NAG Library Function Document

nag_zunghr (f08ntc)

+− 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

nag_zunghr (f08ntc) generates the complex unitary matrix

Q

which was determined by nag_zgehrd (f08nsc) when reducing a complex general matrix

A

to Hessenberg form.

2 Specification

#include <nag.h>

#include <nagf08.h>

void	nag_zunghr (Nag_OrderType order, Integer n, Integer ilo, Integer ihi, Complex a[], Integer pda, const Complex tau[], NagError *fail)

3 Description

nag_zunghr (f08ntc) 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 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: 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: n – IntegerInput

On entry:

n

, the order of the matrix

Q

Constraint:

n \geq 0

3: ilo – IntegerInput

4: ihi – IntegerInput

On entry: these must be the same arguments ilo and ihi, respectively, as supplied to nag_zgehrd (f08nsc).

Constraints:

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

5: a[ $\dim$ ] – ComplexInput/Output

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

\max (1, pda \times n)

On entry: details of the vectors which define the elementary reflectors, as returned by nag_zgehrd (f08nsc).

On exit: the

n

n

unitary matrix

Q

order = Nag_ColMajor

, the

(i, j)

th element of the matrix is stored in

a [(j - 1) \times pda + i - 1]

order = Nag_RowMajor

, the

(i, j)

th element of the matrix is stored in

a [(i - 1) \times pda + j - 1]

6: pda – IntegerInput

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[ $\dim$ ] – const ComplexInput

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

\max (1, n - 1)

On entry: further details of the elementary reflectors, as returned by nag_zgehrd (f08nsc).

8: 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_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.

7 Accuracy

The computed matrix

Q

differs from an exactly unitary matrix by a matrix

E

such that

{‖E‖}_{2} = O (ε),

where

ε

is the machine precision.

8 Parallelism and Performance

nag_zunghr (f08ntc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_zunghr (f08ntc) 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

\frac{16}{3} q^{3}

, where

q = i_{hi} - i_{lo}

The real analogue of this function is nag_dorghr (f08nfc).

10 Example

This example computes the Schur factorization 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}) .

Here

A

is general and must first be reduced to Hessenberg form by nag_zgehrd (f08nsc). The program then calls nag_zunghr (f08ntc) to form

Q

, and passes this matrix to nag_zhseqr (f08psc) which computes the Schur factorization of

A

NAG Library Function Documentnag_zunghr (f08ntc)

+− 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_zunghr (f08ntc)