nag_dgeevx (f08nbc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dgeevx (f08nbc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dgeevx (f08nbc) computes the eigenvalues and, optionally, the left and/or right eigenvectors for an n by n real nonsymmetric matrix A.
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_dgeevx (Nag_OrderType order, Nag_BalanceType balanc, Nag_LeftVecsType jobvl, Nag_RightVecsType jobvr, Nag_RCondType sense, Integer n, double a[], Integer pda, double wr[], double wi[], double vl[], Integer pdvl, double vr[], Integer pdvr, Integer *ilo, Integer *ihi, double scale[], double *abnrm, double rconde[], double rcondv[], NagError *fail)

3  Description

The right eigenvector vj of A satisfies
A vj = λj vj
where λj is the jth eigenvalue of A. The left eigenvector uj of A satisfies
ujH A = λj ujH
where ujH denotes the conjugate transpose of uj.
Balancing a matrix means permuting the rows and columns to make it more nearly upper triangular, and applying a diagonal similarity transformation DAD-1, where D is a diagonal matrix, with the aim of making its rows and columns closer in norm and the condition numbers of its eigenvalues and eigenvectors smaller. The computed reciprocal condition numbers correspond to the balanced matrix. Permuting rows and columns will not change the condition numbers (in exact arithmetic) but diagonal scaling will. For further explanation of balancing, see Section 4.8.1.2 of Anderson et al. (1999).
Following the optional balancing, the matrix A is first reduced to upper Hessenberg form by means of unitary similarity transformations, and the QR algorithm is then used to further reduce the matrix to upper triangular Schur form, T, from which the eigenvalues are computed. Optionally, the eigenvectors of T are also computed and backtransformed to those of A.

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

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: indicates how the input matrix should be diagonally scaled and/or permuted to improve the conditioning of its eigenvalues.
balanc=Nag_NoBalancing
Do not diagonally scale or permute.
balanc=Nag_BalancePermute
Perform permutations to make the matrix more nearly upper triangular. Do not diagonally scale.
balanc=Nag_BalanceScale
Diagonally scale the matrix, i.e., replace A by DAD-1, where D is a diagonal matrix chosen to make the rows and columns of A more equal in norm. Do not permute.
balanc=Nag_BalanceBoth
Both diagonally scale and permute A.
Computed reciprocal condition numbers will be for the matrix after balancing and/or permuting. Permuting does not change condition numbers (in exact arithmetic), but balancing does.
Constraint: balanc=Nag_NoBalancing, Nag_BalancePermute, Nag_BalanceScale or Nag_BalanceBoth.
3:     jobvlNag_LeftVecsTypeInput
On entry: if jobvl=Nag_NotLeftVecs, the left eigenvectors of A are not computed.
If jobvl=Nag_LeftVecs, the left eigenvectors of A are computed.
If sense=Nag_RCondEigVals or Nag_RCondBoth, jobvl must be set to jobvl=Nag_LeftVecs.
Constraint: jobvl=Nag_NotLeftVecs or Nag_LeftVecs.
4:     jobvrNag_RightVecsTypeInput
On entry: if jobvr=Nag_NotRightVecs, the right eigenvectors of A are not computed.
If jobvr=Nag_RightVecs, the right eigenvectors of A are computed.
If sense=Nag_RCondEigVals or Nag_RCondBoth, jobvr must be set to jobvr=Nag_RightVecs.
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 right eigenvectors only.
sense=Nag_RCondBoth
Computed for eigenvalues and right eigenvectors.
If sense=Nag_RCondEigVals or Nag_RCondBoth, both left and right eigenvectors must also be computed (jobvl=Nag_LeftVecs and jobvr=Nag_RightVecs).
Constraint: sense=Nag_NotRCond, Nag_RCondEigVals, Nag_RCondEigVecs or Nag_RCondBoth.
6:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
7:     a[dim]doubleInput/Output
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 n by n matrix A.
On exit: a has been overwritten. If jobvl=Nag_LeftVecs or jobvr=Nag_RightVecs, A contains the real Schur form of the balanced version of the input matrix A.
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:     wr[dim]doubleOutput
10:   wi[dim]doubleOutput
Note: the dimension, dim, of the arrays wr and wi must be at least max1,n.
On exit: wr and wi contain the real and imaginary parts, respectively, of the computed eigenvalues. Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having the positive imaginary part first.
11:   vl[dim]doubleOutput
Note: the dimension, dim, of the array vl must be at least
  • max1,pdvl×n when jobvl=Nag_LeftVecs;
  • 1 otherwise.
Where VLi,j appears in this document, it refers to the array element
  • 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 vl, in the same order as their corresponding eigenvalues. If the jth eigenvalue is real, then uj=VLi,j, for i=1,2,,n. If the jth and j+1st eigenvalues form a complex conjugate pair, then uj=VLi,j+i×VLi,j+1 and uj+1=VLi,j-i×VLi,j+1, for i=1,2,,n.
If jobvl=Nag_NotLeftVecs, vl is not referenced.
12:   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.
13:   vr[dim]doubleOutput
Note: the dimension, dim, of the array vr must be at least
  • max1,pdvr×n when jobvr=Nag_RightVecs;
  • 1 otherwise.
Where VRi,j appears in this document, it refers to the array element
  • 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 vr, in the same order as their corresponding eigenvalues. If the jth eigenvalue is real, then vj=VRi,j, for i=1,2,,n. If the jth and j+1st eigenvalues form a complex conjugate pair, then vj=VRi,j+i×VRi,j+1 and vj+1=VRi,j-i×VRi,j+1, for i=1,2,,n.
If jobvr=Nag_NotRightVecs, vr is not referenced.
14:   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.
15:   iloInteger *Output
16:   ihiInteger *Output
On exit: ilo and ihi are integer values determined when A was balanced. The balanced A has aij=0 if i>j and j=1,2,,ilo-1 or i=ihi+1,,n.
17:   scale[dim]doubleOutput
Note: the dimension, dim, of the array scale must be at least max1,n.
On exit: details of the permutations and scaling factors applied when balancing A.
If pj is the index of the row and column interchanged with row and column j, and dj is the scaling factor applied to row and column j, then
  • scale[j-1]=pj, for j=1,2,,ilo-1;
  • scale[j-1]=dj, for j=ilo,,ihi;
  • scale[j-1]=pj, for j=ihi+1,,n.
The order in which the interchanges are made is n to ihi+1, then 1 to ilo-1.
18:   abnrmdouble *Output
On exit: the 1-norm of the balanced matrix (the maximum of the sum of absolute values of elements of any column).
19:   rconde[dim]doubleOutput
Note: the dimension, dim, of the array rconde must be at least max1,n.
On exit: rconde[j-1] is the reciprocal condition number of the jth eigenvalue.
20:   rcondv[dim]doubleOutput
Note: the dimension, dim, of the array rcondv must be at least max1,n.
On exit: rcondv[j-1] is the reciprocal condition number of the jth right eigenvector.
21:   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_CONVERGENCE
The QR algorithm failed to compute all the eigenvalues, and no eigenvectors or condition numbers have been computed; elements 1 to ilo-1 and value to n of wr and wi contain eigenvalues which have converged.
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, 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.
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

The computed eigenvalues and eigenvectors are exact for a nearby matrix A+E, where
E2 = Oε A2 ,
and ε is the machine precision. See Section 4.8 of Anderson et al. (1999) for further details.

8  Further Comments

Each eigenvector is normalized to have Euclidean norm equal to unity and the element of largest absolute value real and positive.
The total number of floating point operations is proportional to n3.
The complex analogue of this function is nag_zgeevx (f08npc).

9  Example

This example finds all the eigenvalues and right eigenvectors of the matrix
A = 0.35 0.45 -0.14 -0.17 0.09 0.07 -0.54 0.35 -0.44 -0.33 -0.03 0.17 0.25 -0.32 -0.13 0.11 ,
together with estimates of the condition number and forward error bounds for each eigenvalue and eigenvector. The option to balance the matrix is used. In order to compute the condition numbers of the eigenvalues, the left eigenvectors also have to be computed, but they are not printed out in this example.

9.1  Program Text

Program Text (f08nbce.c)

9.2  Program Data

Program Data (f08nbce.d)

9.3  Program Results

Program Results (f08nbce.r)


nag_dgeevx (f08nbc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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