The routine may be called by the names f07aaf, nagf_lapacklin_dgesv or its LAPACK name dgesv.
f07aaf uses the decomposition with partial pivoting and row interchanges to factor as
where is a permutation matrix, is unit lower triangular, and is upper triangular. 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
1: – IntegerInput
On entry: , the number of linear equations, i.e., the order of the matrix .
2: – IntegerInput
On entry: , the number of right-hand sides, i.e., the number of columns of the matrix .
3: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
On entry: the coefficient matrix .
On exit: the factors and from the factorization ; the unit diagonal elements of are not stored.
4: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f07aaf is called.
5: – Integer arrayOutput
On exit: if no constraints are violated, the pivot indices that define the permutation matrix ; at the th step row of the matrix was interchanged with row . indicates a row interchange was not required.
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 f07aaf 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 factor
is exactly singular, so the solution could not be computed.
The computed solution for a single right-hand side, , satisfies the 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.
Following the use of f07aaf, f07agf can be used to estimate the condition number of and f07ahf can be used to obtain approximate error bounds. Alternatives to f07aaf, which return condition and error estimates directly are f04bafandf07abf.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f07aaf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07aaf 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.