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NAG Toolbox

NAG Toolbox: nag_lapack_dsytrs (f07me)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dsytrs (f07me) solves a real symmetric indefinite system of linear equations with multiple right-hand sides,
AX=B ,  
where A has been factorized by nag_lapack_dsytrf (f07md).

Syntax

[b, info] = f07me(uplo, a, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_dsytrs(uplo, a, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_dsytrs (f07me) is used to solve a real symmetric indefinite system of linear equations AX=B, this function must be preceded by a call to nag_lapack_dsytrf (f07md) which computes the Bunch–Kaufman factorization of A.
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:     alda: – double array
The first dimension of the array a must be at least max1,n.
The second dimension of the array a must be at least max1,n.
Details of the factorization of A, as returned by nag_lapack_dsytrf (f07md).
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_dsytrf (f07md).
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 arrays a, b and the second dimension of the arrays a, ipiv.
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_dsyrfs (f07mh), and an estimate for κA (=κ1A) can be obtained by calling nag_lapack_dsycon (f07mg).

Further Comments

The total number of floating-point operations is approximately 2n2r.
This function may be followed by a call to nag_lapack_dsyrfs (f07mh) to refine the solution and return an error estimate.
The complex analogues of this function are nag_lapack_zhetrs (f07ms) for Hermitian matrices and nag_lapack_zsytrs (f07ns) 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 and must first be factorized by nag_lapack_dsytrf (f07md).
function f07me_example


fprintf('f07me example results\n\n');

% Indefinite matrix A (lower triangular part stored)
uplo = 'L';
a = [ 2.07,  0,    0,     0; 
      3.87, -0.21, 0,     0;
      4.20,  1.87, 1.15,  0;
     -1.15,  0.63, 2.06, -1.81];

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

% Factorize
[af, ipiv, info] = f07md( ...
                          uplo, a);

% Solve
[x, info] = f07me( ...
                   uplo, af, ipiv, b);

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


f07me example results

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


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