NAG Library Routine Document
F07PDF (DSPTRF)
1 Purpose
F07PDF (DSPTRF) computes the Bunch–Kaufman factorization of a real symmetric indefinite matrix, using packed storage.
2 Specification
INTEGER |
N, IPIV(N), INFO |
REAL (KIND=nag_wp) |
AP(*) |
CHARACTER(1) |
UPLO |
|
The routine may be called by its
LAPACK
name dsptrf.
3 Description
F07PDF (DSPTRF) 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 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.
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 by 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 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:
N≥0.
- 3: AP(*) – REAL (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 i≤j;
- if UPLO='L', the lower triangle of A must be stored with element Aij in APi+2n-jj-1/2 for i≥j.
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:
- 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
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 (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 (DSPTRF) may be followed by calls to the routines:
The complex analogues of this routine are
F07PRF (ZHPTRF) for Hermitian matrices and
F07QRF (ZSPTRF) for symmetric matrices.
9 Example
This example computes the Bunch–Kaufman factorization of the matrix
A, where
using packed storage.
9.1 Program Text
Program Text (f07pdfe.f90)
9.2 Program Data
Program Data (f07pdfe.d)
9.3 Program Results
Program Results (f07pdfe.r)