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NAG Toolbox

NAG Toolbox: nag_lapack_dsptrs (f07pe)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dsptrs (f07pe) solves a real symmetric indefinite system of linear equations with multiple right-hand sides,
AX=B ,  
where A has been factorized by nag_lapack_dsptrf (f07pd), using packed storage.

Syntax

[b, info] = f07pe(uplo, ap, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_dsptrs(uplo, ap, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_dsptrs (f07pe) is used to solve a real symmetric indefinite system of linear equations AX=B, the function must be preceded by a call to nag_lapack_dsptrf (f07pd) which computes the Bunch–Kaufman factorization of A, using packed storage.
If uplo='U', A=PUDUTPT, where P is a permutation matrix, U is an upper triangular matrix and D is a symmetric block diagonal matrix with 1 by 1 and 2 by 2 blocks; the solution X is computed by solving PUDY=B and then UTPTX=Y.
If uplo='L', A=PLDLTPT, where L is a lower triangular matrix; the solution X is computed by solving PLDY=B and then LTPTX=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=PUDUTPT, where U is upper triangular.
uplo='L'
A=PLDLTPT, where L is lower triangular.
Constraint: uplo='U' or 'L'.
2:     ap: – double array
The dimension of the array ap must be at least max1,n×n+1/2
The factorization of A stored in packed form, as returned by nag_lapack_dsptrf (f07pd).
3:     ipiv: int64int32nag_int array
The dimension of the array ipiv must be at least max1,n
Details of the interchanges and the block structure of D, as returned by nag_lapack_dsptrf (f07pd).
4:     bldb: – double array
The first dimension of the array b must be at least max1,n.
The second dimension of the array b must be at least max1,nrhs_p.
The n by 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: n0.
2:     nrhs_p int64int32nag_int scalar
Default: the second dimension of the array b.
r, the number of right-hand sides.
Constraint: nrhs_p0.

Output Parameters

1:     bldb: – double array
The first dimension of the array b will be max1,n.
The second dimension of the array b will be max1,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+Ex=b, where cn is a modest linear function of n, and ε is the machine precision
If x^ is the true solution, then the computed solution x satisfies a forward error bound of the form
x-x^ x cncondA,xε  
where condA,x=A-1Ax/xcondA=A-1AκA.
Note that condA,x can be much smaller than condA.
Forward and backward error bounds can be computed by calling nag_lapack_dsprfs (f07ph), and an estimate for κA (=κ1A) can be obtained by calling nag_lapack_dspcon (f07pg).

Further Comments

The total number of floating-point operations is approximately 2n2r.
This function may be followed by a call to nag_lapack_dsprfs (f07ph) to refine the solution and return an error estimate.
The complex analogues of this function are nag_lapack_zhptrs (f07ps) for Hermitian matrices and nag_lapack_zsptrs (f07qs) for symmetric matrices.

Example

This example solves the system of equations AX=B, where
A= 2.07 3.87 4.20 -1.15 3.87 -0.21 1.87 0.63 4.20 1.87 1.15 2.06 -1.15 0.63 2.06 -1.81   and   B= -9.50 27.85 -8.38 9.90 -6.07 19.25 -0.96 3.93 .  
Here A is symmetric indefinite, stored in packed form, and must first be factorized by nag_lapack_dsptrf (f07pd).
function f07pe_example


fprintf('f07pe example results\n\n');

% Indefinite matrix A (lower triangular part stored in packed format)
uplo = 'L';
n = int64(4);
ap = [2.07;   3.87;   4.20;   -1.15;
             -0.21;   1.87;    0.63;
                      1.15;    2.06;
                              -1.81];

% Factorize
[apf, ipiv, info] = f07pd( ...
                           uplo, n, ap);

% RHS
b = [-9.50, 27.85;
     -8.38,  9.90;
     -6.07, 19.25;
     -0.96,  3.93];

% Solve
[x, info] = f07pe( ...
                   uplo, apf, ipiv, b);

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


f07pe example results

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


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