, by backward transformation on the computed eigenvectors given by f08ykc. It is necessary to call this function only if the optional balancing function f08whc was previously called to balance the matrix pair

(A, B)

2 Specification

#include <nag.h>

void	f08wjc (Nag_OrderType order, Nag_JobType job, Nag_SideType side, Integer n, Integer ilo, Integer ihi, const double lscale[], const double rscale[], Integer m, double v[], Integer pdv, NagError *fail)

The function may be called by the names: f08wjc, nag_lapackeig_dggbak or nag_dggbak.

3 Description

If the matrix pair has been previously balanced using the function f08whc then f08wjc backtransforms the eigenvector solution given by f08ykc. This is usually the sixth and last step in the solution of the generalized eigenvalue problem.

For a description of balancing, see the document for f08whc.

4 References

Ward R C (1981) Balancing the generalized eigenvalue problem SIAM J. Sci. Stat. Comp. 2 141–152

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: $job$ – Nag_JobType Input

On entry: specifies the backward transformation step required.

$job = Nag_DoNothing$: No transformations are done.
$job = Nag_Permute$: Only do backward transformations based on permutations.
$job = Nag_Scale$: Only do backward transformations based on scaling.
$job = Nag_DoBoth$: Do backward transformations for both permutations and scaling.

Note: this must be the same argument job as supplied to f08whc.

Constraint:

job = Nag_DoNothing

Nag_Permute

Nag_Scale

Nag_DoBoth

3: $side$ – Nag_SideType Input

On entry: indicates whether left or right eigenvectors are to be transformed.

$side = Nag_LeftSide$: The left eigenvectors are transformed.
$side = Nag_RightSide$: The right eigenvectors are transformed.

Constraint:

side = Nag_LeftSide

Nag_RightSide

4: $n$ – Integer Input

On entry:

n

, the order of the matrices

A

and

B

of the generalized eigenvalue problem.

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.

Constraints:

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

7: $lscale [\dim]$ – const double Input

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

\max (1, n)

On entry: details of the permutations and scaling factors applied to the left side of the matrices

A

and

B

, as returned by a previous call to f08whc.

8: $rscale [\dim]$ – const double Input

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

\max (1, n)

On entry: details of the permutations and scaling factors applied to the right side of the matrices

A

and

B

, as returned by a previous call to f08whc.

9: $m$ – Integer Input

On entry:

m

, the required number of left or right eigenvectors.

Constraint:

0 \leq m \leq n

10: $v [\dim]$ – double Input/Output

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

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

The

(i, j)

th element of the matrix

V

is stored in

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

On entry: the matrix of right or left eigenvectors, as returned by f08whc.

On exit: the transformed right or left eigenvectors.

11: $pdv$ – Integer Input

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

Constraints:

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

12: $fail$ – 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_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

On entry, $pdv = ⟨ value ⟩$ .
Constraint: $pdv > 0$ .
NE_INT_2: On entry, $m = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $0 \leq m \leq n$ .

On entry, $pdv = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: $pdv \geq \max (1, m)$ .

On entry, $pdv = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pdv \geq \max (1, n)$ .
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 errors are negligible, compared with the previous computations.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

f08wjc 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.