NISP

NISP Commit Details

Date:2015-03-27 18:42:29 (3 years 2 months ago)
Author:Michael Baudin
Commit:330
Parents: 329
Message:Added Cheby equiosc.
Changes:
M/doc/polychaos/5-particularorthopoly.tex

File differences

doc/polychaos/5-particularorthopoly.tex
921921
922922
923923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983924
984925
985926
......
1052993
1053994
1054995
1055
996
997
998
999
10561000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
10571093
1058
1094
10591095
1060
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
10611115
10621116
10631117
1064
1118
10651119
1066
1120
10671121
10681122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
10691138
10701139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
10711174
10721175
1073
1176
10741177
10751178
10761179
to \ref{eqn-chebyrecurr}.
\end{proof}
\begin{proposition}
(\emph{Orthogonality of Chebyshev polynomials})
The Chebyshev polynomials are orthogonal with
respect to the weight
\begin{eqnarray}
\label{eqn-chebyweight}
w(x)=\frac{1}{\sqrt{1-x^2}},
\end{eqnarray}
for $x\in[-1,1]$.
Moreover,
\begin{eqnarray}
\label{eqn-chebyprod}
(T_n,T_m)
&=& \int_{-1}^1 T_n(x) T_m(x) w(x) dx = \frac{\pi}{2} \delta_{nm},
\end{eqnarray}
for any integers $n$ and $m$.
\end{proposition}
The weight defined by the equation \ref{eqn-chebyweight} does
not correspond to a particular distribution in
probability theory.
\begin{proof}
First, notice that
$$
\arccos'(x)=\frac{1}{\sqrt{1-x^2}},
$$
for $x\in[-1,1]$.
Using the change of variable $\theta=\arccos(x)$
and the equation \ref{eqn-trigocheby}, we get :
\begin{eqnarray*}
(T_n,T_m)
&=& \int_{-1}^1 T_n(x) T_m(x) \frac{1}{\sqrt{1-x^2}} dx \\
&=& - \int_\pi^0 \cos(n\theta) \cos(m\theta) d\theta \\
&=& \int_0^\pi \cos(n\theta) \cos(m\theta) d\theta \\
&=& \frac{1}{2} \int_0^\pi \left( \cos((n+m)\theta) + \cos((n-m)\theta) \right) d\theta.
\end{eqnarray*}
If $n\neq m$, then
\begin{eqnarray*}
(T_n,T_m)
&=& \frac{1}{2} \left[ \frac{\sin((n+m)\theta)}{n+m} + \frac{\sin((n-m)\theta)}{n-m} \right]_0^\pi \\
&=& 0,
\end{eqnarray*}
since $\sin(j\theta)=0$ for $\theta=0$ or $\theta=\pi$,
for any integer $j$.
This shows that the equation \ref{eqn-chebyprod} is true
for $n\neq m$.
If, on the other hand, we have $n=m$, then
\begin{eqnarray*}
(T_n,T_m)
&=& \frac{1}{2} \int_0^\pi \left( \cos((n+m)\theta) + 1 \right) d\theta \\
&=& \frac{1}{2} \left[ \frac{\cos((n+m)\theta)}{n+m} + \theta \right]_0^\pi \\
&=& \frac{1}{2},
\end{eqnarray*}
which concludes the proof.
\end{proof}
The first Chebyshev polynomials are presented in the figure \ref{fig-Chebyshevpoly1-5}.
\begin{figure}
endfunction
\end{lstlisting}
The roots of the Chebyshev polynomial of degree $n$ are
\begin{proposition}
(\emph{Orthogonality of Chebyshev polynomials})
The Chebyshev polynomials are orthogonal with
respect to the weight
\begin{eqnarray}
\label{eqn-chebyweight}
w(x)=\frac{1}{\sqrt{1-x^2}},
\end{eqnarray}
for $x\in[-1,1]$.
Moreover,
\begin{eqnarray}
\label{eqn-chebyprod}
(T_n,T_m)
&=& \int_{-1}^1 T_n(x) T_m(x) w(x) dx = \frac{\pi}{2} \delta_{nm},
\end{eqnarray}
for any integers $n$ and $m$.
\end{proposition}
The weight defined by the equation \ref{eqn-chebyweight} does
not correspond to a particular distribution in
probability theory.
\begin{proof}
First, notice that
\begin{eqnarray}
\label{eqn-arccosp}
\arccos'(x)=\frac{1}{\sqrt{1-x^2}},
\end{eqnarray}
for $x\in[-1,1]$.
Using the change of variable $\theta=\arccos(x)$
and the equation \ref{eqn-trigocheby}, we get :
\begin{eqnarray*}
(T_n,T_m)
&=& \int_{-1}^1 T_n(x) T_m(x) \frac{1}{\sqrt{1-x^2}} dx \\
&=& - \int_\pi^0 \cos(n\theta) \cos(m\theta) d\theta \\
&=& \int_0^\pi \cos(n\theta) \cos(m\theta) d\theta \\
&=& \frac{1}{2} \int_0^\pi \left( \cos((n+m)\theta) + \cos((n-m)\theta) \right) d\theta.
\end{eqnarray*}
If $n\neq m$, then
\begin{eqnarray*}
(T_n,T_m)
&=& \frac{1}{2} \left[ \frac{\sin((n+m)\theta)}{n+m} + \frac{\sin((n-m)\theta)}{n-m} \right]_0^\pi \\
&=& 0,
\end{eqnarray*}
since $\sin(j\theta)=0$ for $\theta=0$ or $\theta=\pi$,
for any integer $j$.
This shows that the equation \ref{eqn-chebyprod} is true
for $n\neq m$.
If, on the other hand, we have $n=m$, then
\begin{eqnarray*}
(T_n,T_m)
&=& \frac{1}{2} \int_0^\pi \left( \cos((n+m)\theta) + 1 \right) d\theta \\
&=& \frac{1}{2} \left[ \frac{\cos((n+m)\theta)}{n+m} + \theta \right]_0^\pi \\
&=& \frac{1}{2},
\end{eqnarray*}
which concludes the proof.
\end{proof}
\begin{proposition}
(\emph{Roots of Chebyshev polynomials})
The Chebyshev polynomial of degree $n$ has $n$
simple real roots in $[-1,1]$, which are :
\begin{eqnarray}
\label{eqn-chebyroot}
x_k=\cos\left(\frac{2k-1}{2n} \pi \right),
\end{eqnarray}
for $k=1,...,n$.
\end{proposition}
\begin{proof}
The equation \ref{eqn-chebyroot} implies :
\begin{eqnarray}
\arccos(x_k)=\frac{2i-1}{2n} \pi,
\end{eqnarray}
which implies
\begin{eqnarray*}
T_n(x_k)
&=& \cos\left(n\arccos\left(x_k\right)\right) \\
&=& \cos\left(n\frac{2i-1}{2n} \pi\right) \\
&=& \cos\left(\frac{2i-1}{2} \pi\right) \\
&=&0,
\end{eqnarray*}
which concludes the proof.
\end{proof}
There is one small issue with the equation \ref{eqn-chebyroot}.
When $k=1,...,n$, the roots $x_1,...,x_n$ are from the end of the
interval $[-1,1]$ to the beginning.
This might be surprising for users, which may expect that the
roots are in increasing order.
It can be proved that the roots of the Chebyshev polynomial
of degree $n$ are :
\begin{eqnarray}
\label{eq-chebynodes}
x_i = -\cos\left(\frac{2i-1}{2n}\pi\right),
x_k = -\cos\left(\frac{2k-1}{2n}\pi\right),
\end{eqnarray}
for $i=1,...,n$.
for $k=1,...,n$.
The roots defined by the equation \ref{eq-chebynodes}
are computed in increasing order when $k=1,...,n$ and are
the same as the roots defined by the equation \ref{eqn-chebyroot}.
Indeed, notice that :
$$
\cos(\theta)=-\cos(\pi-\theta),
$$
for any $\theta\in\RR$.
Hence,
\begin{eqnarray*}
-\cos\left(\frac{2k-1}{2n}\pi\right)
&=& \cos\left(\pi - \frac{2k-1}{2n}\pi\right) \\
&=& \cos\left(\frac{2n-2k+1}{2n}\pi\right) \\
&=& \cos\left(\frac{2(n-k+1)-1}{2n}\pi\right) \\
&=& \cos\left(\frac{2j-1}{2n}\pi\right),
\end{eqnarray*}
with $j=n-i+1$.
The corresponding weights in the quadrature rule are
\begin{eqnarray}
\label{eq-chebyweights}
\alpha_i = \frac{1}{n}\pi,
\alpha_k = \frac{\pi}{n},
\end{eqnarray}
for $i=1,...,n$.
for $k=1,...,n$.
\begin{proposition}
(\emph{Extrema of Chebyshev polynomials})
The extrema of the Chebyshev polynomial of degree $n$ are :
\begin{eqnarray}
\label{eqn-chebyext}
x_k=\cos\left(\frac{k\pi}{n}\right),
\end{eqnarray}
for $k=1,...,n$.
They are so that
\begin{eqnarray}
\label{eqn-chebyext2}
T_n(x_k)=(-1)^k,
\end{eqnarray}
for $k=1,...,n$.
\end{proposition}
\begin{proof}
The equation \ref{eqn-defcheby} implies that the
derivative of $T_n$ is :
\begin{eqnarray*}
T_n'(x)
&=& - n\sin(n \arccos(x)) \arccos'(x) \\
&=& \frac{n\sin(n \arccos(x))}{\sqrt{1-x^2}},
\end{eqnarray*}
where the derivative of $\arccos$ is given by the equation \ref{eqn-arccosp}.
Therefore, we have $T_n'(x_k)=0$ if
\begin{eqnarray*}
\sin(\arccos(x_k))=0,
\end{eqnarray*}
for $k=1,...,n$ and $x_k\neq \pm 1$.
This implies that the argument of $\sin$ must be a multiple of
$\pi$ :
\begin{eqnarray*}
n \arccos(x_k) = k\pi,
\end{eqnarray*}
which implies the equation \ref{eqn-chebyext}.
Moreover, the abscissas defined by the equation \ref{eqn-chebyext}
are different from -1 and 1, so that the derivative
of $T_n$ is well defined for these points.
We plug the equation \ref{eqn-chebyext} into the equation \ref{eqn-defcheby}
and get :
\begin{eqnarray*}
T_n(x_k)
&=& \cos\left(k\pi\right) \\
&=& (-1)^k,
\end{eqnarray*}
for $k=1,...,n$, which concludes the proof.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Accuracy of evaluation}
\subsection{Accuracy of evaluation of the polynomial}
In this section, we present the accuracy issues which
appear when we evaluate the orthogonal polynomials with

Archive Download the corresponding diff file

Revision: 330