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diff --git a/optimization/step_length_cgm.m b/optimization/step_length_cgm.m
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+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