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

Q

in this representation (see Section 9).

The function can take advantage of a previous call to f08nhc, 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 f08nhc can be supplied to the function directly. If f08nhc 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 Arguments

1: $order$ – Nag_OrderType Input

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

3: $ilo$ – Integer Input

4: $ihi$ – Integer Input

On entry: if

A

has been output by f08nhc, ilo and ihi must contain the values returned by that function. 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 $ihi = 0$ .

5: $a [\dim]$ – double Input/Output

Note: the dimension, dim, of the array a must be at least

\max (1, pda \times n)

The

(i, j)

th element of the matrix

A

is stored in

$a [(j - 1) \times pda + i - 1]$ when $order = Nag_ColMajor$ ;
$a [(i - 1) \times pda + j - 1]$ when $order = Nag_RowMajor$ .

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

6: $pda$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array a.

Constraint:

pda \geq \max (1, n)

7: $tau [\dim]$ – double Output

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

\max (1, n - 1)

On exit: further details of the orthogonal matrix

Q

8: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .

On entry, $pda = 〈value〉$ .
Constraint: $pda > 0$ .
NE_INT_2: On entry, $pda = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pda \geq \max (1, n)$ .
NE_INT_3: On entry, $n = 〈value〉$ , $ilo = 〈value〉$ and $ihi = 〈value〉$ .
Constraint: if $n > 0$ , $1 \leq ilo \leq ihi \leq n$ ;
if $n = 0$ , $ilo = 1$ and $ihi = 0$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

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 Parallelism and Performance

f08nec is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08nec 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 function. 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{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

f08nec may be followed by a call to f08nfc:

nag_lapackeig_dorghr(order,n,ilo,ihi,&a,pda,tau,&fail)

To apply

Q

to an

m

n

real matrix

C

f08nec may be followed by a call to f08ngc. For example,

nag_lapackeig_dormhr(order,Nag_LeftSide,Nag_NoTrans,m,n,ilo,ihi,&a,pda,
  tau,&c,pdc,&fail)

forms the matrix product

Q C

The complex analogue of this function is f08nsc.

10 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}) .

Interfaces: FL CL AD

NAG CL Interface Introduction

F08 (Lapackeig) Chapter Contents

F08 (Lapackeig) Chapter Introduction

f08ne: FL CL AD

NAG CL Interfacef08nec (dgehrd)

▸▿ 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 CL Interface
f08nec (dgehrd)