Integer, Intent (In)	::	m, n, lda, ldc, lwork
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (In)	::	tau(*)
Real (Kind=nag_wp), Intent (Inout)	::	a(lda,), c(ldc,)
Real (Kind=nag_wp), Intent (Out)	::	work(max(1,lwork))
Character (1), Intent (In)	::	side, uplo, trans

C Header Interface

#include <nag.h>

void

f08fgf_ (const char *side, const char *uplo, const char *trans, const Integer *m, const Integer *n, double a[], const Integer *lda, const double tau[], double c[], const Integer *ldc, double work[], const Integer *lwork, Integer *info, const Charlen length_side, const Charlen length_uplo, const Charlen length_trans)

The routine may be called by the names f08fgf, nagf_lapackeig_dormtr or its LAPACK name dormtr.

3 Description

f08fgf is intended to be used after a call to f08fef, which reduces a real symmetric matrix

A

to symmetric tridiagonal form

T

by an orthogonal similarity transformation:

A = Q T Q^{T}

. f08fef represents the orthogonal matrix

Q

as a product of elementary reflectors.

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

Q C, Q^{T} C, C Q ​ or ​ C Q^{T},

overwriting the result on

C

(which may be any real rectangular matrix).

A common application of this routine is to transform a matrix

Z

of eigenvectors of

T

to the matrix

QZ

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 Arguments

1: $side$ – Character(1) Input

On entry: indicates how

Q

Q^{T}

is to be applied to

C

$side ='L'$: $Q$ or $Q^{T}$ is applied to $C$ from the left.
$side ='R'$: $Q$ or $Q^{T}$ is applied to $C$ from the right.

Constraint:

side ='L'

'R'

2: $uplo$ – Character(1) Input

On entry: this must be the same argument uplo as supplied to f08fef.

Constraint:

uplo ='U'

'L'

3: $trans$ – Character(1) Input

On entry: indicates whether

Q

Q^{T}

is to be applied to

C

$trans ='N'$: $Q$ is applied to $C$ .
$trans ='T'$: $Q^{T}$ is applied to $C$ .

Constraint:

trans ='N'

'T'

4: $m$ – Integer Input

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: $n$ – Integer Input

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: $a (lda, *)$ – Real (Kind=nag_wp) array Input

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

\max (1, m)

side ='L'

and at least

\max (1, n)

side ='R'

On entry: details of the vectors which define the elementary reflectors, as returned by f08fef.

7: $lda$ – Integer Input

On entry: the first dimension of the array a as declared in the (sub)program from which f08fgf is called.

Constraints:

if $side ='L'$ , $lda \geq \max (1, m)$ ;
if $side ='R'$ , $lda \geq \max (1, n)$ .

8: $tau (*)$ – Real (Kind=nag_wp) array Input

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 f08fef.

9: $c (ldc, *)$ – Real (Kind=nag_wp) array Input/Output

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

\max (1, n)

On entry: the

m \times n

matrix

C

On exit: c is overwritten by

Q C

Q^{T} C

C Q

C Q^{T}

as specified by side and trans.

10: $ldc$ – Integer Input

On entry: the first dimension of the array c as declared in the (sub)program from which f08fgf is called.

Constraint:

ldc \geq \max (1, m)

11: $work (\max (1, lwork))$ – Real (Kind=nag_wp) array Workspace

On exit: if

info = 0

work (1)

contains the minimum value of lwork required for optimal performance.

12: $lwork$ – Integer Input

On entry: the dimension of the array work as declared in the (sub)program from which f08fgf is called.

lwork = −1

, a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.

Suggested value: for optimal performance,

lwork \geq n \times nb

side ='L'

and at least

m \times nb

side ='R'

, where

nb

is the optimal block size.

Constraints:

if $side ='L'$ , $lwork \geq \max (1, n)$ or $lwork = −1$ ;
if $side ='R'$ , $lwork \geq \max (1, m)$ or $lwork = −1$ .

13: $info$ – Integer Output

On exit:

info = 0

unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

$info < 0$: 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 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

f08fgf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08fgf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The total number of floating-point operations is approximately

2 m^{2} n

side ='L'

and

2 m n^{2}

side ='R'

The complex analogue of this routine is f08fuf.

10 Example

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

A

, where

A = (\begin{array}{r} 2.07 & 3.87 & 4.20 & - 1.15 \\ 3.87 & - 0.21 & 1.87 & 0.63 \\ 4.20 & 1.87 & 1.15 & 2.06 \\ - 1.15 & 0.63 & 2.06 & - 1.81 \end{array}) .

Here

A

is symmetric and must first be reduced to tridiagonal form

T

by f08fef. The program then calls f08jjf to compute the requested eigenvalues and f08jkf to compute the associated eigenvectors of

T

. Finally f08fgf is called to transform the eigenvectors to those of

A

f08fg: FL CL CPP AD PY MB

NAG FL Interfacef08fgf (dormtr)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
f08fgf (dormtr)