-function x_star = conjugate_gradient_method(A, b, x0, tolerance)
- ##
- ## Solve,
- ##
- ## Ax = b
- ##
- ## or equivalently,
- ##
- ## min [phi(x) = (1/2)*<Ax,x> + <b,x>]
- ##
- ## using 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.
- ##
- ## OUTPUT:
- ##
- ## - ``x_star`` - The solution to Ax=b.
- ##
- ## NOTES:
- ##
- ## All vectors are assumed to be *column* vectors.
- ##
- zero_vector = zeros(length(x0), 1);
+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.
+ %
+ % 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.
+ %
+ % The rather verbose name of this function was chosen to avoid
+ % conflicts with other implementations.
+ %
+ n = length(x0);
+ M = eye(n);