NAG Library Routine Document

F07HEF (DPBTRS)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

F07HEF (DPBTRS) solves a real symmetric positive definite band system of linear equations with multiple right-hand sides,

A X = B,

where

A

has been factorized by F07HDF (DPBTRF).

2 Specification

SUBROUTINE F07HEF (

UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO)

INTEGER	N, KD, NRHS, LDAB, LDB, INFO
REAL (KIND=nag_wp)	AB(LDAB,), B(LDB,)
CHARACTER(1)	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 Parameters

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

Errors or warnings detected by the routine:

$INFO < 0$: If $INFO = - i$ , the $i$ th parameter 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 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).

9 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 DocumentF07HEF (DPBTRS)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

F07HEF (DPBTRS)