The routine may be called by the names f07baf, nagf_lapacklin_dgbsv or its LAPACK name dgbsv.
f07baf uses the decomposition with partial pivoting and row interchanges to factor as , where is a permutation matrix, is a product of permutation and unit lower triangular matrices with subdiagonals, and is upper triangular with superdiagonals. 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 subdiagonals within the band of the matrix .
3: – IntegerInput
On entry: , the number of superdiagonals within the band of the matrix .
4: – IntegerInput
On entry: , the number of right-hand sides, i.e., the number of columns of the matrix .
5: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array ab
must be at least
On entry: the coefficient matrix .
The matrix is stored in rows to ; the first rows need not be set, more precisely, the element must be stored in
On exit: if , ab is overwritten by details of the factorization.
The upper triangular band matrix , with superdiagonals, is stored in rows to of the array, and the multipliers used to form the matrix are stored in rows to .
6: – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f07baf is called.
7: – 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.
8: – 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 .
9: – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07baf is called.
10: – 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 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.
Following the use of f07baf, f07bgf can be used to estimate the condition number of and f07bhf can be used to obtain approximate error bounds. Alternatives to f07baf, which return condition and error estimates directly are f04bbfandf07bbf.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f07baf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07baf 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:
Array elements marked need not be set and are not referenced by the routine. Array elements marked need not be set, but are defined on exit from the routine and contain the elements , and .
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.