NAG Library Routine Document

f07bsf  (zgbtrs)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

f07bsf (zgbtrs) solves a complex band system of linear equations with multiple right-hand sides,
AX=B ,  ATX=B   or   AHX=B ,  
where A has been factorized by f07brf (zgbtrf).

2
Specification

Fortran Interface
Subroutine f07bsf ( trans, n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info)
Integer, Intent (In):: n, kl, ku, nrhs, ldab, ipiv(*), ldb
Integer, Intent (Out):: info
Complex (Kind=nag_wp), Intent (In):: ab(ldab,*)
Complex (Kind=nag_wp), Intent (Inout):: b(ldb,*)
Character (1), Intent (In):: trans
C Header Interface
#include nagmk26.h
void  f07bsf_ ( const char *trans, const Integer *n, const Integer *kl, const Integer *ku, const Integer *nrhs, const Complex ab[], const Integer *ldab, const Integer ipiv[], Complex b[], const Integer *ldb, Integer *info, const Charlen length_trans)
The routine may be called by its LAPACK name zgbtrs.

3
Description

f07bsf (zgbtrs) is used to solve a complex band system of linear equations AX=B, ATX=B or AHX=B, the routine must be preceded by a call to f07brf (zgbtrf) which computes the LU factorization of A as A=PLU. The solution is computed by forward and backward substitution.
If trans='N', the solution is computed by solving PLY=B and then UX=Y.
If trans='T', the solution is computed by solving UTY=B and then LTPTX=Y.
If trans='C', the solution is computed by solving UHY=B and then LHPTX=Y.

4
References

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

5
Arguments

1:     trans – Character(1)Input
On entry: indicates the form of the equations.
trans='N'
AX=B is solved for X.
trans='T'
ATX=B is solved for X.
trans='C'
AHX=B is solved for X.
Constraint: trans='N', 'T' or 'C'.
2:     n – IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
3:     kl – IntegerInput
On entry: kl, the number of subdiagonals within the band of the matrix A.
Constraint: kl0.
4:     ku – IntegerInput
On entry: ku, the number of superdiagonals within the band of the matrix A.
Constraint: ku0.
5:     nrhs – IntegerInput
On entry: r, the number of right-hand sides.
Constraint: nrhs0.
6:     abldab* – Complex (Kind=nag_wp) arrayInput
Note: the second dimension of the array ab must be at least max1,n.
On entry: the LU factorization of A, as returned by f07brf (zgbtrf).
7:     ldab – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f07bsf (zgbtrs) is called.
Constraint: ldab2×kl+ku+1.
8:     ipiv* – Integer arrayInput
Note: the dimension of the array ipiv must be at least max1,n.
On entry: the pivot indices, as returned by f07brf (zgbtrf).
9:     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: the n by r solution matrix X.
10:   ldb – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07bsf (zgbtrs) is called.
Constraint: ldbmax1,n.
11:   info – IntegerOutput
On exit: info=0 unless the routine detects an error (see Section 6).

6
Error Indicators and Warnings

info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.

7
Accuracy

For each right-hand side vector b, the computed solution x is the exact solution of a perturbed system of equations A+Ex=b, where
EckεLU ,  
ck is a modest linear function of k=kl+ku+1, and ε is the machine precision. This assumes kn.
If x^ is the true solution, then the computed solution x satisfies a forward error bound of the form
x-x^ x ckcondA,xε  
where condA,x=A-1Ax/xcondA=A-1AκA.
Note that condA,x can be much smaller than condA, and condAH (which is the same as condAT) can be much larger (or smaller) than condA.
Forward and backward error bounds can be computed by calling f07bvf (zgbrfs), and an estimate for κA can be obtained by calling f07buf (zgbcon) with norm='I'.

8
Parallelism and Performance

f07bsf (zgbtrs) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07bsf (zgbtrs) 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 total number of real floating-point operations is approximately 8n2kl+kur, assuming nkl and nku.
This routine may be followed by a call to f07bvf (zgbrfs) to refine the solution and return an error estimate.
The real analogue of this routine is f07bef (dgbtrs).

10
Example

This example solves the system of equations AX=B, where
A= -1.65+2.26i -2.05-0.85i 0.97-2.84i 0.00+0.00i 0.00+6.30i -1.48-1.75i -3.99+4.01i 0.59-0.48i 0.00+0.00i -0.77+2.83i -1.06+1.94i 3.33-1.04i 0.00+0.00i 0.00+0.00i 4.48-1.09i -0.46-1.72i  
and
B= -1.06+21.50i 12.85+02.84i -22.72-53.90i -70.22+21.57i 28.24-38.60i -20.70-31.23i -34.56+16.73i 26.01+31.97i .  
Here A is nonsymmetric and is treated as a band matrix, which must first be factorized by f07brf (zgbtrf).

10.1
Program Text

Program Text (f07bsfe.f90)

10.2
Program Data

Program Data (f07bsfe.d)

10.3
Program Results

Program Results (f07bsfe.r)

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