NAG CL Interface
f07pec (dsptrs)

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1 Purpose

f07pec solves a real symmetric indefinite system of linear equations with multiple right-hand sides,
AX=B ,  
where A has been factorized by f07pdc, using packed storage.

2 Specification

#include <nag.h>
void  f07pec (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer nrhs, const double ap[], const Integer ipiv[], double b[], Integer pdb, NagError *fail)
The function may be called by the names: f07pec, nag_lapacklin_dsptrs or nag_dsptrs.

3 Description

f07pec is used to solve a real symmetric indefinite system of linear equations AX=B, the function must be preceded by a call to f07pdc which computes the Bunch–Kaufman factorization of A, using packed storage.
If uplo=Nag_Upper, A=PUDUTPT, where P is a permutation matrix, U is an upper triangular matrix and D is a symmetric block diagonal matrix with 1×1 and 2×2 blocks; the solution X is computed by solving PUDY=B and then UTPTX=Y.
If uplo=Nag_Lower, A=PLDLTPT, where L is a lower triangular matrix; the solution X is computed by solving PLDY=B and then LTPTX=Y.

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 how A has been factorized.
uplo=Nag_Upper
A=PUDUTPT, where U is upper triangular.
uplo=Nag_Lower
A=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: nrhs Integer Input
On entry: r, the number of right-hand sides.
Constraint: nrhs0.
5: ap[dim] const double Input
Note: the dimension, dim, of the array ap must be at least max(1,n×(n+1)/2).
On entry: the factorization of A stored in packed form, as returned by f07pdc.
6: ipiv[dim] const Integer Input
Note: the dimension, dim, of the array ipiv must be at least max(1,n).
On entry: details of the interchanges and the block structure of D, as returned by f07pdc.
7: b[dim] double Input/Output
Note: the dimension, dim, of the array b must be at least
  • max(1,pdb×nrhs) when order=Nag_ColMajor;
  • max(1,n×pdb) when order=Nag_RowMajor.
The (i,j)th element of the matrix B is stored in
  • b[(j-1)×pdb+i-1] when order=Nag_ColMajor;
  • b[(i-1)×pdb+j-1] when order=Nag_RowMajor.
On entry: the n×r right-hand side matrix B.
On exit: the n×r solution matrix X.
8: pdb Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
  • if order=Nag_ColMajor, pdbmax(1,n);
  • if order=Nag_RowMajor, pdbmax(1,nrhs).
9: 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.
On entry, nrhs=value.
Constraint: nrhs0.
On entry, pdb=value.
Constraint: pdb>0.
NE_INT_2
On entry, pdb=value and n=value.
Constraint: pdbmax(1,n).
On entry, pdb=value and nrhs=value.
Constraint: pdbmax(1,nrhs).
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.

7 Accuracy

For each right-hand side vector b, the computed solution x is the exact solution of a perturbed system of equations (A+E)x=b, where c(n) is a modest linear function of n, and ε is the machine precision.
If x^ is the true solution, then the computed solution x satisfies a forward error bound of the form
x-x^ x c(n)cond(A,x)ε  
where cond(A,x)=|A-1||A||x|/xcond(A)=|A-1||A|κ(A).
Note that cond(A,x) can be much smaller than cond(A).
Forward and backward error bounds can be computed by calling f07phc, and an estimate for κ(A) (=κ1(A)) can be obtained by calling f07pgc.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f07pec 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 total number of floating-point operations is approximately 2n2r.
This function may be followed by a call to f07phc to refine the solution and return an error estimate.
The complex analogues of this function are f07psc for Hermitian matrices and f07qsc for symmetric matrices.

10 Example

This example solves the system of equations AX=B, 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 )   and   B= ( -9.50 27.85 -8.38 9.90 -6.07 19.25 -0.96 3.93 ) .  
Here A is symmetric indefinite, stored in packed form, and must first be factorized by f07pdc.

10.1 Program Text

Program Text (f07pece.c)

10.2 Program Data

Program Data (f07pece.d)

10.3 Program Results

Program Results (f07pece.r)