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: $uplo$ – string (length ≥ 1)

uplo ='U'

, the upper triangle of the matrix

A

is stored.

uplo ='L'

, the lower triangle of the matrix

A

is stored.

Constraint:

uplo ='U'

'L'

2: $kd$ – int64int32nag_int scalar

The number of superdiagonals

k

(and the number of subdiagonals) of the band matrix

A

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

n

n

Hermitian band matrix

A

. The upper or lower triangular part of the Hermitian matrix is stored in the first

kd + 1

rows of the array. The

j

th column of

A

is stored in the

j

th column of the array ab as follows:

The matrix is stored in rows

1

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

See Further Comments below for further details.

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

matrix of right-hand sides

B

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the first dimension of the array b.
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: $ab (ldab :)$ – 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 $ifail = 0$ or $n + 1$ , the factor $U$ or $L$ from the Cholesky factorization $A = U^{H} U$ or $A = L L^{H}$ , 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 $ifail = 0$ or $n + 1$ , the $n$ by $r$ solution matrix $X$ .
3: $rcond$ – double scalar: If $ifail = 0$ or $n + 1$ , an estimate of the reciprocal of the condition number of the matrix $A$ , computed as $rcond = 1 / ({‖A‖}_{1} {‖A^{- 1}‖}_{1})$ .
4: $errbnd$ – 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.
5: $ifail$ – 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.

$ifail < 0 and ifail \neq - 999$: If $ifail = - i$ , the $i$ th argument had an illegal value.

$ifail > 0 and ifail \leq n$: If $ifail = i$ , the leading minor of order $i$ of $A$ is not positive definite. The factorization could not be completed, and the solution has not been computed.

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

$ifail = - 99$: 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.

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_complex_posdef_band_solve (f04cf) 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 = 2

, and

uplo ='U'

On entry:

\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} \end{matrix}

On exit:

\begin{matrix} * & * & u_{13} & u_{24} & u_{35} & u_{46} \\ * & u_{12} & u_{23} & u_{34} & u_{45} & u_{56} \\ u_{11} & u_{22} & u_{33} & u_{44} & u_{55} & u_{66} \end{matrix}

Similarly, if

uplo ='L'

the format of ab is as follows:

On entry:

\begin{matrix} a_{11} & a_{22} & a_{33} & a_{44} & a_{55} & a_{66} \\ a_{21} & a_{32} & a_{43} & a_{54} & a_{65} & * \\ a_{31} & a_{42} & a_{53} & a_{64} & * & * \end{matrix}

On exit:

\begin{matrix} l_{11} & l_{22} & l_{33} & l_{44} & l_{55} & l_{66} \\ l_{21} & l_{32} & l_{43} & l_{54} & l_{65} & * \\ l_{31} & l_{42} & l_{53} & l_{64} & * & * \end{matrix}

Array elements marked

*

need not be set and are not referenced by the function.

Assuming that

n ≫ k

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

A X = B

is approximately

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

for the factorization and

4 n k 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 real analogue of nag_linsys_complex_posdef_band_solve (f04cf) is nag_linsys_real_posdef_band_solve (f04bf).

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 & 54.30 - 56.56 i \\ - 9.93 + 0.88 i & 18.32 + 4.76 i \\ - 27.30 - 0.01 i & - 4.40 + 9.97 i \\ 5.31 + 23.63 i & 9.43 + 1.41 i \end{array}) .

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: f04cf_example

function f04cf_example


fprintf('f04cf example results\n\n');

% Solve complex Ax = b for banded Hermitian A with error and condition number
uplo = 'U';
kd = int64(1);
ab = [ 0    + 0i,      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, 54.30 - 56.56i;
      -9.93 +  0.88i, 18.32 +  4.76i;
     -27.30 -  0.01i, -4.40 +  9.97i;
       5.31 + 23.63i,  9.43 +  1.41i];

[ab, x, rcond, errbnd, ifail] = ...
  f04cf(uplo, kd, 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);

f04cf example results

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

Estimate of condition number
     132.2

Estimate of error bound for computed solutions
   1.5e-14

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

Chapter Contents

Chapter Introduction

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