NAG CL Interface
f08qyc (ztrsna)

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1 Purpose

f08qyc estimates condition numbers for specified eigenvalues and/or right eigenvectors of a complex upper triangular matrix.

2 Specification

#include <nag.h>
void  f08qyc (Nag_OrderType order, Nag_JobType job, Nag_HowManyType how_many, const Nag_Boolean select[], Integer n, const Complex t[], Integer pdt, const Complex vl[], Integer pdvl, const Complex vr[], Integer pdvr, double s[], double sep[], Integer mm, Integer *m, NagError *fail)
The function may be called by the names: f08qyc, nag_lapackeig_ztrsna or nag_ztrsna.

3 Description

f08qyc estimates condition numbers for specified eigenvalues and/or right eigenvectors of a complex upper triangular matrix T. These are the same as the condition numbers of the eigenvalues and right eigenvectors of an original matrix A=ZTZH (with unitary Z), from which T may have been derived.
f08qyc computes the reciprocal of the condition number of an eigenvalue λi as
si = |vHu| uEvE ,  
where u and v are the right and left eigenvectors of T, respectively, corresponding to λi. This reciprocal condition number always lies between zero (i.e., ill-conditioned) and one (i.e., well-conditioned).
An approximate error estimate for a computed eigenvalue λi is then given by
εT si ,  
where ε is the machine precision.
To estimate the reciprocal of the condition number of the right eigenvector corresponding to λi, the function first calls f08qtc to reorder the eigenvalues so that λi is in the leading position:
T =Q ( λi cH 0 T22 ) QH.  
The reciprocal condition number of the eigenvector is then estimated as sepi, the smallest singular value of the matrix (T22-λiI). This number ranges from zero (i.e., ill-conditioned) to very large (i.e., well-conditioned).
An approximate error estimate for a computed right eigenvector u corresponding to λi is then given by
εT sepi .  

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: 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 T.
Constraint: n0.
6: t[dim] const Complex Input
Note: the dimension, dim, of the array t must be at least 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 triangular matrix T, as returned by f08psc.
7: 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) .
8: 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.
The (i,j)th element of the matrix 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 the left eigenvectors of T (or of any matrix QTQH with Q unitary) corresponding to the eigenpairs specified by how_many and select. The eigenvectors must be stored in consecutive rows or columns (depending on the value of order) of vl, as returned by f08pxc or f08qxc.
If job=Nag_EigVecs, vl is not referenced and may be NULL.
9: pdvl Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array vl.
Constraints:
  • if order=Nag_ColMajor,
    • if job=Nag_EigVals or Nag_DoBoth, pdvl n ;
    • if job=Nag_EigVecs, vl may be NULL;
  • if order=Nag_RowMajor,
    • if job=Nag_EigVals or Nag_DoBoth, pdvlmm;
    • if job=Nag_EigVecs, vl may be NULL.
10: 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.
The (i,j)th element of the matrix 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 the right eigenvectors of T (or of any matrix QTQH with Q unitary) corresponding to the eigenpairs specified by how_many and select. The eigenvectors must be stored in consecutive rows or columns (depending on the value of order) of vr, as returned by f08pxc or f08qxc.
If job=Nag_EigVecs, vr is not referenced and may be NULL.
11: pdvr Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array vr.
Constraints:
  • if order=Nag_ColMajor,
    • if job=Nag_EigVals or Nag_DoBoth, pdvr n ;
    • if job=Nag_EigVecs, vr may be NULL;
  • if order=Nag_RowMajor,
    • if job=Nag_EigVals or Nag_DoBoth, pdvrmm;
    • if job=Nag_EigVecs, vr may be NULL.
12: 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: the reciprocal condition numbers of the selected eigenvalues if job=Nag_EigVals or Nag_DoBoth, stored in consecutive elements of the array. Thus s[j-1], sep[j-1] and the jth rows or columns of vl and vr all correspond to the same eigenpair (but not in general the jth eigenpair unless all eigenpairs have been selected).
If job=Nag_EigVecs, s is not referenced and may be NULL.
13: sep[dim] double Output
Note: the dimension, dim, of the array sep must be at least
  • mm when job=Nag_EigVecs or Nag_DoBoth;
  • otherwise sep may be NULL.
On exit: the estimated reciprocal condition numbers of the selected right eigenvectors if job=Nag_EigVecs or Nag_DoBoth, stored in consecutive elements of the array.
If job=Nag_EigVals, sep is not referenced and may be NULL.
14: mm Integer Input
On entry: the number of elements in the arrays s and sep, and the number of rows or columns (depending on the value of order) in the arrays vl and vr (if used). The precise number required, m, is n if how_many=Nag_ComputeAll; if how_many=Nag_ComputeSelected, m is the number of selected eigenpairs (see select), in which case 0mn.
Constraints:
  • if how_many=Nag_ComputeAll, mmn;
  • otherwise mmm.
15: m Integer * Output
On exit: m, the number of selected eigenpairs. If how_many=Nag_ComputeAll, m is set to n.
16: 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, mm=value and n=value.
Constraint: if how_many=Nag_ComputeAll, mmn;
otherwise mmm.
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: n0.
On entry, pdt=value.
Constraint: pdt>0.
On entry, pdvl=value.
Constraint: pdvl>0.
On entry, pdvr=value.
Constraint: pdvr>0.
NE_INT_2
On entry, pdt=value and n=value.
Constraint: pdt max(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.

7 Accuracy

The computed values sepi may over estimate the true value, but seldom by a factor of more than 3.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08qyc 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 real analogue of this function is f08qlc.

10 Example

This example computes approximate error estimates for all the eigenvalues and right eigenvectors of the matrix T, where
T = ( -6.0004-6.9999i 0.3637-0.3656i -0.1880+0.4787i 0.8785-0.2539i 0.0000+0.0000i -5.0000+2.0060i -0.0307-0.7217i -0.2290+0.1313i 0.0000+0.0000i 0.0000+0.0000i 7.9982-0.9964i 0.9357+0.5359i 0.0000+0.0000i 0.0000+0.0000i 0.0000+0.0000i 3.0023-3.9998i ) .  

10.1 Program Text

Program Text (f08qyce.c)

10.2 Program Data

Program Data (f08qyce.d)

10.3 Program Results

Program Results (f08qyce.r)