NAG FL Interface
f07pdf (dsptrf)

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1 Purpose

f07pdf computes the Bunch–Kaufman factorization of a real symmetric indefinite matrix, using packed storage.

2 Specification

Fortran Interface
Subroutine f07pdf ( uplo, n, ap, ipiv, info)
Integer, Intent (In) :: n
Integer, Intent (Out) :: ipiv(n), info
Real (Kind=nag_wp), Intent (Inout) :: ap(*)
Character (1), Intent (In) :: uplo
C Header Interface
#include <nag.h>
void  f07pdf_ (const char *uplo, const Integer *n, double ap[], Integer ipiv[], Integer *info, const Charlen length_uplo)
The routine may be called by the names f07pdf, nagf_lapacklin_dsptrf or its LAPACK name dsptrf.

3 Description

f07pdf factorizes a real 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 1×1 and 2×2 diagonal blocks; U (or L) has 2×2 unit diagonal blocks corresponding to the 2×2 blocks of D. 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 A is in fact positive definite, no interchanges are performed and no 2×2 blocks occur in D.

4 References

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

5 Arguments

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 Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
3: ap(*) Real (Kind=nag_wp) array Input/Output
Note: the dimension of the array ap must be at least max(1,n×(n+1)/2).
On entry: the n×n symmetric matrix A, packed by columns.
More precisely,
  • if uplo='U', the upper triangle of A must be stored with element Aij in ap(i+j(j-1)/2) for ij;
  • if uplo='L', the lower triangle of A must be stored with element Aij in ap(i+(2n-j)(j-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 array Output
On exit: details of the interchanges and the block structure of D. More precisely,
  • if ipiv(i)=k>0, dii is a 1×1 pivot block and the ith row and column of A were interchanged with the kth row and column;
  • if uplo='U' and ipiv(i-1)=ipiv(i)=-l<0, (di-1,i-1d¯i,i-1 d¯i,i-1dii ) is a 2×2 pivot block and the (i-1)th row and column of A were interchanged with the lth row and column;
  • if uplo='L' and ipiv(i)=ipiv(i+1)=-m<0, (diidi+1,idi+1,idi+1,i+1) is a 2×2 pivot block and the (i+1)th row and column of A were interchanged with the mth row and column.
5: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
Element value 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.

7 Accuracy

If uplo='U', the computed factors U and D are the exact factors of a perturbed matrix A+E, where
|E|c(n)εP|U||D||UT|PT ,  
c(n) 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 Parallelism and Performance

f07pdf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The elements of D overwrite the corresponding elements of A; if D has 2×2 blocks, only the upper or lower triangle is stored, as specified by uplo.
The unit diagonal elements of U or L and the 2×2 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 ipiv(i)=i, for i=1,2,,n (as is the case when A is positive definite), 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 floating-point operations is approximately 13n3.
A call to f07pdf may be followed by calls to the routines:
The complex analogues of this routine are f07prf for Hermitian matrices and f07qrf for symmetric matrices.

10 Example

This example computes the Bunch–Kaufman factorization of the matrix A, where
A= ( 2.07 3.87 4.20 -1.15 3.87 -0.21 1.87 0.63 4.20 1.87 1.15 2.06 -1.15 0.63 2.06 -1.81 ) ,  
using packed storage.

10.1 Program Text

Program Text (f07pdfe.f90)

10.2 Program Data

Program Data (f07pdfe.d)

10.3 Program Results

Program Results (f07pdfe.r)