void	f08wfc (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)

The function may be called by the names: f08wfc, nag_lapackeig_dgghd3 or nag_dgghd3.

3 Description

f08wfc 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 f08whc. In the second step, matrix

B

is reduced to upper triangular form using the

Q R

factorization function f08aec and this orthogonal transformation

Q

is applied to matrix

A

by calling f08agc. The driver, f08wcc, solves the real generalized eigenvalue problem by combining all the required steps including those just listed.

f08wfc 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 (2012) Matrix Computations (4th 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_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: $compq$ – Nag_ComputeQType Input

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_ComputeZType Input

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: $n$ – Integer Input

On entry:

n

, the order of the matrices

A

and

B

Constraint:

n \geq 0

5: $ilo$ – Integer Input

6: $ihi$ – Integer Input

On entry:

i_{lo}

and

i_{hi}

as determined by a previous call to 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: $a [\dim]$ – 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 matrix

A

of the matrix pair

(A, B)

. Usually, this is the matrix

A

returned by f08agc.

On exit: a is overwritten by the upper Hessenberg matrix

H

8: $pda$ – 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)

9: $b [\dim]$ – double Input/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 f08aec.

On exit: b is overwritten by the upper triangular matrix

T

10: $pdb$ – Integer Input

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: $q [\dim]$ – double Input/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: $pdq$ – Integer Input

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: $z [\dim]$ – double Input/Output

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

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

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: $pdz$ – Integer Input

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

8 Parallelism and Performance

f08wfc 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.