NAG Library Function Document

nag_dgghrd (f08wec)

void	nag_dgghrd (Nag_OrderType order, Nag_ComputeQType compq, Nag_ComputeZType compz, Integer n, Integer ilo, Integer ihi, double a[], Integer pda, double b[], Integer pdb, double q[], Integer pdq, double z[], Integer pdz, NagError *fail)

3 Description

nag_dgghrd (f08wec) is the third step in the solution of the real generalized eigenvalue problem

A x = λ B x .

The (optional) first step balances the two matrices using nag_dggbal (f08whc). In the second step, matrix

B

is reduced to upper triangular form using the

Q R

factorization function nag_dgeqrf (f08aec) and this orthogonal transformation

Q

is applied to matrix

A

by calling nag_dormqr (f08agc).

nag_dgghrd (f08wec) reduces a pair of real matrices

(A, B)

, where

B

is upper triangular, to the generalized upper Hessenberg form using orthogonal transformations. This two-sided transformation is of the form

\begin{matrix} Q^{T} A Z = H \\ Q^{T} B Z = T \end{matrix}

where

H

is an upper Hessenberg matrix,

T

is an upper triangular matrix and

Q

and

Z

are orthogonal matrices determined as products of Givens rotations. They may either be formed explicitly, or they may be postmultiplied into input matrices

Q_{1}

and

Z_{1}

, so that

\begin{matrix} Q_{1} A Z_{1}^{T} = (Q_{1} Q) H {(Z_{1} Z)}^{T}, \\ Q_{1} B Z_{1}^{T} = (Q_{1} Q) T {(Z_{1} Z)}^{T} . \end{matrix}

4 References

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

Moler C B and Stewart G W (1973) An algorithm for generalized matrix eigenproblems SIAM J. Numer. Anal. 10 241–256

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: $compq$ – Nag_ComputeQTypeInput

On entry: specifies the form of the computed orthogonal matrix

Q

$compq = Nag_NotQ$: Do not compute $Q$ .
$compq = Nag_InitQ$: The orthogonal matrix $Q$ is returned.
$compq = Nag_UpdateSchur$: q must contain an orthogonal matrix $Q_{1}$ , and the product $Q_{1} Q$ is returned.

Constraint:

compq = Nag_NotQ

Nag_InitQ

Nag_UpdateSchur

3: $compz$ – Nag_ComputeZTypeInput

On entry: specifies the form of the computed orthogonal matrix

Z

$compz = Nag_NotZ$: Do not compute $Z$ .
$compz = Nag_InitZ$: The orthogonal matrix $Z$ is returned.
$compz = Nag_UpdateZ$: z must contain an orthogonal matrix $Z_{1}$ , and the product $Z_{1} Z$ is returned.

Constraint:

compz = Nag_NotZ

Nag_UpdateZ

Nag_InitZ

4: $n$ – IntegerInput

On entry:

n

, the order of the matrices

A

and

B

Constraint:

n \geq 0

5: $ilo$ – IntegerInput

6: $ihi$ – IntegerInput

On entry:

i_{lo}

and

i_{hi}

as determined by a previous call to nag_dggbal (f08whc). Otherwise, they should be set to

1

and

n

, respectively.

Constraints:

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

7: $a [\dim]$ – doubleInput/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 matrix

A

of the matrix pair

(A, B)

. Usually, this is the matrix

A

returned by nag_dormqr (f08agc).

On exit: a is overwritten by the upper Hessenberg matrix

H

8: $pda$ – 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)

9: $b [\dim]$ – doubleInput/Output

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

\max (1, pdb \times n)

The

(i, j)

th element of the matrix

B

is stored in

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

On entry: the upper triangular matrix

B

of the matrix pair

(A, B)

. Usually, this is the matrix

B

returned by the

Q R

factorization function nag_dgeqrf (f08aec).

On exit: b is overwritten by the upper triangular matrix

T

10: $pdb$ – IntegerInput

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

Constraint:

pdb \geq \max (1, n)

11: $q [\dim]$ – doubleInput/Output

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

$\max (1, pdq \times n)$ when $compq = Nag_InitQ$ or $Nag_UpdateSchur$ ;
$1$ when $compq = Nag_NotQ$ .

The

(i, j)

th element of the matrix

Q

is stored in

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

On entry: if

compq = Nag_UpdateSchur

, q must contain an orthogonal matrix

Q_{1}

compq = Nag_NotQ

, q is not referenced.

On exit: if

compq = Nag_InitQ

, q contains the orthogonal matrix

Q

compq = Nag_UpdateSchur

, q is overwritten by

Q_{1} Q

12: $pdq$ – IntegerInput

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

Constraints:

if $compq = Nag_InitQ$ or $Nag_UpdateSchur$ , $pdq \geq \max (1, n)$ ;
if $compq = Nag_NotQ$ , $pdq \geq 1$ .

13: $z [\dim]$ – doubleInput/Output

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

\max (1, pdz \times n)

when

compz = Nag_UpdateZ

Nag_InitZ

The

(i, j)

th element of the matrix

Z

is stored in

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

On entry: if

compz = Nag_UpdateZ

, z must contain an orthogonal matrix

Z_{1}

compz = Nag_NotZ

, z is not referenced.

On exit: if

compz = Nag_InitZ

, z contains the orthogonal matrix

Z

compz = Nag_UpdateZ

, z is overwritten by

Z_{1} Z

14: $pdz$ – IntegerInput

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

Constraints:

if $compz = Nag_UpdateZ$ or $Nag_InitZ$ , $pdz \geq \max (1, n)$ ;
if $compz = Nag_NotZ$ , $pdz \geq 1$ .

15: $fail$ – 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.
See Section 3.2.1.2 in the Essential Introduction for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_ENUM_INT_2: On entry, $compq = 〈value〉$ , $pdq = 〈value〉$ and $n = 〈value〉$ .
Constraint: if $compq = Nag_InitQ$ or $Nag_UpdateSchur$ , $pdq \geq \max (1, n)$ ;
if $compq = Nag_NotQ$ , $pdq \geq 1$ .

On entry, $compz = 〈value〉$ , $pdz = 〈value〉$ and $n = 〈value〉$ .
Constraint: if $compz = Nag_UpdateZ$ or $Nag_InitZ$ , $pdz \geq \max (1, n)$ ;
if $compz = Nag_NotZ$ , $pdz \geq 1$ .
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .

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

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

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

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

On entry, $pdb = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pdb \geq \max (1, n)$ .
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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.

7 Accuracy

The reduction to the generalized Hessenberg form is implemented using orthogonal transformations which are backward stable.

8 Parallelism and Performance

nag_dgghrd (f08wec) is not threaded by NAG in any implementation.

nag_dgghrd (f08wec) 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.