F07QRF (ZSPTRF) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

NAG Library Routine Document


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

F07QRF (ZSPTRF) computes the Bunch–Kaufman factorization of a complex symmetric matrix, using packed storage.

2  Specification

COMPLEX (KIND=nag_wp)  AP(*)
The routine may be called by its LAPACK name zsptrf.

3  Description

F07QRF (ZSPTRF) 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.

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.
The upper triangular part of A is stored and A is factorized as PUDUTPT, where U is upper triangular.
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:     AP(*) – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array AP must be at least max1,N×N+1/2.
On entry: 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.
On exit: 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:     IPIV(N) – INTEGER arrayOutput
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.
5:     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:
If INFO=-i, the ith parameter had an illegal value. An explanatory message is output, and execution of the program is terminated.
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
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 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 F07QRF (ZSPTRF) may be followed by calls to the routines:
The real analogue of this routine is F07PDF (DSPTRF).

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 ,
using packed storage.

9.1  Program Text

Program Text (f07qrfe.f90)

9.2  Program Data

Program Data (f07qrfe.d)

9.3  Program Results

Program Results (f07qrfe.r)

F07QRF (ZSPTRF) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

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