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### Parameter Estimation and Inverse Problems, 2nd edition, 2011
### by R. Aster, B. Borchers, C. Thurber
### Generate the m by n system matrix for the Shaw problem
###
### G = shaw.G(n,m)
###
###
### m and n must be even.
### Reference: C. B. Shaw, Jr., "Improvements of the resolution of
### an instrument by numerical solution of an integral equation",
### J. Math. Anal. Appl. 37m 83-112, 1972.
shawG <-function(m,n) {
if( (m %% 2) !=0) { print('m must be even'); return() } #
if( (n%%2)!=0) { print('n must be even'); return() } #
### Initialization.
delta = pi/n
G = matrix(rep(0,times=m*n) ,m,n)
vm=((1:m)-1/2)*pi/m-pi/2
vn=((1:n)-1/2)*pi/n-pi/2
for ( i in 1:m ) {
for ( j in 1:n ) {
s=vm[i]
th=vn[j]
x=pi*(sin(s)+sin(th))
if(x==0)
{
G[i,j]=(cos(s)+cos(th))^2
}
else
{
G[i,j]=(cos(s)+cos(th))^2*(sin(x)/x)^2
}
}
}
G=G * delta
return(G)
}
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