void	nag_dormhr (Nag_OrderType order, Nag_SideType side, Nag_TransType trans, Integer m, Integer n, Integer ilo, Integer ihi, const double a[], Integer pda, const double tau[], double c[], Integer pdc, NagError *fail)

3

Description

nag_dormhr (f08ngc) is intended to be used following a call to nag_dgehrd (f08nec), which reduces a real general matrix

A

to upper Hessenberg form

H

by an orthogonal similarity transformation:

A = Q H Q^{T}

. nag_dgehrd (f08nec) represents the matrix

Q

as a product of

i_{hi} - i_{lo}

elementary reflectors. Here

i_{lo}

and

i_{hi}

are values determined by nag_dgebal (f08nhc) when balancing the matrix; if the matrix has not been balanced,

i_{lo} = 1

and

i_{hi} = n

This function 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 function is to transform a matrix

V

of eigenvectors of

H

to the matrix

QV

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: $order$ – Nag_OrderTypeInput

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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $side$ – Nag_SideTypeInput

On entry: indicates how

Q

Q^{T}

is to be applied to

C

$side = Nag_LeftSide$: $Q$ or $Q^{T}$ is applied to $C$ from the left.
$side = Nag_RightSide$: $Q$ or $Q^{T}$ is applied to $C$ from the right.

Constraint:

side = Nag_LeftSide

Nag_RightSide

3: $trans$ – Nag_TransTypeInput

On entry: indicates whether

Q

Q^{T}

is to be applied to

C

$trans = Nag_NoTrans$: $Q$ is applied to $C$ .
$trans = Nag_Trans$: $Q^{T}$ is applied to $C$ .

Constraint:

trans = Nag_NoTrans

Nag_Trans

4: $m$ – IntegerInput

On entry:

m

, the number of rows of the matrix

C

;

m

is also the order of

Q

side = Nag_LeftSide

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 = Nag_RightSide

Constraint:

n \geq 0

6: $ilo$ – IntegerInput

7: $ihi$ – IntegerInput

On entry: these must be the same arguments ilo and ihi, respectively, as supplied to nag_dgehrd (f08nec).

Constraints:

if $side = Nag_LeftSide$ and $m > 0$ , $1 \leq ilo \leq ihi \leq m$ ;
if $side = Nag_LeftSide$ and $m = 0$ , $ilo = 1$ and $ihi = 0$ ;
if $side = Nag_RightSide$ and $n > 0$ , $1 \leq ilo \leq ihi \leq n$ ;
if $side = Nag_RightSide$ and $n = 0$ , $ilo = 1$ and $ihi = 0$ .

8: $a [\dim]$ – const doubleInput

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

$\max (1, pda \times m)$ when $side = Nag_LeftSide$ ;
$\max (1, pda \times n)$ when $side = Nag_RightSide$ .

On entry: details of the vectors which define the elementary reflectors, as returned by nag_dgehrd (f08nec).

9: $pda$ – IntegerInput

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

Constraints:

if $side = Nag_LeftSide$ , $pda \geq \max (1, m)$ ;
if $side = Nag_RightSide$ , $pda \geq \max (1, n)$ .

10: $tau [\dim]$ – const doubleInput

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

$\max (1, m - 1)$ when $side = Nag_LeftSide$ ;
$\max (1, n - 1)$ when $side = Nag_RightSide$ .

On entry: further details of the elementary reflectors, as returned by nag_dgehrd (f08nec).

11: $c [\dim]$ – doubleInput/Output

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

$\max (1, pdc \times n)$ when $order = Nag_ColMajor$ ;
$\max (1, m \times pdc)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

C

is stored in

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

On entry: the

m

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.

12: $pdc$ – IntegerInput

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

Constraints:

if $order = Nag_ColMajor$ , $pdc \geq \max (1, m)$ ;
if $order = Nag_RowMajor$ , $pdc \geq \max (1, n)$ .

13: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_ENUM_INT_3: On entry, $side = 〈value〉$ , $m = 〈value〉$ , $n = 〈value〉$ and $pda = 〈value〉$ .
Constraint: if $side = Nag_LeftSide$ , $pda \geq \max (1, m)$ ;
if $side = Nag_RightSide$ , $pda \geq \max (1, n)$ .

On entry, $side = 〈value〉$ , $pda = 〈value〉$ , $m = 〈value〉$ and $n = 〈value〉$ .
Constraint: if $side = Nag_LeftSide$ , $pda \geq \max (1, m)$ ;
if $side = Nag_RightSide$ , $pda \geq \max (1, n)$ .
NE_ENUM_INT_4: On entry, $side = 〈value〉$ , $m = 〈value〉$ , $n = 〈value〉$ , $ilo = 〈value〉$ and $ihi = 〈value〉$ .
Constraint: if $side = Nag_LeftSide$ and $m > 0$ , $1 \leq ilo \leq ihi \leq m$ ;
if $side = Nag_LeftSide$ and $m = 0$ , $ilo = 1$ and $ihi = 0$ ;
if $side = Nag_RightSide$ and $n > 0$ , $1 \leq ilo \leq ihi \leq n$ ;
if $side = Nag_RightSide$ and $n = 0$ , $ilo = 1$ and $ihi = 0$ .
NE_INT: On entry, $m = 〈value〉$ .
Constraint: $m \geq 0$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .

On entry, $pda = 〈value〉$ .
Constraint: $pda > 0$ .

On entry, $pdc = 〈value〉$ .
Constraint: $pdc > 0$ .
NE_INT_2: On entry, $pdc = 〈value〉$ and $m = 〈value〉$ .
Constraint: $pdc \geq \max (1, m)$ .

On entry, $pdc = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pdc \geq \max (1, n)$ .
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

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

nag_dormhr (f08ngc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_dormhr (f08ngc) 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

2 n q^{2}

side = Nag_LeftSide

and

2 m q^{2}

side = Nag_RightSide

, where

q = i_{hi} - i_{lo}

The complex analogue of this function is nag_zunmhr (f08nuc).

10

Example

This example computes all the eigenvalues 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}),

and those eigenvectors which correspond to eigenvalues

λ

such that

Re (λ) < 0

. Here

A

is general and must first be reduced to upper Hessenberg form

H

by nag_dgehrd (f08nec). The program then calls nag_dhseqr (f08pec) to compute the eigenvalues, and nag_dhsein (f08pkc) to compute the required eigenvectors of

H

by inverse iteration. Finally nag_dormhr (f08ngc) is called to transform the eigenvectors of

H

back to eigenvectors of the original matrix

A

NAG Library Function Document

nag_dormhr (f08ngc)

▸▿ 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