NAG Library Routine Document

F07BSF (ZGBTRS)

+− 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

F07BSF (ZGBTRS) solves a complex band system of linear equations with multiple right-hand sides,

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

where

A

has been factorized by F07BRF (ZGBTRF).

2 Specification

SUBROUTINE F07BSF (

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

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

The routine may be called by its LAPACK name zgbtrs.

3 Description

F07BSF (ZGBTRS) is used to solve a complex band system of linear equations

A X = B

A^{T} X = B

A^{H} X = B

, the routine must be preceded by a call to F07BRF (ZGBTRF) 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'

, the solution is computed by solving

U^{T} Y = B

and then

L^{T} P^{T} X = Y

TRANS ='C'

, the solution is computed by solving

U^{H} Y = B

and then

L^{H} 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 $X$ .
$TRANS ='T'$: $A^{T} X = B$ is solved for $X$ .
$TRANS ='C'$: $A^{H} X = B$ 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, $*$ ) – COMPLEX (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 F07BRF (ZGBTRF).

7: LDAB – INTEGERInput

On entry: the first dimension of the array AB as declared in the (sub)program from which F07BSF (ZGBTRS) 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 F07BRF (ZGBTRF).

9: B(LDB, $*$ ) – COMPLEX (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 F07BSF (ZGBTRS) 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 |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^{H})

(which is the same as

cond (A^{T})

) can be much larger (or smaller) than

cond (A)

Forward and backward error bounds can be computed by calling F07BVF (ZGBRFS), and an estimate for

κ_{\infty} (A)

can be obtained by calling F07BUF (ZGBCON) with

NORM ='I'

8 Further Comments

The total number of real floating point operations is approximately

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

, assuming

n ≫ k_{l}

and

n ≫ k_{u}

This routine may be followed by a call to F07BVF (ZGBRFS) to refine the solution and return an error estimate.

The real analogue of this routine is F07BEF (DGBTRS).

9 Example

This example solves the system of equations

A X = B

, where

A = (\begin{array}{r} - 1.65 + 2.26 i & - 2.05 - 0.85 i & 0.97 - 2.84 i & 0.00 + 0.00 i \\ 0.00 + 6.30 i & - 1.48 - 1.75 i & - 3.99 + 4.01 i & 0.59 - 0.48 i \\ 0.00 + 0.00 i & - 0.77 + 2.83 i & - 1.06 + 1.94 i & 3.33 - 1.04 i \\ 0.00 + 0.00 i & 0.00 + 0.00 i & 4.48 - 1.09 i & - 0.46 - 1.72 i \end{array})

and

B = (\begin{array}{r} - 1.06 + 21.50 i & 12.85 + 2.84 i \\ - 22.72 - 53.90 i & - 70.22 + 21.57 i \\ 28.24 - 38.60 i & - 20.7 - 31.23 i \\ - 34.56 + 16.73 i & 26.01 + 31.97 i \end{array}) .

Here

A

is nonsymmetric and is treated as a band matrix, which must first be factorized by F07BRF (ZGBTRF).

NAG Library Routine DocumentF07BSF (ZGBTRS)

+− 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

F07BSF (ZGBTRS)