NAG CL Interface
f08yyc (ztgsna)

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1 Purpose

f08yyc estimates condition numbers for specified eigenvalues and/or eigenvectors of a complex matrix pair in generalized Schur form.

2 Specification

#include <nag.h>
void  f08yyc (Nag_OrderType order, Nag_JobType job, Nag_HowManyType how_many, const Nag_Boolean select[], Integer n, const Complex a[], Integer pda, const Complex b[], Integer pdb, const Complex vl[], Integer pdvl, const Complex vr[], Integer pdvr, double s[], double dif[], Integer mm, Integer *m, NagError *fail)
The function may be called by the names: f08yyc, nag_lapackeig_ztgsna or nag_ztgsna.

3 Description

f08yyc estimates condition numbers for specified eigenvalues and/or right eigenvectors of an n×n matrix pair (S,T) in generalized Schur form. The function actually returns estimates of the reciprocals of the condition numbers in order to avoid possible overflow.
The pair (S,T) are in generalized Schur form if S and T are upper triangular as returned, for example, by f08xpc or f08xqc, or f08xsc with job=Nag_Schur. The diagonal elements define the generalized eigenvalues (αi,βi), for i=1,2,,n, of the pair (S,T) and the eigenvalues are given by
λi = αi / βi ,  
so that
βi S xi = αi T xi   or   S xi = λi T xi ,  
where xi is the corresponding (right) eigenvector.
If S and T are the result of a generalized Schur factorization of a matrix pair (A,B)
A = QSZH ,   B = QTZH  
then the eigenvalues and condition numbers of the pair (S,T) are the same as those of the pair (A,B).
Let (α,β)(0,0) be a simple generalized eigenvalue of (A,B). Then the reciprocal of the condition number of the eigenvalue λ=α/β is defined as
s(λ)= ( |yHAx| 2 + |yHBx| 2 ) 1/2 (x2y2) ,  
where x and y are the right and left eigenvectors of (A,B) corresponding to λ. If both α and β are zero, then (A,B) is singular and s(λ)=-1 is returned.
If U and V are unitary transformations such that
UH (A,B) V= (S,T) = ( α * 0 S22 ) ( β * 0 T22 ) ,  
where S22 and T22 are (n-1)×(n-1) matrices, then the reciprocal condition number is given by
Dif(x) Dif(y) = Dif((α,β),(S22,T22)) = σmin (Z) ,  
where σmin(Z) denotes the smallest singular value of the 2(n-1)×2(n-1) matrix
Z = ( αI -1S22 βI -1T22 )  
and is the Kronecker product.
See Sections 2.4.8 and 4.11 of Anderson et al. (1999) and Kågström and Poromaa (1996) for further details and information.

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
Kågström B and Poromaa P (1996) LAPACK-style algorithms and software for solving the generalized Sylvester equation and estimating the separation between regular matrix pairs ACM Trans. Math. Software 22 78–103

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: indicates whether condition numbers are required for eigenvalues and/or eigenvectors.
job=Nag_EigVals
Condition numbers for eigenvalues only are computed.
job=Nag_EigVecs
Condition numbers for eigenvectors only are computed.
job=Nag_DoBoth
Condition numbers for both eigenvalues and eigenvectors are computed.
Constraint: job=Nag_EigVals, Nag_EigVecs or Nag_DoBoth.
3: how_many Nag_HowManyType Input
On entry: indicates how many condition numbers are to be computed.
how_many=Nag_ComputeAll
Condition numbers for all eigenpairs are computed.
how_many=Nag_ComputeSelected
Condition numbers for selected eigenpairs (as specified by select) are computed.
Constraint: how_many=Nag_ComputeAll or Nag_ComputeSelected.
4: select[dim] const Nag_Boolean Input
Note: the dimension, dim, of the array select must be at least
  • n when how_many=Nag_ComputeSelected;
  • otherwise select may be NULL.
On entry: specifies the eigenpairs for which condition numbers are to be computed if how_many=Nag_ComputeSelected. To select condition numbers for the eigenpair corresponding to the eigenvalue λj, select[j-1] must be set to Nag_TRUE.
If how_many=Nag_ComputeAll, select is not referenced and may be NULL.
5: n Integer Input
On entry: n, the order of the matrix pair (S,T).
Constraint: n0.
6: a[dim] const Complex Input
Note: the dimension, dim, of the array a must be at least 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 upper triangular matrix S.
7: pda Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: pdan.
8: b[dim] const Complex Input
Note: the dimension, dim, of the array b must be at least pdb×n.
The (i,j)th element of the matrix B is stored in
  • b[(j-1)×pdb+i-1] when order=Nag_ColMajor;
  • b[(i-1)×pdb+j-1] when order=Nag_RowMajor.
On entry: the upper triangular matrix T.
9: pdb Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraint: pdbn.
10: vl[dim] const Complex Input
Note: the dimension, dim, of the array vl must be at least
  • pdvl×mm when job=Nag_EigVals or Nag_DoBoth and order=Nag_ColMajor;
  • n×pdvl when job=Nag_EigVals or Nag_DoBoth and order=Nag_RowMajor;
  • otherwise vl may be NULL.
ith element of the jth vector is stored in
  • vl[(j-1)×pdvl+i-1] when order=Nag_ColMajor;
  • vl[(i-1)×pdvl+j-1] when order=Nag_RowMajor.
On entry: if job=Nag_EigVals or Nag_DoBoth, vl must contain left eigenvectors of (S,T), corresponding to the eigenpairs specified by how_many and select. The eigenvectors must be stored in consecutive columns of vl, as returned by f08wqc or f08yxc.
If job=Nag_EigVecs, vl is not referenced and may be NULL.
11: pdvl Integer Input
On entry: the stride used in the array vl.
Constraints:
  • if order=Nag_ColMajor,
    • if job=Nag_EigVals or Nag_DoBoth, pdvl n ;
    • otherwise pdvl1;
  • if order=Nag_RowMajor,
    • if job=Nag_EigVals or Nag_DoBoth, pdvlmm;
    • otherwise vl may be NULL.
12: vr[dim] const Complex Input
Note: the dimension, dim, of the array vr must be at least
  • pdvr×mm when job=Nag_EigVals or Nag_DoBoth and order=Nag_ColMajor;
  • n×pdvr when job=Nag_EigVals or Nag_DoBoth and order=Nag_RowMajor;
  • otherwise vr may be NULL.
ith element of the jth vector is stored in
  • vr[(j-1)×pdvr+i-1] when order=Nag_ColMajor;
  • vr[(i-1)×pdvr+j-1] when order=Nag_RowMajor.
On entry: if job=Nag_EigVals or Nag_DoBoth, vr must contain right eigenvectors of (S,T), corresponding to the eigenpairs specified by how_many and select. The eigenvectors must be stored in consecutive columns of vr, as returned by f08wqc or f08yxc.
If job=Nag_EigVecs, vr is not referenced and may be NULL.
13: pdvr Integer Input
On entry: the stride used in the array vr.
Constraints:
  • if order=Nag_ColMajor,
    • if job=Nag_EigVals or Nag_DoBoth, pdvr n ;
    • otherwise pdvr1;
  • if order=Nag_RowMajor,
    • if job=Nag_EigVals or Nag_DoBoth, pdvrmm;
    • otherwise vr may be NULL.
14: s[dim] double Output
Note: the dimension, dim, of the array s must be at least
  • mm when job=Nag_EigVals or Nag_DoBoth;
  • otherwise s may be NULL.
On exit: if job=Nag_EigVals or Nag_DoBoth, the reciprocal condition numbers of the selected eigenvalues, stored in consecutive elements of the array.
If job=Nag_EigVecs, s is not referenced and may be NULL.
15: dif[dim] double Output
Note: the dimension, dim, of the array dif must be at least
  • mm when job=Nag_EigVecs or Nag_DoBoth;
  • otherwise dif may be NULL.
On exit: if job=Nag_EigVecs or Nag_DoBoth, the estimated reciprocal condition numbers of the selected eigenvectors, stored in consecutive elements of the array. If the eigenvalues cannot be reordered to compute dif[j-1], dif[j-1] is set to 0; this can only occur when the true value would be very small anyway.
If job=Nag_EigVals, dif is not referenced and may be NULL.
16: mm Integer Input
On entry: the number of elements in the arrays s and dif.
Constraints:
  • if how_many=Nag_ComputeAll, mmn;
  • otherwise mm​ the number of selected eigenvalues.
17: m Integer * Output
On exit: the number of elements of the arrays s and dif used to store the specified condition numbers; for each selected eigenvalue one element is used.
If how_many=Nag_ComputeAll, m is set to n.
18: 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, how_many=value, n=value and mm=value.
Constraint: if how_many=Nag_ComputeAll, mmn;
otherwise mm​ the number of selected eigenvalues.
On entry, job=value, pdvl=value and mm=value.
Constraint: if job=Nag_EigVals or Nag_DoBoth, pdvlmm.
On entry, job=value, pdvl=value and n=value.
Constraint: if job=Nag_EigVals or Nag_DoBoth, pdvl n .
On entry, job=value, pdvr=value and mm=value.
Constraint: if job=Nag_EigVals or Nag_DoBoth, pdvrmm.
On entry, job=value, pdvr=value and n=value.
Constraint: if job=Nag_EigVals or Nag_DoBoth, pdvr n .
NE_INT
On entry, n=value.
Constraint: n>0.
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
On entry, pdb=value.
Constraint: pdb>0.
On entry, pdvl=value.
Constraint: pdvl>0.
On entry, pdvr=value.
Constraint: pdvr>0.
NE_INT_2
On entry, pda=value and n=value.
Constraint: pdan.
On entry, pdb=value and n=value.
Constraint: pdbn.
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

Not applicable.

8 Parallelism and Performance

f08yyc 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

An approximate asymptotic error bound on the chordal distance between the computed eigenvalue λ~ and the corresponding exact eigenvalue λ is
χ(λ~,λ) ε(A,B)F / S(λ)  
where ε is the machine precision.
An approximate asymptotic error bound for the right or left computed eigenvectors x~ or y~ corresponding to the right and left eigenvectors x and y is given by
θ(z~,z) ε (A,B)F / Dif .  
The real analogue of this function is f08ylc.

10 Example

This example estimates condition numbers and approximate error estimates for all the eigenvalues and right eigenvectors of the pair (S,T) given by
S = ( 4.0+4.0i 1.0+1.0i 1.0+1.0i 2.0-1.0i 0.0i+0.0 2.0+1.0i 1.0+1.0i 1.0+1.0i 0.0i+0.0 0.0i+0.0 2.0-1.0i 1.0+1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 6.0-2.0i )  
and
T = ( 2.0 1.0+1.0i 1.0+1.0i 3.0-1.0i 0.0 1.0i+0.0 2.0+1.0i 1.0+1.0i 0.0 0.0i+0.0 1.0i+0.0 1.0+1.0i 0.0 0.0i+0.0 0.0i+0.0 2.0i+0.0 ) .  
The eigenvalues and eigenvectors are computed by calling f08yxc.

10.1 Program Text

Program Text (f08yyce.c)

10.2 Program Data

Program Data (f08yyce.d)

10.3 Program Results

Program Results (f08yyce.r)