NAG Library Routine Document

F08NEF (DGEHRD)

is not formed explicitly, but is represented as a product of elementary reflectors (see the F08 Chapter Introduction for details). Routines are provided to work with

Q

in this representation (see Section 8).

The routine can take advantage of a previous call to F08NHF (DGEBAL), which may produce a matrix with the structure:

(\begin{matrix} A_{11} & A_{12} & A_{13} \\ A_{22} & A_{23} \\ A_{33} \end{matrix})

where

A_{11}

and

A_{33}

are upper triangular. If so, only the central diagonal block

A_{22}

, in rows and columns

i_{lo}

i_{hi}

, needs to be reduced to Hessenberg form (the blocks

A_{12}

and

A_{23}

will also be affected by the reduction). Therefore the values of

i_{lo}

and

i_{hi}

determined by F08NHF (DGEBAL) can be supplied to the routine directly. If F08NHF (DGEBAL) has not previously been called however, then

i_{lo}

must be set to

1

and

i_{hi}

n

4 References

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

5 Parameters

1: N – INTEGERInput

On entry:

n

, the order of the matrix

A

Constraint:

N \geq 0

2: ILO – INTEGERInput

3: IHI – INTEGERInput

On entry: if

A

has been output by F08NHF (DGEBAL), then ILO and IHI must contain the values returned by that routine. Otherwise, ILO must be set to

1

and IHI to N.

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: the

n

n

general matrix

A

On exit: A is overwritten by the upper Hessenberg matrix

H

and details of the orthogonal matrix

Q

5: LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F08NEF (DGEHRD) is called.

Constraint:

LDA \geq \max (1, N)

6: TAU( $*$ ) – REAL (KIND=nag_wp) arrayOutput

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

\max (1, N - 1)

On exit: further details of the orthogonal matrix

Q

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 F08NEF (DGEHRD) 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

, where

nb

is the optimal block size.

Constraint:

LWORK \geq \max (1, N)

LWORK = - 1

9: 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 Hessenberg matrix

H

is exactly similar to a nearby matrix

(A + E)

, where

{‖E‖}_{2} \leq c (n) ε {‖A‖}_{2},

c (n)

is a modestly increasing function of

n

, and

ε

is the machine precision.

The elements of

H

themselves may be sensitive to small perturbations in

A

or to rounding errors in the computation, but this does not affect the stability of the eigenvalues, eigenvectors or Schur factorization.

8 Further Comments

The total number of floating point operations is approximately

\frac{2}{3} q^{2} (2 q + 3 n)

, where

q = i_{hi} - i_{lo}

; if

i_{lo} = 1

and

i_{hi} = n

, the number is approximately

\frac{10}{3} n^{3}

To form the orthogonal matrix

Q

F08NEF (DGEHRD) may be followed by a call to F08NFF (DORGHR):

CALL DORGHR(N,ILO,IHI,A,LDA,TAU,WORK,LWORK,INFO)

To apply

Q

to an

m

n

real matrix

C

F08NEF (DGEHRD) may be followed by a call to F08NGF (DORMHR). For example,

CALL DORMHR('Left','No Transpose',M,N,ILO,IHI,A,LDA,TAU,C,LDC, &
              WORK,LWORK,INFO)

forms the matrix product

Q C

The complex analogue of this routine is F08NSF (ZGEHRD).

9 Example

This example computes the upper Hessenberg form 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}) .

NAG Library Routine DocumentF08NEF (DGEHRD)

+− 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

F08NEF (DGEHRD)