F07BSF (ZGBTRS) (PDF version)
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F07 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F07BSF (ZGBTRS)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ 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

SUBROUTINE F07BSF ( TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO)
INTEGER  N, KL, KU, NRHS, LDAB, IPIV(*), LDB, INFO
COMPLEX (KIND=nag_wp)  AB(LDAB,*), B(LDB,*)
CHARACTER(1)  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  Parameters

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:     AB(LDAB,*) – 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:     B(LDB,*) – 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

Errors or warnings detected by the routine:
INFO<0
If INFO=-i, the ith parameter 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  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).

9  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).

9.1  Program Text

Program Text (f07bsfe.f90)

9.2  Program Data

Program Data (f07bsfe.d)

9.3  Program Results

Program Results (f07bsfe.r)


F07BSF (ZGBTRS) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012