[MUG] eigenvect for real nonsymmetric matrix

[MUG] Re: eigenvect for real nonsymmetric matrix
Author: Maple User Group    Posted: Wed, 15 May 2002 11:27:21 -0400
>> From: Maple User Group "maple_gr" >> From: "jdwright" | Does Maple's linalg[eigenvects] routine handle real non-symmetric | matrices? ... | What method does linalg[eigenvects] use for real non- symmetric | matrices? | I would also like to find an equally good routine for finding the | eigenvalues (eigenvectors maybe too) using the C programming language | on a UNIX platform. -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- Date: Mon, 13 May 2002 16:26:41 +0200 To: "maple-list" From: Juan Manuel de Olazabal "olazabaj" Subject: eigenvect for real nonsymmetric matrix I suggest you http://www.matesco.unican.es/%7Eolazabal/software/maples/algelin/english/ and http://www.matesco.unican.es/%7Eolazabal/software/maples/algelin/engli sh/similarity/autov/autov.html If you like it, I could send you more information about it. Yours sincerely, J.M. de olazabal Manolo Departamento de Matematicas, E. y Computacion Facultad de Ciencias Universidad de Cantabria Avda. Los Castros s/n 39005 Santander Cantabria Spain Tfno. 942 201426 e-mail: "olazabaj" URL: http://www.matesco.unican.es/~olazabal http://gesacapc22.gestion.unican.es:8000/public/4304_2/index.html -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- Date: Tue, 14 May 2002 11:52:54 -0400 (EDT) From: David Linder "dlinder" To: "maple-list" Subject: eigenvect for real nonsymmetric matrix David Wright () asked about the algorithm used by linalg[eigenvects]. Since he mentioned accuracy, I am supposing that he is dealing with floating-point matrices. For a real nonsymmetric floating point matrices the linalg[eigenvects] routine obtains its result from `evalf/EigensR`. For details, see, B.N. Parlett and C.Reinsch, "Balancing a Matrix for Calculation of Eigenvalues and Eigenvectors", Numer. Math. 13, 1969, 293-304. Maple's more recent LinearAlgebra[Eigenvectors] routine will make use of a compiled routine via Maple's external calling mechanism. For example, in Maple 7, > infolevel[LinearAlgebra]:=1: > M:=LinearAlgebra[RandomMatrix](3,outputoptions=[datatype=float]): > LinearAlgebra[Eigenvectors](M): Eigenvectors: "calling external function" Eigenvectors: "NAG" hw_f02ebf In the above example, the "hw_" prefix denotes a computation performed at hardware double precision. If Digits is set higher than evalhf(Digits) then the same algorithm would be used in software float mode, at the higher precision. The LinearAlgebra[Eigenvectors] routine, available in Maple 6 and higher, is recommended over the older linalg[eigenvects] routine for floating-point problems. A generally well-regarded set of routines for such computations are in the LAPACK suite available at, http://www.netlib.org/ One can also find a Fortran-to-C (f2c) translated version of this package, under the name CLAPACK. The CLAPACK package can be linked with either a generic BLAS (Basic Linear Algebra Subprograms), or a tuned set of BLAS such as ATLAS, either of which can also be obtained from the same site. For more details, see, http://www.netlib.org/lapack/double/dgeevx.f http://www.netlib.org/lapack/double/dgeev.f The CLAPACK library can be accessed directly from Maple 7, using the external calling mechanism. Here is a brief example, using the dgeev function to obtain the eigenvalues and eigenvectors. Refer to CLAPACK's own documentation for calling sequence details. Notice that, unlike function dgeevx, the function dgeev does not allow for optional control of scaling and balancing. NB. Adequate workspace must be supplied. The dimensions of the arrays WORK, VR, etc, below must be adequate according to the minimal values specified in the CLAPACK documentation. |\^/| Maple 7 (IBM INTEL LINUX) ._|\| |/|_. Copyright (c) 2001 by Waterloo Maple Inc. \ MAPLE / All rights reserved. Maple is a registered trademark of <____ ____> Waterloo Maple Inc. | Type ? for help. > > A := Matrix(2,2,[[1,1],[2,3]],datatype=float[8]); [1. 1.] A := [ ] [2. 3.] > > copyA := copy(A): # CLAPACK overwrites A. > > extDGEEV := define_external(`dgeev_`, > jobvl::REF(char[1]), > jobvr::REF(char[1]), > n::REF(integer[4]), > a::ARRAY(1..i,1..j,float[8]), > lda::REF(integer[4]), > wr::ARRAY(1..i,1..j,float[8]), > wi::ARRAY(1..i,1..j,float[8]), > vl::ARRAY(1..i,1..j,float[8]), > ldvl::REF(integer[4]), > vr::ARRAY(1..i,1..j,float[8]), > ldvr::REF(integer[4]), > work::ARRAY(1..i,1..j,float[8]), > lwork::REF(integer[4]), > info::REF(integer[4]), > LIB="libclapack.so"): > > JOBVL:="N": > JOBVR:="V": > LDA:=LinearAlgebra:-RowDimension(copyA): > N:=LinearAlgebra:-ColumnDimension(copyA): > WR:=Vector(LDA,datatype=float[8]): > WI:=Vector(LDA,datatype=float[8]): > LDVL:=1: > VL:=Matrix(LDVL,N,datatype=float[8]): > LDVR:=N: > VR:=Matrix(LDVR,N,datatype=float[8]): > LWORK:=`if`( (JOBVR="V" or JOBVL="V"), max(1,4*N), max(1,3*N) ): > WORK:=Vector(LWORK,datatype=float[8]): > INFO:=0: > > extDGEEV(JOBVL,JOBVR,N,copyA,LDA,WR,WI,VL,LDVL,VR,LDVR,WORK,LWORK,INFO); > > INFO; # Zero if successful. See CLAPACK dgeev.f comments. 0 > # See CLAPACK dgeev.f comments for details of output format, # especially with respect to complex conjugate eigenvalues. > WR; [0.267949192431122807] [ ] [3.73205080756887720 ] > WI; [0.] [ ] [0.] > VR; [-0.806898221355073497 -0.343723769333440288] [ ] [0.590690494568872237 -0.939070801588044124] > > Digits := 15; Digits := 15 > LinearAlgebra[Eigenvectors](A); [3.73205080756887720 + 0. I ] [ ], [0.267949192431122807 + 0. I] [0.343723769333440010 + 0. I 0.806898221355073053 + 0. I ] [ ] [0.939070801588044012 + 0. I -0.590690494568872015 + 0. I] - Dave Linder "dlinder" -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- Date: Tue, 14 May 2002 19:56:45 +0300 From: Florin Sabau "fsabau" To: "maple-list" Subject: eigenvect for real nonsymmetric matrix Hello, I'm quite new to this list (and to Maple also) so, apologies from the start if I may be "off topic". 1. Don't know how Maple really computes eigenvals, but (theoretically) it should be the same as following: 2. If you have a matrix A (of an linear operator for instance), then the eigenvalues of the matrix A are the roots of the characteristic polynom (in lambda): det(A-lambda*I)=0 By "real non-symmetric matrix" do you mean a non-symmetric matrix with real elements? If so there should be no problem in finding eigenvalues with the charactertic polynom (only the equation in lambda has real coefficients). | 1 2 -1| for instance: A=| 2 1 1| |-1 -1 1| -- det(A-lambda*I)=-lambda^3+3*lambda^2+lambda-3=0 -- lambda(1,2,3)=-1, 1, 3 If you're interested, I could wrap up a few routines in C to get eigenvalues. PS: I may be wrong about this. (first semester, when I was studying Linear Algebra at school, is so far away in the past!) -- Best regards, Florin "mailto:fsabau"

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