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<p>
Next: <a href="Zeros-Treatment.html#Zeros-Treatment" accesskey="n" rel="next">Zeros Treatment</a>, Previous: <a href="Function-Support.html#Function-Support" accesskey="p" rel="prev">Function Support</a>, Up: <a href="Diagonal-and-Permutation-Matrices.html#Diagonal-and-Permutation-Matrices" accesskey="u" rel="up">Diagonal and Permutation Matrices</a> &nbsp; [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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<hr>
<a name="Examples-of-Usage"></a>
<h3 class="section">21.4 Examples of Usage</h3>

<p>The following can be used to solve a linear system <code>A*x = b</code>
using the pivoted LU&nbsp;factorization:
</p>
<div class="example">
<pre class="example">  [L, U, P] = lu (A); ## now L*U = P*A
  x = U \ (L \ P) * b;
</pre></div>

<p>This is one way to normalize columns of a matrix <var>X</var> to unit norm:
</p>
<div class="example">
<pre class="example">  s = norm (X, &quot;columns&quot;);
  X /= diag (s);
</pre></div>

<p>The same can also be accomplished with broadcasting (see <a href="Broadcasting.html#Broadcasting">Broadcasting</a>):
</p>
<div class="example">
<pre class="example">  s = norm (X, &quot;columns&quot;);
  X ./= s;
</pre></div>

<p>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.
</p>
<div class="example">
<pre class="example">  det (eye (length (p))(p, :))
</pre></div>

<p>Finally, here&rsquo;s how to solve a linear system <code>A*x = b</code>
with Tikhonov regularization (ridge regression) using SVD (a skeleton
only):
</p>
<div class="example">
<pre class="example">  m = rows (A); n = columns (A);
  [U, S, V] = svd (A);
  ## determine the regularization factor alpha
  ## alpha = &hellip;
  ## 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></div>





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