NAG FL Interface
f07bnf (zgbsv)

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1 Purpose

f07bnf computes the solution to a complex system of linear equations
AX=B ,  
where A is an n×n band matrix, with kl subdiagonals and ku superdiagonals, and X and B are n×r matrices.

2 Specification

Fortran Interface
Subroutine f07bnf ( n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info)
Integer, Intent (In) :: n, kl, ku, nrhs, ldab, ldb
Integer, Intent (Out) :: ipiv(n), info
Complex (Kind=nag_wp), Intent (Inout) :: ab(ldab,*), b(ldb,*)
C Header Interface
#include <nag.h>
void  f07bnf_ (const Integer *n, const Integer *kl, const Integer *ku, const Integer *nrhs, Complex ab[], const Integer *ldab, Integer ipiv[], Complex b[], const Integer *ldb, Integer *info)
The routine may be called by the names f07bnf, nagf_lapacklin_zgbsv or its LAPACK name zgbsv.

3 Description

f07bnf uses the LU decomposition with partial pivoting and row interchanges to factor A as A=PLU, where P is a permutation matrix, L is a product of permutation and unit lower triangular matrices with kl subdiagonals, and U is upper triangular with (kl+ku) superdiagonals. The factored form of A is then used to solve the system of equations AX=B.

4 References

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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Arguments

1: n Integer Input
On entry: n, the number of linear equations, i.e., the order of the matrix A.
Constraint: n0.
2: kl Integer Input
On entry: kl, the number of subdiagonals within the band of the matrix A.
Constraint: kl0.
3: ku Integer Input
On entry: ku, the number of superdiagonals within the band of the matrix A.
Constraint: ku0.
4: nrhs Integer Input
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs0.
5: ab(ldab,*) Complex (Kind=nag_wp) array Input/Output
Note: the second dimension of the array ab must be at least max(1,n).
On entry: the n×n coefficient matrix A.
The matrix is stored in rows kl+1 to 2kl+ku+1; the first kl rows need not be set, more precisely, the element Aij must be stored in
ab(kl+ku+1+i-j,j)=Aij  for ​max(1,j-ku)imin(n,j+kl).  
See Section 9 for further details.
On exit: if info0, ab is overwritten by details of the factorization.
The upper triangular band matrix U, with kl+ku superdiagonals, is stored in rows 1 to kl+ku+1 of the array, and the multipliers used to form the matrix L are stored in rows kl+ku+2 to 2kl+ku+1.
6: ldab Integer Input
On entry: the first dimension of the array ab as declared in the (sub)program from which f07bnf is called.
Constraint: ldab2×kl+ku+1.
7: ipiv(n) Integer array Output
On exit: if no constraints are violated, the pivot indices that define the permutation matrix P; at the ith step row i of the matrix was interchanged with row ipiv(i). ipiv(i)=i indicates a row interchange was not required.
8: b(ldb,*) Complex (Kind=nag_wp) array Input/Output
Note: the second dimension of the array b must be at least max(1,nrhs).
On entry: the n×r right-hand side matrix B.
On exit: if info=0, the n×r solution matrix X.
9: ldb Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f07bnf is called.
Constraint: ldbmax(1,n).
10: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
Element value of the diagonal is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution could not be computed.

7 Accuracy

The computed solution for a single right-hand side, x^ , satisfies an equation of the form
(A+E) x^ = b ,  
E1 = O(ε) A1  
and ε is the machine precision. An approximate error bound for the computed solution is given by
x^-x1 x1 κ(A) E1 A1 ,  
where κ(A) = A-11 A1 , the condition number of A 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 f07bnf, f07buf can be used to estimate the condition number of A and f07bvf can be used to obtain approximate error bounds. Alternatives to f07bnf, which return condition and error estimates directly are f04cbf and f07bpf.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f07bnf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07bnf 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.

9 Further Comments

The band storage scheme for the array ab is illustrated by the following example, when n=6 , kl=1 , and ku=2 . Storage of the band matrix A in the array ab:
* * * + + + * * a13 a24 a35 a46 * a12 a23 a34 a45 a56 a11 a22 a33 a44 a55 a66 a21 a32 a43 a54 a65 *  
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 u14 , u25 and u36 .
The total number of floating-point operations required to solve the equations AX=B depends upon the pivoting required, but if nkl + ku then it is approximately bounded by O( nkl ( kl + ku ) ) for the factorization and O( n ( 2 kl + ku ) r ) for the solution following the factorization.
The real analogue of this routine is f07baf.

10 Example

This example solves the equations
Ax=b ,  
where A is the band matrix
A = ( -1.65+2.26i -2.05-0.85i 0.97-2.84i 0.00i+0.00 6.30i -1.48-1.75i -3.99+4.01i 0.59-0.48i 0.00i+0.00 -0.77+2.83i -1.06+1.94i 3.33-1.04i 0.00i+0.00 0.00i+0.00 4.48-1.09i -0.46-1.72i )  
b = ( -1.06+21.50i -22.72-53.90i 28.24-38.60i -34.56+16.73i ) .  
Details of the LU factorization of A are also output.

10.1 Program Text

Program Text (f07bnfe.f90)

10.2 Program Data

Program Data (f07bnfe.d)

10.3 Program Results

Program Results (f07bnfe.r)