Dgges Dgges_r Dggesx Dggesx_r Dggev Dggevx

Dggev() Sub Dggev ( Jobvl As  String, Jobvr As  String, N As  Long, A() As  Double, B() As  Double, Alphar() As  Double, Alphai() As  Double, Beta() As  Double, Vl() As  Double, Vr() As  Double, Info As  Long  ) (Simple driver) Generalized eigenvalue problem of general matrices Purpose This routine computes for a pair of n x n real 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] Alphar() Array Alphar(LAlphar - 1) (LAlphar >= N) [out] Alphai() Array Alphai(LAlphai - 1) (LAlphai >= N) [out] Beta() Array Beta(LBeta - 1) (LBeta >= N) (Alphar(j) + Alphai(j)*i)/Beta(j), j = 0, ..., N-1, will be the generalized eigenvalues. If Alphai(j) is zero, then the j-th eigenvalue is real. If positive, then the j-th and (j+1)-st eigenvalues are a complex conjugate pair, with Alphai(j+1) negative. 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 eigenvectors u(j) are stored one after another in the columns of Vl(), in the same order as their eigenvalues.   If the j-th eigenvalue is real, then u(j) = (j-th column of Vl()).   If the j-th and (j+1)-th eigenvalues form a complex conjugate pair, then u(j) = (j-th column of Vl()) + ((j+1)-th column of Vl())*i and u(j+1) = (j-th column of Vl()) - ((j+1)-st column of Vl())*i.   Each eigenvector is scaled so the largest component has |real part| + |imaginary part| = 1. Jobvl = "N": Vl() is not referenced. [out] Vr() Array Vr(LVr1 - 1, LVr2 - 1) (LVr1 >= N, LVr2 >= N) Jobvr = "V": The right eigenvectors v(j) are stored one after another in the columns of Vr() in the same order as their eigenvalues.   If the j-th eigenvalue is real, then v(j) = (j-th column of Vr()). If the j-th and (j+1)-st eigenvalues form a complex conjugate pair, then v(j) = (j-th column of Vr()) + ((j+1)-st column of Vr())*i and v(j+1) = (j-th column of Vr()) - ((j+1)-st column of Vr())*i.   Each eigenvector is scaled so the largest component has |real part| + |imaginary part| = 1. Jobvr = "N": Vr() is 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 Alphar() is invalid. = -7: The argument Alphai() is invalid. = -8: The argument Beta() is invalid. = -9: The argument Vl() is invalid. = -10: The argument Vr() is invalid. = i (0 < i <= N): The QZ iteration failed. No eigenvectors have been calculated, but Alphar(j), Alphai(j), and Beta(j) should be correct for j = i, ..., N-1. = N+1: Other than QZ iteration failed in Dhgeqz. = N+2: Error return from Dtgevc. Reference LAPACK Example Program Compute for a pair of matrices (A, B) the generalized eigenvalues and the left and right generalized eigenvectors, where ( 0.20 -0.11 -0.93 ) ( -0.58 -0.79 0.82 ) A = ( -0.32 0.81 0.37 ), B = ( 0.77 0.71 -0.55 ) ( -0.80 -0.92 -0.29 ) ( -1.36 -1.22 1.66 ) Sub Ex_Dggev() Const N = 3 Dim A(N - 1, N - 1) As Double, B(N - 1, N - 1) As Double Dim Alphar(N - 1) As Double, Alphai(N - 1) As Double, Beta(N - 1) As Double Dim Vl(N - 1, N - 1) As Double, Vr(N - 1, N - 1) As Double, Info As Long A(0, 0) = 0.2: A(0, 1) = -0.11: A(0, 2) = -0.93 A(1, 0) = -0.32: A(1, 1) = 0.81: A(1, 2) = 0.37 A(2, 0) = -0.8: A(2, 1) = -0.92: A(2, 2) = -0.29 B(0, 0) = -0.58: B(0, 1) = -0.79: B(0, 2) = 0.82 B(1, 0) = 0.77: B(1, 1) = 0.71: B(1, 2) = -0.55 B(2, 0) = -1.36: B(2, 1) = -1.22: B(2, 2) = 1.66 Call Dggev("V", "V", N, A(), B(), Alphar(), Alphai(), Beta(), Vl(), Vr(), Info) Debug.Print "Eigenvalues =" Debug.Print " (r)", Alphar(0) / Beta(0), Alphar(1) / Beta(1), Alphar(2) / Beta(2) Debug.Print " (i)", Alphai(0) / Beta(0), Alphai(1) / Beta(1), Alphai(2) / Beta(2) Debug.Print "Eigenvectors (L) =" Debug.Print Vl(0, 0), Vl(0, 1), Vl(0, 2) Debug.Print Vl(1, 0), Vl(1, 1), Vl(1, 2) Debug.Print Vl(2, 0), Vl(2, 1), Vl(2, 2) Debug.Print "Eigenvectors (R) =" Debug.Print Vr(0, 0), Vr(0, 1), Vr(0, 2) Debug.Print Vr(1, 0), Vr(1, 1), Vr(1, 2) Debug.Print Vr(2, 0), Vr(2, 1), Vr(2, 2) Debug.Print "Info =", Info End Sub Example Results Eigenvalues = (r) 1.10765548065266 1.10765548065266 -1.74329621491312 (i) 1.40252005479606 -1.40252005479606 0 Eigenvectors (L) = 0.828456655019124 0.171543344980876 -1 -0.174134648086565 0.261212076890975 -0.66437599243234 -0.601718475294568 -0.175396675418769 4.15956472218708E-02 Eigenvectors (R) = -9.54330235945138E-02 0.502674461662748 -1 -0.945775638065751 5.42243619342491E-02 0.224694817103864 -0.659251386021417 5.30923334667921E-02 -0.954838851339161 Info = 0