optimization/step_length_cgm.m

   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   ##
  36
  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