NAG Library Routine Document

f07hef  (dpbtrs)


    1  Purpose
    7  Accuracy


f07hef (dpbtrs) solves a real symmetric positive definite band system of linear equations with multiple right-hand sides,
AX=B ,  
where A has been factorized by f07hdf (dpbtrf).


Fortran Interface
Subroutine f07hef ( uplo, 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
C Header Interface
#include nagmk26.h
void  f07hef_ ( const char *uplo, 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)
The routine may be called by its LAPACK name dpbtrs.


f07hef (dpbtrs) is used to solve a real symmetric positive definite band system of linear equations AX=B, the routine must be preceded by a call to f07hdf (dpbtrf) which computes the Cholesky factorization of A. The solution X is computed by forward and backward substitution.
If uplo='U', A=UTU, where U is upper triangular; the solution X is computed by solving UTY=B and then UX=Y.
If uplo='L', A=LLT, where L is lower triangular; the solution X is computed by solving LY=B and then LTX=Y.


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


1:     uplo – Character(1)Input
On entry: specifies how A has been factorized.
A=UTU, where U is upper triangular.
A=LLT, where L is lower triangular.
Constraint: uplo='U' or 'L'.
2:     n – IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
3:     kd – IntegerInput
On entry: kd, the number of superdiagonals or subdiagonals of the matrix A.
Constraint: kd0.
4:     nrhs – IntegerInput
On entry: r, the number of right-hand sides.
Constraint: nrhs0.
5:     abldab* – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array ab must be at least max1,n.
On entry: the Cholesky factor of A, as returned by f07hdf (dpbtrf).
6:     ldab – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f07hef (dpbtrs) is called.
Constraint: ldabkd+1.
7:     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.
8:     ldb – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07hef (dpbtrs) is called.
Constraint: ldbmax1,n.
9:     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.


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 ck+1 is a modest linear function of k+1, 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 ck+1condA,xε  
where condA,x=A-1Ax/xcondA=A-1AκA. Note that condA,x can be much smaller than condA.
Forward and backward error bounds can be computed by calling f07hhf (dpbrfs), and an estimate for κA (=κ1A) can be obtained by calling f07hgf (dpbcon).

Parallelism and Performance

f07hef (dpbtrs) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07hef (dpbtrs) 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 4nkr, assuming nk.
This routine may be followed by a call to f07hhf (dpbrfs) to refine the solution and return an error estimate.
The complex analogue of this routine is f07hsf (zpbtrs).


This example solves the system of equations AX=B, where
A= 5.49 2.68 0.00 0.00 2.68 5.63 -2.39 0.00 0.00 -2.39 2.60 -2.22 0.00 0.00 -2.22 5.17   and   B= 22.09 5.10 9.31 30.81 -5.24 -25.82 11.83 22.90 .  
Here A is symmetric and positive definite, and is treated as a band matrix, which must first be factorized by f07hdf (dpbtrf).

Program Text

Program Text (f07hefe.f90)

Program Data

Program Data (f07hefe.d)

Program Results

Program Results (f07hefe.r)

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