nag_dtgsna (f08ylc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dtgsna (f08ylc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dtgsna (f08ylc) estimates condition numbers for specified eigenvalues and/or eigenvectors of a matrix pair in generalized real Schur form.

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_dtgsna (Nag_OrderType order, Nag_JobType job, Nag_HowManyType howmny, const Nag_Boolean select[], Integer n, const double a[], Integer pda, const double b[], Integer pdb, const double vl[], Integer pdvl, const double vr[], Integer pdvr, double s[], double dif[], Integer mm, Integer *m, NagError *fail)

3  Description

nag_dtgsna (f08ylc) estimates condition numbers for specified eigenvalues and/or right eigenvectors of an n by n matrix pair S,T in real 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 real generalized Schur form if S is block upper triangular with 1 by 1 and 2 by 2 diagonal blocks and T is upper triangular as returned, for example, by nag_dgges (f08xac), or nag_dhgeqz (f08xec) with job=Nag_Schur. The diagonal elements, or blocks, 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 = QSZT ,   B = QTZT
then the eigenvalues and condition numbers of the pair S,T are the same as those of the pair A,B.
Let α,β0,0 be a simple generalized eigenvalue of A,B. Then the reciprocal of the condition number of the eigenvalue λ=α/β is defined as
sλ= yTAx 2 + yTBx 2 1/2 x2 y2 ,
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 sλ=-1 is returned.
The definition of the reciprocal of the estimated condition number of the right eigenvector x and the left eigenvector y corresponding to the simple eigenvalue λ depends upon whether λ is a real eigenvalue, or one of a complex conjugate pair.
If the eigenvalue λ is real and U and V are orthogonal transformations such that
UT A,B V= S,T = α * 0 S22 β * 0 T22 ,
where S22 and T22 are n-1 by n-1 matrices, then the reciprocal condition number is given by
Difx Dify = Difα,β,S22,T22 = σmin Z ,
where σminZ denotes the smallest singular value of the 2n-1 by 2n-1 matrix
Z = αI -1S22 βI -1T22
and  is the Kronecker product.
If λ is part of a complex conjugate pair and U and V are orthogonal transformations such that
UT A,B V = S,T = S11 * 0 S22 T11 * 0 T22 ,
where S11 and T11 are two by two matrices, S22 and T22 are n-2 by n-2 matrices, and S11,T11 corresponds to the complex conjugate eigenvalue pair λ, λ-, then there exist unitary matrices U1 and V1 such that
U1H S11 V1 = s11 s12 0 s22   and   U1H T11 V1 = t11 t12 0 t22 .
The eigenvalues are given by λ=s11/t11 and λ-=s22/t22. Then the Frobenius norm-based, estimated reciprocal condition number is bounded by
Difx Dify mind1,max1, Res11 / Res22 ,d2
where Rez denotes the real part of z, d1=Difs11,t11,s22,t22=σminZ1, Z1 is the complex two by two matrix
Z1 = s11 -s22 t11 -t22 ,
and d2 is an upper bound on DifS11,T11,S22,T22; i.e., an upper bound on σminZ2, where Z2 is the 2n-2 by 2n-2 matrix
Z2 = S11TI -IS22 T11TI -IT22 .
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 http://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:     orderNag_OrderTypeInput
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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     jobNag_JobTypeInput
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:     howmnyNag_HowManyTypeInput
On entry: indicates how many condition numbers are to be computed.
howmny=Nag_ComputeAll
Condition numbers for all eigenpairs are computed.
howmny=Nag_ComputeSelected
Condition numbers for selected eigenpairs (as specified by select) are computed.
Constraint: howmny=Nag_ComputeAll or Nag_ComputeSelected.
4:     select[dim]const Nag_BooleanInput
Note: the dimension, dim, of the array select must be at least
  • max1,n when howmny=Nag_ComputeSelected;
  • 1 otherwise.
On entry: specifies the eigenpairs for which condition numbers are to be computed if howmny=Nag_ComputeSelected. To select condition numbers for the eigenpair corresponding to the real eigenvalue λj, select[j-1] must be set Nag_TRUE. To select condition numbers corresponding to a complex conjugate pair of eigenvalues λj and λj+1, select[j-1] and/or select[j] must be set to Nag_TRUE.
If howmny=Nag_ComputeAll, select is not referenced.
5:     nIntegerInput
On entry: n, the order of the matrix pair S,T.
Constraint: n0.
6:     a[dim]const doubleInput
Note: the dimension, dim, of the array a must be at least max1,pda×n.
The i,jth 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 quasi-triangular matrix S.
7:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: pdamax1,n.
8:     b[dim]const doubleInput
Note: the dimension, dim, of the array b must be at least max1,pdb×n.
The i,jth 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:     pdbIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraint: pdbmax1,n.
10:   vl[dim]const doubleInput
Note: the dimension, dim, of the array vl must be at least
  • max1,pdvl×mm when job=Nag_EigVals or Nag_DoBoth and order=Nag_ColMajor;
  • max1,n×pdvl when job=Nag_EigVals or Nag_DoBoth and order=Nag_RowMajor;
  • 1 otherwise.
The i,jth 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 left eigenvectors of S,T, corresponding to the eigenpairs specified by howmny and select. The eigenvectors must be stored in consecutive columns of vl, as returned by nag_dggev (f08wac) or nag_dtgevc (f08ykc).
If job=Nag_EigVecs, vl is not referenced.
11:   pdvlIntegerInput
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 max1,n ;
    • otherwise pdvl1;
  • if order=Nag_RowMajor,
    • if job=Nag_EigVals or Nag_DoBoth, pdvlmax1,mm;
    • otherwise pdvl1.
12:   vr[dim]const doubleInput
Note: the dimension, dim, of the array vr must be at least
  • max1,pdvr×mm when job=Nag_EigVals or Nag_DoBoth and order=Nag_ColMajor;
  • max1,n×pdvr when job=Nag_EigVals or Nag_DoBoth and order=Nag_RowMajor;
  • 1 otherwise.
The i,jth 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 right eigenvectors of S,T, corresponding to the eigenpairs specified by howmny and select. The eigenvectors must be stored in consecutive columns of vr, as returned by nag_dggev (f08wac) or nag_dtgevc (f08ykc).
If job=Nag_EigVecs, vr is not referenced.
13:   pdvrIntegerInput
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 max1,n ;
    • otherwise pdvr1;
  • if order=Nag_RowMajor,
    • if job=Nag_EigVals or Nag_DoBoth, pdvrmax1,mm;
    • otherwise pdvr1.
14:   s[dim]doubleOutput
Note: the dimension, dim, of the array s must be at least max1,mm.
On exit: if job=Nag_EigVals or Nag_DoBoth, the reciprocal condition numbers of the selected eigenvalues, stored in consecutive elements of the array. For a complex conjugate pair of eigenvalues two consecutive elements of s are set to the same value. Thus s[j-1], dif[j-1], and the jth columns of VL and VR all correspond to the same eigenpair (but not in general the jth eigenpair, unless all eigenpairs are selected).
If job=Nag_EigVecs, s is not referenced.
15:   dif[dim]doubleOutput
Note: the dimension, dim, of the array dif must be at least max1,mm.
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. For a complex eigenvector two consecutive elements of dif are set to the same value. 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.
16:   mmIntegerInput
On entry: the number of elements in the arrays s and dif.
Constraint: mmn.
17:   mInteger *Output
On exit: the number of elements of the arrays s and dif used to store the specified condition numbers; for each selected real eigenvalue one element is used, and for each selected complex conjugate pair of eigenvalues, two elements are used. If howmny=Nag_ComputeAll, m is set to n.
18:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_ENUM_INT_2
On entry, job=value, pdvl=value, mm=value.
Constraint: if job=Nag_EigVals or Nag_DoBoth, pdvlmax1,mm;
otherwise pdvl1.
On entry, job=value, pdvl=value and n=value.
Constraint: if job=Nag_EigVals or Nag_DoBoth, pdvl max1,n ;
otherwise pdvl1.
On entry, job=value, pdvr=value, mm=value.
Constraint: if job=Nag_EigVals or Nag_DoBoth, pdvrmax1,mm;
otherwise pdvr1.
On entry, job=value, pdvr=value and n=value.
Constraint: if job=Nag_EigVals or Nag_DoBoth, pdvr max1,n ;
otherwise pdvr1.
NE_INT
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, n=value and mm=value.
Constraint: mmn.
On entry, pda=value and n=value.
Constraint: pdamax1,n.
On entry, pdb=value and n=value.
Constraint: pdbmax1,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.

7  Accuracy

None.

8  Further Comments

An approximate asymptotic error bound on the chordal distance between the computed eigenvalue λ~ and the corresponding exact eigenvalue λ is
χλ~,λ εA,BF / Sλ
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
θz~,z ε A,BF / Dif .
The complex analogue of this function is nag_ztgsna (f08yyc).

9  Example

This example estimates condition numbers and approximate error estimates for all the eigenvalues and eigenvalues and right eigenvectors of the pair S,T given by
S = 4.0 1.0 1.0 2.0 0.0 3.0 -1.0 1.0 0.0 1.0 3.0 1.0 0.0 0.0 0.0 6.0   and   T= 2.0 1.0 1.0 3.0 0.0 1.0 0.0 1.0 0.0 0.0 1.0 1.0 0.0 0.0 0.0 2.0 .
The eigenvalues and eigenvectors are computed by calling nag_dtgevc (f08ykc).

9.1  Program Text

Program Text (f08ylce.c)

9.2  Program Data

Program Data (f08ylce.d)

9.3  Program Results

Program Results (f08ylce.r)


nag_dtgsna (f08ylc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012