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<a name="Tridiagonal-Systems"></a>
<div class="header">
<p>
Next: <a href="Triangular-Systems.html#Triangular-Systems" accesskey="n" rel="next">Triangular Systems</a>, Previous: <a href="Householder-solver-for-linear-systems.html#Householder-solver-for-linear-systems" accesskey="p" rel="previous">Householder solver for linear systems</a>, Up: <a href="Linear-Algebra.html#Linear-Algebra" accesskey="u" rel="up">Linear Algebra</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<hr>
<a name="Tridiagonal-Systems-1"></a>
<h3 class="section">14.17 Tridiagonal Systems</h3>
<a name="index-tridiagonal-systems"></a>

<p>The functions described in this section efficiently solve symmetric,
non-symmetric and cyclic tridiagonal systems with minimal storage.
Note that the current implementations of these functions use a variant
of Cholesky decomposition, so the tridiagonal matrix must be positive
definite.  For non-positive definite matrices, the functions return
the error code <code>GSL_ESING</code>.
</p>
<dl>
<dt><a name="index-gsl_005flinalg_005fsolve_005ftridiag"></a>Function: <em>int</em> <strong>gsl_linalg_solve_tridiag</strong> <em>(const gsl_vector * <var>diag</var>, const gsl_vector * <var>e</var>, const gsl_vector * <var>f</var>, const gsl_vector * <var>b</var>, gsl_vector * <var>x</var>)</em></dt>
<dd><p>This function solves the general <em>N</em>-by-<em>N</em> system <em>A x =
b</em> where <var>A</var> is tridiagonal (<em>N &gt;= 2</em>). The super-diagonal and
sub-diagonal vectors <var>e</var> and <var>f</var> must be one element shorter
than the diagonal vector <var>diag</var>.  The form of <var>A</var> for the 4-by-4
case is shown below,
</p>
<div class="example">
<pre class="example">A = ( d_0 e_0  0   0  )
    ( f_0 d_1 e_1  0  )
    (  0  f_1 d_2 e_2 )
    (  0   0  f_2 d_3 )
</pre></div>
</dd></dl>

<dl>
<dt><a name="index-gsl_005flinalg_005fsolve_005fsymm_005ftridiag"></a>Function: <em>int</em> <strong>gsl_linalg_solve_symm_tridiag</strong> <em>(const gsl_vector * <var>diag</var>, const gsl_vector * <var>e</var>, const gsl_vector * <var>b</var>, gsl_vector * <var>x</var>)</em></dt>
<dd><p>This function solves the general <em>N</em>-by-<em>N</em> system <em>A x =
b</em> where <var>A</var> is symmetric tridiagonal (<em>N &gt;= 2</em>).  The off-diagonal vector
<var>e</var> must be one element shorter than the diagonal vector <var>diag</var>.
The form of <var>A</var> for the 4-by-4 case is shown below,
</p>
<div class="example">
<pre class="example">A = ( d_0 e_0  0   0  )
    ( e_0 d_1 e_1  0  )
    (  0  e_1 d_2 e_2 )
    (  0   0  e_2 d_3 )
</pre></div>
</dd></dl>

<dl>
<dt><a name="index-gsl_005flinalg_005fsolve_005fcyc_005ftridiag"></a>Function: <em>int</em> <strong>gsl_linalg_solve_cyc_tridiag</strong> <em>(const gsl_vector * <var>diag</var>, const gsl_vector * <var>e</var>, const gsl_vector * <var>f</var>, const gsl_vector * <var>b</var>, gsl_vector * <var>x</var>)</em></dt>
<dd><p>This function solves the general <em>N</em>-by-<em>N</em> system <em>A x =
b</em> where <var>A</var> is cyclic tridiagonal (<em>N &gt;= 3</em>).  The cyclic super-diagonal and
sub-diagonal vectors <var>e</var> and <var>f</var> must have the same number of
elements as the diagonal vector <var>diag</var>.  The form of <var>A</var> for the
4-by-4 case is shown below,
</p>
<div class="example">
<pre class="example">A = ( d_0 e_0  0  f_3 )
    ( f_0 d_1 e_1  0  )
    (  0  f_1 d_2 e_2 )
    ( e_3  0  f_2 d_3 )
</pre></div>
</dd></dl>


<dl>
<dt><a name="index-gsl_005flinalg_005fsolve_005fsymm_005fcyc_005ftridiag"></a>Function: <em>int</em> <strong>gsl_linalg_solve_symm_cyc_tridiag</strong> <em>(const gsl_vector * <var>diag</var>, const gsl_vector * <var>e</var>, const gsl_vector * <var>b</var>, gsl_vector * <var>x</var>)</em></dt>
<dd><p>This function solves the general <em>N</em>-by-<em>N</em> system <em>A x =
b</em> where <var>A</var> is symmetric cyclic tridiagonal (<em>N &gt;= 3</em>).  The cyclic
off-diagonal vector <var>e</var> must have the same number of elements as the
diagonal vector <var>diag</var>.  The form of <var>A</var> for the 4-by-4 case is
shown below,
</p>
<div class="example">
<pre class="example">A = ( d_0 e_0  0  e_3 )
    ( e_0 d_1 e_1  0  )
    (  0  e_1 d_2 e_2 )
    ( e_3  0  e_2 d_3 )
</pre></div>
</dd></dl>

<hr>
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Next: <a href="Triangular-Systems.html#Triangular-Systems" accesskey="n" rel="next">Triangular Systems</a>, Previous: <a href="Householder-solver-for-linear-systems.html#Householder-solver-for-linear-systems" accesskey="p" rel="previous">Householder solver for linear systems</a>, Up: <a href="Linear-Algebra.html#Linear-Algebra" accesskey="u" rel="up">Linear Algebra</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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