+function alpha = step_length_cgm(r, A, p)
+ ##
+ ## Compute the step length for the conjugate gradient method (CGM).
+ ## The CGM attempts to solve,
+ ##
+ ## Ax = b
+ ##
+ ## or equivalently,
+ ##
+ ## min[phi(x) = (1/2)<Ax,x> - <b,x>]
+ ##
+ ## where ``A`` is positive-definite. In the process, we need to
+ ## compute a number of search directions ``p`` and optimal step
+ ## lengths ``alpha``; i.e.,
+ ##
+ ## x_{k+1} = x_{k} + alpha_{k}*p_{k}
+ ##
+ ## This function computes alpha_{k} in the formula above.
+ ##
+ ## INPUT:
+ ##
+ ## - ``r`` -- The residual, Ax - b, at the current step.
+ ##
+ ## - ``A`` -- The matrix ``A`` in the formulation above.
+ ##
+ ## - ``p`` -- The current search direction.
+ ##
+ ## OUTPUT:
+ ##
+ ## - ``alpha`` -- The minimizer of ``f(x) = x + alpha*p`` along ``p`.
+ ##
+ ## NOTES:
+ ##
+ ## All vectors are assumed to be *column* vectors.
+ ##
+
+ ## A simple calculation should convince you that the gradient of
+ ## phi(x) above is Ax - b == r.
+ alpha = step_length_positive_definite(r, A, p);
+end