The routine may be called by the names f07bef, nagf_lapacklin_dgbtrs or its LAPACK name dgbtrs.
3Description
f07bef is used to solve a real band system of linear equations or , the routine must be preceded by a call to f07bdf which computes the factorization of as . The solution is computed by forward and backward substitution.
If , the solution is computed by solving and then .
If or , the solution is computed by solving and then .
4References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5Arguments
1: – Character(1)Input
On entry: indicates the form of the equations.
is solved for .
or
is solved for .
Constraint:
, or .
2: – IntegerInput
On entry: , the order of the matrix .
Constraint:
.
3: – IntegerInput
On entry: , the number of subdiagonals within the band of the matrix .
Constraint:
.
4: – IntegerInput
On entry: , the number of superdiagonals within the band of the matrix .
Constraint:
.
5: – IntegerInput
On entry: , the number of right-hand sides.
Constraint:
.
6: – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array ab
must be at least
.
On entry: the factorization of , as returned by f07bdf.
7: – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f07bef is called.
Constraint:
.
8: – Integer arrayInput
Note: the dimension of the array ipiv
must be at least
.
On entry: the pivot indices, as returned by f07bdf.
9: – 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: the solution matrix .
10: – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07bef is called.
Constraint:
.
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.
7Accuracy
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. This assumes .
If is the true solution, then the computed solution satisfies a forward error bound of the form
where .
Note that can be much smaller than , and can be much larger (or smaller) than .
Forward and backward error bounds can be computed by calling f07bhf, and an estimate for can be obtained by calling f07bgf with .
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f07bef is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07bef 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.
9Further Comments
The total number of floating-point operations is approximately , assuming and .
This routine may be followed by a call to f07bhf to refine the solution and return an error estimate.