Move conjugate_gradient_method.m to vanilla_cgm.m.

[octave.git] / optimization / vanilla_cgm.m

diff --git a/optimization/vanilla_cgm.m b/optimization/vanilla_cgm.m
new file mode 100644 (file)
index 0000000..2c94487
--- /dev/null
+++ b/optimization/vanilla_cgm.m
@@ -0,0 +1,63 @@
+function [x, k] = conjugate_gradient_method(A, b, x0, tolerance, max_iterations)
+  %
+  % Solve,
+  %
+  %   Ax = b
+  %
+  % or equivalently,
+  %
+  %   min [phi(x) = (1/2)*<Ax,x> + <b,x>]
+  %
+  % using the conjugate_gradient_method (Algorithm 5.2 in Nocedal and
+  % Wright).
+  %
+  % INPUT:
+  %
+  %   - ``A`` -- The coefficient matrix of the system to solve. Must
+  %     be positive definite.
+  %
+  %   - ``b`` -- The right-hand-side of the system to solve.
+  %
+  %   - ``x0`` -- The starting point for the search.
+  %
+  %   - ``tolerance`` -- How close ``Ax`` has to be to ``b`` (in
+  %     magnitude) before we stop.
+  %
+  %   - ``max_iterations`` -- The maximum number of iterations to perform.
+  %
+  % OUTPUT:
+  %
+  %   - ``x`` - The solution to Ax=b.
+  %
+  %   - ``k`` - The ending value of k; that is, the number of iterations that
+  %     were performed.
+  %
+  % NOTES:
+  %
+  % All vectors are assumed to be *column* vectors.
+  %
+  zero_vector = zeros(length(x0), 1);
+
+  k = 0;
+  x = x0; % Eschew the 'k' suffix on 'x' for simplicity.
+  rk = A*x - b; % The first residual must be computed the hard way.
+  pk = -rk;
+
+  for k = [ 0 : max_iterations ]
+    if (norm(rk) < tolerance)
+       % Success.
+       return;
+    end
+
+    alpha_k = step_length_cgm(rk, A, pk);
+    x_next = x + alpha_k*pk;
+    r_next = rk + alpha_k*A*pk;
+    beta_next = (r_next' * r_next)/(rk' * rk);
+    p_next = -r_next + beta_next*pk;
+
+    k = k + 1;
+    x = x_next;
+    rk = r_next;
+    pk = p_next;
+  end
+end