F07NRF (ZSYTRF) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F07NRF (ZSYTRF)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

F07NRF (ZSYTRF) computes the Bunch–Kaufman factorization of a complex symmetric matrix.

2  Specification

SUBROUTINE F07NRF ( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)
INTEGER  N, LDA, IPIV(*), LWORK, INFO
COMPLEX (KIND=nag_wp)  A(LDA,*), WORK(max(1,LWORK))
CHARACTER(1)  UPLO
The routine may be called by its LAPACK name zsytrf.

3  Description

F07NRF (ZSYTRF) 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.

4  References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5  Parameters

1:     UPLO – CHARACTER(1)Input
On entry: 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 – INTEGERInput
On entry: n, the order of the matrix A.
Constraint: N0.
3:     A(LDA,*) – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array A must be at least max1,N.
On entry: 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.
On exit: the upper or lower triangle of A is overwritten by details of the block diagonal matrix D and the multipliers used to obtain the factor U or L as specified by UPLO.
4:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F07NRF (ZSYTRF) is called.
Constraint: LDAmax1,N.
5:     IPIV(*) – INTEGER arrayOutput
Note: the dimension of the array IPIV must be at least max1,N.
On exit: 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.
6:     WORK(max1,LWORK) – COMPLEX (KIND=nag_wp) arrayWorkspace
On exit: if INFO=0, WORK1 contains the minimum value of LWORK required for optimum performance.
7:     LWORK – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which F07NRF (ZSYTRF) is called, unless LWORK=-1, in which case a workspace query is assumed and the routine only calculates the optimal dimension of WORK (using the formula given below).
Suggested value: for optimum performance LWORK should be at least N×nb, where nb is the block size.
Constraint: LWORK1 or LWORK=-1.
8:     INFO – INTEGEROutput
On exit: INFO=0 unless the routine detects an error (see Section 6).

6  Error Indicators and Warnings

Errors or warnings detected by the routine:
INFO<0
If INFO=-i, the ith parameter had an illegal value. An explanatory message is output, and execution of the program is terminated.
INFO>0
If INFO=i, di,i 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.

7  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.

8  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 F07NRF (ZSYTRF) may be followed by calls to the routines:
The real analogue of this routine is F07MDF (DSYTRF).

9  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 .

9.1  Program Text

Program Text (f07nrfe.f90)

9.2  Program Data

Program Data (f07nrfe.d)

9.3  Program Results

Program Results (f07nrfe.r)


F07NRF (ZSYTRF) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012