NAG Library Function Document

nag_zunmhr (f08nuc)

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

3 Description

nag_zunmhr (f08nuc) is intended to be used following a call to nag_zgehrd (f08nsc), which reduces a complex general matrix

A

to upper Hessenberg form

H

by a unitary similarity transformation:

A = Q H Q^{H}

. nag_zgehrd (f08nsc) 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_zgebal (f08nvc) 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^{H} C, C Q ​ or ​ C Q^{H},

overwriting the result on

C

(which may be any complex 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 3.2.1.3 in the Essential Introduction 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^{H}

is to be applied to

C

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

Constraint:

side = Nag_LeftSide

Nag_RightSide

3: trans – Nag_TransTypeInput

On entry: indicates whether

Q

Q^{H}

is to be applied to

C

$trans = Nag_NoTrans$: $Q$ is applied to $C$ .
$trans = Nag_ConjTrans$: $Q^{H}$ is applied to $C$ .

Constraint:

trans = Nag_NoTrans

Nag_ConjTrans

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_zgehrd (f08nsc).

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 ComplexInput

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_zgehrd (f08nsc).

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 ComplexInput

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_zgehrd (f08nsc).

11: c[ $\dim$ ] – ComplexInput/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^{H} C

C Q

C Q^{H}

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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
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.

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_zunmhr (f08nuc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_zunmhr (f08nuc) 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 Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The total number of real floating-point operations is approximately

8 n q^{2}

side = Nag_LeftSide

and

8 m q^{2}

side = Nag_RightSide

, where

q = i_{hi} - i_{lo}

The real analogue of this function is nag_dormhr (f08ngc).

10 Example

This example computes all the eigenvalues of the matrix

A

, where

A = (\begin{array}{r} - 3.97 - 5.04 i & - 4.11 + 3.70 i & - 0.34 + 1.01 i & 1.29 - 0.86 i \\ 0.34 - 1.50 i & 1.52 - 0.43 i & 1.88 - 5.38 i & 3.36 + 0.65 i \\ 3.31 - 3.85 i & 2.50 + 3.45 i & 0.88 - 1.08 i & 0.64 - 1.48 i \\ - 1.10 + 0.82 i & 1.81 - 1.59 i & 3.25 + 1.33 i & 1.57 - 3.44 i \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_zgehrd (f08nsc). The program then calls nag_zhseqr (f08psc) to compute the eigenvalues, and nag_zhsein (f08pxc) to compute the required eigenvectors of

H

by inverse iteration. Finally nag_zunmhr (f08nuc) is called to transform the eigenvectors of

H

back to eigenvectors of the original matrix

A

NAG Library Function Documentnag_zunmhr (f08nuc)

+− 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 Library Function Document

nag_zunmhr (f08nuc)