NAG CL Interface
f08wjc (dggbak)

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1 Purpose

f08wjc forms the right or left eigenvectors of the real generalized eigenvalue problem Ax=λBx, 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 or 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 or 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 or Nag_RightSide.
4: n Integer Input
On entry: n, the order of the matrices A and B of the generalized eigenvalue problem.
Constraint: n0.
5: ilo Integer Input
6: ihi Integer Input
On entry: ilo and ihi as determined by a previous call to f08whc.
Constraints:
  • if n>0, 1 ilo ihi 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: 0mn.
10: v[dim] double Input/Output
Note: the dimension, dim, of the array v must be at least
  • max(1,pdv×m) when order=Nag_ColMajor;
  • max(1,n×pdv) when order=Nag_RowMajor.
The (i,j)th element of the matrix V is stored in
  • v[(j-1)×pdv+i-1] when order=Nag_ColMajor;
  • v[(i-1)×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 max(1,n) ;
  • if order=Nag_RowMajor, pdvmax(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: n0.
On entry, pdv=value.
Constraint: pdv>0.
NE_INT_2
On entry, m=value and n=value.
Constraint: 0mn.
On entry, pdv=value and m=value.
Constraint: pdvmax(1,m).
On entry, pdv=value and n=value.
Constraint: pdv max(1,n) .
NE_INT_3
On entry, n=value, ilo=value and ihi=value.
Constraint: if n>0, 1 ilo ihi 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.

9 Further Comments

The number of operations is proportional to n2.
The complex analogue of this function is f08wwc.

10 Example

See Section 10 in f08xec and f08ykc.