nag_dggevx (f08wbc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dggevx (f08wbc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dggevx (f08wbc) computes for a pair of n by n real nonsymmetric matrices A,B the generalized eigenvalues and, optionally, the left and/or right generalized eigenvectors using the QZ algorithm.
Optionally it also computes a balancing transformation to improve the conditioning of the eigenvalues and eigenvectors, reciprocal condition numbers for the eigenvalues, and reciprocal condition numbers for the right eigenvectors.

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_dggevx (Nag_OrderType order, Nag_BalanceType balanc, Nag_LeftVecsType jobvl, Nag_RightVecsType jobvr, Nag_RCondType sense, Integer n, double a[], Integer pda, double b[], Integer pdb, double alphar[], double alphai[], double beta[], double vl[], Integer pdvl, double vr[], Integer pdvr, Integer *ilo, Integer *ihi, double lscale[], double rscale[], double *abnrm, double *bbnrm, double rconde[], double rcondv[], NagError *fail)

3  Description

A generalized eigenvalue for a pair of matrices A,B is a scalar λ or a ratio α/β=λ, such that A-λB is singular. It is usually represented as the pair α,β, as there is a reasonable interpretation for β=0, and even for both being zero.
The right eigenvector vj corresponding to the eigenvalue λj of A,B satisfies
A vj = λj B vj .
The left eigenvector uj corresponding to the eigenvalue λj of A,B satisfies
ujH A = λj ujH B ,
where ujH is the conjugate-transpose of uj.
All the eigenvalues and, if required, all the eigenvectors of the generalized eigenproblem Ax=λBx, where A and B are real, square matrices, are determined using the QZ algorithm. The QZ algorithm consists of four stages:
  1. A is reduced to upper Hessenberg form and at the same time B is reduced to upper triangular form.
  2. A is further reduced to quasi-triangular form while the triangular form of B is maintained. This is the real generalized Schur form of the pair A,B .
  3. The quasi-triangular form of A is reduced to triangular form and the eigenvalues extracted. This function does not actually produce the eigenvalues λj, but instead returns αj and βj such that
    λj=αj/βj,  j=1,2,,n.
    The division by βj becomes your responsibility, since βj may be zero, indicating an infinite eigenvalue. Pairs of complex eigenvalues occur with αj/βj and αj+1/βj+1 complex conjugates, even though αj and αj+1 are not conjugate.
  4. If the eigenvectors are required they are obtained from the triangular matrices and then transformed back into the original coordinate system.
For details of the balancing option, see Section 3 in nag_dggbal (f08whc).

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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Wilkinson J H (1979) Kronecker's canonical form and the QZ algorithm Linear Algebra Appl. 28 285–303

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:     balancNag_BalanceTypeInput
On entry: specifies the balance option to be performed.
balanc=Nag_NoBalancing
Do not diagonally scale or permute.
balanc=Nag_BalancePermute
Permute only.
balanc=Nag_BalanceScale
Scale only.
balanc=Nag_BalanceBoth
Both permute and scale.
Computed reciprocal condition numbers will be for the matrices after permuting and/or balancing. Permuting does not change condition numbers (in exact arithmetic), but balancing does. In the absence of other information, balanc=Nag_BalanceBoth is recommended.
Constraint: balanc=Nag_NoBalancing, Nag_BalancePermute, Nag_BalanceScale or Nag_BalanceBoth.
3:     jobvlNag_LeftVecsTypeInput
On entry: if jobvl=Nag_NotLeftVecs, do not compute the left generalized eigenvectors.
If jobvl=Nag_LeftVecs, compute the left generalized eigenvectors.
Constraint: jobvl=Nag_NotLeftVecs or Nag_LeftVecs.
4:     jobvrNag_RightVecsTypeInput
On entry: if jobvr=Nag_NotRightVecs, do not compute the right generalized eigenvectors.
If jobvr=Nag_RightVecs, compute the right generalized eigenvectors.
Constraint: jobvr=Nag_NotRightVecs or Nag_RightVecs.
5:     senseNag_RCondTypeInput
On entry: determines which reciprocal condition numbers are computed.
sense=Nag_NotRCond
None are computed.
sense=Nag_RCondEigVals
Computed for eigenvalues only.
sense=Nag_RCondEigVecs
Computed for eigenvectors only.
sense=Nag_RCondBoth
Computed for eigenvalues and eigenvectors.
Constraint: sense=Nag_NotRCond, Nag_RCondEigVals, Nag_RCondEigVecs or Nag_RCondBoth.
6:     nIntegerInput
On entry: n, the order of the matrices A and B.
Constraint: n0.
7:     a[dim]doubleInput/Output
Note: the dimension, dim, of the array a must be at least max1,pda×n.
Where Ai,j appears in this document, it refers to the array element
  • a[j-1×pda+i-1] when order=Nag_ColMajor;
  • a[i-1×pda+j-1] when order=Nag_RowMajor.
On entry: the matrix A in the pair A,B.
On exit: a has been overwritten. If jobvl=Nag_LeftVecs or jobvr=Nag_RightVecs or both, then A contains the first part of the real Schur form of the ‘balanced’ versions of the input A and B.
8:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: pdamax1,n.
9:     b[dim]doubleInput/Output
Note: the dimension, dim, of the array b must be at least max1,pdb×n.
Where Bi,j appears in this document, it refers to the array element
  • b[j-1×pdb+i-1] when order=Nag_ColMajor;
  • b[i-1×pdb+j-1] when order=Nag_RowMajor.
On entry: the matrix B in the pair A,B.
On exit: b has been overwritten.
10:   pdbIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraint: pdbmax1,n.
11:   alphar[n]doubleOutput
On exit: the element alphar[j-1] contains the real part of αj.
12:   alphai[n]doubleOutput
On exit: the element alphai[j-1] contains the imaginary part of αj.
13:   beta[n]doubleOutput
On exit: alphar[j-1]+alphai[j-1]×i/beta[j-1], for j=1,2,,n, will be the generalized eigenvalues.
If alphai[j-1] is zero, then the jth eigenvalue is real; if positive, then the jth and j+1st eigenvalues are a complex conjugate pair, with alphai[j] negative.
Note:  the quotients alphar[j-1]/beta[j-1] and alphai[j-1]/beta[j-1] may easily overflow or underflow, and beta[j-1] may even be zero. Thus, you should avoid naively computing the ratio αj/βj. However, maxαj will always be less than and usually comparable with a2 in magnitude, and maxβj will always be less than and usually comparable with b2.
14:   vl[dim]doubleOutput
Note: the dimension, dim, of the array vl must be at least
  • max1,pdvl×n when jobvl=Nag_LeftVecs;
  • 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 exit: if jobvl=Nag_LeftVecs, the left eigenvectors uj are stored one after another in the columns of vl, in the same order as the corresponding eigenvalues.
If the jth eigenvalue is real, then uj=VL:,j, the jth column of vl.
If the jth and j+1th eigenvalues form a complex conjugate pair, then uj=VL:,j+i×VL:,j+1 and uj+1=VL:,j-i×VL:,j+1. Each eigenvector will be scaled so the largest component has real part+imag. part=1.
If jobvl=Nag_NotLeftVecs, vl is not referenced.
15:   pdvlIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array vl.
Constraints:
  • if jobvl=Nag_LeftVecs, pdvl max1,n ;
  • otherwise pdvl1.
16:   vr[dim]doubleOutput
Note: the dimension, dim, of the array vr must be at least
  • max1,pdvr×n when jobvr=Nag_RightVecs;
  • 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 exit: if jobvr=Nag_RightVecs, the right eigenvectors vj are stored one after another in the columns of vr, in the same order as their eigenvalues.
If the jth eigenvalue is real, then vj=VR:,j, the jth column of VR.
If the jth and j+1th eigenvalues form a complex conjugate pair, then vj=VR:,j+i×VR:,j+1 and vj+1 = VR:,j -i×VR:,j+1.
Each eigenvector will be scaled so the largest component has real part+imag. part=1.
If jobvr=Nag_NotRightVecs, vr is not referenced.
17:   pdvrIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array vr.
Constraints:
  • if jobvr=Nag_RightVecs, pdvr max1,n ;
  • otherwise pdvr1.
18:   iloInteger *Output
19:   ihiInteger *Output
On exit: ilo and ihi are integer values such that Ai,j=0 and Bi,j=0 if i>j and j=1,2,,ilo-1 or i=ihi+1,,n.
If balanc=Nag_NoBalancing or Nag_BalanceScale, ilo=1 and ihi=n.
20:   lscale[n]doubleOutput
On exit: details of the permutations and scaling factors applied to the left side of A and B.
If plj  is the index of the row interchanged with row j, and dlj  is the scaling factor applied to row j, then:
  • lscale[j-1] = plj , for j=1,2,,ilo-1;
  • lscale = dlj , for j=ilo,,ihi;
  • lscale = plj , for j=ihi+1,,n.
The order in which the interchanges are made is n to ihi+1, then 1 to ilo-1.
21:   rscale[n]doubleOutput
On exit: details of the permutations and scaling factors applied to the right side of A and B.
If prj is the index of the column interchanged with column j, and drj is the scaling factor applied to column j, then:
  • rscale[j-1]=prj, for j=1,2,,ilo-1;
  • if rscale=drj, for j=ilo,,ihi;
  • if rscale=prj, for j=ihi+1,,n.
The order in which the interchanges are made is n to ihi+1, then 1 to ilo-1.
22:   abnrmdouble *Output
On exit: the 1-norm of the balanced matrix A.
23:   bbnrmdouble *Output
On exit: the 1-norm of the balanced matrix B.
24:   rconde[dim]doubleOutput
Note: the dimension, dim, of the array rconde must be at least max1,n.
On exit: if sense=Nag_RCondEigVals or Nag_RCondBoth, the reciprocal condition numbers of the eigenvalues, stored in consecutive elements of the array. For a complex conjugate pair of eigenvalues two consecutive elements of rconde are set to the same value. Thus rconde[j-1], rcondv[j-1], and the jth columns of vl and vr all correspond to the jth eigenpair.
If sense=Nag_RCondEigVecs, rconde is not referenced.
25:   rcondv[dim]doubleOutput
Note: the dimension, dim, of the array rcondv must be at least max1,n.
On exit: if sense=Nag_RCondEigVecs or Nag_RCondBoth, the estimated reciprocal condition numbers of the eigenvectors, stored in consecutive elements of the array. For a complex eigenvector two consecutive elements of rcondv are set to the same value.
If sense=Nag_RCondEigVals, rcondv is not referenced.
26:   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_EIGENVECTORS
A failure occurred in nag_dtgevc (f08ykc) while computing generalized eigenvectors.
NE_ENUM_INT_2
On entry, jobvl=value, pdvl=value and n=value.
Constraint: if jobvl=Nag_LeftVecs, pdvl max1,n ;
otherwise pdvl1.
On entry, jobvr=value, pdvr=value and n=value.
Constraint: if jobvr=Nag_RightVecs, 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, 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.
NE_ITERATION_QZ
The QZ iteration failed. No eigenvectors have been calculated but alphar[j], alphai[j], and beta[j] should be correct from element value.
The QZ iteration failed with an unexpected error, please contact NAG.

7  Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrices A+E and B+F, where
E,F F = Oε A,B F ,
and ε is the machine precision.
An approximate error bound on the chordal distance between the ith computed generalized eigenvalue w and the corresponding exact eigenvalue λ  is
ε × abnrm,bbnrm2 / rconde[i-1] .
An approximate error bound for the angle between the ith computed eigenvector VLi  or VRi  is given by
ε × abnrm,bbnrm2 / rcondv[i-1] .
For further explanation of the reciprocal condition numbers rconde and rcondv, see Section 4.11 of Anderson et al. (1999).
Note:  interpretation of results obtained with the QZ algorithm often requires a clear understanding of the effects of small changes in the original data. These effects are reviewed in Wilkinson (1979), in relation to the significance of small values of αj and βj. It should be noted that if αj and βj are both small for any j, it may be that no reliance can be placed on any of the computed eigenvalues λi=αi/βi. You are recommended to study Wilkinson (1979) and, if in difficulty, to seek expert advice on determining the sensitivity of the eigenvalues to perturbations in the data.

8  Further Comments

The total number of floating point operations is proportional to n3.
The complex analogue of this function is nag_zggevx (f08wpc).

9  Example

This example finds all the eigenvalues and right eigenvectors of the matrix pair A,B, where
A = 3.9 12.5 -34.5 -0.5 4.3 21.5 -47.5 7.5 4.3 21.5 -43.5 3.5 4.4 26.0 -46.0 6.0   and   B= 1.0 2.0 -3.0 1.0 1.0 3.0 -5.0 4.0 1.0 3.0 -4.0 3.0 1.0 3.0 -4.0 4.0 ,
together with estimates of the condition number and forward error bounds for each eigenvalue and eigenvector. The option to balance the matrix pair is used.

9.1  Program Text

Program Text (f08wbce.c)

9.2  Program Data

Program Data (f08wbce.d)

9.3  Program Results

Program Results (f08wbce.r)


nag_dggevx (f08wbc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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