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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zsytrf (f07nr)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zsytrf (f07nr) computes the Bunch–Kaufman factorization of a complex symmetric matrix.

Syntax

[a, ipiv, info] = f07nr(uplo, a, 'n', n)
[a, ipiv, info] = nag_lapack_zsytrf(uplo, a, 'n', n)

Description

nag_lapack_zsytrf (f07nr) factorizes a complex symmetric matrix A, using the Bunch–Kaufman diagonal pivoting method. A is factorized as either A=PUDUTPT if uplo='U' or A=PLDLTPT if uplo='L', where P is a permutation matrix, U (or L) is a unit upper (or lower) triangular matrix and D is a symmetric block diagonal matrix with 1 by 1 and 2 by 2 diagonal blocks; U (or L) has 2 by 2 unit diagonal blocks corresponding to the 2 by 2 blocks of D. Row and column interchanges are performed to ensure numerical stability while preserving symmetry.

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 whether the upper or lower triangular part of A is stored and how A is to be factorized.
uplo='U'
The upper triangular part of A is stored and A is factorized as PUDUTPT, where U is upper triangular.
uplo='L'
The lower triangular part of A is stored and A is factorized as PLDLTPT, where L is lower triangular.
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 symmetric indefinite 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.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the array a and the second dimension of the array a.
n, the order of the matrix A.
Constraint: n0.

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.
The upper or lower triangle of A stores details of the block diagonal matrix D and the multipliers used to obtain the factor U or L as specified by uplo.
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:     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, and division by zero will occur if it is used to solve a system of equations.

Accuracy

If uplo='U', the computed factors U and D are the exact factors of a perturbed matrix A+E, where
EcnεPUDUTPT ,  
cn is a modest linear function of n, and ε is the machine precision.
If uplo='L', a similar statement holds for the computed factors L and D.

Further Comments

The elements of D overwrite the corresponding elements of A; if D has 2 by 2 blocks, only the upper or lower triangle is stored, as specified by uplo.
The unit diagonal elements of U or L and the 2 by 2 unit diagonal blocks are not stored. The remaining elements of U or L are stored in the corresponding columns of the array a, but additional row interchanges must be applied to recover U or L explicitly (this is seldom necessary). If ipivi=i, for i=1,2,,n, then U or L is stored explicitly (except for its unit diagonal elements which are equal to 1).
The total number of real floating-point operations is approximately 43n3.
A call to nag_lapack_zsytrf (f07nr) may be followed by calls to the functions:
The real analogue of this function is nag_lapack_dsytrf (f07md).

Example

This example computes the Bunch–Kaufman factorization of the matrix A, where
A= -0.39-0.71i 5.14-0.64i -7.86-2.96i 3.80+0.92i 5.14-0.64i 8.86+1.81i -3.52+0.58i 5.32-1.59i -7.86-2.96i -3.52+0.58i -2.83-0.03i -1.54-2.86i 3.80+0.92i 5.32-1.59i -1.54-2.86i -0.56+0.12i .  
function f07nr_example


fprintf('f07nr example results\n\n');

% Symmetric indefinite matrix A (Upper triangular part stored)
uplo = 'L';
a = [-0.39 - 0.71i,  0    + 0i,     0    + 0i,     0    + 0i;
      5.14 - 0.64i,  8.86 + 1.81i,  0    + 0i,     0    + 0i;
     -7.86 - 2.96i, -3.52 + 0.58i, -2.83 - 0.03i,  0    + 0i;
      3.80 + 0.92i,  5.32 - 1.59i, -1.54 - 2.86i, -0.56 + 0.12i];

[af, ipiv, info] = f07nr( ...
                         uplo, a);

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

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


f07nr example results

 Details of factorization
             1          2          3          4
 1     -0.3900
       -0.7100

 2     -7.8600    -2.8300
       -2.9600    -0.0300

 3      0.5279    -0.6078     4.4079
       -0.3715     0.2811     5.3991

 4      0.4426    -0.4823    -0.1071    -2.0954
        0.1936     0.0150    -0.3157    -2.2011

Pivot indices
            -3         -3          3          4

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