function [x , normrn] = gmres(A,b,maxiter)
% solves A x = b using gmres
% input: A - m by m matrix
%
b - m by 1 vector
% output: x - approximate solution
%
normrn - norm(b-A*x) at each step of algorithm
% Also
%
plot normrn/ norm(b) versus n (step number)
% Example:
%
m = 200;
%
A = eigmat(2,1/2,m);
%
xtrue = randn(m,1);
%
b = A*xtrue;
%
[x , normrn] = gmres(A,b,20);
% CAUTION: Matlab has a built-in function with the same
%
name (gmres). If you want to use the new version
%
of gmres make sure your active folder contains the
%
new version of gmres in file gmres.m.
Q = [];
H = 0;
m = length(A);
normb = norm(b);
normrn=normb;
Q(:,1) = b / normb;
for n = 1:maxiter
% Arnoldi step calculations (Algorithm 33.1 for Trefethen and Bau)
v = A*Q(:,n);
for j = 1:n
H(j,n) = Q(:,j)'* v;
v = v - H(j,n)*Q(:,j);
end
Hn = H(1:n,1:n);
H(n+1,n) = norm(v);
if H(n+1,n) == 0, break, end
% breakdown so stop
Q(:,n+1) = v / H(n+1,n);
e1 = [1;zeros(n,1)];
y = H \ (normb*e1); % This can be done much more quickly
% using Givens rotations.
% For simplicity we just use Matlab's \
normrn = [normrn,norm(H*y-normb*e1)];
% remember residual norm
end
x = Q(:,1:n)*y;
clf
semilogy(0:n,normrn/normb,'--o'),shg
xlabel('step number of gmres algorithm')
ylabel(' norm( b - A * xn ) / norm(b)')
title('convergence of residual in gmres')
grid
% file example35p1.m
% example 35.1 from Trefethen + Bau
m=200;
maxiter = 20;
A=eigmat(2,.5,m);
lam=eig(A);
figure(1)
clf
plot(lam,'x');
axis square
title('eigenvalues of A')
ylabel('imag part')
xlabel('real part')
grid
xtrue=randn(m,1);
b = A*xtrue;
figure(2)
[x , normrn] = gmres(A,b,maxiter);
eigenvalues of A
0.5
0.4
0.3
0.2
imag part
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
1.4
1.6
1.8
2
2.2
real part
2.4
2.6
2.8
convergence of residual in gmres
0
10
-2
norm( b - A * xn ) / norm(b)
10
-4
10
-6
10
-8
10
-10
10
-12
10
-14
10
0
2
4
6
8
10
12
14
step number of gmres algorithm
16
18
20
% file example35p2.m
% example 35.2 from Trefethen + Bau
m=200;
maxiter = 20;
th = (0:m-1)*pi / (m-1);
d=(-2 +2* sin(th))+sqrt(-1)*cos(th);
A=eigmat(2,.5,m)+diag(d);
lam=eig(A);
figure(1)
clf
plot(lam,'x');
axis square
title('eigenvalues of A')
ylabel('imag part')
xlabel('real part')
grid
xtrue=randn(m,1);
b = A*xtrue;
figure(2)
[x , normrn] = gmres(A,b,maxiter);
eigenvalues of A
1.5
1
imag part
0.5
0
-0.5
-1
-1.5
-0.5
0
0.5
1
real part
1.5
2
2.5
convergence of residual in gmres
0
norm( b - A * xn ) / norm(b)
10
-1
10
-2
10
-3
10
0
2
4
6
8
10
12
14
step number of gmres algorithm
16
18
20