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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 52 "     Gram-Schmidt Orthogonalization: Function Spaces" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "     1) Define \+
inner product \"INN\" below" }}{PARA 0 "" 0 "" {TEXT -1 72 "     2) De
fine elements X[k], k=1...N, and N then repeatedly hit return." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 " \+
     The following inner product and ordered set of basis functions yi
elds normalized" }}{PARA 0 "" 0 "" {TEXT -1 28 "      Chebychev polyno
mials:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 47 "INN:=(F,G)->int(F(x)*G(x)/sqrt(1-x^2),x=-1..1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$INNG:6$%\"FG%\"GG6\"6$%)operatorG%&
arrowGF)-%$intG6$*(-9$6#%\"xG\"\"\"-9%F3F5-%%sqrtG6#,&F5F5*$F4\"\"#!\"
\"F>/F4;F>F5F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "X[1]:=x
->1:X[2]:=x->x:X[3]:=x->x^2:X[4]:=x->x^3:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "N:=4:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 70 "     Define an order for the X[k] on whic
h to base the G-S algorithm. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Y:=[X[1],X[2],X[3],X[4]]:Y(x
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"\"%\"xG*$F%\"\"#*$F%\"\"$
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
85 "     Gram-Schmidt for your choice yields phi[k] which are linear c
ombinations of X[j]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 13 "phi[1]:=Y[1]:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 20 "for i from 2 to N do" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 73 "  phi[i]:=Y[i]-sum(INN(Y[i],phi[n])/INN(phi[n],phi[n])*phi[n],
n=1..(i-1))" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 34 "for j from 1 to N do phi[j](x) od;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$%\"xG\"\"#\"\"\"#!
\"\"F&F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$%\"xG\"\"$\"\"\"F%#!\"
$\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "To obtain an orthonorma
l set we normalize the orthogonal vectors:" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "for k from 1 to N
 do psi[k]:=phi[k](x)/sqrt(INN(phi[k],phi[k])) od:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 31 "for j from 1 to N do psi[j] od;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#*$%#PiG#!\"\"\"\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*(%\"xG\"\"\"\"\"##F%F&%#PiG#!\"\"F&" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,$*(,&*$%\"xG\"\"#\"\"\"#!\"\"F(F)F)F(#F)F(%#PiGF*F(
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,&*$%\"xG\"\"$\"\"\"F'#!\"$\"
\"%F)\"\"##F)F-%#PiG#!\"\"F-F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}}{MARK "13 0 0" 
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