NAG CL Interface
f07pdc (dsptrf)

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1 Purpose

f07pdc computes the Bunch–Kaufman factorization of a real symmetric indefinite matrix, using packed storage.

2 Specification

#include <nag.h>
void  f07pdc (Nag_OrderType order, Nag_UploType uplo, Integer n, double ap[], Integer ipiv[], NagError *fail)
The function may be called by the names: f07pdc, nag_lapacklin_dsptrf or nag_dsptrf.

3 Description

f07pdc 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=Nag_Upper or A=PLDLTPT if uplo=Nag_Lower, 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: order Nag_OrderType Input
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: uplo Nag_UploType Input
On entry: specifies whether the upper or lower triangular part of A is stored and how A is to be factorized.
uplo=Nag_Upper
The upper triangular part of A is stored and A is factorized as PUDUTPT, where U is upper triangular.
uplo=Nag_Lower
The lower triangular part of A is stored and A is factorized as PLDLTPT, where L is lower triangular.
Constraint: uplo=Nag_Upper or Nag_Lower.
3: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
4: ap[dim] double Input/Output
Note: the dimension, dim, of the array ap must be at least max(1,n×(n+1)/2).
On entry: the n×n symmetric matrix A, packed by rows or columns.
The storage of elements Aij depends on the order and uplo arguments as follows:
if order=Nag_ColMajor and uplo=Nag_Upper,
Aij is stored in ap[(j-1)×j/2+i-1], for ij;
if order=Nag_ColMajor and uplo=Nag_Lower,
Aij is stored in ap[(2n-j)×(j-1)/2+i-1], for ij;
if order=Nag_RowMajor and uplo=Nag_Upper,
Aij is stored in ap[(2n-i)×(i-1)/2+j-1], for ij;
if order=Nag_RowMajor and uplo=Nag_Lower,
Aij is stored in ap[(i-1)×i/2+j-1], 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.
5: ipiv[n] Integer Output
On exit: details of the interchanges and the block structure of D. More precisely,
  • if ipiv[i-1]=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=Nag_Upper and ipiv[i-2]=ipiv[i-1]=-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=Nag_Lower and ipiv[i-1]=ipiv[i]=-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.
6: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_SINGULAR
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=Nag_Upper, 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=Nag_Lower, a similar statement holds for the computed factors L and D.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f07pdc 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 function. 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-1]=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 f07pdc may be followed by calls to the functions:
The complex analogues of this function are f07prc for Hermitian matrices and f07qrc 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 (f07pdce.c)

10.2 Program Data

Program Data (f07pdce.d)

10.3 Program Results

Program Results (f07pdce.r)