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<h3 class="section">20.4 Some Examples of Usage</h3>
<p>The following can be used to solve a linear system <code>A*x = b</code>
using the pivoted LU factorization:
<pre class="example"> [L, U, P] = lu (A); ## now L*U = P*A
x = U \ L \ P*b;
</pre>
<p class="noindent">This is how you normalize columns of a matrix <var>X</var> to unit norm:
<pre class="example"> s = norm (X, "columns");
X = diag (s) \ X;
</pre>
<p class="noindent">The following expression is a way to efficiently calculate the sign of a
permutation, given by a permutation vector <var>p</var>. It will also work
in earlier versions of Octave, but slowly.
<pre class="example"> det (eye (length (p))(p, :))
</pre>
<p class="noindent">Finally, here's how you solve a linear system <code>A*x = b</code>
with Tikhonov regularization (ridge regression) using SVD (a skeleton only):
<pre class="example"> 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;
</pre>
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