The routine may be called by the names f07paf, nagf_lapacklin_dspsv or its LAPACK name dspsv.
f07paf uses the diagonal pivoting method to factor as if or if , where (or ) is a product of permutation and unit upper (lower) triangular matrices, is symmetric and block diagonal with and diagonal blocks. The factored form of is then used to solve the system of equations .
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 https://www.netlib.org/lapack/lug
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
1: – Character(1)Input
On entry: if , the upper triangle of is stored.
If , the lower triangle of is stored.
2: – IntegerInput
On entry: , the number of linear equations, i.e., the order of the matrix .
3: – IntegerInput
On entry: , the number of right-hand sides, i.e., the number of columns of the matrix .
4: – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array ap
must be at least
On entry: the symmetric matrix , packed by columns.
if , the upper triangle of must be stored with element in for ;
if , the lower triangle of must be stored with element in for .
On exit: the block diagonal matrix and the multipliers used to obtain the factor or from the factorization or as computed by f07pdf, stored as a packed triangular matrix in the same storage format as .
5: – Integer arrayOutput
On exit: details of the interchanges and the block structure of . More precisely,
if , is a pivot block and the th row and column of were interchanged with the th row and column;
if and , is a pivot block and the th row and column of were interchanged with the th row and column;
if and , is a pivot block and the th row and column of were interchanged with the th row and column.
6: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array b
must be at least
On entry: the right-hand side matrix .
On exit: if , the solution matrix .
7: – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07paf is called.
8: – IntegerOutput
On exit: unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
Element of the diagonal is exactly zero.
The factorization has been completed, but the block diagonal matrix
is exactly singular, so the solution could not be computed.
The computed solution for a single right-hand side, , satisfies an equation of the form
and is the machine precision. An approximate error bound for the computed solution is given by
where , the condition number of 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 and returns a forward error bound and condition estimate. f04bjf calls f07paf to solve the equations.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
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.
The total number of floating-point operations is approximately , where is the number of right-hand sides.
The complex analogues of f07paf are f07pnf for Hermitian matrices, and f07qnf for symmetric matrices.