NAG Library Routine Document

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.

3

Description

f07hef (dpbtrs) is used to solve a real symmetric positive definite band system of linear equations

A X = 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.

uplo ='U'

A = U^{T} U

, where

U

is upper triangular; the solution

X

is computed by solving

U^{T} Y = B

and then

U X = Y

uplo ='L'

A = L L^{T}

, where

L

is lower triangular; the solution

X

is computed by solving

L Y = B

and then

L^{T} X = Y

4

References

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

5

Arguments

1: $uplo$ – Character(1)Input

On entry: specifies how

A

has been factorized.

$uplo ='U'$: $A = U^{T} U$ , where $U$ is upper triangular.
$uplo ='L'$: $A = L L^{T}$ , where $L$ is lower triangular.

Constraint:

uplo ='U'

'L'

2: $n$ – IntegerInput

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

3: $kd$ – IntegerInput

On entry:

k_{d}

, the number of superdiagonals or subdiagonals of the matrix

A

Constraint:

kd \geq 0

4: $nrhs$ – IntegerInput

On entry:

r

, the number of right-hand sides.

Constraint:

nrhs \geq 0

5: $ab (ldab *)$ – Real (Kind=nag_wp) arrayInput

Note: the second dimension of the array ab must be at least

\max (1, 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:

ldab \geq kd + 1

7: $b (ldb *)$ – Real (Kind=nag_wp) arrayInput/Output

Note: the second dimension of the array b must be at least

\max (1, nrhs)

On entry: the

n

r

right-hand side matrix

B

On exit: the

n

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:

ldb \geq \max (1, n)

9: $info$ – IntegerOutput

On exit:

info = 0

unless the routine detects an error (see Section 6).

6

Error Indicators and Warnings

$info < 0$: If $info = - i$ , argument $i$ 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 + E) x = b

, where

if $uplo ='U'$ , $|E| \leq c (k + 1) ε |U^{T}| |U|$ ;
if $uplo ='L'$ , $|E| \leq c (k + 1) ε |L| |L^{T}|$ ,

c (k + 1)

is a modest linear function of

k + 1

, and

ε

is the machine precision.

\hat{x}

is the true solution, then the computed solution

x

satisfies a forward error bound of the form

\frac{{‖x - \hat{x}‖}_{\infty}}{{‖x‖}_{\infty}} \leq c (k + 1) cond (A, x) ε

where

cond (A, x) = {‖|A^{- 1}| |A| |x|‖}_{\infty} / {‖x‖}_{\infty} \leq cond (A) = {‖|A^{- 1}| |A|‖}_{\infty} \leq κ_{\infty} (A)

. Note that

cond (A, x)

can be much smaller than

cond (A)

Forward and backward error bounds can be computed by calling f07hhf (dpbrfs), and an estimate for

κ_{\infty} (A)

(

= κ_{1} (A)

) can be obtained by calling f07hgf (dpbcon).

8

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.

9

Further Comments

The total number of floating-point operations is approximately

4 n k r

, assuming

n ≫ 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).

10

Example

This example solves the system of equations

A X = B

, where

A = (\begin{matrix} 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 \end{matrix}) and B = (\begin{matrix} 22.09 & 5.10 \\ 9.31 & 30.81 \\ - 5.24 & - 25.82 \\ 11.83 & 22.90 \end{matrix}) .

Here

A

is symmetric and positive definite, and is treated as a band matrix, which must first be factorized by f07hdf (dpbtrf).

NAG Library Routine Document

f07hef (dpbtrs)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

10.2

Program Data

10.3

Program Results