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: $ab (ldab :)$ – complex 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 Hermitian 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: $b (ldb :)$ – complex 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: $ab (ldab :)$ – complex 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 $A = U^{H} U$ or $A = L L^{H}$ of the band matrix $A$ , in the same storage format as $A$ .
2: $b (ldb :)$ – complex 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_zpbsvx (f07hp) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, nag_linsys_complex_posdef_band_solve (f04cf) solves

A x = b

and returns a forward error bound and condition estimate. nag_linsys_complex_posdef_band_solve (f04cf) calls nag_lapack_zpbsv (f07hn) to solve the equations.

Further Comments

When

n ≫ k

, the total number of floating-point operations is approximately

4 n {(k + 1)}^{2} + 16 n k r

, where

k

is the number of superdiagonals and

r

is the number of right-hand sides.

The real analogue of this function is nag_lapack_dpbsv (f07ha).

Example

This example solves the equations

A x = b,

where

A

is the Hermitian positive definite band matrix

A = (\begin{array}{r} 9.39 & 1.08 - 1.73 i & 0 & 0 \\ 1.08 + 1.73 i & 1.69 & - 0.04 + 0.29 i & 0 \\ 0 & - 0.04 - 0.29 i & 2.65 & - 0.33 + 2.24 i \\ 0 & 0 & - 0.33 - 2.24 i & 2.17 \end{array})

and

b = (\begin{array}{r} - 12.42 + 68.42 i \\ - 9.93 + 0.88 i \\ - 27.30 - 0.01 i \\ 5.31 + 23.63 i \end{array}) .

Details of the Cholesky factorization of

A

are also output.

Open in the MATLAB editor: f07hn_example

function f07hn_example


fprintf('f07hn example results\n\n');

uplo = 'U';
kd = int64(1);
n  = int64(4);
ab = [0,          1.08 - 1.73i,  -0.04 + 0.29i,  -0.33 + 2.24i;
      9.39 + 0i,  1.69 + 0i,      2.65 + 0i,      2.17 + 0i];

b = [ -12.42 + 68.42i;
       -9.93 +  0.88i;
      -27.3  -  0.01i;
        5.31 + 23.63i];

[abf, x, info] = f07hn( ...
                        uplo, kd, ab, b);

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

kl = int64(0);
[ifail] = x04de( ...
                 n, n, kl, kd, abf, 'Cholesky factor');

f07hn example results

Solution
  -1.0000 + 8.0000i
   2.0000 - 3.0000i
  -4.0000 - 5.0000i
   7.0000 + 6.0000i

 Cholesky factor
             1          2          3          4
 1      3.0643     0.3524
        0.0000    -0.5646

 2                 1.1167    -0.0358
                   0.0000     0.2597

 3                            1.6066    -0.2054
                              0.0000     1.3942

 4                                       0.4289
                                         0.0000

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

Chapter Contents

Chapter Introduction

NAG Toolbox