The routine may be called by the names f07maf, nagf_lapacklin_dsysv or its LAPACK name dsysv.
f07maf uses the diagonal pivoting method to factor as
if or if ,
where (or ) is a product of permutation and unit upper (lower) triangular matrices, and is symmetric and block diagonal with and diagonal blocks. The factored form of is then used to solve the system of equations .
Note that, in general, different permutations (pivot sequences) and diagonal block structures are obtained for or
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
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 second dimension of the array a
must be at least
On entry: the symmetric matrix .
If , the upper triangular part of must be stored and the elements of the array below the diagonal are not referenced.
If , the lower triangular part of must be stored and the elements of the array above the diagonal are not referenced.
On exit: if , the block diagonal matrix and the multipliers used to obtain the factor or from the factorization or as computed by f07mdf.
5: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f07maf is called.
6: – Integer arrayOutput
Note: the dimension of the array ipiv
must be at least
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.
7: – 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 .
8: – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07maf is called.
On entry: the dimension of the array work as declared in the (sub)program from which f07maf is called.
, and for best performance , where is the optimal block size for f07mdf.
If , a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.
11: – 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) for further details.
f07mbf is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, f04bhf solves and returns a forward error bound and condition estimate. f04bhf calls f07maf to solve the equations.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f07maf 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 f07maf are f07mnf for Hermitian matrices, and f07nnf for symmetric matrices.