NAG CL Interface
f08qfc (dtrexc)

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1 Purpose

f08qfc reorders the Schur factorization of a real general matrix.

2 Specification

#include <nag.h>
void  f08qfc (Nag_OrderType order, Nag_ComputeQType compq, Integer n, double t[], Integer pdt, double q[], Integer pdq, Integer *ifst, Integer *ilst, NagError *fail)
The function may be called by the names: f08qfc, nag_lapackeig_dtrexc or nag_dtrexc.

3 Description

f08qfc reorders the Schur factorization of a real general matrix A=QTQT, so that the diagonal element or block of T with row index ifst is moved to row ilst.
The reordered Schur form T~ is computed by an orthogonal similarity transformation: T~=ZTTZ. Optionally the updated matrix Q~ of Schur vectors is computed as Q~=QZ, giving A=Q~T~Q~T.

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_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: compq Nag_ComputeQType Input
On entry: indicates whether the matrix Q of Schur vectors is to be updated.
compq=Nag_UpdateSchur
The matrix Q of Schur vectors is updated.
compq=Nag_NotQ
No Schur vectors are updated.
Constraint: compq=Nag_UpdateSchur or Nag_NotQ.
3: n Integer Input
On entry: n, the order of the matrix T.
Constraint: n0.
4: t[dim] double Input/Output
Note: the dimension, dim, of the array t must be at least max(1,pdt×n).
The (i,j)th element of the matrix T is stored in
  • t[(j-1)×pdt+i-1] when order=Nag_ColMajor;
  • t[(i-1)×pdt+j-1] when order=Nag_RowMajor.
On entry: the n×n upper quasi-triangular matrix T in canonical Schur form, as returned by f08pec.
On exit: t is overwritten by the updated matrix T~. See also Section 9.
5: pdt Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array t.
Constraint: pdt max(1,n) .
6: q[dim] double Input/Output
Note: the dimension, dim, of the array q must be at least
  • max(1,pdq×n) when compq=Nag_UpdateSchur;
  • 1 when compq=Nag_NotQ.
The (i,j)th element of the matrix Q is stored in
  • q[(j-1)×pdq+i-1] when order=Nag_ColMajor;
  • q[(i-1)×pdq+j-1] when order=Nag_RowMajor.
On entry: if compq=Nag_UpdateSchur, q must contain the n×n orthogonal matrix Q of Schur vectors.
On exit: if compq=Nag_UpdateSchur, q contains the updated matrix of Schur vectors.
If compq=Nag_NotQ, q is not referenced.
7: 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_UpdateSchur, pdq max(1,n) ;
  • if compq=Nag_NotQ, pdq1.
8: ifst Integer * Input/Output
9: ilst Integer * Input/Output
On entry: ifst and ilst must specify the reordering of the diagonal elements or blocks of T. The element or block with row index ifst is moved to row ilst by a sequence of exchanges between adjacent elements or blocks.
On exit: if ifst pointed to the second row of a 2×2 block on entry, it is changed to point to the first row. ilst always points to the first row of the block in its final position (which may differ from its input value by ±1).
Constraint: 1ifstn and 1ilstn.
10: 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_UpdateSchur, pdq max(1,n) ;
if compq=Nag_NotQ, pdq1.
NE_EXCHANGE
Two adjacent diagonal elements or blocks could not be successfully exchanged. This error can only occur if the exchange involves at least one 2×2 block; it implies that the problem is very ill-conditioned, and that the eigenvalues of the two blocks are very close. On exit, T may have been partially reordered, and ilst points to the first row of the current position of the block being moved; Q (if requested) is updated consistently with T.
NE_INT
On entry, n=value.
Constraint: n0.
On entry, pdq=value.
Constraint: pdq>0.
On entry, pdt=value.
Constraint: pdt>0.
NE_INT_2
On entry, pdt=value and n=value.
Constraint: pdt max(1,n) .
NE_INT_3
On entry, n=value, ifst=value and ilst=value.
Constraint: 1ifstn and 1ilstn.
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 computed matrix T~ is exactly similar to a matrix (T+E), where
E2 = O(ε) T2 ,  
and ε is the machine precision.
Note that if a 2×2 diagonal block is involved in the reordering, its off-diagonal elements are in general changed; the diagonal elements and the eigenvalues of the block are unchanged unless the block is sufficiently ill-conditioned, in which case they may be noticeably altered. It is possible for a 2×2 block to break into two 1×1 blocks, i.e., for a pair of complex eigenvalues to become purely real. The values of real eigenvalues however are never changed by the reordering.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08qfc 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 total number of floating-point operations is approximately 6nr if compq=Nag_NotQ, and 12nr if compq=Nag_UpdateSchur, where r=|ifst-ilst|.
The input matrix T must be in canonical Schur form, as is the output matrix T~. This has the following structure.
If all the computed eigenvalues are real, T is upper triangular and its diagonal elements are the eigenvalues.
If some of the computed eigenvalues form complex conjugate pairs, then T has 2×2 diagonal blocks. Each diagonal block has the form
( tii ti,i+1 ti+1,i ti+1,i+1 )= ( α β γ α )  
where βγ<0. The corresponding eigenvalues are α±βγ.
The complex analogue of this function is f08qtc.

10 Example

This example reorders the Schur factorization of the matrix T so that the 2×2 block with row index 2 is moved to row 1, where
T = ( 0.80 -0.11 0.01 0.03 0.00 -0.10 0.25 0.35 0.00 -0.65 -0.10 0.20 0.00 0.00 0.00 -0.10 ) .  

10.1 Program Text

Program Text (f08qfce.c)

10.2 Program Data

Program Data (f08qfce.d)

10.3 Program Results

Program Results (f08qfce.r)