NAG Library Routine Document

F07TSF (ZTRTRS)

+− 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

F07TSF (ZTRTRS) solves a complex triangular system of linear equations with multiple right-hand sides,

A X = B

A^{T} X = B

A^{H} X = B

2 Specification

SUBROUTINE F07TSF (

UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, INFO)

INTEGER	N, NRHS, LDA, LDB, INFO
COMPLEX (KIND=nag_wp)	A(LDA,), B(LDB,)
CHARACTER(1)	UPLO, TRANS, DIAG

The routine may be called by its LAPACK name ztrtrs.

3 Description

F07TSF (ZTRTRS) solves a complex triangular system of linear equations

A X = B

A^{T} X = B

A^{H} X = B

4 References

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

Higham N J (1989) The accuracy of solutions to triangular systems SIAM J. Numer. Anal. 26 1252–1265

5 Parameters

1: UPLO – CHARACTER(1)Input

On entry: specifies whether

A

is upper or lower triangular.

$UPLO ='U'$: $A$ is upper triangular.
$UPLO ='L'$: $A$ is lower triangular.

Constraint:

UPLO ='U'

'L'

2: TRANS – CHARACTER(1)Input

On entry: indicates the form of the equations.

$TRANS ='N'$: The equations are of the form $A X = B$ .
$TRANS ='T'$: The equations are of the form $A^{T} X = B$ .
$TRANS ='C'$: The equations are of the form $A^{H} X = B$ .

Constraint:

TRANS ='N'

'T'

'C'

3: DIAG – CHARACTER(1)Input

On entry: indicates whether

A

is a nonunit or unit triangular matrix.

$DIAG ='N'$: $A$ is a nonunit triangular matrix.
$DIAG ='U'$: $A$ is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be $1$ .

Constraint:

DIAG ='N'

'U'

4: N – INTEGERInput

On entry:

n

, the order of the matrix

A

Constraint:

N \geq 0

5: NRHS – INTEGERInput

On entry:

r

, the number of right-hand sides.

Constraint:

NRHS \geq 0

6: A(LDA, $*$ ) – COMPLEX (KIND=nag_wp) arrayInput

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

\max (1, N)

On entry: the

n

n

triangular matrix

A

If $UPLO ='U'$ , $A$ is upper triangular and the elements of the array below the diagonal are not referenced.
If $UPLO ='L'$ , $A$ is lower triangular and the elements of the array above the diagonal are not referenced.
If $DIAG ='U'$ , the diagonal elements of $A$ are assumed to be $1$ , and are not referenced.

7: LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F07TSF (ZTRTRS) is called.

Constraint:

LDA \geq \max (1, N)

8: 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

9: LDB – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F07TSF (ZTRTRS) is called.

Constraint:

LDB \geq \max (1, N)

10: 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.

$INFO > 0$: If $INFO = i$ , $a (i, i)$ is exactly zero; $A$ is singular and the solution has not been computed.

7 Accuracy

The solutions of triangular systems of equations are usually computed to high accuracy. See Higham (1989).

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 (n) ε |A|,

c (n)

is a modest linear function of

n

, 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 (n) cond (A, x) ε,   provided c (n) cond (A, x) ε < 1,

where

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

Note that

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

;

cond (A, x)

can be much smaller than

cond (A)

and it is also possible for

cond (A^{H})

, which is the same as

cond (A^{T})

, to be much larger (or smaller) than

cond (A)

Forward and backward error bounds can be computed by calling F07TVF (ZTRRFS), and an estimate for

κ_{\infty} (A)

can be obtained by calling F07TUF (ZTRCON) with

NORM ='I'

8 Further Comments

The total number of real floating point operations is approximately

4 n^{2} r

The real analogue of this routine is F07TEF (DTRTRS).

9 Example

This example solves the system of equations

A X = B

, where

A = (\begin{array}{r} 4.78 + 4.56 i & 0.00 + 0.00 i & 0.00 + 0.00 i & 0.00 + 0.00 i \\ 2.00 - 0.30 i & - 4.11 + 1.25 i & 0.00 + 0.00 i & 0.00 + 0.00 i \\ 2.89 - 1.34 i & 2.36 - 4.25 i & 4.15 + 0.80 i & 0.00 + 0.00 i \\ - 1.89 + 1.15 i & 0.04 - 3.69 i & - 0.02 + 0.46 i & 0.33 - 0.26 i \end{array})

and

B = (\begin{array}{r} - 14.78 - 32.36 i & - 18.02 + 28.46 i \\ 2.98 - 2.14 i & 14.22 + 15.42 i \\ - 20.96 + 17.06 i & 5.62 + 35.89 i \\ 9.54 + 9.91 i & - 16.46 - 1.73 i \end{array}) .

NAG Library Routine DocumentF07TSF (ZTRTRS)

+− 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

F07TSF (ZTRTRS)