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

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Parameters

Compulsory Input Parameters

1: $kl$ – int64int32nag_int scalar

k_{l}

, the number of subdiagonals within the band of the matrix

A

Constraint:

kl \geq 0

2: $ku$ – int64int32nag_int scalar

k_{u}

, the number of superdiagonals within the band of the matrix

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

coefficient 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

right-hand side matrix

B

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the first dimension of the array b.
$n$ , the number of linear equations, 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.
$r$ , the number of right-hand sides, 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 $info \geq 0$ , 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 no constraints are violated, 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 $info = 0$ , the $n$ by $r$ solution matrix $X$ .
4: $info$ – int64int32nag_int scalar: $info = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

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

$info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

W $info > 0$: Element $_$ of the diagonal is exactly zero. The factorization has been completed, but the factor $U$ is exactly singular, so the solution could not be 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. See Section 4.4 of Anderson et al. (1999) for further details.

Following the use of nag_lapack_dgbsv (f07ba), nag_lapack_dgbcon (f07bg) can be used to estimate the condition number of

A

and nag_lapack_dgbrfs (f07bh) can be used to obtain approximate error bounds. Alternatives to nag_lapack_dgbsv (f07ba), which return condition and error estimates directly are nag_linsys_real_band_solve (f04bb) and nag_lapack_dgbsvx (f07bb).

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 complex analogue of this function is nag_lapack_zgbsv (f07bn).

Example

This example solves the equations

A x = b,

where

A

is the band matrix

A = (\begin{array}{r} - 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{array}) and b = (\begin{matrix} 4.42 \\ 27.13 \\ - 6.14 \\ 10.50 \end{matrix}) .

Details of the

L U

factorization of

A

are also output.

Open in the MATLAB editor: f07ba_example

function f07ba_example


fprintf('f07ba example results\n\n');

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;
     27.13;
     -6.14;
     10.50];

% Solve Ax = B
[LU, ipiv, x, info] = f07ba( ...
                             kl, ku, ab, b);

disp('Solution');
disp(x');
mtitle = 'Details of factorization';
n = int64(size(b,1));
[ifail] = x04ce( ...
                 n, n, kl, kl+ku, LU, mtitle);
fprintf('\n');
disp('Pivot indices');
disp(double(ipiv'));

f07ba example results

Solution
   -2.0000    3.0000    1.0000   -4.0000

 Details of factorization
             1          2          3          4
 1     -6.9800     2.4600    -2.7300    -2.1300
 2      0.0330     2.5600     2.4600     4.0700
 3                 0.9605    -5.9329    -3.8391
 4                            0.8057    -0.7269

Pivot indices
     2     3     3     4

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

Chapter Contents

Chapter Introduction

NAG Toolbox