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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_linsys_real_tridiag_solve (f04bc)

Purpose

nag_linsys_real_tridiag_solve (f04bc) computes the solution to a real system of linear equations $AX=B$, where $A$ is an $n$ by $n$ tridiagonal matrix and $X$ and $B$ are $n$ by $r$ matrices. An estimate of the condition number of $A$ and an error bound for the computed solution are also returned.

Syntax

[dl, d, du, du2, ipiv, b, rcond, errbnd, ifail] = f04bc(dl, d, du, b, 'n', n, 'nrhs_p', nrhs_p)
[dl, d, du, du2, ipiv, b, rcond, errbnd, ifail] = nag_linsys_real_tridiag_solve(dl, d, du, b, 'n', n, 'nrhs_p', nrhs_p)

Description

The $LU$ decomposition with partial pivoting and row interchanges is used to factor $A$ as $A=PLU$, where $P$ is a permutation matrix, $L$ is unit lower triangular with at most one nonzero subdiagonal element, and $U$ is an upper triangular band matrix with two superdiagonals. The factored form of $A$ is then used to solve the system of equations $AX=B$.
Note that the equations ${A}^{\mathrm{T}}X=B$ may be solved by interchanging the order of the arguments du and dl.

References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

Parameters

Compulsory Input Parameters

1:     $\mathrm{dl}\left(:\right)$ – double array
The dimension of the array dl must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$
Must contain the $\left(n-1\right)$ subdiagonal elements of the matrix $A$.
2:     $\mathrm{d}\left(:\right)$ – double array
The dimension of the array d must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
Must contain the $n$ diagonal elements of the matrix $A$.
3:     $\mathrm{du}\left(:\right)$ – double array
The dimension of the array du must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$
Must contain the $\left(n-1\right)$ superdiagonal elements of the matrix $A$
4:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – double array
The first dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
The $n$ by $r$ matrix of right-hand sides $B$.

Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the first dimension of the array b and the dimension of the array d.
The number of linear equations $n$, i.e., the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathrm{nrhs_p}$int64int32nag_int scalar
Default: the second dimension of the array b.
The number of right-hand sides $r$, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{nrhs_p}}\ge 0$.

Output Parameters

1:     $\mathrm{dl}\left(:\right)$ – double array
The dimension of the array dl will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$
If ${\mathbf{ifail}}\ge {\mathbf{0}}$, dl stores the $\left(n-1\right)$ multipliers that define the matrix $L$ from the $LU$ factorization of $A$.
2:     $\mathrm{d}\left(:\right)$ – double array
The dimension of the array d will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
If ${\mathbf{ifail}}\ge {\mathbf{0}}$, d stores the $n$ diagonal elements of the upper triangular matrix $U$ from the $LU$ factorization of $A$.
3:     $\mathrm{du}\left(:\right)$ – double array
The dimension of the array du will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$
If ${\mathbf{ifail}}\ge {\mathbf{0}}$, du stores the $\left(n-1\right)$ elements of the first superdiagonal of $U$.
4:     $\mathrm{du2}\left({\mathbf{n}}-2\right)$ – double array
If ${\mathbf{ifail}}\ge {\mathbf{0}}$, du2 returns the $\left(n-2\right)$ elements of the second superdiagonal of $U$.
5:     $\mathrm{ipiv}\left({\mathbf{n}}\right)$int64int32nag_int array
If ${\mathbf{ifail}}\ge {\mathbf{0}}$, the pivot indices that define the permutation matrix $P$; at the $i$th step row $i$ of the matrix was interchanged with row ${\mathbf{ipiv}}\left(i\right)$. ${\mathbf{ipiv}}\left(i\right)$ will always be either $i$ or $\left(i+1\right)$; ${\mathbf{ipiv}}\left(i\right)=i$ indicates a row interchange was not required.
6:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – double array
The first dimension of the array b will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array b will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
If ${\mathbf{ifail}}={\mathbf{0}}$ or $\mathbf{n}+{\mathbf{1}}$, the $n$ by $r$ solution matrix $X$.
7:     $\mathrm{rcond}$ – double scalar
If no constraints are violated, an estimate of the reciprocal of the condition number of the matrix $A$, computed as ${\mathbf{rcond}}=1/\left({‖A‖}_{1}{‖{A}^{-1}‖}_{1}\right)$.
8:     $\mathrm{errbnd}$ – double scalar
If ${\mathbf{ifail}}={\mathbf{0}}$ or $\mathbf{n}+{\mathbf{1}}$, an estimate of the forward error bound for a computed solution $\stackrel{^}{x}$, such that ${‖\stackrel{^}{x}-x‖}_{1}/{‖x‖}_{1}\le {\mathbf{errbnd}}$, where $\stackrel{^}{x}$ is a column of the computed solution returned in the array b and $x$ is the corresponding column of the exact solution $X$. If rcond is less than machine precision, then errbnd is returned as unity.
9:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W  ${\mathbf{ifail}}>0 \text{and} {\mathbf{ifail}}={\mathbf{n}}$
Diagonal element $_$ of the upper triangular factor is zero. The factorization has been completed, but the solution could not be computed.
W  ${\mathbf{ifail}}={\mathbf{n}}+1$
A solution has been computed, but rcond is less than machine precision so that the matrix $A$ is numerically singular.
${\mathbf{ifail}}=-1$
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-2$
Constraint: ${\mathbf{nrhs_p}}\ge 0$.
${\mathbf{ifail}}=-9$
Constraint: $\mathit{ldb}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
The integer allocatable memory required is n, and the double allocatable memory required is $2×{\mathbf{n}}$. In this case the factorization and the solution $X$ have been computed, but rcond and errbnd have not been computed.

Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 $A+E x^=b,$
where
 $E1 = Oε A1$
and $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 $x^-x1 x1 ≤ κA E1 A1 ,$
where $\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, the condition number of $A$ with respect to the solution of the linear equations. nag_linsys_real_tridiag_solve (f04bc) uses the approximation ${‖E‖}_{1}=\epsilon {‖A‖}_{1}$ to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

The total number of floating-point operations required to solve the equations $AX=B$ is proportional to $nr$. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.
The complex analogue of nag_linsys_real_tridiag_solve (f04bc) is nag_linsys_complex_tridiag_solve (f04cc).

Example

This example solves the equations
 $AX=B,$
where $A$ is the tridiagonal matrix
 $A= 3.0 2.1 0 0 0 3.4 2.3 -1.0 0 0 0 3.6 -5.0 1.9 0 0 0 7.0 -0.9 8.0 0 0 0 -6.0 7.1 and B= 2.7 6.6 -0.5 10.8 2.6 -3.2 0.6 -11.2 2.7 19.1 .$
An estimate of the condition number of $A$ and an approximate error bound for the computed solutions are also printed.
```function f04bc_example

fprintf('f04bc example results\n\n');

% Solve Ax = b for tridiagonal A with error bound and condition number
du = [          2.1;     -1;      1.9;     8.0];
d  = [ 3.0;     2.3;     -5;     -0.9;     7.1];
dl = [ 3.4;     3.6;      7;     -6];

b  = [ 2.7,     6.6;
-0.5,    10.8;
2.6,    -3.2;
0.6,   -11.2;
2.7,    19.1];

[dl, d, du, du2, ipiv, x, rcond, errbnd, ifail] = ...
f04bc(dl, d, du, b);

fprintf('Solution is x:\n');
disp(x);
fprintf('\nApproximate condition number = %8.3f\n',1/rcond);
fprintf('Error bound on solution      = %11.3e\n',errbnd);

```
```f04bc example results

Solution is x:
-4.0000    5.0000
7.0000   -4.0000
3.0000   -3.0000
-4.0000   -2.0000
-3.0000    1.0000

Approximate condition number =   92.745
Error bound on solution      =   1.030e-14
```