## Used throughout. The PCGM uses the infinity norm as the stopping
## condition, so we had better also.
-max_iterations = 100000;
-tolerance = 1e-11;
+max_iterations = 10000;
+tolerance = 1e-10;
## First a simple example.
A = [5,1,2; ...
# Test again Octave's pcg() function.
for n = [ 5, 10, 25, 50, 100 ]
- A = random_positive_definite_matrix(5, 1000);
- C = random_positive_definite_matrix(5, 1000);
- M = C*C';
+ A = random_positive_definite_matrix(n, 100);
+
+ # Use the cholesky factorization as a preconditioner.
+ Ct = perturb(chol(A));
+ C = Ct';
+ M = Ct*C;
# Assumed by Octave's implementation when you don't supply a
# preconditioner.
- x0 = zeros(5, 1);
- b = unifrnd(-1000, 1000, 5, 1);
- [o_x, o_flag, o_relres, o_iter] = pcg(A, b, tolerance, max_iterations, C, C');
+ x0 = zeros(n, 1);
+ b = unifrnd(-100, 100, n, 1);
+ g = @(x) A*x - b;
+
+ ## pcg() stops when the /relative/ norm falls below tolerance. To
+ ## eliminate the relativity, we divide the tolerance by the
+ ## quantity that pcg() will divide by.
+ [o_x, o_flag, o_relres, o_iter] = pcg(A, b, tolerance/norm(g(x0)), ...
+ max_iterations, C, C');
[x, k] = preconditioned_conjugate_gradient_method(A,
M,
- b,
- x0,
- tolerance,
- max_iterations);
+ b,
+ x0,
+ tolerance,
+ max_iterations);
+
diff = norm(o_x - x, 'inf');
msg = sprintf("Our PCGM agrees with Octave's, n=%d.", n);
- unit_test_equals(msg, true, diff < 2*tolerance);
+ ## There's no good way to choose the tolerance here, since each
+ ## individual algorithm terminates based on the (2,infinity)-norm of
+ ## the gradient. So we use two orders of magnitude.
+ unit_test_equals(msg, true, diff <= sqrt(tolerance));
end