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The following can be used to solve a linear system A*x = b
using the pivoted LU factorization:
[L, U, P] = lu (A); ## now L*U = P*A x = U \ (L \ P) * b;
This is one way to normalize columns of a matrix X to unit norm:
s = norm (X, "columns"); X /= diag (s);
The same can also be accomplished with broadcasting (see Broadcasting):
s = norm (X, "columns"); X ./= s;
The following expression is a way to efficiently calculate the sign of a permutation, given by a permutation vector p. It will also work in earlier versions of Octave, but slowly.
det (eye (length (p))(p, :))
Finally, here’s how to solve a linear system A*x = b
with Tikhonov regularization (ridge regression) using SVD (a skeleton
only):
m = rows (A); n = columns (A); [U, S, V] = svd (A); ## determine the regularization factor alpha ## alpha = … ## transform to orthogonal basis b = U'*b; ## Use the standard formula, replacing A with S. ## S is diagonal, so the following will be very fast and accurate. x = (S'*S + alpha^2 * eye (n)) \ (S' * b); ## transform to solution basis x = V*x;