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20.4 Some Examples of Usage

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 how you normalize columns of a matrix X to unit norm:

 ``` s = norm (X, "columns"); X = diag (s) \ X; ```

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 you 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; ```

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