shaw

(PublicToolbox/regutools/shaw.m in BrainStorm 2.0 (Alpha))

Function Synopsis

[A,b,x] = shaw(n)

Help Text

SHAW Test problem: one-dimensional iimage restoration model.

 [A,b,x] = shaw(n)

 Discretization of a first kind Fredholm integral equation with
 [-pi/2,pi/2] as both integration intervals.  The kernel K and
 the solution f, which are given by
    K(s,t) = (cos(s) + cos(t))*(sin(u)/u)^2
    u = pi*(sin(s) + sin(t))
    f(t) = a1*exp(-c1*(t - t1)^2) + a2*exp(-c2*(t - t2)^2) ,
 are discretized by simple quadrature to produce A and x.
 Then the discrete right-hand b side is produced as b = A*x.

 The order n must be even.

Cross-Reference Information

regudemo                           C:\BrainStorm_2001\PublicToolbox\regutools\regudemo.m

Listing of function C:\BrainStorm_2001\PublicToolbox\regutools\shaw.m

function [A,b,x] = shaw(n)
%SHAW Test problem: one-dimensional iimage restoration model.
%
% [A,b,x] = shaw(n)
%
% Discretization of a first kind Fredholm integral equation with
% [-pi/2,pi/2] as both integration intervals.  The kernel K and
% the solution f, which are given by
%    K(s,t) = (cos(s) + cos(t))*(sin(u)/u)^2
%    u = pi*(sin(s) + sin(t))
%    f(t) = a1*exp(-c1*(t - t1)^2) + a2*exp(-c2*(t - t2)^2) ,
% are discretized by simple quadrature to produce A and x.
% Then the discrete right-hand b side is produced as b = A*x.
%
% The order 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. 37 (1972), 83-112.

% Per Christian Hansen, UNI-C, 08/20/91.

% Check input.
if (rem(n,2)~=0), error('The order n must be even'), end

% Initialization.
h = pi/n; A = zeros(n,n);

% Compute the matrix A.
co = cos(-pi/2 + [.5:n-.5]*h);
psi = pi*sin(-pi/2 + [.5:n-.5]*h);
for i=1:n/2
  for j=i:n-i
    ss = psi(i) + psi(j);
    A(i,j) = ((co(i) + co(j))*sin(ss)/ss)^2;
    A(n-j+1,n-i+1) = A(i,j);
  end
  A(i,n-i+1) = (2*co(i))^2;
end
A = A + triu(A,1)'; A = A*h;

% Compute the vectors x and b.
a1 = 2; c1 = 6; t1 =  .8;
a2 = 1; c2 = 2; t2 = -.5;
if (nargout>1)
  x =   a1*exp(-c1*(-pi/2 + [.5:n-.5]'*h - t1).^2) ...
      + a2*exp(-c2*(-pi/2 + [.5:n-.5]'*h - t2).^2);
  b = A*x;
end