Zgges Zgges_r Zggesx Zggesx_r Zggev Zggevx

Zggev() Sub Zggev ( Jobvl As  String, Jobvr As  String, N As  Long, A() As  Complex, B() As  Complex, Alpha() As  Complex, Beta() As  Complex, Vl() As  Complex, Vr() As  Complex, Info As  Long  ) (Simple driver) Generalized eigenvalue problem of complex matrices Purpose This routine computes for a pair of N x N complex nonsymmetric matrices (A, B), the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors. A generalized eigenvalue for a pair of matrices (A, B) is a scalar λ or a ratio α/β = λ, such that A - λ*B 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 generalized eigenvector v(j) corresponding to the generalized eigenvalue λ(j) of (A, B) satisfies A * v(j) = λ(j) * B * v(j) The left generalized eigenvector u(j) corresponding to the generalized eigenvalue λ(j) of (A, B) satisfies u(j)^H * A = λ(j) * u(j)^H * B where u(j)^H is the conjugate-transpose of u(j). Parameters [in] JobVl = "N": Do not compute the left generalized eigenvectors. = "V": Compute the left generalized eigenvectors. [in] JobVr = "N": Do not compute the right generalized eigenvectors. = "V": Compute the right generalized eigenvectors. [in] N Order of the matrices A, B, VL and VR. (N >= 0) (If N = 0, returns without computation) [in,out] A() Array A(LA1 - 1, LA2 - 1) (LA1 >= N, LA2 >= N) [in] Matrix A in the pair (A, B). [out] A() has been overwritten. [in,out] B() Array B(LB1 - 1, LB2 - 1) (LB1 >= N, LB2 >= N) [in] Matrix B in the pair (A, B). [out] B() has been overwritten. [out] Alpha() Array Alpha(LAlpha - 1) (LAlpha >= N) [out] Beta() Array Beta(LBeta - 1) (LBeta >= N) Alphar(j)/Beta(j), j = 0, ..., N-1, will be the generalized eigenvalues. Note: The quotients Alphar(j)/Beta(j) and Alphai(j)/Beta(j) may easily over- or underflow, and Beta(j) may even be zero. Thus, the user should avoid naively computing the ratio α/β. However, Alphar and Alphai will be always less than and usually comparable with norm(A) in magnitude, and Beta always less than and usually comparable with norm(B). [out] Vl() Array Vl(LVl1 - 1, LVl2 - 1) (LVl1 >= N, LVl2 >= N) Jobvl = "V": The left generalized eigenvectors u(j) are stored one after another in the columns of Vl(), in the same order as their eigenvalues. Each eigenvector is scaled so the largest component has |real part| + |imaginary part| = 1. Jobvl = "N": Not referenced. [out] Vr() Array Vr(LVr1 - 1, LVr2 - 1) (LVr1 >= N, LVr2 >= N) jobvr = "V": The right generalized eigenvectors v(j) are stored one after another in the columns of Vr() in the same order as their eigenvalues. Each eigenvector is scaled so the largest component has |real part| + |imaginary part| = 1. jobvr = "N": Not referenced. [out] Info = 0: Successful exit. = -1: The argument Jobvl had an illegal value. (Jobvl <> "V" nor "N") = -2: The argument Jobvr had an illegal value. (Jobvr <> "V" nor "N") = -3: The argument N had an illegal value. (N < 0) = -4: The argument A() is invalid. = -5: The argument B() is invalid. = -6: The argument Alpha() is invalid. = -7: The argument Beta() is invalid. = -8: The argument Vl() is invalid. = -9: The argument Vr() is invalid. = i (0 < i <= N): The QZ iteration failed. No eigenvectors have been calculated, but Alpha(j) and Beta(j) should be correct for j = i, ..., N-1. = N+1: Other than QZ iteration failed in Zhgeqz. = N+2: Error return from Ztgevc. Reference LAPACK Example Program Compute for a pair of matrices (A, B) the generalized eigenvalues and the left and right generalized eigenvectors, where ( 0.2-0.11i -0.93-0.32i 0.81+0.37i ) A = ( -0.8-0.92i -0.29+0.86i 0.64+0.51i ) ( 0.71+0.59i -0.15+0.19i 0.2+0.94i ) ( 0.57-0.91i -0.28-0.45i 0.25+0.91i ) B = ( 0.83-0.46i 0.63-0.19i -0.69+0.09i ) ( 0.24-1.33i -0.56-0.67i 0.9+1.25i ) Sub Ex_Zggev() Const N = 3 Dim A(N - 1, N - 1) As Complex, B(N - 1, N - 1) As Complex Dim Alpha(N - 1) As Complex, Beta(N - 1) As Complex Dim Vl(N - 1, N - 1) As Complex, Vr(N - 1, N - 1) As Complex, Info As Long A(0, 0) = Cmplx(0.2, -0.11): A(0, 1) = Cmplx(-0.93, -0.32): A(0, 2) = Cmplx(0.81, 0.37) A(1, 0) = Cmplx(-0.8, -0.92): A(1, 1) = Cmplx(-0.29, 0.86): A(1, 2) = Cmplx(0.64, 0.51) A(2, 0) = Cmplx(0.71, 0.59): A(2, 1) = Cmplx(-0.15, 0.19): A(2, 2) = Cmplx(0.2, 0.94) B(0, 0) = Cmplx(0.57, -0.91): B(0, 1) = Cmplx(-0.28, -0.45): B(0, 2) = Cmplx(0.25, 0.91) B(1, 0) = Cmplx(0.83, -0.46): B(1, 1) = Cmplx(0.63, -0.19): B(1, 2) = Cmplx(-0.69, 0.09) B(2, 0) = Cmplx(0.24, -1.33): B(2, 1) = Cmplx(-0.56, -0.67): B(2, 2) = Cmplx(0.9, 1.25) Call Zggev("V", "V", N, A(), B(), Alpha(), Beta(), Vl(), Vr(), Info) Debug.Print "Eigenvalues =" Debug.Print Creal(Cdiv(Alpha(0), Beta(0))), Cimag(Cdiv(Alpha(0), Beta(0))), Debug.Print Creal(Cdiv(Alpha(1), Beta(1))), Cimag(Cdiv(Alpha(1), Beta(1))) Debug.Print Creal(Cdiv(Alpha(2), Beta(2))), Cimag(Cdiv(Alpha(2), Beta(2))) Debug.Print "Eigenvectors (L) =" Debug.Print Creal(Vl(0, 0)), Cimag(Vl(0, 0)), Creal(Vl(0, 1)), Cimag(Vl(0, 1)) Debug.Print Creal(Vl(1, 0)), Cimag(Vl(1, 0)), Creal(Vl(1, 1)), Cimag(Vl(1, 1)) Debug.Print Creal(Vl(2, 0)), Cimag(Vl(2, 0)), Creal(Vl(2, 1)), Cimag(Vl(2, 1)) Debug.Print Creal(Vl(0, 2)), Cimag(Vl(0, 2)) Debug.Print Creal(Vl(1, 2)), Cimag(Vl(1, 2)) Debug.Print Creal(Vl(2, 2)), Cimag(Vl(2, 2)) Debug.Print "Eigenvectors (R) =" Debug.Print Creal(Vr(0, 0)), Cimag(Vr(0, 0)), Creal(Vr(0, 1)), Cimag(Vr(0, 1)) Debug.Print Creal(Vr(1, 0)), Cimag(Vr(1, 0)), Creal(Vr(1, 1)), Cimag(Vr(1, 1)) Debug.Print Creal(Vr(2, 0)), Cimag(Vr(2, 0)), Creal(Vr(2, 1)), Cimag(Vr(2, 1)) Debug.Print Creal(Vr(0, 2)), Cimag(Vr(0, 2)) Debug.Print Creal(Vr(1, 2)), Cimag(Vr(1, 2)) Debug.Print Creal(Vr(2, 2)), Cimag(Vr(2, 2)) Debug.Print "Info =", Info End Sub Example Results Eigenvalues = -0.4784814787767 -0.640760182519056 1.62433774044009 1.10642263894432 -0.149178317709927 5.63446258373312 Eigenvectors (L) = -0.488182728920006 0.511817271079994 0.112214636564846 -0.78086108170552 -0.493389319144031 1.33311637129139E-02 -0.132533171028514 -0.347190877300084 0.339735049171677 -6.82051937200691E-02 0.672543628409034 0.327456371590966 0.902635460243872 9.73645397561283E-02 -0.178648471173719 0.129164561360569 -0.730361156069673 7.60898456031052E-02 Eigenvectors (R) = -0.573248979253811 -0.426751020746189 0.644105768156504 -0.298230968832631 -0.249087460228498 -9.31247915124554E-02 -0.701765691258405 -0.298234308741595 0.271914164828682 -0.477707190075455 0.274425774522266 -0.280815941661082 2.08352227991442E-02 2.31726206010056E-02 -0.737220272842649 -0.262779727157351 -0.474367740415068 -4.82054262401725E-02 Info = 0