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<title>Tridiagonal Systems - GNU Scientific Library -- Reference Manual</title>
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<h3 class="section">13.13 Tridiagonal Systems</h3>
<p><a name="index-tridiagonal-systems-1323"></a>
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>.
<div class="defun">
— Function: int <b>gsl_linalg_solve_tridiag</b> (<var>const gsl_vector * diag, const gsl_vector * e, const gsl_vector * f, const gsl_vector * b, gsl_vector * x</var>)<var><a name="index-gsl_005flinalg_005fsolve_005ftridiag-1324"></a></var><br>
<blockquote><p>This function solves the general N-by-N system A x =
b where <var>A</var> is tridiagonal (<!-- {$N\geq 2$} -->
N >= 2). 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,
<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>
<p class="noindent"></p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_linalg_solve_symm_tridiag</b> (<var>const gsl_vector * diag, const gsl_vector * e, const gsl_vector * b, gsl_vector * x</var>)<var><a name="index-gsl_005flinalg_005fsolve_005fsymm_005ftridiag-1325"></a></var><br>
<blockquote><p>This function solves the general N-by-N system A x =
b where <var>A</var> is symmetric tridiagonal (<!-- {$N\geq 2$} -->
N >= 2). 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,
<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>
</blockquote></div>
<div class="defun">
— Function: int <b>gsl_linalg_solve_cyc_tridiag</b> (<var>const gsl_vector * diag, const gsl_vector * e, const gsl_vector * f, const gsl_vector * b, gsl_vector * x</var>)<var><a name="index-gsl_005flinalg_005fsolve_005fcyc_005ftridiag-1326"></a></var><br>
<blockquote><p>This function solves the general N-by-N system A x =
b where <var>A</var> is cyclic tridiagonal (<!-- {$N\geq 3$} -->
N >= 3). 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,
<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>
</blockquote></div>
<div class="defun">
— Function: int <b>gsl_linalg_solve_symm_cyc_tridiag</b> (<var>const gsl_vector * diag, const gsl_vector * e, const gsl_vector * b, gsl_vector * x</var>)<var><a name="index-gsl_005flinalg_005fsolve_005fsymm_005fcyc_005ftridiag-1327"></a></var><br>
<blockquote><p>This function solves the general N-by-N system A x =
b where <var>A</var> is symmetric cyclic tridiagonal (<!-- {$N\geq 3$} -->
N >= 3). 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,
<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>
</blockquote></div>
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