NAG FL Interface
f07paf (dspsv)

1 Purpose

f07paf computes the solution to a real system of linear equations
AX=B ,  
where A is an n by n symmetric matrix stored in packed format and X and B are n by r matrices.

2 Specification

Fortran Interface
Subroutine f07paf ( uplo, n, nrhs, ap, ipiv, b, ldb, info)
Integer, Intent (In) :: n, nrhs, ldb
Integer, Intent (Out) :: ipiv(n), info
Real (Kind=nag_wp), Intent (Inout) :: ap(*), b(ldb,*)
Character (1), Intent (In) :: uplo
C Header Interface
#include <nag.h>
void  f07paf_ (const char *uplo, const Integer *n, const Integer *nrhs, double ap[], Integer ipiv[], double b[], const Integer *ldb, Integer *info, const Charlen length_uplo)
The routine may be called by the names f07paf, nagf_lapacklin_dspsv or its LAPACK name dspsv.

3 Description

f07paf uses the diagonal pivoting method to factor A as A=UDUT if uplo='U' or A=LDLT if uplo='L', where U (or L) is a product of permutation and unit upper (lower) triangular matrices, D is symmetric and block diagonal with 1 by 1 and 2 by 2 diagonal blocks. The factored form of A is then used to solve the system of equations AX=B.

4 References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

5 Arguments

1: uplo Character(1) Input
On entry: if uplo='U', the upper triangle of A is stored.
If uplo='L', the lower triangle of A is stored.
Constraint: uplo='U' or 'L'.
2: n Integer Input
On entry: n, the number of linear equations, i.e., the order of the matrix A.
Constraint: n0.
3: nrhs Integer Input
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs0.
4: ap* Real (Kind=nag_wp) array Input/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: the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A=UDUT or A=LDLT as computed by f07pdf, stored as a packed triangular matrix in the same storage format as A.
5: ipivn Integer array Output
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.
6: bldb* Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array b must be at least max1,nrhs.
On entry: the n by r right-hand side matrix B.
On exit: if info=0, the n by r solution matrix X.
7: ldb Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f07paf is called.
Constraint: ldbmax1,n.
8: 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, so the solution could not be computed.

7 Accuracy

The computed solution for a single right-hand side, x^ , satisfies an equation of the form
A+E x^=b ,  
E1 = Oε A1  
and ε is the machine precision. An approximate error bound for the computed solution is given by
x^-x1 x1 κA E1 A1 ,  
where κA = A-11 A1 , the condition number of A with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) and Chapter 11 of Higham (2002) for further details.
f07pbf is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, f04bjf solves AX=B and returns a forward error bound and condition estimate. f04bjf calls f07paf to solve the equations.

8 Parallelism and Performance

f07paf 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 total number of floating-point operations is approximately 13 n3 + 2n2r , where r is the number of right-hand sides.
The complex analogues of f07paf are f07pnf for Hermitian matrices, and f07qnf for symmetric matrices.

10 Example

This example solves the equations
Ax=b ,  
where A is the symmetric matrix
A = -1.81 2.06 0.63 -1.15 2.06 1.15 1.87 4.20 0.63 1.87 -0.21 3.87 -1.15 4.20 3.87 2.07   and   b = 0.96 6.07 8.38 9.50 .  
Details of the factorization of A are also output.

10.1 Program Text

Program Text (f07pafe.f90)

10.2 Program Data

Program Data (f07pafe.d)

10.3 Program Results

Program Results (f07pafe.r)