NAG Library Routine Document

F08GUF (ZUPMTR)

+− 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

F08GUF (ZUPMTR) multiplies an arbitrary complex matrix

C

by the complex unitary matrix

Q

which was determined by F08GSF (ZHPTRD) when reducing a complex Hermitian matrix to tridiagonal form.

2 Specification

SUBROUTINE F08GUF (

SIDE, UPLO, TRANS, M, N, AP, TAU, C, LDC, WORK, INFO)

INTEGER	M, N, LDC, INFO
COMPLEX (KIND=nag_wp)	AP(), TAU(), C(LDC,), WORK()
CHARACTER(1)	SIDE, UPLO, TRANS

The routine may be called by its LAPACK name zupmtr.

3 Description

F08GUF (ZUPMTR) is intended to be used after a call to F08GSF (ZHPTRD), which reduces a complex Hermitian matrix

A

to real symmetric tridiagonal form

T

by a unitary similarity transformation:

A = Q T Q^{H}

. F08GSF (ZHPTRD) represents the unitary matrix

Q

as a product of elementary reflectors.

This routine may be used to form one of the matrix products

Q C, Q^{H} C, C Q ​ or ​ C Q^{H},

overwriting the result on

C

(which may be any complex rectangular matrix).

A common application of this routine is to transform a matrix

Z

of eigenvectors of

T

to the matrix

Q Z

of eigenvectors of

A

4 References

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

5 Parameters

1: SIDE – CHARACTER(1)Input

On entry: indicates how

Q

Q^{H}

is to be applied to

C

$SIDE ='L'$: $Q$ or $Q^{H}$ is applied to $C$ from the left.
$SIDE ='R'$: $Q$ or $Q^{H}$ is applied to $C$ from the right.

Constraint:

SIDE ='L'

'R'

2: UPLO – CHARACTER(1)Input

On entry: this must be the same parameter UPLO as supplied to F08GSF (ZHPTRD).

Constraint:

UPLO ='U'

'L'

3: TRANS – CHARACTER(1)Input

On entry: indicates whether

Q

Q^{H}

is to be applied to

C

$TRANS ='N'$: $Q$ is applied to $C$ .
$TRANS ='C'$: $Q^{H}$ is applied to $C$ .

Constraint:

TRANS ='N'

'C'

4: M – INTEGERInput

On entry:

m

, the number of rows of the matrix

C

;

m

is also the order of

Q

SIDE ='L'

Constraint:

M \geq 0

5: N – INTEGERInput

On entry:

n

, the number of columns of the matrix

C

;

n

is also the order of

Q

SIDE ='R'

Constraint:

N \geq 0

6: AP( $*$ ) – COMPLEX (KIND=nag_wp) arrayInput/Output

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

\max (1, M \times (M + 1) / 2)

SIDE ='L'

and at least

\max (1, N \times (N + 1) / 2)

SIDE ='R'

On entry: details of the vectors which define the elementary reflectors, as returned by F08GSF (ZHPTRD).

On exit: is used as internal workspace prior to being restored and hence is unchanged.

7: TAU( $*$ ) – COMPLEX (KIND=nag_wp) arrayInput

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

\max (1, M - 1)

SIDE ='L'

and at least

\max (1, N - 1)

SIDE ='R'

On entry: further details of the elementary reflectors, as returned by F08GSF (ZHPTRD).

8: C(LDC, $*$ ) – COMPLEX (KIND=nag_wp) arrayInput/Output

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

\max (1, N)

On entry: the

m

n

matrix

C

On exit: C is overwritten by

Q C

Q^{H} C

C Q

C Q^{H}

as specified by SIDE and TRANS.

9: LDC – INTEGERInput

On entry: the first dimension of the array C as declared in the (sub)program from which F08GUF (ZUPMTR) is called.

Constraint:

LDC \geq \max (1, M)

10: WORK( $*$ ) – COMPLEX (KIND=nag_wp) arrayWorkspace

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

\max (1, N)

SIDE ='L'

and at least

\max (1, M)

SIDE ='R'

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 $INFO = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

7 Accuracy

The computed result differs from the exact result by a matrix

E

such that

{‖E‖}_{2} = O (ε) {‖C‖}_{2},

where

ε

is the machine precision.

8 Further Comments

The total number of real floating point operations is approximately

8 m^{2} n

SIDE ='L'

and

8 m n^{2}

SIDE ='R'

The real analogue of this routine is F08GGF (DOPMTR).

9 Example

This example computes the two smallest eigenvalues, and the associated eigenvectors, of the matrix

A

, where

A = (\begin{array}{r} - 2.28 + 0.00 i & 1.78 - 2.03 i & 2.26 + 0.10 i & - 0.12 + 2.53 i \\ 1.78 + 2.03 i & - 1.12 + 0.00 i & 0.01 + 0.43 i & - 1.07 + 0.86 i \\ 2.26 - 0.10 i & 0.01 - 0.43 i & - 0.37 + 0.00 i & 2.31 - 0.92 i \\ - 0.12 - 2.53 i & - 1.07 - 0.86 i & 2.31 + 0.92 i & - 0.73 + 0.00 i \end{array}),

using packed storage. Here

A

is Hermitian and must first be reduced to tridiagonal form

T

by F08GSF (ZHPTRD). The program then calls F08JJF (DSTEBZ) to compute the requested eigenvalues and F08JXF (ZSTEIN) to compute the associated eigenvectors of

T

. Finally F08GUF (ZUPMTR) is called to transform the eigenvectors to those of

A

NAG Library Routine DocumentF08GUF (ZUPMTR)

+− 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

F08GUF (ZUPMTR)