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: $uplo$ – string (length ≥ 1)

uplo ='U'

, the upper triangle of

A

is stored.

uplo ='L'

, the lower triangle of

A

is stored.

Constraint:

uplo ='U'

'L'

2: $kd$ – int64int32nag_int scalar

k_{d}

, the number of superdiagonals of the matrix

A

uplo ='U'

, or the number of subdiagonals if

uplo ='L'

Constraint:

kd \geq 0

3: $ab (ldab :)$ – double array

The first dimension of the array ab must be at least

kd + 1

The second dimension of the array ab must be at least

\max (1, n)

The upper or lower triangle of the symmetric band matrix

A

The matrix is stored in rows

1

k_{d} + 1

, more precisely,

if $uplo ='U'$ , the elements of the upper triangle of $A$ within the band must be stored with element $A_{i j}$ in $ab (k_{d} + 1 + i - j, j) for \max (1, j - k_{d}) \leq i \leq j$ ;
if $uplo ='L'$ , the elements of the lower triangle of $A$ within the band must be stored with element $A_{i j}$ in $ab (1 + i - j, j) for j \leq i \leq \min (n, j + k_{d}) .$

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 and the second dimension of the array ab.
$n$ , 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 $kd + 1$ .
The second dimension of the array ab will be $\max (1, n)$ .

If $info = 0$ , the triangular factor $U$ or $L$ from the Cholesky factorization $A = U^{T} U$ or $A = L L^{T}$ of the band matrix $A$ , in the same storage format as $A$ .
2: $b (ldb :)$ – 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 $X$ .
3: $info$ – int64int32nag_int scalar: $info = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

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

$info > 0$: The leading minor of order $_$ of $A$ is not positive definite, so the factorization could not be completed, and the solution has 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. See Section 4.4 of Anderson et al. (1999) for further details.

nag_lapack_dpbsvx (f07hb) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, nag_linsys_real_posdef_band_solve (f04bf) solves

A x = b

and returns a forward error bound and condition estimate. nag_linsys_real_posdef_band_solve (f04bf) calls nag_lapack_dpbsv (f07ha) to solve the equations.

Further Comments

When

n ≫ k

, the total number of floating-point operations is approximately

n {(k + 1)}^{2} + 4 n k r

, where

k

is the number of superdiagonals and

r

is the number of right-hand sides.

The complex analogue of this function is nag_lapack_zpbsv (f07hn).

Example

This example solves the equations

A x = b,

where

A

is the symmetric positive definite band matrix

A = (\begin{array}{r} 5.49 & 2.68 & 0 & 0 \\ 2.68 & 5.63 & - 2.39 & 0 \\ 0 & - 2.39 & 2.60 & - 2.22 \\ 0 & 0 & - 2.22 & 5.17 \end{array}) and b = (\begin{matrix} 22.09 \\ 9.31 \\ - 5.24 \\ 11.83 \end{matrix}) .

Details of the Cholesky factorization of

A

are also output.

Open in the MATLAB editor: f07ha_example

function f07ha_example


fprintf('f07ha example results\n\n');

% Symmetric A (one lower/upper off-diagonal) in banded form 
uplo = 'U';
kd = int64(1);
m  = int64(4);
ab = [0,    2.68, -2.39, -2.22;
      5.49, 5.63,  2.6,   5.17];

% RHS
b = [22.09;
      9.31;
     -5.24;
     11.83];

% Solve Ax = b
[abf, x, info] = f07ha( ...
                        uplo, kd, ab, b);

disp('Solution');
disp(x');

kl = int64(0);
[ifail] = x04ce( ...
                 m, m, kl, kd, abf, 'Cholesky factor U');

f07ha example results

Solution
    5.0000   -2.0000   -3.0000    1.0000

 Cholesky factor U
             1          2          3          4
 1      2.3431     1.1438
 2                 2.0789    -1.1497
 3                            1.1306    -1.9635
 4                                       1.1465

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

Chapter Contents

Chapter Introduction

NAG Toolbox