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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zhesv (f07mn)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zhesv (f07mn) computes the solution to a complex system of linear equations
AX=B ,  
where A is an n by n Hermitian matrix and X and B are n by r matrices.

Syntax

[a, ipiv, b, info] = f07mn(uplo, a, b, 'n', n, 'nrhs_p', nrhs_p)
[a, ipiv, b, info] = nag_lapack_zhesv(uplo, a, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_zhesv (f07mn) uses the diagonal pivoting method to factor A as A=UDUH if uplo='U' or A=LDLH if uplo='L', where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is Hermitian and block diagonal with 1 by 1 and 2 by 2 diagonal blocks. The factored form of A is then used to solve the system of equations AX=B.

References

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)
If uplo='U', the upper triangle of A is stored.
If uplo='L', the lower triangle of A is stored.
Constraint: uplo='U' or 'L'.
2:     alda: – complex 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.
The n by n Hermitian matrix A.
  • If uplo='U', the upper triangular part of a must be stored and the elements of the array below the diagonal are not referenced.
  • If uplo='L', the lower triangular part of a must be stored and the elements of the array above the diagonal are not referenced.
3:     bldb: – complex 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.
Note: to solve the equations Ax=b, where b is a single right-hand side, b may be supplied as a one-dimensional array with length ldb=max1,n.
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 array a.
n, the number of linear equations, i.e., 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, i.e., the number of columns of the matrix B.
Constraint: nrhs_p0.

Output Parameters

1:     alda: – complex array
The first dimension of the array a will be max1,n.
The second dimension of the array a will be max1,n.
If info=0, the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A=UDUH or A=LDLH as computed by nag_lapack_zhetrf (f07mr).
2:     ipiv: int64int32nag_int array
The dimension of the array ipiv will be max1,n
Details of the interchanges and the block structure of D. More precisely,
  • if ipivi=k>0, dii is a 1 by 1 pivot block and the ith row and column of A were interchanged with the kth row and column;
  • if uplo='U' and ipivi-1=ipivi=-l<0, di-1,i-1d-i,i-1 d-i,i-1dii is a 2 by 2 pivot block and the i-1th row and column of A were interchanged with the lth row and column;
  • if uplo='L' and ipivi=ipivi+1=-m<0, diidi+1,idi+1,idi+1,i+1 is a 2 by 2 pivot block and the i+1th row and column of A were interchanged with the mth row and column.
3:     bldb: – complex array
The first dimension of the array b will be max1,n.
The second dimension of the array b will be max1,nrhs_p.
Note: to solve the equations Ax=b, where b is a single right-hand side, b may be supplied as a one-dimensional array with length ldb=max1,n.
If info=0, the n by r solution matrix X.
4:     info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

   info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
W  info>0
Element _ of the diagonal is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, so the solution could not be computed.

Accuracy

The computed solution for a single right-hand side, x^ , satisfies an equation of the form
A+E x^=b ,  
where
E1 = Oε A1  
and ε  is the machine precision. An approximate error bound for the computed solution is given by
x^-x1 x1 κA E1 A1 ,  
where κA = A-11 A1 , 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_zhesvx (f07mp) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, nag_linsys_complex_herm_solve (f04ch) solves Ax=b  and returns a forward error bound and condition estimate. nag_linsys_complex_herm_solve (f04ch) calls nag_lapack_zhesv (f07mn) to solve the equations.

Further Comments

The total number of floating-point operations is approximately 43 n3 + 8n2r , where r  is the number of right-hand sides.
The real analogue of this function is nag_lapack_dsysv (f07ma). The complex symmetric analogue of this function is nag_lapack_zsysv (f07nn).

Example

This example solves the equations
Ax=b ,  
where A  is the Hermitian matrix
A = -1.84i+0.00 0.11-0.11i -1.78-1.18i 3.91-1.50i 0.11+0.11i -4.63i+0.00 -1.84+0.03i 2.21+0.21i -1.78+1.18i -1.84-0.03i -8.87i+0.00 1.58-0.90i 3.91+1.50i 2.21-0.21i 1.58+0.90i -1.36i+0.00  
and
b = 2.98-10.18i -9.58+03.88i -0.77-16.05i 7.79+05.48i .  
Details of the factorization of A  are also output.
function f07mn_example


fprintf('f07mn example results\n\n');

% Hermitian indefinite matrix A (Upper triangular part stored)
uplo = 'Upper';
a = [-1.84 + 0i,  0.11 - 0.11i, -1.78 - 1.18i,  3.91 - 1.5i;
      0    + 0i, -4.63 + 0i,    -1.84 + 0.03i,  2.21 + 0.21i;
      0    + 0i,  0    + 0i,    -8.87 + 0i,     1.58 - 0.9i;
      0    + 0i,  0    + 0i,     0    + 0i,    -1.36 + 0i];

% RHS
b = [ 2.98 - 10.18i;
     -9.58 +  3.88i;
     -0.77 - 16.05i;
      7.79 +  5.48i];

% Solve
[af, ipiv, x, info] = f07mn( ...
                             uplo, a, b);

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

[ifail] = x04da( ...
                 uplo, 'Non-unit', af, 'Details of factorization');

fprintf('\nPivot indices\n   ');
fprintf('%11d', ipiv);
fprintf('\n');


f07mn example results

Solution
   2.0000 + 1.0000i
   3.0000 - 2.0000i
  -1.0000 + 2.0000i
   1.0000 - 1.0000i

 Details of factorization
             1          2          3          4
 1     -7.1028     0.2997     0.3397    -0.1518
        0.0000     0.1578     0.0303     0.3743

 2                -5.4176     0.5637     0.3100
                   0.0000     0.2850     0.0433

 3                           -1.8400     3.9100
                              0.0000    -1.5000

 4                                      -1.3600
                                         0.0000

Pivot indices
             1          2         -1         -1

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