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NAG Toolbox: nag_lapack_zsptrf (f07qr)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zsptrf (f07qr) computes the Bunch–Kaufman factorization of a complex symmetric matrix, using packed storage.

Syntax

[ap, ipiv, info] = f07qr(uplo, n, ap)
[ap, ipiv, info] = nag_lapack_zsptrf(uplo, n, ap)

Description

nag_lapack_zsptrf (f07qr) factorizes a complex symmetric matrix A, using the Bunch–Kaufman diagonal pivoting method and packed storage. 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:     n int64int32nag_int scalar
n, the order of the matrix A.
Constraint: n0.
3:     ap: – complex array
The dimension of the array ap must be at least max1,n×n+1/2
The n by n symmetric matrix A, packed by columns.
More precisely,
  • if uplo='U', the upper triangle of A must be stored with element Aij in api+jj-1/2 for ij;
  • if uplo='L', the lower triangle of A must be stored with element Aij in api+2n-jj-1/2 for ij.

Optional Input Parameters

None.

Output Parameters

1:     ap: – complex array
The dimension of the array ap will be max1,n×n+1/2
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:     ipivn int64int32nag_int array
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 overwrite elements in the corresponding columns of 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 are stored explicitly in packed form (except for their unit diagonal elements which are equal to 1).
The total number of real floating-point operations is approximately 43n3.
A call to nag_lapack_zsptrf (f07qr) may be followed by calls to the functions:
The real analogue of this function is nag_lapack_dsptrf (f07pd).

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 ,  
using packed storage.
function f07qr_example


fprintf('f07qr example results\n\n');

% complex symmetric matrix, upper triangle stored in packed format
uplo = 'U';
n    = int64(4);
ap = [-0.56 + 0.12i;
      -1.54 - 2.86i; -2.83 - 0.03i;
       5.32 - 1.59i; -3.52 + 0.58i;  8.86 + 1.81i;
       3.80 + 0.92i; -7.86 - 2.96i;  5.14 - 0.64i; -0.39 - 0.71i];

[apf, ipiv, info] = f07qr( ...
                           uplo, n, ap);

[ifail] = x04dc( ...
                 uplo, 'Non-unit', n, apf, 'Details of factorization');

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


f07qr example results

 Details of factorization
             1          2          3          4
 1     -2.0954    -0.1071    -0.4823     0.4426
       -2.2011    -0.3157     0.0150     0.1936

 2                 4.4079    -0.6078     0.5279
                   5.3991     0.2811    -0.3715

 3                           -2.8300    -7.8600
                             -0.0300    -2.9600

 4                                      -0.3900
                                        -0.7100

Pivot indices
             1          2         -2         -2

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