1 function alpha = step_length_cgm(r, A, p)
2 ##
3 ## Compute the step length for the conjugate gradient method (CGM).
4 ## The CGM attempts to solve,
5 ##
6 ## Ax = b
7 ##
8 ## or equivalently,
9 ##
10 ## min[phi(x) = (1/2)<Ax,x> - <b,x>]
11 ##
12 ## where ``A`` is positive-definite. In the process, we need to
13 ## compute a number of search directions ``p`` and optimal step
14 ## lengths ``alpha``; i.e.,
15 ##
16 ## x_{k+1} = x_{k} + alpha_{k}*p_{k}
17 ##
18 ## This function computes alpha_{k} in the formula above.
19 ##
20 ## INPUT:
21 ##
22 ## - ``r`` -- The residual, Ax - b, at the current step.
23 ##
24 ## - ``A`` -- The matrix ``A`` in the formulation above.
25 ##
26 ## - ``p`` -- The current search direction.
27 ##
28 ## OUTPUT:
29 ##
30 ## - ``alpha`` -- The minimizer of ``f(x) = x + alpha*p`` along ``p`.
31 ##
32 ## NOTES:
33 ##
34 ## All vectors are assumed to be *column* vectors.
35 ##
37 ## A simple calculation should convince you that the gradient of
38 ## phi(x) above is Ax - b == r.
39 alpha = step_length_positive_definite(r, A, p);
40 end