hide long namesshow long names
hide short namesshow short names
Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

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

NAG Toolbox: nag_lapack_dsytrf (f07md)

Purpose

nag_lapack_dsytrf (f07md) computes the Bunch–Kaufman factorization of a real symmetric indefinite matrix.

Syntax

[a, ipiv, info] = f07md(uplo, a, 'n', n)
[a, ipiv, info] = nag_lapack_dsytrf(uplo, a, 'n', n)

Description

nag_lapack_dsytrf (f07md) factorizes a real symmetric matrix AA, using the Bunch–Kaufman diagonal pivoting method. AA is factorized as either A = PUDUTPTA=PUDUTPT if uplo = 'U'uplo='U' or A = PLDLTPTA=PLDLTPT 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 a symmetric 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 preserving symmetry.
This method is suitable for symmetric 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 PUDUTPTPUDUTPT, where UU is upper triangular.
uplo = 'L'uplo='L'
The lower triangular part of AA is stored and AA is factorized as PLDLTPTPLDLTPT, where LL is lower triangular.
Constraint: uplo = 'U'uplo='U' or 'L''L'.
2:     a(lda, : :) – double array
The first dimension of the array a must be at least max (1,n)max(1,n)
The second dimension of the array must be at least max (1,n)max(1,n)
The nn by nn symmetric indefinite matrix AA.
  • If uplo = 'U'uplo='U', the upper triangular part of aa must be stored and the elements of the array below the diagonal are not referenced.
  • If uplo = 'L'uplo='L', the lower triangular part of aa 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 The second dimension of the array a.
nn, the order of the matrix AA.
Constraint: n0n0.

Input Parameters Omitted from the MATLAB Interface

lda work lwork

Output Parameters

1:     a(lda, : :) – double array
The first dimension of the array a will be max (1,n)max(1,n)
The second dimension of the array will be max (1,n)max(1,n)
ldamax (1,n)ldamax(1,n).
The upper or lower triangle of 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( : :) – int64int32nag_int array
Note: the dimension of the array ipiv must be at least max (1,n)max(1,n).
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: a, 4: lda, 5: ipiv, 6: work, 7: lwork, 8: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.
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||UT|PT ,
|E|c(n)εP|U||D||UT|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 or LL are stored in the corresponding columns of the array a, 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 is stored explicitly (except for its unit diagonal elements which are equal to 11).
The total number of floating point operations is approximately (1/3)n313n3.
A call to nag_lapack_dsytrf (f07md) may be followed by calls to the functions:
The complex analogues of this function are nag_lapack_zhetrf (f07mr) for Hermitian matrices and nag_lapack_zsytrf (f07nr) for symmetric matrices.

Example

function nag_lapack_dsytrf_example
uplo = 'L';
a = [2.07, 0, 0, 0;
     3.87, -0.21, 0, 0;
     4.2, 1.87, 1.15, 0;
     -1.15, 0.63, 2.06, -1.81];
[aOut, ipiv, info] = nag_lapack_dsytrf(uplo, a)
 

aOut =

    2.0700         0         0         0
    4.2000    1.1500         0         0
    0.2230    0.8115   -2.5907         0
    0.6537   -0.5960    0.3031    0.4074


ipiv =

                   -3
                   -3
                    3
                    4


info =

                    0


function f07md_example
uplo = 'L';
a = [2.07, 0, 0, 0;
     3.87, -0.21, 0, 0;
     4.2, 1.87, 1.15, 0;
     -1.15, 0.63, 2.06, -1.81];
[aOut, ipiv, info] = f07md(uplo, a)
 

aOut =

    2.0700         0         0         0
    4.2000    1.1500         0         0
    0.2230    0.8115   -2.5907         0
    0.6537   -0.5960    0.3031    0.4074


ipiv =

                   -3
                   -3
                    3
                    4


info =

                    0



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

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2013