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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_lapack_zsptrs (f07qs)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_lapack_zsptrs (f07qs) solves a complex symmetric system of linear equations with multiple right-hand sides,

A X = B,

where

A

has been factorized by nag_lapack_zsptrf (f07qr), using packed storage.

Syntax

[b, info] = f07qs(uplo, ap, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)

[b, info] = nag_lapack_zsptrs(uplo, ap, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_zsptrs (f07qs) is used to solve a complex symmetric system of linear equations

A X = B

, the function must be preceded by a call to nag_lapack_zsptrf (f07qr) which computes the Bunch–Kaufman factorization of

A

, using packed storage.

uplo ='U'

A = P U D U^{T} P^{T}

, where

P

is a permutation matrix,

U

is an upper triangular matrix and

D

is a symmetric block diagonal matrix with

1

1

and

2

2

blocks; the solution

X

is computed by solving

P U D Y = B

and then

U^{T} P^{T} X = Y

uplo ='L'

A = P L D L^{T} P^{T}

, where

L

is a lower triangular matrix; the solution

X

is computed by solving

P L D Y = B

and then

L^{T} P^{T} X = Y

References

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)

Specifies how

A

has been factorized.

$uplo ='U'$: $A = P U D U^{T} P^{T}$ , where $U$ is upper triangular.
$uplo ='L'$: $A = P L D L^{T} P^{T}$ , where $L$ is lower triangular.

Constraint:

uplo ='U'

'L'

2: $ap (:)$ – complex array

The dimension of the array ap must be at least

\max (1, n \times (n + 1) / 2)

The factorization of

A

stored in packed form, as returned by nag_lapack_zsptrf (f07qr).

3: $ipiv (:)$ – int64int32nag_int array

The dimension of the array ipiv must be at least

\max (1, n)

Details of the interchanges and the block structure of

D

, as returned by nag_lapack_zsptrf (f07qr).

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 ap and the second dimension of the array ap. (An error is raised if these dimensions are not equal.)
$n$ , 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.

Constraint: $nrhs_p \geq 0$ .

Output Parameters

1: $b (ldb :)$ – complex array: The first dimension of the array b will be $\max (1, n)$ .
The second dimension of the array b will be $\max (1, nrhs_p)$ .

The $n$ by $r$ solution matrix $X$ .
2: $info$ – 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.

Accuracy

For each right-hand side vector

b

, the computed solution

x

is the exact solution of a perturbed system of equations

(A + E) x = b

, where

if $uplo ='U'$ , $|E| \leq c (n) ε P |U| |D| |U^{T}| P^{T}$ ;
if $uplo ='L'$ , $|E| \leq c (n) ε P |L| |D| |L^{T}| P^{T}$ ,

c (n)

is a modest linear function of

n

, and

ε

is the machine precision

\hat{x}

is the true solution, then the computed solution

x

satisfies a forward error bound of the form

\frac{{‖x - \hat{x}‖}_{\infty}}{{‖x‖}_{\infty}} \leq c (n) cond (A, x) ε

where

cond (A, x) = {‖|A^{- 1}| |A| |x|‖}_{\infty} / {‖x‖}_{\infty} \leq cond (A) = {‖|A^{- 1}| |A|‖}_{\infty} \leq κ_{\infty} (A)

Note that

cond (A, x)

can be much smaller than

cond (A)

Forward and backward error bounds can be computed by calling nag_lapack_zsprfs (f07qv), and an estimate for

κ_{\infty} (A)

(

= κ_{1} (A)

) can be obtained by calling nag_lapack_zspcon (f07qu).

Further Comments

The total number of real floating-point operations is approximately

8 n^{2} r

This function may be followed by a call to nag_lapack_zsprfs (f07qv) to refine the solution and return an error estimate.

The real analogue of this function is nag_lapack_dsptrs (f07pe).

Example

This example solves the system of equations

A X = B

, where

A = (\begin{array}{r} - 0.39 - 0.71 i & 5.14 - 0.64 i & - 7.86 - 2.96 i & 3.80 + 0.92 i \\ 5.14 - 0.64 i & 8.86 + 1.81 i & - 3.52 + 0.58 i & 5.32 - 1.59 i \\ - 7.86 - 2.96 i & - 3.52 + 0.58 i & - 2.83 - 0.03 i & - 1.54 - 2.86 i \\ 3.80 + 0.92 i & 5.32 - 1.59 i & - 1.54 - 2.86 i & - 0.56 + 0.12 i \end{array})

and

B = (\begin{array}{r} - 55.64 + 41.22 i & - 19.09 - 35.97 i \\ - 48.18 + 66.00 i & - 12.08 - 27.02 i \\ - 0.49 - 1.47 i & 6.95 + 20.49 i \\ - 6.43 + 19.24 i & - 4.59 - 35.53 i \end{array}) .

Here

A

is symmetric, stored in packed form, and must first be factorized by nag_lapack_zsptrf (f07qr).

Open in the MATLAB editor: f07qs_example

function f07qs_example


fprintf('f07qs example results\n\n');

% Solve Ax = B, where A is complex symmetric matrix such that the
% lower triangular part is stored in packed format
uplo = 'L';
n = int64(4);
ap = [ -0.39 -  0.71i,   5.14 -  0.64i,  -7.86 - 2.96i,   3.80 + 0.92i, ...
        8.86 +  1.81i,  -3.52 +  0.58i,   5.32 - 1.59i,   ...
       -2.83 -  0.03i,  -1.54 -  2.86i,  ...
       -0.56 + 0.12i];
b  = [-55.64 + 41.22i, -19.09 - 35.97i;
      -48.18 + 66.00i, -12.08 - 27.02i;
       -0.49 -  1.47i,   6.95 + 20.49i;
       -6.43 + 19.24i,  -4.59 - 35.53i];

% Factorize
[apf, ipiv, info] = f07qr( ...
                           uplo, n, ap);
% Solve
[x, info] = f07qs(uplo, apf, ipiv, b);

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

f07qs example results

Solution(s)
   1.0000 - 1.0000i  -2.0000 - 1.0000i
  -2.0000 + 5.0000i   1.0000 - 3.0000i
   3.0000 - 2.0000i   3.0000 + 2.0000i
  -4.0000 + 3.0000i  -1.0000 + 1.0000i

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

Chapter Contents

Chapter Introduction

NAG Toolbox