Integer, Intent (In)	::	n, ilo, ihi, lda, lwork
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (In)	::	tau(*)
Real (Kind=nag_wp), Intent (Inout)	::	a(lda,*)
Real (Kind=nag_wp), Intent (Out)	::	work(max(1,lwork))

C Header Interface

#include nagmk26.h

void	f08nff_ ( const Integer n, const Integer ilo, const Integer ihi, double a[], const Integer lda, const double tau[], double work[], const Integer lwork, Integer info)

The routine may be called by its LAPACK name dorghr.

3

Description

f08nff (dorghr) is intended to be used following a call to f08nef (dgehrd), which reduces a real general matrix

A

to upper Hessenberg form

H

by an orthogonal similarity transformation:

A = Q H Q^{T}

. f08nef (dgehrd) represents the matrix

Q

as a product of

i_{hi} - i_{lo}

elementary reflectors. Here

i_{lo}

and

i_{hi}

are values determined by f08nhf (dgebal) when balancing the matrix; if the matrix has not been balanced,

i_{lo} = 1

and

i_{hi} = n

This routine may be used to generate

Q

explicitly as a square matrix.

Q

has the structure:

Q = (\begin{matrix} I & 0 & 0 \\ 0 & Q_{22} & 0 \\ 0 & 0 & I \end{matrix})

where

Q_{22}

occupies rows and columns

i_{lo}

i_{hi}

4

References

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

5

Arguments

1: $n$ – IntegerInput

On entry:

n

, the order of the matrix

Q

Constraint:

n \geq 0

2: $ilo$ – IntegerInput

3: $ihi$ – IntegerInput

On entry: these must be the same arguments ilo and ihi, respectively, as supplied to f08nef (dgehrd).

Constraints:

if $n > 0$ , $1 \leq ilo \leq ihi \leq n$ ;
if $n = 0$ , $ilo = 1$ and $ihi = 0$ .

4: $a (lda *)$ – Real (Kind=nag_wp) arrayInput/Output

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

\max (1, n)

On entry: details of the vectors which define the elementary reflectors, as returned by f08nef (dgehrd).

On exit: the

n

n

orthogonal matrix

Q

5: $lda$ – IntegerInput

On entry: the first dimension of the array a as declared in the (sub)program from which f08nff (dorghr) is called.

Constraint:

lda \geq \max (1, n)

6: $tau (*)$ – Real (Kind=nag_wp) arrayInput

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

\max (1, n - 1)

On entry: further details of the elementary reflectors, as returned by f08nef (dgehrd).

7: $work (\max (1, lwork))$ – Real (Kind=nag_wp) arrayWorkspace

On exit: if

info = 0

work (1)

contains the minimum value of lwork required for optimal performance.

8: $lwork$ – IntegerInput

On entry: the dimension of the array work as declared in the (sub)program from which f08nff (dorghr) is called, unless

lwork = - 1

, in which case a workspace query is assumed and the routine only calculates the optimal dimension of work (using the formula given below).

Suggested value: for optimal performance lwork should be at least

(ihi - ilo) \times n b

, where

n b

is the block size.

Constraint:

lwork \geq \max (1, ihi - ilo)

lwork = - 1

9: $info$ – IntegerOutput

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 matrix

Q

differs from an exactly orthogonal matrix by a matrix

E

such that

{‖E‖}_{2} = O (ε),

where

ε

is the machine precision.

8

Parallelism and Performance

f08nff (dorghr) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08nff (dorghr) 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

\frac{4}{3} q^{3}

, where

q = i_{hi} - i_{lo}

The complex analogue of this routine is f08ntf (zunghr).

10

Example

This example computes the Schur factorization of the matrix

A

, where

A = (\begin{array}{r} 0.35 & 0.45 & - 0.14 & - 0.17 \\ 0.09 & 0.07 & - 0.54 & 0.35 \\ - 0.44 & - 0.33 & - 0.03 & 0.17 \\ 0.25 & - 0.32 & - 0.13 & 0.11 \end{array}) .

Here

A

is general and must first be reduced to Hessenberg form by f08nef (dgehrd). The program then calls f08nff (dorghr) to form

Q

, and passes this matrix to f08pef (dhseqr) which computes the Schur factorization of

A

NAG Library Routine Document

f08nff (dorghr)

▸▿ 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

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