NAG Library Routine Document

f07bnf  (zgbsv)


    1  Purpose
    7  Accuracy


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


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 nagmk26.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 its LAPACK name zgbsv.


f07bnf (zgbsv) 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.


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


1:     n – IntegerInput
On entry: n, the number of linear equations, i.e., the order of the matrix A.
Constraint: n0.
2:     kl – IntegerInput
On entry: kl, the number of subdiagonals within the band of the matrix A.
Constraint: kl0.
3:     ku – IntegerInput
On entry: ku, the number of superdiagonals within the band of the matrix A.
Constraint: ku0.
4:     nrhs – IntegerInput
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs0.
5:     abldab* – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array ab must be at least max1,n.
On entry: the n by 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
abkl+ku+1+i-jj=Aij  for ​max1,j-kuiminn,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 – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f07bnf (zgbsv) is called.
Constraint: ldab2×kl+ku+1.
7:     ipivn – Integer arrayOutput
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 ipivi. ipivi=i indicates a row interchange was not required.
8:     bldb* – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array b must be at least max1,nrhs.
On entry: the n by r right-hand side matrix B.
On exit: if info=0, the n by r solution matrix X.
9:     ldb – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07bnf (zgbsv) is called.
Constraint: ldbmax1,n.
10:   info – IntegerOutput
On exit: info=0 unless the routine detects an error (see Section 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.


The computed solution for a single right-hand side, x^ , satisfies an equation of the form
A+E x^ = b ,  
E1 = Oε A1  
and ε  is the machine precision. An approximate error bound for the computed solution is given by
x^-x 1 x1 κA E1 A1 ,  
where κA = 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 (zgbsv), f07buf (zgbcon) can be used to estimate the condition number of A  and f07bvf (zgbrfs) can be used to obtain approximate error bounds. Alternatives to f07bnf (zgbsv), which return condition and error estimates directly are f04cbf and f07bpf (zgbsvx).

Parallelism and Performance

f07bnf (zgbsv) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07bnf (zgbsv) 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.

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 (dgbsv).


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.

Program Text

Program Text (f07bnfe.f90)

Program Data

Program Data (f07bnfe.d)

Program Results

Program Results (f07bnfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017