Add some step length functions, untested.

[octave.git] / optimization / step_length_positive_definite.m

diff --git a/optimization/step_length_positive_definite.m b/optimization/step_length_positive_definite.m
new file mode 100644 (file)
index 0000000..b1e5c48
--- /dev/null
+++ b/optimization/step_length_positive_definite.m
@@ -0,0 +1,36 @@
+function alpha = step_length_positive_definite(g, Q, p)
+  ##
+  ## Find the minimizer alpha of,
+  ##
+  ##   phi(alpha) = f(x + alpha*p)
+  ##
+  ## where ``p`` is a descent direction,
+  ##
+  ##   f(x) = (1/2)<Qx,x> - <b,x>
+  ##
+  ## and ``Q`` is positive-definite.
+  ##
+  ## The closed-form solution to this problem is given in Nocedal and
+  ## Wright, (3.55).
+  ##
+  ## INPUT:
+  ##
+  ##   - ``g`` -- The gradient of f.
+  ##
+  ##   - ``Q`` -- The positive-definite matrix in the definition of
+  ##     ``f`` above.
+  ##
+  ##   - ``p`` -- The direction in which ``f`` decreases. The line
+  ##     along which we minimize f(x + alpha*p).
+  ##
+  ## OUTPUT:
+  ##
+  ##   - ``alpha`` -- The value which causes ``f`` to decrease the
+  ##     most.
+  ##
+  ## NOTES:
+  ##
+  ## All vectors are assumed to be *column* vectors.
+  ##
+  alpha = -(g' * p)/(p' * Q * p)
+end