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: $kl$ – int64int32nag_int scalar

The number of subdiagonals

k_{l}

, within the band of

A

Constraint:

kl \geq 0

2: $ku$ – int64int32nag_int scalar

The number of superdiagonals

k_{u}

, within the band of

A

Constraint:

ku \geq 0

3: $ab (ldab :)$ – double array

The first dimension of the array ab must be at least

2 \times kl + ku + 1

The second dimension of the array ab must be at least

\max (1, n)

The

n

n

matrix

A

The matrix is stored in rows

k_{l} + 1

2 k_{l} + k_{u} + 1

; the first

k_{l}

rows need not be set, more precisely, the element

A_{i j}

must be stored in

ab (k_{l} + k_{u} + 1 + i - j, j) = A_{i j} for ​ \max (1, j - k_{u}) \leq i \leq \min (n, j + k_{l}) .

See Further Comments for further details.

4: $b (ldb :)$ – double array

The first dimension of the array b must be at least

\max (1, n)

The second dimension of the array b must be at least

\max (1, nrhs_p)

The

n

r

matrix of right-hand sides

B

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the first dimension of the array b and the second dimension of the array ab.
The number of linear equations $n$ , i.e., the order of the matrix $A$ .

Constraint: $n \geq 0$ .
2: $nrhs_p$ – 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: $nrhs_p \geq 0$ .

Output Parameters

1: $ab (ldab :)$ – double array: The first dimension of the array ab will be $2 \times kl + ku + 1$ .
The second dimension of the array ab will be $\max (1, n)$ .

If $ifail \geq 0$ , ab stores details of the factorization.
The upper triangular band matrix $U$ , with $k_{l} + k_{u}$ superdiagonals, is stored in rows $1$ to $k_{l} + k_{u} + 1$ of the array, and the multipliers used to form the matrix $L$ are stored in rows $k_{l} + k_{u} + 2$ to $2 k_{l} + k_{u} + 1$ .
2: $ipiv (n)$ – int64int32nag_int array: If $ifail \geq 0$ , the pivot indices that define the permutation matrix $P$ ; at the $i$ th step row $i$ of the matrix was interchanged with row $ipiv (i)$ . $ipiv (i) = i$ indicates a row interchange was not required.
3: $b (ldb :)$ – double array: The first dimension of the array b will be $\max (1, n)$ .
The second dimension of the array b will be $\max (1, nrhs_p)$ .

If $ifail = 0$ or $n + 1$ , the $n$ by $r$ solution matrix $X$ .
4: $rcond$ – double scalar: If no constraints are violated, an estimate of the reciprocal of the condition number of the matrix $A$ , computed as $rcond = 1 / ({‖A‖}_{1} {‖A^{- 1}‖}_{1})$ .
5: $errbnd$ – double scalar: If $ifail = 0$ or $n + 1$ , an estimate of the forward error bound for a computed solution $\hat{x}$ , such that ${‖\hat{x} - x‖}_{1} / {‖x‖}_{1} \leq errbnd$ , where $\hat{x}$ 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.
6: $ifail$ – int64int32nag_int scalar: $ifail = 0$ 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 $ifail > 0 and ifail \leq n$: Diagonal element $_$ of the upper triangular factor is zero. The factorization has been completed, but the solution could not be computed.

W $ifail = n + 1$: A solution has been computed, but rcond is less than machine precision so that the matrix $A$ is numerically singular.

$ifail = - 1$: Constraint: $n \geq 0$ .

$ifail = - 2$: Constraint: $kl \geq 0$ .

$ifail = - 3$: Constraint: $ku \geq 0$ .

$ifail = - 4$: Constraint: $nrhs_p \geq 0$ .

$ifail = - 6$: Constraint: $ldab \geq 2 \times kl + ku + 1$ .

$ifail = - 9$: Constraint: $ldb \geq \max (1, n)$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.
The integer allocatable memory required is n, and the double allocatable memory required is $3 \times n$ . 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,

\hat{x}

, satisfies an equation of the form

(A + E) \hat{x} = b,

where

{‖E‖}_{1} = O (ε) {‖A‖}_{1}

and

ε

is the machine precision. An approximate error bound for the computed solution is given by

\frac{{‖\hat{x} - x‖}_{1}}{{‖x‖}_{1}} \leq κ (A) \frac{{‖E‖}_{1}}{{‖A‖}_{1}},

where

κ (A) = {‖A^{- 1}‖}_{1} {‖A‖}_{1}

, the condition number of

A

with respect to the solution of the linear equations. nag_linsys_real_band_solve (f04bb) uses the approximation

{‖E‖}_{1} = ε {‖A‖}_{1}

to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

Further Comments

The band storage scheme for the array ab is illustrated by the following example, when

n = 6

k_{l} = 1

, and

k_{u} = 2

. Storage of the band matrix

A

in the array ab:

\begin{matrix} * & * & * & + & + & + \\ * & * & a_{13} & a_{24} & a_{35} & a_{46} \\ * & a_{12} & a_{23} & a_{34} & a_{45} & a_{56} \\ a_{11} & a_{22} & a_{33} & a_{44} & a_{55} & a_{66} \\ a_{21} & a_{32} & a_{43} & a_{54} & a_{65} & * \end{matrix}

Array elements marked

*

need not be set and are not referenced by the function. Array elements marked + need not be set, but are defined on exit from the function and contain the elements

u_{14}

u_{25}

and

u_{36}

The total number of floating-point operations required to solve the equations

A X = B

depends upon the pivoting required, but if

n ≫ k_{l} + k_{u}

then it is approximately bounded by

O (n k_{l} (k_{l} + k_{u}))

for the factorization and

O (n (2 k_{l} + k_{u}) r)

for the solution following the factorization. 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_band_solve (f04bb) is nag_linsys_complex_band_solve (f04cb).

Example

This example solves the equations

A X = B,

where

A

is the band matrix

A = (\begin{matrix} - 0.23 & 2.54 & - 3.66 & 0 \\ - 6.98 & 2.46 & - 2.73 & - 2.13 \\ 0 & 2.56 & 2.46 & 4.07 \\ 0 & 0 & - 4.78 & - 3.82 \end{matrix}) and B = (\begin{matrix} 4.42 & - 36.01 \\ 27.13 & - 31.67 \\ - 6.14 & - 1.16 \\ 10.50 & - 25.82 \end{matrix}) .

An estimate of the condition number of

A

and an approximate error bound for the computed solutions are also printed.

Open in the MATLAB editor: f04bb_example

function f04bb_example


fprintf('f04bb example results\n\n');

% Solve Ax = b for banded A with error bound and condition number
kl = int64(1);
ku = int64(2);
ab = [ 0,      0,      0,      0;
       0,      0,     -3.66,  -2.13;
       0,      2.54,  -2.73,   4.07;
      -0.23,   2.46,   2.46,  -3.82;
      -6.98,   2.56,  -4.78,   0];
b = [  4.42, -36.01;
      27.13, -31.67;
      -6.14,  -1.16;
      10.5,  -25.82];

[ab, ipiv, x, rcond, errbnd, ifail] = ...
  f04bb(kl, ku, ab, b);

disp('Solution');
disp(x);
disp('Estimate of condition number');
fprintf('%10.1f\n\n',1/rcond);
disp('Estimate of error bound for computed solutions');
fprintf('%10.1e\n\n',errbnd);

f04bb example results

Solution
   -2.0000    1.0000
    3.0000   -4.0000
    1.0000    7.0000
   -4.0000   -2.0000

Estimate of condition number
      56.4

Estimate of error bound for computed solutions
   6.3e-15

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox