F07VSF (ZTBTRS) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F07VSF (ZTBTRS)

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

F07VSF (ZTBTRS) solves a complex triangular band system of linear equations with multiple right-hand sides, AX=B, ATX=B or AHX=B.

2  Specification

SUBROUTINE F07VSF ( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B, LDB, INFO)
INTEGER  N, KD, NRHS, LDAB, LDB, INFO
COMPLEX (KIND=nag_wp)  AB(LDAB,*), B(LDB,*)
CHARACTER(1)  UPLO, TRANS, DIAG
The routine may be called by its LAPACK name ztbtrs.

3  Description

F07VSF (ZTBTRS) solves a complex triangular band system of linear equations AX=B, ATX=B or AHX=B.

4  References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Higham N J (1989) The accuracy of solutions to triangular systems SIAM J. Numer. Anal. 26 1252–1265

5  Parameters

1:     UPLO – CHARACTER(1)Input
On entry: specifies whether A is upper or lower triangular.
UPLO='U'
A is upper triangular.
UPLO='L'
A is lower triangular.
Constraint: UPLO='U' or 'L'.
2:     TRANS – CHARACTER(1)Input
On entry: indicates the form of the equations.
TRANS='N'
The equations are of the form AX=B.
TRANS='T'
The equations are of the form ATX=B.
TRANS='C'
The equations are of the form AHX=B.
Constraint: TRANS='N', 'T' or 'C'.
3:     DIAG – CHARACTER(1)Input
On entry: indicates whether A is a nonunit or unit triangular matrix.
DIAG='N'
A is a nonunit triangular matrix.
DIAG='U'
A is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be 1.
Constraint: DIAG='N' or 'U'.
4:     N – INTEGERInput
On entry: n, the order of the matrix A.
Constraint: N0.
5:     KD – INTEGERInput
On entry: kd, the number of superdiagonals of the matrix A if UPLO='U', or the number of subdiagonals if UPLO='L'.
Constraint: KD0.
6:     NRHS – INTEGERInput
On entry: r, the number of right-hand sides.
Constraint: NRHS0.
7:     AB(LDAB,*) – COMPLEX (KIND=nag_wp) arrayInput
Note: the second dimension of the array AB must be at least max1,N.
On entry: the n by n triangular band matrix A.
The matrix is stored in rows 1 to kd+1, more precisely,
  • if UPLO='U', the elements of the upper triangle of A within the band must be stored with element Aij in ABkd+1+i-jj​ for ​max1,j-kdij;
  • if UPLO='L', the elements of the lower triangle of A within the band must be stored with element Aij in AB1+i-jj​ for ​jiminn,j+kd.
If DIAG='U', the diagonal elements of A are assumed to be 1, and are not referenced.
8:     LDAB – INTEGERInput
On entry: the first dimension of the array AB as declared in the (sub)program from which F07VSF (ZTBTRS) is called.
Constraint: LDABKD+1.
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 F07VSF (ZTBTRS) 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.
INFO>0
If INFO=i, ai,i is exactly zero; A is singular and the solution has not been computed.

7  Accuracy

The solutions of triangular systems of equations are usually computed to high accuracy. See Higham (1989).
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
EckεA ,
ck is a modest linear function of k, and ε is the machine precision.
If x^ is the true solution, then the computed solution x satisfies a forward error bound of the form
x-x^ x ckcondA,xε ,   provided   ckcondA,xε<1 ,
where condA,x=A-1Ax/x.
Note that condA,xcondA=A-1AκA; condA,x can be much smaller than condA and it is also possible for condAH, which is the same as condAT, to be much larger (or smaller) than condA.
Forward and backward error bounds can be computed by calling F07VVF (ZTBRFS), and an estimate for κA can be obtained by calling F07VUF (ZTBCON) with NORM='I'.

8  Further Comments

The total number of real floating point operations is approximately 8nkr if kn.
The real analogue of this routine is F07VEF (DTBTRS).

9  Example

This example solves the system of equations AX=B, where
A= -1.94+4.43i 0.00+0.00i 0.00+0.00i 0.00+0.00i -3.39+3.44i 4.12-4.27i 0.00+0.00i 0.00+0.00i 1.62+3.68i -1.84+5.53i 0.43-2.66i 0.00+0.00i 0.00+0.00i -2.77-1.93i 1.74-0.04i 0.44+0.10i
and
B= -8.86-03.88i -24.09-05.27i -15.57-23.41i -57.97+08.14i -7.63+22.78i 19.09-29.51i -14.74-02.40i 19.17+21.33i .
Here A is treated as a lower triangular band matrix with two subdiagonals.

9.1  Program Text

Program Text (f07vsfe.f90)

9.2  Program Data

Program Data (f07vsfe.d)

9.3  Program Results

Program Results (f07vsfe.r)


F07VSF (ZTBTRS) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

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