325.328,687.849 324.81,688.867 324.301,689.886 323.802,690.905 323.312,691.924 \n",
" 322.832,692.945 322.362,693.965 321.901,694.986 321.449,696.008 321.007,697.03 320.575,698.052 320.152,699.075 319.739,700.099 319.335,701.123 318.941,702.147 \n",
" 318.556,703.171 318.181,704.197 317.816,705.222 317.46,706.248 317.114,707.274 316.777,708.301 316.45,709.328 316.132,710.355 315.824,711.382 315.526,712.41 \n",
" 315.237,713.439 314.958,714.467 314.688,715.496 314.428,716.525 314.178,717.555 313.937,718.584 313.706,719.614 313.484,720.644 313.272,721.675 313.07,722.705 \n",
" 312.877,723.736 312.694,724.767 312.521,725.798 312.357,726.829 312.203,727.861 312.058,728.893 311.923,729.924 311.798,730.956 311.682,731.988 311.576,733.021 \n",
" 311.48,734.053 311.393,735.085 311.316,736.118 311.249,737.15 311.191,738.183 311.143,739.216 311.104,740.249 311.075,741.281 311.056,742.314 311.046,743.347 \n",
" 311.046,744.38 311.056,745.413 311.075,746.445 311.104,747.478 311.143,748.511 311.191,749.544 311.249,750.576 311.316,751.609 311.393,752.641 311.48,753.674 \n",
" 311.576,754.706 311.682,755.738 311.798,756.77 311.923,757.802 312.058,758.834 312.203,759.866 312.357,760.897 312.521,761.929 312.694,762.96 312.877,763.991 \n",
" 313.07,765.022 313.272,766.052 313.484,767.083 313.706,768.113 313.937,769.143 314.178,770.172 314.428,771.202 314.688,772.231 314.958,773.26 315.237,774.288 \n",
" 315.526,775.316 315.824,776.344 316.132,777.372 316.45,778.399 316.777,779.426 317.114,780.453 317.46,781.479 317.816,782.505 318.181,783.53 318.556,784.555 \n",
" 318.941,785.58 319.335,786.604 319.739,787.628 320.152,788.651 320.575,789.674 321.007,790.697 321.449,791.719 321.901,792.74 322.362,793.762 322.832,794.782 \n",
" 323.312,795.802 323.802,796.822 324.301,797.841 324.81,798.859 325.328,799.877 325.855,800.895 326.392,801.912 326.939,802.928 327.495,803.944 328.061,804.959 \n",
" 328.636,805.973 329.22,806.987 329.814,808.001 330.417,809.013 331.03,810.025 331.653,811.036 332.284,812.047 332.925,813.057 333.576,814.067 334.236,815.075 \n",
" 334.905,816.083 335.584,817.09 336.273,818.097 336.97,819.103 337.677,820.108 338.394,821.112 339.119,822.115 339.854,823.118 340.599,824.12 341.353,825.121 \n",
" 342.116,826.121 342.888,827.121 343.67,828.12 344.461,829.118 345.262,830.115 346.071,831.111 346.89,832.106 347.719,833.101 348.556,834.094 349.403,835.087 \n",
" 350.259,836.079 351.125,837.069 351.999,838.059 352.883,839.048 353.776,840.036 354.679,841.024 355.59,842.01 356.511,842.995 357.441,843.979 358.38,844.962 \n",
" 359.328,845.945 360.286,846.926 361.252,847.906 362.228,848.885 363.213,849.863 364.207,850.84 365.21,851.816 366.222,852.791 367.244,853.765 368.274,854.738 \n",
" 369.313,855.71 370.362,856.68 371.42,857.65 372.486,858.618 373.562,859.585 374.647,860.552 375.74,861.516 376.843,862.48 377.955,863.443 379.075,864.404 \n",
" 380.205,865.364 381.344,866.323 382.491,867.281 383.648,868.238 384.813,869.193 385.987,870.147 387.171,871.1 388.363,872.052 389.564,873.002 390.774,873.951 \n",
" 391.992,874.899 393.22,875.846 394.456,876.791 395.701,877.735 396.955,878.677 398.218,879.618 399.49,880.558 400.77,881.497 402.059,882.434 403.357,883.37 \n",
" 404.664,884.304 405.979,885.237 407.303,886.169 408.636,887.099 409.977,888.028 411.327,888.955 412.686,889.881 414.053,890.806 415.429,891.729 416.813,892.651 \n",
" 418.206,893.571 419.608,894.489 421.018,895.407 422.437,896.322 423.865,897.236 425.3,898.149 426.745,899.06 428.198,899.97 429.659,900.878 431.129,901.784 \n",
" 432.607,902.689 434.094,903.593 435.589,904.495 437.093,905.395 438.605,906.293 440.125,907.19 441.654,908.086 443.191,908.98 444.736,909.872 446.29,910.762 \n",
" 447.852,911.651 449.423,912.538 451.001,913.424 452.588,914.308 454.183,915.19 455.786,916.07 457.398,916.949 459.018,917.826 460.646,918.702 462.282,919.575 \n",
" 463.926,920.447 465.578,921.317 467.239,922.186 468.907,923.052 470.584,923.917 472.269,924.78 473.961,925.641 475.662,926.501 477.371,927.359 479.088,928.214 \n",
" 480.813,929.069 482.545,929.921 484.286,930.771 486.035,931.62 487.791,932.466 489.556,933.311 491.328,934.154 493.108,934.995 494.896,935.834 496.692,936.672 \n",
" 498.495,937.507 500.307,938.341 502.126,939.172 503.953,940.002 505.788,940.829 507.63,941.655 509.48,942.479 511.338,943.301 513.203,944.121 515.077,944.938 \n",
" 516.957,945.754 518.846,946.568 520.741,947.38 522.645,948.19 524.556,948.998 526.475,949.804 528.401,950.607 530.334,951.409 532.275,952.209 534.224,953.007 \n",
" 536.18,953.802 538.143,954.596 540.114,955.387 542.092,956.176 544.077,956.964 546.07,957.749 548.07,958.532 550.078,959.313 552.092,960.091 554.114,960.868 \n",
" 556.144,961.643 558.18,962.415 560.224,963.185 562.274,963.953 564.332,964.719 566.397,965.483 568.47,966.244 570.549,967.003 572.635,967.76 574.729,968.515 \n",
" 576.829,969.268 578.937,970.018 581.051,970.766 583.173,971.512 585.301,972.256 587.437,972.997 589.579,973.737 591.728,974.473 593.884,975.208 596.047,975.94 \n",
" 598.217,976.67 600.393,977.398 602.576,978.124 604.766,978.847 606.963,979.567 609.167,980.286 611.377,981.002 613.594,981.716 615.817,982.427 618.047,983.136 \n",
" 620.284,983.843 622.527,984.547 624.777,985.249 627.034,985.949 629.297,986.646 631.566,987.341 633.842,988.033 636.124,988.723 638.413,989.411 640.708,990.096 \n",
" 643.01,990.779 645.318,991.459 647.632,992.137 649.953,992.812 652.28,993.485 654.613,994.155 656.952,994.823 659.298,995.489 661.65,996.152 664.008,996.812 \n",
" 666.372,997.47 668.742,998.126 671.119,998.779 673.501,999.429 675.89,1000.08 678.285,1000.72 680.685,1001.37 683.092,1002.01 685.504,1002.64 687.923,1003.28 \n",
" 690.347,1003.91 692.778,1004.54 695.214,1005.17 697.656,1005.79 700.104,1006.42 702.557,1007.04 705.017,1007.65 707.482,1008.27 709.953,1008.88 712.429,1009.49 \n",
" 714.912,1010.1 717.4,1010.7 719.893,1011.3 722.392,1011.9 724.897,1012.5 727.407,1013.09 729.923,1013.68 732.444,1014.27 734.971,1014.85 737.503,1015.44 \n",
" 740.041,1016.02 742.584,1016.59 745.133,1017.17 747.686,1017.74 750.245,1018.31 752.81,1018.88 755.38,1019.44 757.954,1020 760.535,1020.56 763.12,1021.12 \n",
" 765.711,1021.67 768.306,1022.22 770.907,1022.77 773.513,1023.31 776.124,1023.85 778.74,1024.39 781.361,1024.93 783.987,1025.46 786.618,1025.99 789.254,1026.52 \n",
" 791.895,1027.05 794.541,1027.57 797.191,1028.09 799.847,1028.61 802.507,1029.12 805.172,1029.63 807.842,1030.14 810.517,1030.65 813.196,1031.15 815.88,1031.65 \n",
" 818.568,1032.15 821.262,1032.64 823.959,1033.13 826.662,1033.62 829.369,1034.1 832.08,1034.59 834.796,1035.07 837.517,1035.54 840.242,1036.02 842.971,1036.49 \n",
" 845.704,1036.96 848.442,1037.42 851.185,1037.89 853.931,1038.35 856.682,1038.8 859.438,1039.26 862.197,1039.71 864.961,1040.16 867.728,1040.6 870.5,1041.04 \n",
" 873.276,1041.48 876.056,1041.92 878.841,1042.35 881.629,1042.78 884.421,1043.21 887.217,1043.63 890.017,1044.06 892.821,1044.47 895.629,1044.89 898.441,1045.3 \n",
" 901.257,1045.71 904.076,1046.12 906.899,1046.52 909.726,1046.92 912.557,1047.32 915.392,1047.72 918.23,1048.11 921.071,1048.5 923.917,1048.88 926.766,1049.26 \n",
" 929.618,1049.64 932.474,1050.02 935.334,1050.39 938.197,1050.76 941.063,1051.13 943.933,1051.5 946.806,1051.86 949.683,1052.22 952.563,1052.57 955.446,1052.92 \n",
" 958.332,1053.27 961.222,1053.62 964.115,1053.96 967.011,1054.3 969.911,1054.64 972.813,1054.97 975.719,1055.3 978.627,1055.63 981.539,1055.96 984.454,1056.28 \n",
" 987.371,1056.6 990.292,1056.91 993.216,1057.23 996.142,1057.53 999.071,1057.84 1002,1058.14 1004.94,1058.44 1007.88,1058.74 1010.82,1059.03 1013.76,1059.32 \n",
" 1016.71,1059.61 1019.65,1059.9 1022.61,1060.18 1025.56,1060.46 1028.52,1060.73 1031.48,1061 1034.44,1061.27 1037.4,1061.54 1040.37,1061.8 1043.34,1062.06 \n",
" 1046.31,1062.32 1049.28,1062.57 1052.26,1062.82 1055.24,1063.07 1058.22,1063.31 1061.2,1063.55 1064.18,1063.79 1067.17,1064.02 1070.16,1064.25 1073.15,1064.48 \n",
" 1076.15,1064.7 1079.14,1064.93 1082.14,1065.14 1085.14,1065.36 1088.14,1065.57 1091.15,1065.78 1094.15,1065.99 1097.16,1066.19 1100.17,1066.39 1103.18,1066.58 \n",
" 1106.19,1066.78 1109.21,1066.97 1112.23,1067.15 1115.24,1067.33 1118.27,1067.51 1121.29,1067.69 1124.31,1067.87 1127.34,1068.04 1130.36,1068.2 1133.39,1068.37 \n",
" 1136.42,1068.53 1139.45,1068.69 1142.48,1068.84 1145.52,1068.99 1148.55,1069.14 1151.59,1069.28 1154.63,1069.43 1157.67,1069.56 1160.71,1069.7 1163.75,1069.83 \n",
" 1166.79,1069.96 1169.84,1070.09 1172.89,1070.21 1175.93,1070.33 1178.98,1070.44 1182.03,1070.56 1185.08,1070.66 1188.13,1070.77 1191.18,1070.87 1194.23,1070.97 \n",
" 1197.29,1071.07 1200.34,1071.16 1203.4,1071.25 1206.45,1071.34 1209.51,1071.42 1212.57,1071.5 1215.63,1071.58 1218.69,1071.65 1221.75,1071.72 1224.81,1071.79 \n",
" 1227.87,1071.86 1230.93,1071.92 1233.99,1071.97 1237.06,1072.03 1240.12,1072.08 1243.18,1072.13 1246.25,1072.17 1249.31,1072.21 1252.38,1072.25 1255.44,1072.29 \n",
" 1258.51,1072.32 1261.58,1072.35 1264.64,1072.37 1267.71,1072.39 1270.78,1072.41 1273.84,1072.43 1276.91,1072.44 1279.98,1072.45 1283.04,1072.45 1286.11,1072.46 \n",
" 1289.18,1072.46 1292.25,1072.45 1295.31,1072.45 1298.38,1072.43 1301.45,1072.42 1304.51,1072.4 1307.58,1072.38 1310.65,1072.36 1313.71,1072.33 1316.78,1072.3 \n",
" 1319.84,1072.27 1322.91,1072.23 1325.97,1072.19 1329.04,1072.15 1332.1,1072.1 1335.17,1072.05 1338.23,1072 1341.29,1071.95 1344.35,1071.89 1347.42,1071.82 \n",
" 1350.48,1071.76 1353.54,1071.69 1356.6,1071.62 1359.66,1071.54 1362.71,1071.46 1365.77,1071.38 1368.83,1071.3 1371.88,1071.21 1374.94,1071.12 1377.99,1071.02 \n",
" 1381.05,1070.92 1384.1,1070.82 1387.15,1070.72 1390.2,1070.61 1393.25,1070.5 1396.3,1070.39 1399.35,1070.27 1402.39,1070.15 1405.44,1070.02 1408.48,1069.9 \n",
" 1411.52,1069.77 1414.57,1069.63 1417.61,1069.5 1420.64,1069.36 1423.68,1069.21 1426.72,1069.07 1429.75,1068.92 1432.79,1068.76 1435.82,1068.61 1438.85,1068.45 \n",
" 1441.88,1068.29 1444.91,1068.12 1447.93,1067.95 1450.96,1067.78 1453.98,1067.6 1457,1067.43 1460.02,1067.24 1463.04,1067.06 1466.05,1066.87 1469.07,1066.68 \n",
" 1472.08,1066.49 1475.09,1066.29 1478.1,1066.09 1481.11,1065.88 1484.11,1065.68 1487.11,1065.47 1490.11,1065.25 1493.11,1065.03 1496.11,1064.82 1499.11,1064.59 \n",
" 1502.1,1064.37 1505.09,1064.14 1508.08,1063.9 1511.06,1063.67 1514.05,1063.43 1517.03,1063.19 1520.01,1062.94 1522.99,1062.69 1525.96,1062.44 1528.93,1062.19 \n",
" 1531.9,1061.93 1534.87,1061.67 1537.84,1061.41 1540.8,1061.14 1543.76,1060.87 1546.72,1060.59 1549.67,1060.32 1552.62,1060.04 1555.57,1059.76 1558.52,1059.47 \n",
" 1561.47,1059.18 1564.41,1058.89 1567.35,1058.59 1570.28,1058.29 1573.22,1057.99 1576.15,1057.69 1579.08,1057.38 1582,1057.07 1584.92,1056.76 1587.84,1056.44 \n",
" 1590.76,1056.12 1593.67,1055.8 1596.58,1055.47 1599.49,1055.14 1602.39,1054.81 1605.29,1054.47 1608.19,1054.13 1611.09,1053.79 1613.98,1053.45 1616.87,1053.1 \n",
" 1619.75,1052.75 1622.63,1052.39 1625.51,1052.04 1628.39,1051.68 1631.26,1051.31 1634.13,1050.95 1636.99,1050.58 1639.85,1050.21 1642.71,1049.83 1645.56,1049.45 \n",
" 1648.41,1049.07 1651.26,1048.69 1654.1,1048.3 1656.94,1047.91 1659.78,1047.52 1662.61,1047.12 1665.44,1046.72 1668.27,1046.32 1671.09,1045.92 1673.91,1045.51 \n",
" 1676.72,1045.1 1679.53,1044.68 1682.34,1044.27 1685.14,1043.85 1687.94,1043.42 1690.73,1043 1693.52,1042.57 1696.31,1042.14 1699.09,1041.7 1701.87,1041.26 \n",
" 1704.64,1040.82 1707.41,1040.38 1710.18,1039.93 1712.94,1039.48 1715.7,1039.03 1718.45,1038.58 1721.2,1038.12 1723.94,1037.66 1726.68,1037.19 1729.42,1036.72 \n",
" 1732.15,1036.25 1734.88,1035.78 1737.6,1035.31 1740.32,1034.83 1743.03,1034.35 1745.74,1033.86 1748.44,1033.38 1751.14,1032.89 1753.84,1032.39 1756.53,1031.9 \n",
" 1759.22,1031.4 1761.9,1030.9 1764.58,1030.39 1767.25,1029.89 1769.92,1029.38 1772.58,1028.86 1775.24,1028.35 1777.89,1027.83 1780.54,1027.31 1783.18,1026.78 \n",
" 1785.82,1026.26 1788.45,1025.73 1791.08,1025.2 1793.7,1024.66 1796.32,1024.12 1798.94,1023.58 1801.55,1023.04 1804.15,1022.49 1806.75,1021.94 1809.34,1021.39 \n",
" 1811.93,1020.84 1814.51,1020.28 1817.09,1019.72 1819.66,1019.16 1822.23,1018.59 1824.79,1018.03 1827.35,1017.46 1829.9,1016.88 1832.44,1016.31 1834.98,1015.73 \n",
" 1837.52,1015.15 1840.05,1014.56 1842.57,1013.97 1845.09,1013.39 1847.6,1012.79 1850.11,1012.2 1852.61,1011.6 1855.11,1011 1857.6,1010.4 1860.09,1009.79 \n",
" 1862.56,1009.18 1865.04,1008.57 1867.51,1007.96 1869.97,1007.34 1872.43,1006.73 1874.88,1006.11 1877.32,1005.48 1879.76,1004.86 1882.19,1004.23 1884.62,1003.6 \n",
" 1887.04,1002.96 1889.46,1002.32 1891.87,1001.69 1894.27,1001.04 1896.67,1000.4 1899.06,999.753 1901.45,999.104 1903.82,998.452 1906.2,997.798 1908.57,997.142 \n",
" 1910.93,996.482 1913.28,995.821 1915.63,995.156 1917.97,994.49 1920.31,993.82 1922.64,993.149 1924.96,992.475 1927.28,991.798 1929.59,991.119 1931.9,990.438 \n",
" 1934.19,989.754 1936.49,989.067 1938.77,988.379 1941.05,987.687 1943.32,986.994 1945.59,986.298 1947.85,985.599 1950.1,984.899 1952.35,984.195 1954.59,983.49 \n",
" 1956.82,982.782 1959.05,982.072 1961.27,981.359 1963.48,980.644 1965.69,979.927 1967.89,979.207 1970.08,978.485 1972.27,977.761 1974.45,977.035 1976.62,976.306 \n",
" 1978.79,975.574 1980.95,974.841 1983.1,974.105 1985.25,973.367 1987.39,972.627 1989.52,971.884 1991.64,971.14 1993.76,970.393 1995.87,969.643 1997.98,968.892 \n",
" 2000.07,968.138 2002.16,967.382 2004.25,966.624 2006.32,965.864 2008.39,965.101 2010.45,964.336 2012.51,963.569 2014.55,962.8 2016.59,962.029 2018.63,961.256 \n",
" 2020.65,960.48 2022.67,959.702 2024.68,958.922 2026.69,958.14 2028.68,957.356 2030.67,956.57 2032.65,955.782 2034.63,954.992 2036.59,954.199 2038.55,953.405 \n",
" 2040.51,952.608 2042.45,951.809 2044.39,951.009 2046.32,950.206 2048.24,949.401 2050.16,948.594 2052.06,947.785 2053.96,946.974 2055.85,946.161 2057.74,945.347 \n",
" 2059.62,944.53 2061.49,943.711 2063.35,942.89 2065.2,942.067 2067.05,941.242 2068.89,940.416 2070.72,939.587 2072.54,938.757 2074.35,937.924 2076.16,937.09 \n",
" 2077.96,936.253 2079.75,935.415 2081.54,934.575 2083.31,933.733 2085.08,932.889 2086.84,932.043 2088.6,931.196 2090.34,930.346 2092.08,929.495 2093.81,928.642 \n",
" 2095.53,927.787 2097.24,926.93 2098.94,926.071 2100.64,925.211 2102.33,924.349 2104.01,923.485 2105.68,922.619 2107.35,921.752 2109,920.882 2110.65,920.011 \n",
" 2112.29,919.139 2113.92,918.264 2115.55,917.388 2117.16,916.51 2118.77,915.63 2120.37,914.749 2121.96,913.866 2123.54,912.981 2125.12,912.095 2126.68,911.207 \n",
" 2128.24,910.317 2129.79,909.426 2131.33,908.533 2132.87,907.638 2134.39,906.742 2135.91,905.844 2137.41,904.945 2138.91,904.044 2140.41,903.141 2141.89,902.237 \n",
" 2143.36,901.331 2144.83,900.424 2146.28,899.515 2147.73,898.605 2149.17,897.693 2150.61,896.78 2152.03,895.865 2153.44,894.948 2154.85,894.03 2156.25,893.111 \n",
" 2157.63,892.19 2159.02,891.268 2160.39,890.344 2161.75,889.419 2163.1,888.492 2164.45,887.564 2165.79,886.634 2167.12,885.703 2168.43,884.771 2169.75,883.837 \n",
" 2171.05,882.902 2172.34,881.966 2173.63,881.028 2174.9,880.089 2176.17,879.148 2177.43,878.206 2178.68,877.263 2179.92,876.318 2181.15,875.373 2182.37,874.425 \n",
" 2183.59,873.477 2184.79,872.527 2185.99,871.576 2187.18,870.624 2188.36,869.67 2189.53,868.716 2190.69,867.76 2191.84,866.803 2192.98,865.844 2194.12,864.884 \n",
" 2195.24,863.924 2196.36,862.962 2197.46,861.998 2198.56,861.034 2199.65,860.069 2200.73,859.102 2201.8,858.134 2202.87,857.165 2203.92,856.195 2204.96,855.224 \n",
" 2206,854.252 2207.02,853.278 2208.04,852.304 2209.05,851.329 2210.05,850.352 2211.04,849.374 2212.02,848.396 2212.99,847.416 2213.95,846.435 2214.9,845.454 \n",
" 2215.85,844.471 2216.78,843.487 2217.71,842.502 2218.62,841.517 2219.53,840.53 2220.43,839.543 2221.31,838.554 2222.19,837.565 2223.06,836.574 2223.92,835.583 \n",
" 2224.78,834.591 2225.62,833.597 2226.45,832.603 2227.28,831.609 2228.09,830.613 2228.89,829.616 2229.69,828.619 2230.48,827.62 2231.25,826.621 2232.02,825.621 \n",
" 2232.78,824.621 2233.53,823.619 2234.27,822.617 2235,821.614 2235.72,820.61 2236.43,819.605 2237.13,818.6 2237.83,817.594 2238.51,816.587 2239.19,815.579 \n",
" 2239.85,814.571 2240.51,813.562 2241.15,812.552 2241.79,811.542 2242.41,810.531 2243.03,809.519 2243.64,808.507 2244.24,807.494 2244.83,806.48 2245.41,805.466 \n",
" 2245.98,804.451 2246.54,803.436 2247.09,802.42 2247.63,801.403 2248.16,800.386 2248.69,799.368 2249.2,798.35 2249.7,797.331 2250.2,796.312 2250.68,795.292 \n",
" 2251.16,794.272 2251.62,793.251 2252.08,792.23 2252.53,791.208 2252.96,790.186 2253.39,789.163 2253.81,788.14 2254.22,787.116 2254.62,786.092 2255.01,785.068 \n",
" 2255.39,784.043 2255.76,783.017 2256.12,781.992 2256.47,780.966 2256.81,779.939 2257.14,778.913 2257.47,777.885 2257.78,776.858 2258.08,775.83 2258.37,774.802 \n",
" 2258.66,773.774 2258.93,772.745 2259.2,771.716 2259.45,770.687 2259.7,769.657 2259.93,768.628 2260.16,767.598 2260.38,766.567 2260.58,765.537 2260.78,764.506 \n",
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" 2262.32,753.158 2262.4,752.125 2262.47,751.093 2262.54,750.06 2262.59,749.027 2262.63,747.995 2262.67,746.962 2262.69,745.929 2262.71,744.896 2262.71,743.863 \n",
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"execution_count": 34,
"metadata": {},
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}
],
"source": [
"scatter([1/5im,-1/5im]; label=\"singularities\")\n",
"θ = range(0; stop=2π, length=2000)\n",
"a = 2; b= 0.1\n",
"plot!(a * cos.(θ) + im*b * sin.(θ); label=\"ellipse\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Thus we can still use Cauchy's integral formula:"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {},
"outputs": [
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" 408.474,315.368 410.428,325.32 412.382,349.23 414.335,311.794 416.289,340.615 418.243,323.686 420.197,389.334 422.151,313.994 424.105,445.953 426.059,325.767 \n",
" 428.013,345.222 429.966,319.285 431.92,353.418 433.874,330.684 435.828,333.2 437.782,327.163 439.736,336.947 441.69,338.323 443.644,328.325 445.597,337.763 \n",
" 447.551,330.535 449.505,349.198 451.459,326.862 453.413,352.022 455.367,328.472 457.321,365.009 459.275,327.518 461.228,373.005 463.182,328.999 465.136,392.085 \n",
" 467.09,329.71 469.044,417.927 470.998,331.341 472.952,457.131 474.906,333.138 476.859,407.31 478.813,335.094 480.767,392.502 482.721,337.642 484.675,382.455 \n",
" 486.629,340.037 488.583,377.533 490.537,343.146 492.49,373.042 494.444,346.056 496.398,370.945 498.352,349.629 500.306,368.958 502.26,353.111 504.214,368.241 \n",
" 506.168,357.12 508.121,367.751 510.075,361.232 512.029,367.908 513.983,365.705 515.937,368.44 517.891,370.523 519.845,369.252 521.799,375.549 523.752,370.54 \n",
" 525.706,381.196 527.66,371.898 529.614,386.945 531.568,373.781 533.522,393.649 535.476,375.629 537.43,400.445 539.384,378.005 541.337,408.676 543.291,380.312 \n",
" 545.245,417.208 547.199,383.122 549.153,428.133 551.107,385.872 553.061,440.352 555.015,389.085 556.968,458.375 558.922,392.277 560.876,486.041 562.83,395.885 \n",
" 564.784,529.548 566.738,399.528 568.692,481.299 570.646,403.544 572.599,465.479 574.553,407.663 576.507,457.915 578.461,412.124 580.415,453.219 582.369,416.763 \n",
" 584.323,450.728 586.277,421.735 588.23,449.408 590.184,426.971 592.138,449.121 594.092,432.563 596.046,449.563 598,438.529 599.954,450.629 601.908,444.924 \n",
" 603.861,452.253 605.815,451.859 607.769,454.331 609.723,459.397 611.677,456.89 613.631,467.777 615.585,459.845 617.539,477.185 619.492,463.255 621.446,488.172 \n",
" 623.4,467.07 625.354,501.53 627.308,471.36 629.262,519.482 631.216,476.127 633.17,549.777 635.123,481.456 637.077,589.738 639.031,487.431 640.985,542.504 \n",
" 642.939,494.187 644.893,528.311 646.847,501.98 648.801,521.173 650.754,511.15 652.708,517.392 654.662,522.506 656.616,515.526 658.57,537.592 660.524,515.063 \n",
" 662.478,561.85 664.432,515.578 666.385,709.64 668.339,516.971 670.293,567.732 672.247,519.052 674.201,548.795 676.155,521.836 678.109,538.985 680.063,525.236 \n",
" 682.016,533.054 683.97,529.328 685.924,529.229 687.878,534.108 689.832,526.819 691.786,539.727 693.74,525.385 695.694,546.314 697.647,524.717 699.601,554.209 \n",
" 701.555,524.644 703.509,563.936 705.463,525.077 707.417,576.693 709.371,525.933 711.325,595.618 713.278,527.165 715.232,637.473 717.186,528.733 719.14,625.941 \n",
" 721.094,530.609 723.048,597.75 725.002,532.77 726.956,585.035 728.909,535.202 730.863,577.437 732.817,537.895 734.771,572.462 736.725,540.841 738.679,569.117 \n",
" 740.633,544.039 742.587,566.895 744.54,547.488 746.494,565.511 748.448,551.193 750.402,564.781 752.356,555.163 754.31,564.584 756.264,559.409 758.218,564.836 \n",
" 760.171,563.95 762.125,565.474 764.079,568.809 766.033,566.454 767.987,574.02 769.941,567.739 771.895,579.626 773.849,569.303 775.803,585.689 777.756,571.126 \n",
" 779.71,592.29 781.664,573.191 783.618,599.553 785.572,575.486 787.526,607.658 789.48,578.002 791.434,616.903 793.387,580.732 795.341,627.799 797.295,583.673 \n",
" 799.249,641.382 801.203,586.821 803.157,660.265 805.111,590.178 807.065,695.86 809.018,593.745 810.972,707.043 812.926,597.528 814.88,673.465 816.834,601.531 \n",
" 818.788,660.689 820.742,605.764 822.696,653.805 824.649,610.241 826.603,649.816 828.557,614.976 830.511,647.588 832.465,619.992 834.419,646.572 836.373,625.314 \n",
" 838.327,646.463 840.28,630.979 842.234,647.075 844.188,637.032 846.142,648.289 848.096,643.536 850.05,650.029 852.004,650.577 853.958,652.243 855.911,658.277 \n",
" 857.865,654.901 859.819,666.82 861.773,657.988 863.727,676.5 865.681,661.502 867.635,687.828 869.589,665.454 871.542,701.824 873.496,669.872 875.45,721.055 \n",
" 877.404,674.8 879.358,756.5 881.312,680.307 883.266,770.612 885.22,686.5 887.173,736.639 889.127,693.544 891.081,724.279 893.035,701.706 894.989,717.948 \n",
" 896.943,711.455 898.897,714.586 900.851,723.717 902.804,713.052 904.758,740.753 906.712,712.8 908.666,771.358 910.62,713.536 912.574,799.725 914.528,715.087 \n",
" 916.482,756.998 918.435,717.358 920.389,742.068 922.343,720.3 924.297,733.784 926.251,723.901 928.205,728.616 930.159,728.187 932.113,725.295 934.066,733.224 \n",
" 936.02,723.214 937.974,739.134 939.928,722.038 941.882,746.128 943.836,721.562 945.79,754.579 947.744,721.652 949.697,765.204 951.651,722.217 953.605,779.622 \n",
" 955.559,723.19 957.513,802.966 959.467,724.524 961.421,921.667 963.375,726.184 965.328,808.893 967.282,728.143 969.236,789.107 971.19,730.383 973.144,778.744 \n",
" 975.098,732.889 977.052,772.263 979.006,735.653 980.959,767.951 982.913,738.668 984.867,765.045 986.821,741.934 988.775,763.142 990.729,745.451 992.683,762 \n",
" 994.637,749.225 996.59,761.463 998.544,753.266 1000.5,761.424 1002.45,757.586 1004.41,761.81 1006.36,762.206 1008.31,762.563 1010.27,767.151 1012.22,763.643 \n",
" 1014.18,772.456 1016.13,765.018 1018.08,778.168 1020.04,766.664 1021.99,784.353 1023.94,768.562 1025.9,791.101 1027.85,770.697 1029.81,798.546 1031.76,773.057 \n",
" 1033.71,806.89 1035.67,775.635 1037.62,816.465 1039.58,778.425 1041.53,827.866 1043.48,781.423 1045.44,842.341 1047.39,784.628 1049.35,863.316 1051.3,788.041 \n",
" 1053.25,910.578 1055.21,791.664 1057.16,891.879 1059.11,795.501 1061.07,866.921 1063.02,799.561 1064.98,856.009 1066.93,803.853 1068.88,849.96 1070.84,808.39 \n",
" 1072.79,846.455 1074.75,813.188 1076.7,844.553 1078.65,818.271 1080.61,843.778 1082.56,823.667 1084.51,843.859 1086.47,829.412 1088.42,844.629 1090.38,835.555 \n",
" 1092.33,845.982 1094.28,842.162 1096.24,847.848 1098.19,849.325 1100.15,850.181 1102.1,857.174 1104.05,852.955 1106.01,865.909 1107.96,856.158 1109.92,875.851 \n",
" 1111.87,859.793 1113.82,887.571 1115.78,863.875 1117.73,902.246 1119.68,868.435 1121.64,923.018 1123.59,873.525 1125.55,966.223 1127.5,879.224 1129.45,957.908 \n",
" 1131.41,885.654 1133.36,931.318 1135.32,893.006 1137.27,920.48 1139.22,901.595 1141.18,914.835 1143.13,911.993 1145.09,911.884 1147.04,925.392 1148.99,910.638 \n",
" 1150.95,945.016 1152.9,910.614 1154.85,988.212 1156.81,911.543 1158.76,975.355 1160.72,913.273 1162.67,948.089 1164.62,915.718 1166.58,935.952 1168.53,918.839 \n",
" 1170.49,928.857 1172.44,922.634 1174.39,924.357 1176.35,927.141 1178.3,921.469 1180.25,932.443 1182.21,919.694 1184.16,938.687 1186.12,918.751 1188.07,946.131 \n",
" 1190.02,918.459 1191.98,955.244 1193.93,918.7 1195.89,966.969 1197.84,919.392 1199.79,983.652 1201.75,920.476 1203.7,1014.73 1205.66,921.909 1207.61,1037.1 \n",
" 1209.56,923.658 1211.52,996.607 1213.47,925.699 1215.42,981.552 1217.38,928.016 1219.33,972.922 1221.29,930.595 1223.24,967.353 1225.19,933.429 1227.15,963.606 \n",
" 1229.1,936.512 1231.06,961.09 1233.01,939.846 1234.96,959.475 1236.92,943.431 1238.87,958.555 1240.83,947.274 1242.78,958.198 1244.73,951.386 1246.69,958.308 \n",
" 1248.64,955.781 1250.59,958.819 1252.55,960.482 1254.5,959.683 1256.46,965.513 1258.41,960.86 1260.36,970.916 1262.32,962.322 1264.27,976.736 1266.23,964.047 \n",
" 1268.18,983.047 1270.13,966.018 1272.09,989.949 1274.04,968.22 1275.99,997.586 1277.95,970.645 1279.9,1006.18 1281.86,973.284 1283.81,1016.12 1285.76,976.132 \n",
" 1287.72,1028.09 1289.67,979.187 1291.63,1043.6 1293.58,982.447 1295.53,1067.31 1297.49,985.915 1299.44,1146.42 1301.4,989.593 1303.35,1080.71 1305.3,993.489 \n",
" 1307.26,1061.01 1309.21,997.605 1311.16,1051.58 1313.12,1001.95 1315.07,1046.26 1317.03,1006.55 1318.98,1043.19 1320.93,1011.41 1322.89,1041.59 1324.84,1016.57 \n",
" 1326.8,1041.04 1328.75,1022.03 1330.7,1041.3 1332.66,1027.86 1334.61,1042.22 1336.56,1034.09 1338.52,1043.71 1340.47,1040.8 1342.43,1045.7 1344.38,1048.09 \n",
" 1346.33,1048.16 1348.29,1056.09 1350.24,1051.05 1352.2,1065.04 1354.15,1054.37 1356.1,1075.24 1358.06,1058.13 1360.01,1087.4 1361.97,1062.31 1363.92,1102.84 \n",
" 1365.87,1067.04 1367.83,1125.38 1369.78,1072.32 1371.73,1181.78 1373.69,1078.21 1375.64,1148.23 1377.6,1084.9 1379.55,1126.51 1381.5,1092.61 1383.46,1116.82 \n",
" 1385.41,1101.68 1387.37,1111.87 1389.32,1112.91 1391.27,1109.28 1393.23,1127.6 1395.18,1108.27 1397.14,1151 1399.09,1108.53 1401.04,1247.95 1403,1109.56 \n",
" 1404.95,1160.84 1406.9,1111.55 1408.86,1140.6 1410.81,1114.19 1412.77,1130.33 1414.72,1117.5 1416.67,1124.17 1418.63,1121.46 1420.58,1120.33 1422.54,1126.26 \n",
" 1424.49,1117.67 1426.44,1131.76 1428.4,1116.32 1430.35,1138.59 1432.3,1115.54 1434.26,1146.61 1436.21,1115.37 1438.17,1156.35 1440.12,1115.72 1442.07,1169.64 \n",
" 1444.03,1116.76 1445.98,1188.09 1447.94,1117.79 1449.89,1236.45 1451.84,1119.21 1453.8,1211.32 1455.75,1121.22 1457.71,1186.86 1459.66,1123.23 1461.61,1174.42 \n",
" 1463.57,1125.76 1465.52,1167.99 1467.47,1128.29 1469.43,1163.22 1471.38,1131.28 1473.34,1159.26 1475.29,1134.5 1477.24,1156.75 1479.2,1137.57 1481.15,1155.49 \n",
" 1483.11,1141.3 1485.06,1155.01 1487.01,1145.42 1488.97,1154.92 1490.92,1149.39 1492.88,1155.14 1494.83,1154.09 1496.78,1156 1498.74,1159.01 1500.69,1157.21 \n",
" 1502.64,1163.88 1504.6,1158.1 1506.55,1169.25 1508.51,1159.51 1510.46,1176.03 1512.41,1161.75 1514.37,1182.22 1516.32,1163.79 1518.28,1187.91 1520.23,1165.89 \n",
" 1522.18,1197.9 1524.14,1168.61 1526.09,1205.19 1528.04,1170.66 1530,1215.97 1531.95,1173.79 1533.91,1226.79 1535.86,1176.59 1537.81,1240.82 1539.77,1180.07 \n",
" 1541.72,1266.28 1543.68,1184.38 1545.63,1307.63 1547.58,1188.55 1549.54,1273.51 1551.49,1193.04 1553.45,1250.46 1555.4,1195.44 1557.35,1241.91 1559.31,1200.73 \n",
" 1561.26,1243.44 1563.21,1205.82 1565.17,1245.62 1567.12,1208.8 1569.08,1242.94 1571.03,1214.07 1572.98,1233.27 1574.94,1221.96 1576.89,1239.96 1578.85,1228.06 \n",
" 1580.8,1242.94 1582.75,1232.15 1584.71,1243.46 1586.66,1241.43 1588.61,1239.53 1590.57,1250.33 1592.52,1244.51 1594.48,1247.93 1596.43,1243.98 1598.38,1253.57 \n",
" 1600.34,1244.52 1602.29,1282.99 1604.25,1259.47 1606.2,1277.6 1608.15,1267.19 1610.11,1306.08 1612.06,1268.24 1614.02,1296.71 1615.97,1265.01 1617.92,1306.57 \n",
" 1619.88,1281.25 1621.83,1312.71 1623.78,1279.2 1625.74,1280.04 1627.69,1317.77 1629.65,1349.83 1631.6,1310.84 1633.55,1270.78 1635.51,1310.46 1637.46,1288.28 \n",
" 1639.42,1298.59 1641.37,1319.92 1643.32,1349.76 1645.28,1315.26 1647.23,1321.26 1649.19,1299.47 1651.14,1319.72 1653.09,1293.98 1655.05,1323 1657,1327.81 \n",
" 1658.95,1363.42 1660.91,1288.36 1662.86,1305.99 1664.82,1299.42 1666.77,1343.8 1668.72,1310.83 1670.68,1309.62 1672.63,1299.82 1674.59,1330.23 1676.54,1310.9 \n",
" 1678.49,1291.25 1680.45,1356.34 1682.4,1320.92 1684.35,1289.38 1686.31,1320.7 1688.26,1334.19 1690.22,1298.84 1692.17,1336.72 1694.12,1301.92 1696.08,1293.99 \n",
" 1698.03,1310.51 1699.99,1309.59 1701.94,1317.47 1703.89,1326.88 1705.85,1332.28 1707.8,1324.31 1709.76,1306.18 1711.71,1300.89 1713.66,1310.18 1715.62,1310.34 \n",
" 1717.57,1306 1719.52,1320.94 1721.48,1319.71 1723.43,1319.67 1725.39,1326.72 1727.34,1349.9 1729.29,1331.54 1731.25,1320.21 1733.2,1349.3 1735.16,1296.61 \n",
" 1737.11,1320.94 1739.06,1327.89 1741.02,1321.45 1742.97,1306.55 1744.93,1321.92 1746.88,1309.75 1748.83,1319.64 1750.79,1334.43 1752.74,1328.4 1754.69,1371.96 \n",
" 1756.65,1337.02 1758.6,1318.39 1760.56,1296.61 1762.51,1308.08 1764.46,1327.2 1766.42,1328.13 1768.37,1328.47 1770.33,1328.08 1772.28,1346.58 1774.23,1385.33 \n",
" 1776.19,1340.98 1778.14,1293.89 1780.09,1286.94 1782.05,1336.99 1784,1279.56 1785.96,1296.17 1787.91,1350.21 1789.86,1310.52 1791.82,1305.8 1793.77,1320.35 \n",
" 1795.73,1302.61 1797.68,1321.29 1799.63,1344.64 1801.59,1326.82 1803.54,1279.55 1805.5,1367.41 1807.45,1334.19 1809.4,1346.77 1811.36,1337.04 1813.31,1342.57 \n",
" 1815.26,1310.52 1817.22,1322.8 1819.17,1349.99 1821.13,1314.76 1823.08,1386.88 1825.03,1321.05 1826.99,1355.14 1828.94,1305.77 1830.9,1310.27 1832.85,1341.32 \n",
" 1834.8,1308.01 1836.76,1299.7 1838.71,1290.96 1840.66,1289.09 1842.62,1322.58 1844.57,1281.16 1846.53,1302.9 1848.48,1343.18 1850.43,1306.4 1852.39,1310.81 \n",
" 1854.34,1369.27 1856.3,1342.07 1858.25,1321.35 1860.2,1296.7 1862.16,1347.96 1864.11,1350.21 1866.07,1321.12 1868.02,1383.87 1869.97,1328.16 1871.93,1310.73 \n",
" 1873.88,1293.36 1875.83,1371.92 1877.79,1314.91 1879.74,1310.49 1881.7,1328.41 1883.65,1320.26 1885.6,1296.43 1887.56,1336.66 1889.51,1304.95 1891.47,1327.67 \n",
" 1893.42,1321.15 1895.37,1327.74 1897.33,1333.27 1899.28,1299.24 1901.24,1344.91 1903.19,1321.46 1905.14,1317.66 1907.1,1284.89 1909.05,1345.77 1911,1293.97 \n",
" 1912.96,1321.45 1914.91,1279.55 1916.87,1303.24 1918.82,1336.82 1920.77,1329.04 1922.73,1301.49 1924.68,1315.71 1926.64,1334.6 1928.59,1313.29 1930.54,1295.76 \n",
" 1932.5,1327.9 1934.45,1317.62 1936.4,1345.58 1938.36,1349.59 1940.31,1315.16 1942.27,1282.78 1944.22,1305.85 1946.17,1300.81 1948.13,1310.25 1950.08,1321.46 \n",
" 1952.04,1315.42 1953.99,1345.82 1955.94,1288.88 1957.9,1318.29 1959.85,1347.34 1961.81,1290.3 1963.76,1310.75 1965.71,1371.85 1967.67,1309.69 1969.62,1302.88 \n",
" 1971.57,1328.24 1973.53,1294.91 1975.48,1328.01 1977.44,1286.54 1979.39,1317.98 1981.34,1349.89 1983.3,1337.43 1985.25,1339.33 1987.21,1330.85 1989.16,1302.82 \n",
" 1991.11,1298.24 1993.07,1326.44 1995.02,1332.43 1996.98,1348.45 1998.93,1332.62 2000.88,1276.3 2002.84,1321.45 2004.79,1351.98 2006.74,1356.14 2008.7,1326.89 \n",
" 2010.65,1315.56 2012.61,1384.56 2014.56,1349.92 2016.51,1288.93 2018.47,1337.5 2020.42,1321.45 2022.38,1293.76 2024.33,1310.81 2026.28,1346.76 2028.24,1298.83 \n",
" 2030.19,1313.47 2032.14,1326.09 2034.1,1332.92 2036.05,1320.87 2038.01,1335.49 2039.96,1327.28 2041.91,1321.42 2043.87,1305.87 2045.82,1350.08 2047.78,1350.2 \n",
" 2049.73,1310.75 2051.68,1333.34 2053.64,1291.47 2055.59,1310.9 2057.55,1315.98 2059.5,1343.88 2061.45,1379.32 2063.41,1321.37 2065.36,1321.58 2067.31,1317.17 \n",
" 2069.27,1343.96 2071.22,1298.64 2073.18,1321.44 2075.13,1307.45 2077.08,1313.43 2079.04,1299.23 2080.99,1315.18 2082.95,1336.12 2084.9,1291.46 2086.85,1299.68 \n",
" 2088.81,1312.1 2090.76,1317.3 2092.71,1328.25 2094.67,1331.69 2096.62,1314.54 2098.58,1311.19 2100.53,1328.26 2102.48,1328.47 2104.44,1358.49 2106.39,1326.35 \n",
" 2108.35,1343.48 2110.3,1293.7 2112.25,1345.18 2114.21,1358.03 2116.16,1336.36 2118.12,1310.82 2120.07,1289.03 2122.02,1293.47 2123.98,1293.98 2125.93,1313.91 \n",
" 2127.88,1288.62 2129.84,1303.01 2131.79,1310.63 2133.75,1332.39 2135.7,1302.92 2137.65,1371.26 2139.61,1289.12 2141.56,1310.81 2143.52,1306.62 2145.47,1314.07 \n",
" 2147.42,1333.75 2149.38,1314.65 2151.33,1328.43 2153.29,1301.13 2155.24,1303.01 2157.19,1306.32 2159.15,1315.17 2161.1,1321.47 2163.05,1313.05 2165.01,1337.32 \n",
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" \n",
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},
"execution_count": 35,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x = 0.1\n",
"f = z -> 1/(25z^2 + 1)\n",
"\n",
"f_ellipse = [((z, w) = ellipse_rule(n, a, b); sum(f.(z)./(z.-x).*w)/(2π*im)) for n=1:1000]\n",
"plot(abs.(f_ellipse .- f(x)); yscale=:log10, title=\"convergence of n-point ellipse approximation\", legend=false, xlabel=\"n\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Taylor series versus Cauchy integral formula\n",
"\n",
"\n",
"The Taylor series gives a polynomial approximation to $f$. The Cauchy's integral formula discretisation gives a rational approximation, which is more adaptiple and does not require knowning the derivatives of $f$:"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {},
"outputs": [
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},
"execution_count": 36,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f = z -> sqrt(z)\n",
"\n",
"function sqrtⁿ(n,z,z₀) \n",
" ret = sqrt(z₀)\n",
" c = 0.5/ret*(z-z₀)\n",
" for k=1:n\n",
" ret += c\n",
" c *= -(2k-1)/(2*(k+1)*z₀)*(z-z₀)\n",
" end\n",
" ret\n",
"end\n",
"\n",
"\n",
"z₀ = 0.3\n",
"n = 40\n",
"phaseplot(-2..2, -2..2, z -> sqrtⁿ.(n,z,z₀))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Here we see that the approximation is vallid on the expected ellipse:"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
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"source": [
"(ζ, w) = ellipse_rule(20, 4.0, 1.0);\n",
"ζ .= ζ .+ 4.5;\n",
"f_c = z -> sum(f.(ζ)./(ζ.-z).*w)/(2π*im)\n",
"phaseplot(-2..10, -2 .. 2, f_c)"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [
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"data": {
"text/plain": [
"-1.3987033753437572e-11 + 2.459960261444439e-16im"
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"source": [
"(ζ, w) = ellipse_rule(100, 4.0, 1.0);\n",
"ζ .= ζ .+ 4.5;\n",
"f_c = z -> sum(f.(ζ)./(ζ.-z).*w)/(2π*im)\n",
"f_c(5.3) - sqrt(5.3)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Application: calculating matrix exponentials\n",
"\n",
"Let $A \\in {\\mathbb C}^{d \\times d}$, ${\\mathbf u}_0 \\in {\\mathbb C}^d$ and consider the constant coefficient linear ODE\n",
"\n",
"$$\n",
" {\\mathbf u}'(t) = A {\\mathbf u}(t)\\qquad\\hbox{and}\\qquad {\\mathbf u}(0) = {\\mathbf u}_0(0)\n",
" $$\n",
" \n",
"The solution to this is given by the _matrix exponential_\n",
"$$\n",
" {\\mathbf u}(t) = \\exp(A t) {\\mathbf u}_0\n",
" $$\n",
" where the matrix exponential is defined by it's Taylor series:\n",
"$$\n",
" \\exp(A) = \\sum_{k=0}^\\infty {A^k \\over k!}\n",
" $$\n",
"This has stability problems, so a more convenient form is as follows:\n",
"\n",
"**Theorem (Cauchy integral representation for matrix exponential)**\n",
"Let $A \\in {\\mathbb C}^{n \\times n}$ be a diagonalizable matrix:\n",
"$$\n",
" A = V \\Lambda V^{-1} \\qquad\\hbox{for}\\qquad\\Lambda =\n",
" \\begin{pmatrix}\\lambda_1 \\cr &\\ddots \\cr && \\lambda_d\\end{pmatrix}\n",
"$$\n",
"Let $\\gamma$ be a contour that surrounds the spectrum of $A$. Then we have\n",
"$$\n",
" \\exp(A) = {1 \\over 2 \\pi i} \\oint_\\gamma e^z (z I - A)^{-1} dz\n",
" $$\n",
" \n",
"**Proof** Note by definition\n",
"$$\n",
"\\begin{align*}\n",
"\\exp(A) &= \\sum_{k=0}^\\infty {A^k \\over k!} = V \\sum_{k=0}^\\infty {\\Lambda^k \\over k!}V^{-1} = V \\begin{pmatrix} \\E^{\\lambda_1}\\\\ & \\ddots \\\\ && \\E^{\\lambda_d} \\end{pmatrix} V^{-1} \\\\\n",
"&= {1 \\over 2\\pi \\I} V \\begin{pmatrix} \\oint_\\gamma {\\E^\\zeta \\over \\zeta - \\lambda_1 } \\D\\zeta \\\\ & \\ddots \\\\ && \\oint_\\gamma {\\E^\\zeta \\over \\zeta - \\lambda_d } \\D\\zeta\\end{pmatrix} V^{-1} \n",
"\\end{align*}\n",
"$$\n",
"It was important here that $\\gamma$ surrounded all $\\zeta_j$.\n",
"\n",
"We now take out the integration from the matrix, the easiest way to see this is to apply the Trapezium rule approximation:\n",
"$$\n",
"\\begin{align*}\n",
"V\\begin{pmatrix} \\oint_\\gamma {\\E^\\zeta \\over \\zeta - \\lambda_1 } \\D\\zeta \\\\ & \\ddots \\\\ && \\oint_\\gamma {\\E^\\zeta \\over \\zeta - \\lambda_d } \\D\\zeta\\end{pmatrix} V^{-1} &= V\\lim_{n \\rightarrow \\infty} \\begin{pmatrix} \\sum_{j=1}^n {w_j \\E^\\zeta_j \\over \\zeta_j - \\lambda_1 } \\\\ & \\ddots \\\\ && \\sum_{j=1}^n {w_j \\E^\\zeta_j \\over \\zeta_j - \\lambda_d } \\end{pmatrix} V^{-1} \\\\\n",
"&= \\lim_{n \\rightarrow \\infty} V\\sum_{j=1}^n w_j \\E^\\zeta_j \\begin{pmatrix} {1 \\over \\zeta_j - \\lambda_1 } \\\\ & \\ddots \\\\ && {1 \\over \\zeta_j - \\lambda_d } \\end{pmatrix} V^{-1} \\\\\n",
"&= \\lim_{n \\rightarrow \\infty} \\sum_{j=1}^n w_j \\E^\\zeta_j V(I \\zeta_j - \\Lambda)^{-1}V^{-1} \\\\\n",
"&= \\oint_\\gamma \\E^\\zeta V(I \\zeta - \\Lambda)^{-1} V^{-1} \\D \\zeta \n",
"= \\oint_\\gamma \\E^\\zeta (I \\zeta - V\\Lambda V^{-1})^{-1} \\D \\zeta \\\\\n",
"& = \\oint_\\gamma \\E^\\zeta (I \\zeta - A)^{-1} \\D \\zeta \\\\\n",
"\\end{align*}\n",
"$$\n",
"\n",
"\n",
"■"
]
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"cell_type": "markdown",
"metadata": {},
"source": [
"_Demonstration_ we use this formula alongside the complex trapezium rule to calculate matrix exponentials. Begin by creating a random symmetric matrix (which only has real eigenvalues):"
]
},
{
"cell_type": "code",
"execution_count": 42,
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{
"data": {
"text/plain": [
"5-element Array{Float64,1}:\n",
" -1.1701131879499997\n",
" -0.8307847323168219\n",
" -0.2011088438314149\n",
" 1.7390356067088892\n",
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"source": [
"A = randn(5,5)\n",
"A = A + A'\n",
"λ = eigvals(A)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can now by hand create a circle that surrounds all the eigenvalues:"
]
},
{
"cell_type": "code",
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},
"execution_count": 49,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"z,w = circle_rule(100,maximum(abs.(λ))+1)\n",
"\n",
"plot(z)\n",
"scatter!(λ,zeros(5); label = \"eigenvalues of A\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Here we wrap this up into a function `circle_exp` that calculates the matrix exponential:"
]
},
{
"cell_type": "code",
"execution_count": 50,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
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]
},
"execution_count": 50,
"metadata": {},
"output_type": "execute_result"
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],
"source": [
"function circle_exp(A, n, z₀, r)\n",
" z,w = circle_rule(n,r)\n",
" z .+= z₀\n",
"\n",
" ret = zero(A)\n",
" for j=1:n\n",
" ret += w[j]*exp(z[j])*inv(z[j]*I - A)\n",
" end\n",
"\n",
" ret/(2π*im)\n",
"end\n",
" \n",
"circle_exp(A, 100, 0, 8.0) -exp(A) |>norm"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In this case, it is beneficial to use an ellipse:"
]
},
{
"cell_type": "code",
"execution_count": 51,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
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]
},
"execution_count": 51,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function ellipse_exp(A, n, z₀, a, b)\n",
" z,w = ellipse_rule(n,a,b)\n",
" z .+= z₀\n",
"\n",
" ret = zero(A)\n",
" for j=1:n\n",
" ret += w[j]*exp(z[j])*inv(z[j]*I - A)\n",
" end\n",
" ret/(2π*im)\n",
"end\n",
"\n",
"\n",
"ellipse_exp(A, 50, 0, 8.0, 5.0) -exp(A) |>norm"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"For matrices with large negative eigenvalues (For example, discretisations of the Laplacian), complex quadrature can lead to much better accuracy than Taylor series:"
]
},
{
"cell_type": "code",
"execution_count": 52,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"4.813209010780333e-8"
]
},
"execution_count": 52,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function taylor_exp(A,n)\n",
" ret = Matrix(I, size(A))\n",
" for k=1:n\n",
" ret += A^k/factorial(1.0k)\n",
" end\n",
" ret\n",
"end\n",
"\n",
"B = A - 20I\n",
"\n",
"taylor_exp(B, 200) -exp(B) |>norm"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can use an ellpise to surround the spectrum:"
]
},
{
"cell_type": "code",
"execution_count": 53,
"metadata": {},
"outputs": [
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},
"execution_count": 53,
"metadata": {},
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}
],
"source": [
"scatter(complex.(eigvals(B)))\n",
"plot!(ellipse_rule(50,8,5)[1] .- 20)"
]
},
{
"cell_type": "code",
"execution_count": 54,
"metadata": {},
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],
"source": [
"norm(ellipse_exp(B, 50, -20.0, 8.0, 5.0) - exp(B))"
]
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"### Gershgorin circle theorem\n",
"\n",
"If we only know $A$, how do we know how big to make the contour? Gershgorin's circle theorem gives the answer:\n",
"\n",
"**Theorem (Gershgorin)** Let $A \\in {\\mathbb C}^{d \\times d}$ and define \n",
"$$R_k = \\sum_{j=1 \\atop j \\neq k}^d |a_{kj}| $$\n",
"Then \n",
"$$\n",
"\\rho(A) \\subset \\bigcup_{k=1}^d \\bar B(a_{kk}, R_k)\n",
"$$\n",
"where $\\bar B(z_0, r)$ is the closed disk of radius $r$ and $\\rho(A)$ is the set of eigenvalues.\n",
"\n",
"**Proof **\n",
"\n",
"\n",
"■\n",
"\n",
"_Demonstration_ Here we apply this to a particular matrix:"
]
},
{
"cell_type": "code",
"execution_count": 55,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3×3 Array{Int64,2}:\n",
" 1 2 3\n",
" 1 5 2\n",
" -4 1 6"
]
},
"execution_count": 55,
"metadata": {},
"output_type": "execute_result"
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"source": [
"A = [1 2 3; 1 5 2; -4 1 6]"
]
},
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"metadata": {},
"source": [
"The following calculates the row sums:"
]
},
{
"cell_type": "code",
"execution_count": 56,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3×1 Array{Int64,2}:\n",
" 5\n",
" 3\n",
" 5"
]
},
"execution_count": 56,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"R = sum(abs.(A - Diagonal(diag(A))),dims=2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Gershgorin's theorem tells us that the spectrum lies in the union of the circles surrounding the diagonals:"
]
},
{
"cell_type": "code",
"execution_count": 57,
"metadata": {},
"outputs": [
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},
"execution_count": 57,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"drawcircle!(z0, R) = plot!(θ-> real(z0) + R[1]*cos(θ), θ-> imag(z0) + R[1]*sin(θ), 0, 2π, fill=(0,:red), α = 0.2, legend=false)\n",
"\n",
"λ = eigvals(A)\n",
"p = plot()\n",
"for k = 1:size(A,1)\n",
" drawcircle!(A[k,k], R[k])\n",
"end\n",
"scatter!(complex.(λ); label=\"eigenvalues\")\n",
"scatter!(complex.(diag(A)); label=\"diagonals\")\n",
"p"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can therefore use this to choose a contour big enough to surround all the circles. Here's a fairly simplistic construction for our case where everything is real:"
]
},
{
"cell_type": "code",
"execution_count": 58,
"metadata": {},
"outputs": [
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"z₀ = (maximum(diag(A) .+ R) + minimum(diag(A) .- R)) /2 # average edges of circle\n",
"r = max(abs.(diag(A) .- R .- z₀)..., abs.(diag(A) .+ R .- z₀)...)\n",
"\n",
"plot!(Circle(z₀, r))"
]
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"## Fourier series and Laurent series \n",
"\n",
"**Definition (Fourier series)** On $[-\\pi, \\pi)$, _Fourier series_ is an expansion of the form\n",
"$$\n",
" g(\\theta) = \\sum_{k=-\\infty}^\\infty g_k e^{i k \\theta}\n",
"$$\n",
"where\n",
"$$\n",
"g_k = {1\\over 2 \\pi} \\int_{-\\pi}^\\pi g(\\theta) e^{-i k \\theta} d \\theta\n",
"$$\n",
"\n",
"**Definition (Laurent series)** In the complex plane, _Laurent series around $z_0$_ is an expansion of the form \n",
"$$ \n",
" f(z) = \\sum_{k=-\\infty}^\\infty f_k (z-z_0)^k \n",
"$$ \n",
"\n",
"**Lemma (Fourier series = Laurent series)** On a circle in the complex plane \n",
"$$\n",
" \\gamma_r = \\{z_0 + re^{i \\theta} : -\\pi \\leq \\theta < \\pi \\},$$ \n",
"Laurent series of $f(z)$ around $z_0$ converges for $z \\in C_r$ if the Fourier series of $g(\\theta) = f(z_0 + r e^{i \\theta})$ converges, and the coefficients are given by\n",
"\n",
"$$ \n",
"f_k = {g_k \\over r^k} = {1 \\over 2 \\pi i} \\oint_{\\gamma_r} {f(\\zeta) \\over (\\zeta - z_0)^{k+1}} d \\zeta.\n",
"$$\n",
"\n",
"**Proof** This follows immediately from the change of variables $z = r \\E^{\\I \\theta} + z_0$. ■"
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"Interestingly, analytic properties of $f$ can be used to show decaying properties in $g$:\n",
"\n",
"**Theorem (Decay in Fourier/Laurent coefficients)** Suppose $f(z)$ is analytic in a closed annulus $A_{r,R}$ around $0$:\n",
"$$A_r(z_0) = \\{z : r ≤ | z| ≤ R\\}$$\n",
"Then for all $k$\n",
"$$|f_k | \\leq M\\min\\left\\{{1 \\over R^k} , {1 \\over r^k}\\right\\}$$\n",
"where $M = \\sup_{z \\in A} |f(z)|$.\n",
"\n",
"**Proof**\n",
"This is a simple application of the ML lemma: \n",
"$$\n",
"|f_k| = {1 \\over 2 \\pi } \\left|\\oint_{\\gamma_1} {f(\\zeta) \\over \\zeta^{k+1}} \\D \\zeta\\right| = {1 \\over 2 \\pi}\\left|\\oint_{\\gamma_r} {f(\\zeta) \\over \\zeta^{k+1}} \\D \\zeta\\right| \\leq \\sup_{\\zeta \\in \\gamma_r} |f(\\zeta)| R^{-k} \\leq M R^{-k}\n",
"$$\n",
"which similar applies deforming to $\\gamma_r$.\n",
"■\n",
"\n",
"\n",
"\n",
"_Demonstration_ The Fourier coefficients of\n",
"$$g(\\theta) = {1 \\over 2 - \\cos \\theta}$$\n",
"satisfies for $k \\geq 0$\n",
"$$\n",
" |g_k| \\leq {2 \\over 4 - R -R^{-1}} R^{-k}\n",
"$$\n",
"for all $R \\leq 2 + \\sqrt{3}$:"
]
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{
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"<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 516.222, 1195.53)\" x=\"516.222\" y=\"1195.53\">R = 1.1</text>\n",
"</g>\n",
"<circle clip-path=\"url(#clip8601)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"432.222\" cy=\"1238.51\" r=\"25\"/>\n",
"<circle clip-path=\"url(#clip8601)\" style=\"fill:#3da44d; stroke:none; fill-opacity:1\" cx=\"432.222\" cy=\"1238.51\" r=\"21\"/>\n",
"<g clip-path=\"url(#clip8601)\">\n",
"<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 516.222, 1256.01)\" x=\"516.222\" y=\"1256.01\">R = 3.5</text>\n",
"</g>\n",
"<circle clip-path=\"url(#clip8601)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"432.222\" cy=\"1298.99\" r=\"25\"/>\n",
"<circle clip-path=\"url(#clip8601)\" style=\"fill:#c271d2; stroke:none; fill-opacity:1\" cx=\"432.222\" cy=\"1298.99\" r=\"21\"/>\n",
"<g clip-path=\"url(#clip8601)\">\n",
"<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 516.222, 1316.49)\" x=\"516.222\" y=\"1316.49\">R = 3.632050807568877</text>\n",
"</g>\n",
"</svg>\n"
]
},
"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"g =Fun(θ -> 1/(2-cos(θ)), Laurent(-π .. π))\n",
"g₊ = g.coefficients[1:2:end]\n",
"scatter(abs.(g₊); yscale=:log10, label=\"|g_k|\", legend=:bottomleft)\n",
"R = 1.1\n",
"scatter!(2/(4-R-inv(R))*R.^(-(0:length(g₊))), label = \"R = $R\")\n",
"R = 3.5\n",
"scatter!(2/(4-R-inv(R))*R.^(-(0:length(g₊))), label = \"R = $R\")\n",
"R = 2+sqrt(3)-0.1\n",
"scatter!(2/(4-R-inv(R))*R.^(-(0:length(g₊))), label = \"R = $R\")"
]
}
],
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"name": "julia-1.0"
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"language_info": {
"file_extension": ".jl",
"mimetype": "application/julia",
"name": "julia",
"version": "1.0.1"
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