NAG Library Routine Document

1Purpose

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).

2Specification

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
#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.

3Description

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 ${\mathbf{uplo}}=\text{'U'}$, $A={U}^{\mathrm{T}}U$, where $U$ is upper triangular; the solution $X$ is computed by solving ${U}^{\mathrm{T}}Y=B$ and then $UX=Y$.
If ${\mathbf{uplo}}=\text{'L'}$, $A=L{L}^{\mathrm{T}}$, where $L$ is lower triangular; the solution $X$ is computed by solving $LY=B$ and then ${L}^{\mathrm{T}}X=Y$.

4References

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

5Arguments

1:     $\mathbf{uplo}$ – Character(1)Input
On entry: specifies how $A$ has been factorized.
${\mathbf{uplo}}=\text{'U'}$
$A={U}^{\mathrm{T}}U$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
$A=L{L}^{\mathrm{T}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathbf{kd}$ – IntegerInput
On entry: ${k}_{d}$, the number of superdiagonals or subdiagonals of the matrix $A$.
Constraint: ${\mathbf{kd}}\ge 0$.
4:     $\mathbf{nrhs}$ – IntegerInput
On entry: $r$, the number of right-hand sides.
Constraint: ${\mathbf{nrhs}}\ge 0$.
5:     $\mathbf{ab}\left({\mathbf{ldab}},*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array ab must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the Cholesky factor of $A$, as returned by f07hdf (dpbtrf).
6:     $\mathbf{ldab}$ – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f07hef (dpbtrs) is called.
Constraint: ${\mathbf{ldab}}\ge {\mathbf{kd}}+1$.
7:     $\mathbf{b}\left({\mathbf{ldb}},*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
On entry: the $n$ by $r$ right-hand side matrix $B$.
On exit: the $n$ by $r$ solution matrix $X$.
8:     $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07hef (dpbtrs) is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
9:     $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

6Error Indicators and Warnings

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

7Accuracy

For each right-hand side vector $b$, the computed solution $x$ is the exact solution of a perturbed system of equations $\left(A+E\right)x=b$, where
• if ${\mathbf{uplo}}=\text{'U'}$, $\left|E\right|\le c\left(k+1\right)\epsilon \left|{U}^{\mathrm{T}}\right|\left|U\right|$;
• if ${\mathbf{uplo}}=\text{'L'}$, $\left|E\right|\le c\left(k+1\right)\epsilon \left|L\right|\left|{L}^{\mathrm{T}}\right|$,
$c\left(k+1\right)$ is a modest linear function of $k+1$, and $\epsilon$ is the machine precision.
If $\stackrel{^}{x}$ is the true solution, then the computed solution $x$ satisfies a forward error bound of the form
 $x-x^∞ x∞ ≤ck+1condA,xε$
where $\mathrm{cond}\left(A,x\right)={‖\left|{A}^{-1}\right|\left|A\right|\left|x\right|‖}_{\infty }/{‖x‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖\left|{A}^{-1}\right|\left|A\right|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$. Note that $\mathrm{cond}\left(A,x\right)$ can be much smaller than $\mathrm{cond}\left(A\right)$.
Forward and backward error bounds can be computed by calling f07hhf (dpbrfs), and an estimate for ${\kappa }_{\infty }\left(A\right)$ ($\text{}={\kappa }_{1}\left(A\right)$) can be obtained by calling f07hgf (dpbcon).

8Parallelism 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.

The total number of floating-point operations is approximately $4nkr$, assuming $n\gg k$.
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).

10Example

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).

10.1Program Text

Program Text (f07hefe.f90)

10.2Program Data

Program Data (f07hefe.d)

10.3Program Results

Program Results (f07hefe.r)

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