1 function alpha = step_length_positive_definite(g, Q, p)
2 %
3 % Find the minimizer alpha of,
4 %
5 % phi(alpha) = f(x + alpha*p)
6 %
7 % where ``p`` is a descent direction,
8 %
9 % f(x) = (1/2)<Qx,x> - <b,x>
10 %
11 % and ``Q`` is positive-definite.
12 %
13 % The closed-form solution to this problem is given in Nocedal and
14 % Wright, (3.55).
15 %
16 % INPUT:
17 %
18 % - ``g`` -- The gradient of f at x.
19 %
20 % - ``Q`` -- The positive-definite matrix in the definition of
21 % ``f`` above.
22 %
23 % - ``p`` -- The direction in which ``f`` decreases. The line
24 % along which we minimize f(x + alpha*p).
25 %
26 % OUTPUT:
27 %
28 % - ``alpha`` -- The value which causes ``f`` to decrease the
29 % most.
30 %
31 % NOTES:
32 %
33 % All vectors are assumed to be *column* vectors.
34 %
35 denom = (p' * Q * p);
37 % denom is non-negative, since it's a Q-norm. No need to abs() it.
38 if (denom < eps)
39 % Catch divide-by-zeros. If denom is effectively zero, set it to
40 % something tiny instead. This trick is also used in the PCGM.
41 denom = eps;
42 end
44 alpha = -(g' * p)/denom;
45 end