NAG Library Routine Document

f07vef (dtbtrs)


f07vef (dtbtrs) solves a real triangular band system of linear equations with multiple right-hand sides, AX=B or ATX=B.


Fortran Interface
Subroutine f07vef ( uplo, trans, diag, n, kd, nrhs, ab, ldab, b, ldb, info)
Integer, Intent (In):: n, kd, nrhs, ldab, ldb
Integer, Intent (Out):: info
Real (Kind=nag_wp), Intent (In):: ab(ldab,*)
Real (Kind=nag_wp), Intent (Inout):: b(ldb,*)
Character (1), Intent (In):: uplo, trans, diag
C Header Interface
#include <nagmk26.h>
void  f07vef_ (const char *uplo, const char *trans, const char *diag, const Integer *n, const Integer *kd, const Integer *nrhs, const double ab[], const Integer *ldab, double b[], const Integer *ldb, Integer *info, const Charlen length_uplo, const Charlen length_trans, const Charlen length_diag)
The routine may be called by its LAPACK name dtbtrs.


f07vef (dtbtrs) solves a real triangular band system of linear equations AX=B or ATX=B.


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


1:     uplo – Character(1)Input
On entry: specifies whether A is upper or lower triangular.
A is upper triangular.
A is lower triangular.
Constraint: uplo='U' or 'L'.
2:     trans – Character(1)Input
On entry: indicates the form of the equations.
The equations are of the form AX=B.
trans='T' or 'C'
The equations are of the form ATX=B.
Constraint: trans='N', 'T' or 'C'.
3:     diag – Character(1)Input
On entry: indicates whether A is a nonunit or unit triangular matrix.
A is a nonunit triangular matrix.
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:     abldab* – Real (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 f07vef (dtbtrs) is called.
Constraint: ldabkd+1.
9:     bldb* – Real (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 f07vef (dtbtrs) is called.
Constraint: ldbmax1,n.
11:   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. A is singular and the solution has not been computed.


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 condAT to be much larger (or smaller) than condA.
Forward and backward error bounds can be computed by calling f07vhf (dtbrfs), and an estimate for κA can be obtained by calling f07vgf (dtbcon) with norm='I'.

Parallelism and Performance

f07vef (dtbtrs) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07vef (dtbtrs) 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 total number of floating-point operations is approximately 2nkr if kn.
The complex analogue of this routine is f07vsf (ztbtrs).


This example solves the system of equations AX=B, where
A= -4.16 0.00 0.00 0.00 -2.25 4.78 0.00 0.00 0.00 5.86 6.32 0.00 0.00 0.00 -4.82 0.16   and   B= -16.64 -4.16 -13.78 -16.59 13.10 -4.94 -14.14 -9.96 .  
Here A is treated as a lower triangular band matrix with one subdiagonal.

Program Text

Program Text (f07vefe.f90)

Program Data

Program Data (f07vefe.d)

Program Results

Program Results (f07vefe.r)