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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;