NAG Library Routine Document

F07BEF (DGBTRS)

+− 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

F07BEF (DGBTRS) solves a real band system of linear equations with multiple right-hand sides,

A X = B or A^{T} X = B,

where

A

has been factorized by F07BDF (DGBTRF).

2 Specification

SUBROUTINE F07BEF (

TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO)

INTEGER	N, KL, KU, NRHS, LDAB, IPIV(*), LDB, INFO
REAL (KIND=nag_wp)	AB(LDAB,), B(LDB,)
CHARACTER(1)	TRANS

The routine may be called by its LAPACK name dgbtrs.

3 Description

F07BEF (DGBTRS) is used to solve a real band system of linear equations

A X = B

A^{T} X = B

, the routine must be preceded by a call to F07BDF (DGBTRF) which computes the

L U

factorization of

A

A = P L U

. The solution is computed by forward and backward substitution.

TRANS ='N'

, the solution is computed by solving

P L Y = B

and then

U X = Y

TRANS ='T'

'C'

, the solution is computed by solving

U^{T} Y = B

and then

L^{T} P^{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: TRANS – CHARACTER(1)Input

On entry: indicates the form of the equations.

$TRANS ='N'$: $A X = B$ is solved for $X$ .
$TRANS ='T'$ or $'C'$: $A^{T} X = B$ is solved for $X$ .

Constraint:

TRANS ='N'

'T'

'C'

2: N – INTEGERInput

On entry:

n

, the order of the matrix

A

Constraint:

N \geq 0

3: KL – INTEGERInput

On entry:

k_{l}

, the number of subdiagonals within the band of the matrix

A

Constraint:

KL \geq 0

4: KU – INTEGERInput

On entry:

k_{u}

, the number of superdiagonals within the band of the matrix

A

Constraint:

KU \geq 0

5: NRHS – INTEGERInput

On entry:

r

, the number of right-hand sides.

Constraint:

NRHS \geq 0

6: 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

L U

factorization of

A

, as returned by F07BDF (DGBTRF).

7: LDAB – INTEGERInput

On entry: the first dimension of the array AB as declared in the (sub)program from which F07BEF (DGBTRS) is called.

Constraint:

LDAB \geq 2 \times KL + KU + 1

8: IPIV( $*$ ) – INTEGER arrayInput

Note: the dimension of the array IPIV must be at least

\max (1, N)

On entry: the pivot indices, as returned by F07BDF (DGBTRF).

9: 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

10: LDB – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F07BEF (DGBTRS) is called.

Constraint:

LDB \geq \max (1, N)

11: 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

|E| \leq c (k) ε P |L| |U|,

c (k)

is a modest linear function of

k = k_{l} + k_{u} + 1

, and

ε

is the machine precision. This assumes

k ≪ n

\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) 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)

, and

cond (A^{T})

can be much larger (or smaller) than

cond (A)

Forward and backward error bounds can be computed by calling F07BHF (DGBRFS), and an estimate for

κ_{\infty} (A)

can be obtained by calling F07BGF (DGBCON) with

NORM ='I'

8 Further Comments

The total number of floating point operations is approximately

2 n (2 k_{l} + k_{u}) r

, assuming

n ≫ k_{l}

and

n ≫ k_{u}

This routine may be followed by a call to F07BHF (DGBRFS) to refine the solution and return an error estimate.

The complex analogue of this routine is F07BSF (ZGBTRS).

9 Example

This example solves the system of equations

A X = B

, where

A = (\begin{matrix} - 0.23 & 2.54 & - 3.66 & 0.00 \\ - 6.98 & 2.46 & - 2.73 & - 2.13 \\ 0.00 & 2.56 & 2.46 & 4.07 \\ 0.00 & 0.00 & - 4.78 & - 3.82 \end{matrix}) and B = (\begin{matrix} 4.42 & - 36.01 \\ 27.13 & - 31.67 \\ - 6.14 & - 1.16 \\ 10.50 & - 25.82 \end{matrix}) .

Here

A

is nonsymmetric and is treated as a band matrix, which must first be factorized by F07BDF (DGBTRF).

NAG Library Routine DocumentF07BEF (DGBTRS)

+− 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

F07BEF (DGBTRS)