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<title>Cholesky Decomposition - GNU Scientific Library -- Reference Manual</title>
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<h3 class="section">13.5 Cholesky Decomposition</h3>
<p><a name="index-Cholesky-decomposition-1279"></a><a name="index-square-root-of-a-matrix_002c-Cholesky-decomposition-1280"></a><a name="index-matrix-square-root_002c-Cholesky-decomposition-1281"></a>
A symmetric, positive definite square matrix A has a Cholesky
decomposition into a product of a lower triangular matrix L and
its transpose L^T,
<pre class="example"> A = L L^T
</pre>
<p class="noindent">This is sometimes referred to as taking the square-root of a matrix. The
Cholesky decomposition can only be carried out when all the eigenvalues
of the matrix are positive. This decomposition can be used to convert
the linear system A x = b into a pair of triangular systems
(L y = b, L^T x = y), which can be solved by forward and
back-substitution.
<div class="defun">
— Function: int <b>gsl_linalg_cholesky_decomp</b> (<var>gsl_matrix * A</var>)<var><a name="index-gsl_005flinalg_005fcholesky_005fdecomp-1282"></a></var><br>
— Function: int <b>gsl_linalg_complex_cholesky_decomp</b> (<var>gsl_matrix_complex * A</var>)<var><a name="index-gsl_005flinalg_005fcomplex_005fcholesky_005fdecomp-1283"></a></var><br>
<blockquote><p>These functions factorize the positive-definite square matrix
<var>A</var> into the Cholesky decomposition A = L L^T (or
<!-- {$A = L L^{\dagger}$} -->
A = L L^H
for the complex case). On input
only the diagonal and lower-triangular part of the matrix <var>A</var> are
needed. On output the diagonal and lower triangular part of the input
matrix <var>A</var> contain the matrix L. The upper triangular part
of the input matrix contains L^T, the diagonal terms being
identical for both L and L^T. If the matrix is not
positive-definite then the decomposition will fail, returning the
error code <code>GSL_EDOM</code>.
<p>When testing whether a matrix is positive-definite, disable the error
handler first to avoid triggering an error.
</p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_linalg_cholesky_solve</b> (<var>const gsl_matrix * cholesky, const gsl_vector * b, gsl_vector * x</var>)<var><a name="index-gsl_005flinalg_005fcholesky_005fsolve-1284"></a></var><br>
— Function: int <b>gsl_linalg_complex_cholesky_solve</b> (<var>const gsl_matrix_complex * cholesky, const gsl_vector_complex * b, gsl_vector_complex * x</var>)<var><a name="index-gsl_005flinalg_005fcomplex_005fcholesky_005fsolve-1285"></a></var><br>
<blockquote><p>These functions solve the system A x = b using the Cholesky
decomposition of A into the matrix <var>cholesky</var> given by
<code>gsl_linalg_cholesky_decomp</code> or
<code>gsl_linalg_complex_cholesky_decomp</code>.
</p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_linalg_cholesky_svx</b> (<var>const gsl_matrix * cholesky, gsl_vector * x</var>)<var><a name="index-gsl_005flinalg_005fcholesky_005fsvx-1286"></a></var><br>
— Function: int <b>gsl_linalg_complex_cholesky_svx</b> (<var>const gsl_matrix_complex * cholesky, gsl_vector_complex * x</var>)<var><a name="index-gsl_005flinalg_005fcomplex_005fcholesky_005fsvx-1287"></a></var><br>
<blockquote><p>These functions solve the system A x = b in-place using the
Cholesky decomposition of A into the matrix <var>cholesky</var> given
by <code>gsl_linalg_cholesky_decomp</code> or
<code>gsl_linalg_complex_cholesky_decomp</code>. On input <var>x</var> should contain
the right-hand side b, which is replaced by the solution on
output.
</p></blockquote></div>
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