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NAG Toolbox: nag_lapack_zhptrf (f07pr)

Purpose

nag_lapack_zhptrf (f07pr) computes the Bunch–Kaufman factorization of a complex Hermitian indefinite matrix, using packed storage.

Syntax

[ap, ipiv, info] = f07pr(uplo, n, ap)
[ap, ipiv, info] = nag_lapack_zhptrf(uplo, n, ap)

Description

nag_lapack_zhptrf (f07pr) factorizes a complex Hermitian matrix AA, using the Bunch–Kaufman diagonal pivoting method and packed storage. AA is factorized as either A = PUDUHPTA=PUDUHPT if uplo = 'U'uplo='U' or A = PLDLHPTA=PLDLHPT if uplo = 'L'uplo='L', where PP is a permutation matrix, UU (or LL) is a unit upper (or lower) triangular matrix and DD is an Hermitian block diagonal matrix with 11 by 11 and 22 by 22 diagonal blocks; UU (or LL) has 22 by 22 unit diagonal blocks corresponding to the 22 by 22 blocks of DD. Row and column interchanges are performed to ensure numerical stability while keeping the matrix Hermitian.
This method is suitable for Hermitian matrices which are not known to be positive definite. If AA is in fact positive definite, no interchanges are performed and no 22 by 22 blocks occur in DD.

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 AA is stored and how AA is to be factorized.
uplo = 'U'uplo='U'
The upper triangular part of AA is stored and AA is factorized as PUDUHPTPUDUHPT, where UU is upper triangular.
uplo = 'L'uplo='L'
The lower triangular part of AA is stored and AA is factorized as PLDLHPTPLDLHPT, where LL is lower triangular.
Constraint: uplo = 'U'uplo='U' or 'L''L'.
2:     n – int64int32nag_int scalar
nn, the order of the matrix AA.
Constraint: n0n0.
3:     ap( : :) – complex array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)max(1,n×(n+1)/2).
The nn by nn Hermitian matrix AA, packed by columns.
More precisely,
  • if uplo = 'U'uplo='U', the upper triangle of AA must be stored with element AijAij in ap(i + j(j1) / 2)api+j(j-1)/2 for ijij;
  • if uplo = 'L'uplo='L', the lower triangle of AA must be stored with element AijAij in ap(i + (2nj)(j1) / 2)api+(2n-j)(j-1)/2 for ijij.

Optional Input Parameters

None.

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     ap( : :) – complex array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)max(1,n×(n+1)/2).
AA stores details of the block diagonal matrix DD and the multipliers used to obtain the factor UU or LL as specified by uplo.
2:     ipiv(n) – int64int32nag_int array
Details of the interchanges and the block structure of DD. More precisely,
  • if ipiv(i) = k > 0ipivi=k>0, diidii is a 11 by 11 pivot block and the iith row and column of AA were interchanged with the kkth row and column;
  • if uplo = 'U'uplo='U' and ipiv(i1) = ipiv(i) = l < 0ipivi-1=ipivi=-l<0,
    (di1,i1di,i1)
    di,i1dii
    di-1,i-1d-i,i-1 d-i,i-1dii is a 22 by 22 pivot block and the (i1)(i-1)th row and column of AA were interchanged with the llth row and column;
  • if uplo = 'L'uplo='L' and ipiv(i) = ipiv(i + 1) = m < 0ipivi=ipivi+1=-m<0,
    (diidi + 1,i)
    di + 1,idi + 1,i + 1
    diidi+1,idi+1,idi+1,i+1 is a 22 by 22 pivot block and the (i + 1)(i+1)th row and column of AA were interchanged with the mmth row and column.
3:     info – int64int32nag_int scalar
info = 0info=0 unless the function detects an error (see Section [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 = iinfo=-i
If info = iinfo=-i, parameter ii had an illegal value on entry. The parameters are numbered as follows:
1: uplo, 2: n, 3: ap, 4: ipiv, 5: info.
W INFO > 0INFO>0
If info = iinfo=i, d(i,i)d(i,i) is exactly zero. The factorization has been completed, but the block diagonal matrix DD is exactly singular, and division by zero will occur if it is used to solve a system of equations.

Accuracy

If uplo = 'U'uplo='U', the computed factors UU and DD are the exact factors of a perturbed matrix A + EA+E, where
|E|c(n)εP|U||D||UH|PT ,
|E|c(n)εP|U||D||UH|PT ,
c(n)c(n) is a modest linear function of nn, and εε is the machine precision.
If uplo = 'L'uplo='L', a similar statement holds for the computed factors LL and DD.

Further Comments

The elements of DD overwrite the corresponding elements of AA; if DD has 22 by 22 blocks, only the upper or lower triangle is stored, as specified by uplo.
The unit diagonal elements of UU or LL and the 22 by 22 unit diagonal blocks are not stored. The remaining elements of UU and LL are stored in the corresponding columns of the array ap, but additional row interchanges must be applied to recover UU or LL explicitly (this is seldom necessary). If ipiv(i) = iipivi=i, for i = 1,2,,ni=1,2,,n (as is the case when AA is positive definite), then UU or LL are stored explicitly in packed form (except for their unit diagonal elements which are equal to 11).
The total number of real floating point operations is approximately (4/3)n343n3.
A call to nag_lapack_zhptrf (f07pr) may be followed by calls to the functions:
The real analogue of this function is nag_lapack_dsptrf (f07pd).

Example

function nag_lapack_zhptrf_example
uplo = 'L';
n = int64(4);
ap = [-1.36;
      1.58 - 0.9i;
      2.21 + 0.21i;
      3.91 - 1.5i;
      -8.87 + 0i;
      -1.84 + 0.03i;
      -1.78 - 1.18i;
      -4.63 + 0i;
      0.11 - 0.11i;
      -1.84 + 0i];
[apOut, ipiv, info] = nag_lapack_zhptrf(uplo, n, ap)
 

apOut =

  -1.3600 + 0.0000i
   3.9100 - 1.5000i
   0.3100 + 0.0433i
  -0.1518 + 0.3743i
  -1.8400 + 0.0000i
   0.5637 + 0.2850i
   0.3397 + 0.0303i
  -5.4176 + 0.0000i
   0.2997 + 0.1578i
  -7.1028 + 0.0000i


ipiv =

                   -4
                   -4
                    3
                    4


info =

                    0


function f07pr_example
uplo = 'L';
n = int64(4);
ap = [-1.36;
      1.58 - 0.9i;
      2.21 + 0.21i;
      3.91 - 1.5i;
      -8.87 + 0i;
      -1.84 + 0.03i;
      -1.78 - 1.18i;
      -4.63 + 0i;
      0.11 - 0.11i;
      -1.84 + 0i];
[apOut, ipiv, info] = f07pr(uplo, n, ap)
 

apOut =

  -1.3600 + 0.0000i
   3.9100 - 1.5000i
   0.3100 + 0.0433i
  -0.1518 + 0.3743i
  -1.8400 + 0.0000i
   0.5637 + 0.2850i
   0.3397 + 0.0303i
  -5.4176 + 0.0000i
   0.2997 + 0.1578i
  -7.1028 + 0.0000i


ipiv =

                   -4
                   -4
                    3
                    4


info =

                    0



PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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