NAG CL Interface
f08pnc (zgees)

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1 Purpose

f08pnc computes the eigenvalues, the Schur form T, and, optionally, the matrix of Schur vectors Z for an n×n complex nonsymmetric matrix A.

2 Specification

#include <nag.h>
void  f08pnc (Nag_OrderType order, Nag_JobType jobvs, Nag_SortEigValsType sort,
Nag_Boolean (*select)(Complex w),
Integer n, Complex a[], Integer pda, Integer *sdim, Complex w[], Complex vs[], Integer pdvs, NagError *fail)
The function may be called by the names: f08pnc, nag_lapackeig_zgees or nag_zgees.

3 Description

The Schur factorization of A is given by
A = Z T ZH ,  
where Z, the matrix of Schur vectors, is unitary and T is the Schur form. A complex matrix is in Schur form if it is upper triangular.
Optionally, f08pnc also orders the eigenvalues on the diagonal of the Schur form so that selected eigenvalues are at the top left. The leading columns of Z form an orthonormal basis for the invariant subspace corresponding to the selected eigenvalues.

4 References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug
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: jobvs Nag_JobType Input
On entry: if jobvs=Nag_DoNothing, Schur vectors are not computed.
If jobvs=Nag_Schur, Schur vectors are computed.
Constraint: jobvs=Nag_DoNothing or Nag_Schur.
3: sort Nag_SortEigValsType Input
On entry: specifies whether or not to order the eigenvalues on the diagonal of the Schur form.
sort=Nag_NoSortEigVals
Eigenvalues are not ordered.
sort=Nag_SortEigVals
Eigenvalues are ordered (see select).
Constraint: sort=Nag_NoSortEigVals or Nag_SortEigVals.
4: select function, supplied by the user External Function
If sort=Nag_SortEigVals, select is used to select eigenvalues to sort to the top left of the Schur form.
If sort=Nag_NoSortEigVals, select is not referenced and f08pnc may be specified as NULLFN.
An eigenvalue w[j-1] is selected if select(w[j-1]) is Nag_TRUE.
The specification of select is:
Nag_Boolean  select (Complex w)
1: w Complex Input
On entry: the real and imaginary parts of the eigenvalue.
5: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
6: a[dim] Complex Input/Output
Note: the dimension, dim, of the array a must be at least max(1,pda×n).
The (i,j)th element of the matrix A is stored in
  • a[(j-1)×pda+i-1] when order=Nag_ColMajor;
  • a[(i-1)×pda+j-1] when order=Nag_RowMajor.
On entry: the n×n matrix A.
On exit: a is overwritten by its Schur form T.
7: pda Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: pdamax(1,n).
8: sdim Integer * Output
On exit: if sort=Nag_NoSortEigVals, sdim=0.
If sort=Nag_SortEigVals, sdim= number of eigenvalues for which select is Nag_TRUE.
9: w[dim] Complex Output
Note: the dimension, dim, of the array w must be at least max(1,n).
On exit: contains the computed eigenvalues, in the same order that they appear on the diagonal of the output Schur form T.
10: vs[dim] Complex Output
Note: the dimension, dim, of the array vs must be at least
  • max(1,pdvs×n) when jobvs=Nag_Schur;
  • 1 otherwise.
ith element of the jth vector is stored in
  • vs[(j-1)×pdvs+i-1] when order=Nag_ColMajor;
  • vs[(i-1)×pdvs+j-1] when order=Nag_RowMajor.
On exit: if jobvs=Nag_Schur, vs contains the unitary matrix Z of Schur vectors.
If jobvs=Nag_DoNothing, vs is not referenced.
11: pdvs Integer Input
On entry: the stride used in the array vs.
Constraints:
  • if jobvs=Nag_Schur, pdvs max(1,n) ;
  • otherwise pdvs1.
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_CONVERGENCE
The QR algorithm failed to compute all the eigenvalues.
NE_ENUM_INT_2
On entry, jobvs=value, pdvs=value and n=value.
Constraint: if jobvs=Nag_Schur, pdvs max(1,n) ;
otherwise pdvs1.
NE_INT
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
On entry, pdvs=value.
Constraint: pdvs>0.
NE_INT_2
On entry, pda=value and n=value.
Constraint: pdamax(1,n).
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.
NE_SCHUR_REORDER
The eigenvalues could not be reordered because some eigenvalues were too close to separate (the problem is very ill-conditioned).
NE_SCHUR_REORDER_SELECT
After reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Schur form no longer satisfy select=Nag_TRUE. This could also be caused by underflow due to scaling.

7 Accuracy

The computed Schur factorization satisfies
A+E=ZT ZH ,  
where
E2 = O(ε) A2 ,  
and ε is the machine precision. See Section 4.8 of Anderson et al. (1999) for further details.

8 Parallelism and Performance

f08pnc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08pnc 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 proportional to n3.
The real analogue of this function is f08pac.

10 Example

This example finds the Schur factorization of the matrix
A = ( -3.97-5.04i -4.11+3.70i -0.34+1.01i 1.29-0.86i 0.34-1.50i 1.52-0.43i 1.88-5.38i 3.36+0.65i 3.31-3.85i 2.50+3.45i 0.88-1.08i 0.64-1.48i -1.10+0.82i 1.81-1.59i 3.25+1.33i 1.57-3.44i ) .  

10.1 Program Text

Program Text (f08pnce.c)

10.2 Program Data

Program Data (f08pnce.d)

10.3 Program Results

Program Results (f08pnce.r)