# NISP Commit Details

Date: 2015-03-31 21:26:29 (3 years 9 months ago) Michael Baudin 334 333 Added Chebyshev figure for T11.

## File differences

doc/polychaos/1-orthopoly.tex
 965 965 966 966 967 967 968 968 969 969 970 970 971 971 972 972 973 973 974 ... ... 1232 1233 1233 1234 1234 1235 1235 1236 1236 1237 1238 1237 1239 1238 1240 1239 1241 ... ... 1291 1293 1292 1294 1293 1295 1294 1296 1295 1297 1296 1298 1297 1299 1298 1300 1299 1301 1300 1301 1302 1303 1304 1305 1306 1307 1308 1302 1309 1303 1310 1304 1311 1305 1312 1313 1306 1307 1308 1309 1310 1311 1314 1312 1315 1313 1316 1314
 \label{prop-threetermgen}␍␊ Assume $\{p_k\}_{k=-1,0,1,...,n}$ is a family of orthogonal ␍␊ polynomials, with ␍␊ $$␍␊ \begin{eqnarray}␍␊ \label{eq-threetermgen0}␍␊ p_{-1}=0, \qquad p_0=\frac{1}{\gamma_0},␍␊$$␍␊ \end{eqnarray}␍␊ Therefore, ␍␊ \begin{eqnarray}␍␊ p_{k+1}(x)=\frac{\gamma_k}{\gamma_{k+1}} (x-\alpha_k)p_k(x) ␍␊ & \ddots & \ddots & \ddots & \\␍␊ & & \sqrt{\beta_{n-1}} & \alpha_{n-1} & \sqrt{\beta_{n}}\\␍␊ 0 & & & \sqrt{\beta_{n}} & \alpha_n␍␊ \end{pmatrix}␍␊ \end{pmatrix},␍␊ $$␍␊ for n\geq 0.␍␊ \end{definition}␍␊ ␍␊ Notice that the previous matrix is symmetric. ␍␊ \end{pmatrix} \\␍␊ &=& ␍␊ \begin{pmatrix}␍␊ x p_0 \\␍␊ x p_0 - \sqrt{\beta_{0}} p_{-1} \\␍␊ x p_1 \\␍␊ \vdots \\␍␊ x p_{n-1} \\␍␊ x p_{n}␍␊ x p_{n} - \sqrt{\beta_{n+1}} p_{n+1}␍␊ \end{pmatrix}␍␊ -␍␊ \begin{pmatrix}␍␊ 0 \\␍␊ 0 \\␍␊ \vdots \\␍␊ 0 \\␍␊ \sqrt{\beta_{n+1}} p_{n+1}␍␊ \end{pmatrix}␍␊ ,␍␊ \end{eqnarray*}␍␊ where we have used the equation \ref{eq-eigenjacobi3} in ␍␊ the last equality. ␍␊ If x is a root of p_{n+1}, then p_{n+1}(x)=0, ␍␊ and the previous equation implies the equation \ref{eq-eigenjacobi1}.␍␊ First, the equation \ref{eq-threetermgen0} implies that ␍␊ the first row is equal to x p_0. ␍␊ Second, if x is a root of p_{n+1}, then p_{n+1}(x)=0 ␍␊ which implies that the last row is equal to x p_{n}. ␍␊ This is why the previous equation implies the equation \ref{eq-eigenjacobi1}, ␍␊ which concludes the proof.␍␊ \end{proof}␍␊ ␍␊ \begin{proposition}␍␊ doc/polychaos/figures/cheby11.pdf doc/polychaos/5-particularorthopoly.tex  206 206 207 207 208 208 209 209 210 210 211 211 212 212 ... ... 228 228 229 229 230 230 231 231 232 232 233 233 234 234 235 235 236 236 237 237 238 238 239 ... ... 261 262 262 263 263 264 264 265 265 266 266 267 267 268 268 269 269 270 270 271 271 272 272 273 273 274 274 275 ... ... 284 285 285 286 286 287 287 288 288 289 289 290 290 291 291 292 293 294 292 295 293 296 294 297 295 298 296 299 297 300 298 301 299 302 ... ... 452 455 453 456 454 457 455 458 456 459 457 460 458 461 ... ... 477 480 478 481 479 482 480 483 484 481 485 482 486 483 487 ... ... 510 514 511 515 512 516 513 517 514 518 515 519 516 520 ... ... 520 524 521 525 522 526 523 524 525 527 528 529 530 526 531 527 532 528 533 529 534 530 531 535 536 537 532 538 533 539 534 540 ... ... 537 543 538 544 539 545 540 541 546 547 542 548 543 549 544 550 545 551 546 552 547 553 548 554 549 555 550 556 ... ... 555 561 556 562 557 563 558 564 559 565 560 566 561 567 ... ... 730 736 731 737 732 738 733 739 734 740 735 741 736 742 ... ... 759 765 760 766 761 767 762 768 763 769 764 770 765 771 ... ... 773 779 774 780 775 781 776 782 777 783 778 784 779 785 ... ... 781 787 782 788 783 789 784 790 785 791 786 792 787 793 ... ... 820 826 821 827 822 828 823 829 824 830 825 831 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 832 833 834 843 835 844 836 837 845 838 846 839 847 840 848 849 850 851 852 853 854 841 855 842 856 843 857 844 ... ... 1076 1063 1077 1064 1078 1065 1079 1066 1067 1080 1068 1081 1069 1082 1070 1083 1071 1084 1072 1085 1086 1087 1073 1088 1074 1089 1075 ... ... 1114 1100 1115 1101 1116 1102 1117 1118 1119 1120 1103 1104 1105 1106 1107 1108 1121 1109 1122 1110 1123 1111 1124 1112 1125 1113 1114 1115 1126 1116 1127 1117 1128 1118 1129 1119 1120 1121 1130 1122 1131 1123 1132 1124 ... ... 1140 1132 1141 1133 1142 1134 1135 1143 1136 1144 1137 1145 1138 ... ... 1148 1141 1149 1142 1150 1143 1151 1152 1144 1153 1145 1154 1146 ... ... 1210 1202 1211 1203 1212 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1213 1241 1214 1242 1215 1243 ... ... 1290 1318 1291 1319 1292 1320 1293 1321 1294 1322 1295 1323 1296 1324  \end{proposition}␍␊ ␍␊ ␍␊ The previous proposition will not be proved here.␍␊ The previous proposition will not be proved in this document.␍␊ ␍␊ \begin{proposition}␍␊ \label{prop-orthoher}␍␊ \begin{proof}␍␊ Rodrigues' formula for Hermite polynomials imply :␍␊ \begin{eqnarray}␍␊ \int_{-\infty}^\infty He_n(x) He_m(x) w(x) dx ␍␊ (He_n,He_m) ␍␊ &=& \int_{-\infty}^\infty He_n(x) He_m(x) w(x) dx \nonumber \\␍␊ &=& (-1)^{n} ␍␊ \int_{-\infty}^\infty e^{\frac{x^2}{2}} \frac{d^n}{dx^n} \left(e^{-\frac{x^2}{2}}\right) He_m(x) e^{-\frac{x^2}{2}} dx \nonumber \\␍␊ &=& (-1)^{n} ␍␊ \int_{-\infty}^\infty \frac{d^n}{dx^n} \left(e^{-\frac{x^2}{2}}\right) He_m(x) dx␍␊ \int_{-\infty}^\infty \frac{d^n}{dx^n} \left(e^{-\frac{x^2}{2}}\right) He_m(x) dx.␍␊ \label{eqn-orthoherm1}␍␊ \end{eqnarray}␍␊ ␍␊ \int_{-\infty}^\infty \frac{d^n}{dx^n} \left(e^{-\frac{x^2}{2}}\right) He_m(x) dx ␍␊ &=&- \int_{-\infty}^\infty \frac{d^{n-1}}{dx^{n-1}} \left(e^{-\frac{x^2}{2}}\right) \frac{d}{dx} He_m(x) dx␍␊ \end{eqnarray*}␍␊ Integrating n-1 more times by part, we get :␍␊ Integrating by part n-1 more times, we get :␍␊ \begin{eqnarray*}␍␊ \int_{-\infty}^\infty \frac{d^n}{dx^n} \left(e^{-\frac{x^2}{2}}\right) He_m(x) dx ␍␊ &=&(-1)^n \int_{-\infty}^\infty e^{-\frac{x^2}{2}} \frac{d^n}{dx^n} He_m(x) dx␍␊ \end{eqnarray*}␍␊ We plug the previous equation into \ref{eqn-orthoherm1} and get :␍␊ \begin{eqnarray}␍␊ \int_{-\infty}^\infty He_n(x) He_m(x) w(x) dx ␍␊ (He_n,He_m) ␍␊ &=& (-1)^{n} (-1)^n \int_{-\infty}^\infty e^{-\frac{x^2}{2}} \frac{d^n}{dx^n} He_m(x) dx \nonumber \\␍␊ &=& \int_{-\infty}^\infty e^{-\frac{x^2}{2}} \frac{d^n}{dx^n} He_m(x) dx␍␊ \label{eqn-orthoherm2}␍␊ Secondly, assume that n=m. ␍␊ The equation \ref{eqn-orthoherm2} implies :␍␊ \begin{eqnarray*}␍␊ \int_{-\infty}^\infty He_n(x) He_m(x) w(x) dx ␍␊ (He_n,He_n) ␍␊ &=& \int_{-\infty}^\infty e^{-\frac{x^2}{2}} \frac{d^n}{dx^n} He_n(x) dx.␍␊ \end{eqnarray*}␍␊ The equation \ref{eqn-hermiterec} makes clear that He_n is a monic degree n polynomial. ␍␊ In other words, its leading term is x^n. ␍␊ The equation \ref{eqn-hermiterec} implies that He_n is a monic degree ␍␊ n polynomial since the monomial with highest exponent ␍␊ is xHe_n(x). ␍␊ In other words, the leading term of He_n is x^n. ␍␊ The n-th derivative of this monomial is n! and the n-th derivative of lower degree ␍␊ monomials is zero. ␍␊ Hence, ␍␊ \begin{eqnarray*}␍␊ \int_{-\infty}^\infty He_n(x) He_m(x) w(x) dx ␍␊ (He_n,He_n) ␍␊ &=& n! \int_{-\infty}^\infty e^{-\frac{x^2}{2}} dx \\␍␊ &=& n! \sqrt{2\pi},␍␊ \end{eqnarray*}␍␊ for x\in[-1,1].␍␊ \end{proposition}␍␊ ␍␊ The previous proposition will not be proved here.␍␊ The previous proposition will not be proved in this document.␍␊ ␍␊ ␍␊ \begin{proposition}␍␊ ␍␊ Rodrigues' formula \ref{eqn-rodrileg} implies :␍␊ \begin{eqnarray*}␍␊ \int_{-1}^1 P_n(x) P_m(x) dx ␍␊ (P_n,P_m)␍␊ &=&\int_{-1}^1 P_n(x) P_m(x) w(x) dx \\␍␊ &=& \frac{1}{2^{n+m} n! m!} \int_{-1}^1 \frac{d^n}{dx^n} ((x^2-1)^n) \frac{d^m}{dx^m} ((x^2-1)^m) dx.␍␊ \end{eqnarray*}␍␊ Integration by part implies :␍␊ \end{eqnarray*}␍␊ We continue to integrate by part n-1 more times, and get :␍␊ \begin{eqnarray}␍␊ \int_{-1}^1 P_n(x)P_m(x) ␍␊ (P_n,P_m)␍␊ =\frac{(-1)^n}{2^{n+m} n! m!} \int_{-1}^1 (x^2-1)^n \frac{d^{m+n}}{dx^{m+n}} ((x^2-1)^m) dx,␍␊ \label{eqn-ortholeg2}␍␊ \end{eqnarray}␍␊ Now, suppose that mn, then we just switch m and ␍␊ n). ␍␊ This implies 2m doc/polychaos/scripts/chebyshev.sce  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16  // Copyright (C) 2015 - Michael Baudin␊ //␊ // This file must be used under the terms of the ␊ // GNU Lesser General Public License license :␊ // http://www.gnu.org/copyleft/lesser.html␊ ␊ n=11;␊ x=linspace(-1,1,100);␊ y=chebyshev_eval(x,n);␊ plot(x,y)␊ [r,w]=chebyshev_quadrature(n);␊ plot(r,zeros(r),"rx")␊ xlabel("x")␊ ylabel("P(x)")␊ title(msprintf("Chebyshev polynomial - Degree %d",n))␊ ␊ doc/polychaos/4-gaussintegral.tex  250 250 251 251 252 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290  \end{eqnarray*}␍␊ which shows that the recurrence is true for n and concludes the proof.␍␊ \end{proof}␍␊ ␍␊ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%␍␊ \subsection{A Laguerre integral}␍␊ \label{sec-laguint}␍␊ ␍␊ The goal of this section is to prove the following proposition. ␍␊ ␍␊ \begin{proposition}␍␊ \label{prop-laguint}␍␊ We have ␍␊$$␍␊ \int_{0}^\infty x^n e^{-x} dx = n!,␍␊ $$␍␊ for n\geq 0.␍␊ \end{proposition}␍␊ ␍␊ \begin{proof}␍␊ Let ␍␊$$␍␊ I_n = \int_{0}^\infty x^n e^{-x} dx,␍␊ $$␍␊ for n=0,1,....␍␊ Obviously, ␍␊$$␍␊ I_0= \int_{0}^\infty e^{-x} dx = 1.␍␊ $$␍␊ Integrating by part, we get :␍␊ \begin{eqnarray*}␍␊ I_{n+1}␍␊ &=& \int_{0}^\infty x^{n+1} e^{-x} dx \\␍␊ &=& - \left[x^{n+1} e^{-x} \right] + (n+1) \int_{0}^\infty x^n e^{-x} dx \\␍␊ &=& (n+1)I_n.␍␊ \end{eqnarray*}␍␊ By induction on n, we get : ␍␊$$␍␊ I_n = n!.␍␊ ␍␊ \end{proof}␍␊

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