The routine may be called by the names f07bbf, nagf_lapacklin_dgbsvx or its LAPACK name dgbsvx.
f07bbf performs the following steps:
The linear system to be solved may be badly scaled. However, the system can be equilibrated as a first stage by setting . In this case, real scaling factors are computed and these factors then determine whether the system is to be equilibrated. Equilibrated forms of the systems and are
respectively, where and are diagonal matrices, with positive diagonal elements, formed from the computed scaling factors.
When equilibration is used, will be overwritten by and will be overwritten by (or when the solution of is sought).
The matrix , or its scaled form, is copied and factored using the decomposition
where is a permutation matrix, is a unit lower triangular matrix, and is upper triangular.
This stage can be by-passed when a factored matrix (with scaled matrices and scaling factors) are supplied; for example, as provided by a previous call to f07bbf with the same matrix .
3.Condition Number Estimation
The factorization of determines whether a solution to the linear system exists. If some diagonal element of is zero, then is exactly singular, no solution exists and the routine returns with a failure. Otherwise the factorized form of is used to estimate the condition number of the matrix . If the reciprocal of the condition number is less than machine precision then a warning code is returned on final exit.
The (equilibrated) system is solved for ( or ) using the factored form of ().
Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for the computed solution.
6.Construct Solution Matrix
If equilibration was used, the matrix is premultiplied by (if ) or (if or ) so that it solves the original system before equilibration.
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: specifies whether or not the factorized form of the matrix is supplied on entry, and if not, whether the matrix should be equilibrated before it is factorized.
afb and ipiv contain the factorized form of . If , the matrix has been equilibrated with scaling factors given by r and c. ab, afb and ipiv are not modified.
Otherwise, if no constraints are violated, then if , afb returns details of the factorization of the band matrix , and if , afb returns details of the factorization of the equilibrated band matrix (see the description of ab for the form of the equilibrated matrix).
10: – IntegerInput
On entry: the first dimension of the array afb as declared in the (sub)program from which f07bbf is called.
11: – Integer arrayInput/Output
Note: the dimension of the array ipiv
must be at least
Otherwise, if no constraints are violated, ipiv contains 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.
If , the pivot indices are those corresponding to the factorization of the original matrix .
If , the pivot indices are those corresponding to the factorization of of the equilibrated matrix .
On entry: the first dimension of the array b as declared in the (sub)program from which f07bbf is called.
17: – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array x
must be at least
On exit: if or , the solution matrix to the original system of equations. Note that the arrays and are modified on exit if , and the solution to the equilibrated system is if and or , or if or and or .
18: – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which f07bbf is called.
19: – Real (Kind=nag_wp)Output
On exit: if no constraints are violated, an estimate of the reciprocal condition number of the matrix (after equilibration if that is performed), computed as .
20: – Real (Kind=nag_wp) arrayOutput
On exit: if or , an estimate of the forward error bound for each computed solution vector, such that where is the th column of the computed solution returned in the array x and is the corresponding column of the exact solution . The estimate is as reliable as the estimate for rcond, and is almost always a slight overestimate of the true error.
21: – Real (Kind=nag_wp) arrayOutput
On exit: if or , an estimate of the component-wise relative backward error of each computed solution vector (i.e., the smallest relative change in any element of or that makes an exact solution).
22: – Real (Kind=nag_wp) arrayOutput
On exit: if , contains the reciprocal pivot growth factor . If is much less than , then the stability of the factorization of the (equilibrated) matrix could be poor. This also means that the solution , condition estimator rcond, and forward error bound ferr could be unreliable. If the factorization fails with , contains the reciprocal pivot growth factor for the leading info columns of .
23: – Integer arrayWorkspace
24: – 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 and error bounds could not be computed.
is nonsingular, but rcond is less than
machine precision, meaning that the matrix is singular to working precision.
Nevertheless, the solution and error bounds are computed because there
are a number of situations where the computed solution can be more accurate
than the value of rcond would suggest.
For each right-hand side vector , the computed solution is the exact solution of a perturbed system of equations , where
is a modest linear function of , and is the machine precision. See Section 9.3 of Higham (2002) for further details.
If is the true solution, then the computed solution satisfies a forward error bound of the form
If is the th column of , then is returned in and a bound on is returned in . See Section 4.4 of Anderson et al. (1999) for further details.
8Parallelism and Performance
f07bbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07bbf 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 band storage scheme for the array ab is illustrated by the following example, when , , and . Storage of the band matrix in the array ab:
The total number of floating-point operations required to solve the equations depends upon the pivoting required, but if then it is approximately bounded by for the factorization and for the solution following the factorization. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization. The solution is then refined, and the errors estimated, using iterative refinement; see f07bhf for information on the floating-point operations required.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.