Zgges Zgges_r Zggesx Zggesx_r Zggev Zggevx

Zgges() Sub Zgges ( Jobvsl As  String, Jobvsr As  String, Sort As  String, Selctg As  LongPtr, N As  Long, A() As  Complex, B() As  Complex, Sdim As  Long, Alpha() As  Complex, Beta() As  Complex, Vsl() As  Complex, Vsr() As  Complex, Info As  Long  ) (Simple driver) Generalized Schur factorization of complex matrices Purpose This routine computes for a pair of n x n complex nonsymmetric matrices (A, B), the generalized eigenvalues, the generalized complex Schur form (S, T), and optionally the left and/or right matrices of Schur vectors (VSL and VSR). This gives the generalized Schur factorization (A, B) = ((VSL)*S*(VSR)^H, (VSL)*T*(VSR)^H) where (VSR)^H is the conjugate-transpose of VSR. Optionally, it also orders the eigenvalues so that a selected cluster of eigenvalues appear in the leading diagonal blocks of the upper triangular matrix S and the upper triangular matrix T. The leading columns of VSL and VSR then form an unitary basis for the corresponding left and right eigenspaces (deflating subspaces). (If only the generalized eigenvalues are needed, use the driver Zggev instead, which is faster.) A generalized eigenvalue for a pair of matrices (A, B) is a scalar w or a ratio α/β = w, such that A - w*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. A pair of matrices (S, T) is in generalized complex Schur form if S and T are upper triangular and, in addition, the diagonal elements of T are non-negative real numbers. Parameters [in] Jobvsl = "N": Do not compute the left Schur vectors. = "V": Compute the left Schur vectors. [in] Jobvsr = "N": Do not compute the right Schur vectors. = "V": Compute the right Schur vectors. [in] Sort Specifies whether or not to order the eigenvalues on the diagonal of the generalized Schur form. = "N": Eigenvalues are not ordered. = "S": Eigenvalues are ordered (see selctg). [in] Selctg Sort = "S": Selctg is used to select eigenvalues to sort to the top left of the Schur form.   An eigenvalue Alpha(j)/Beta(j) is selected if Selctg(Alpha(j), Beta(j)) is true (= 1).   Note that a selected complex eigenvalue may no longer satisfy Selctg(Alpha(j), Beta(j)) = true after ordering, since ordering may change the value of complex eigenvalues (especially if the eigenvalue is ill-conditioned), in this case Info is to be set to N+2 (see Info below). Sort = "N": Selctg is not referenced. [in] N Order of the matrices A, B, VSL and VSR. (N >= 0) (If N = 0, returns without computation) [in,out] A() Array A(LA1 - 1, LA2 - 1) (LA1 >= N, LA2 >= N) [in] The first of the pair of matrices. [out] A() has been overwritten by its generalized Schur form S. [in,out] B() Array B(LB1 - 1, LB2 - 1) (LB1 >= N, LB2 >= N) [in] The second of the pair of matrices. [out] B() has been overwritten by its generalized Schur form T. [in] Sdim Sort = "N": Sdim = 0. Sort = "S": Sdim = number of eigenvalues (after sorting) for which Selctg is true. [out] Alpha() Array Alpha(LAlpha - 1) (LAlpha >= N) [out] Beta() Array Beta(LBeta - 1) (LBeta >= N) Alpha(j)/Beta(j), j = 0, ..., N-1, will be the generalized eigenvalues. Alpha(j) and Beta(j), j = 0, ..., N-1 are the diagonals of the complex Schur form (S, T). The Beta(j) will be non-negative real. Note: The quotients Alpha(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, Alpha 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] Vsl() Array Vsl(LVsl1 - 1, LVsl2 - 1) (LVsl1 >= N, LVsl2 >= N) Jobvsl = "V": Vsl() will contain the left Schur vectors. Jobvsl = "N": Not referenced. [out] Vsr() Array Vsr(LVsr1 - 1, LVsr2 - 1) (LVsr1 >= N, LVsr2 >= N) Jobvsr = "V": Vsr() will contain the right Schur vectors. Jobvsr = "N": Not referenced. [out] Info = 0: Successful exit. = -1: The argument Jobvsl had an illegal value. (Jobvsl <> "V" nor "N") = -2: The argument Jobvsr had an illegal value. (Jobvsr <> "V" nor "N") = -3: The argument Sort had an illegal value. (Sort <> "S" nor "N") = -5: The argument N had an illegal value. (N < 0) = -6: The argument A() is invalid. = -7: The argument B() is invalid. = -9: The argument Alpha() is invalid. = -10: The argument Beta() is invalid. = -11: The argument Vsl() is invalid. = -12: The argument Vsr() is invalid. = i (0 < i <= N): The QZ iteration failed. (A, B) are not in Schur form, 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: After reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the generalized Schur form no longer satisfy Selctg = true. This could also be caused due to scaling. = N+3: Reordering failed in Ztgsen. Reference LAPACK Example Program Compute for a pair of matrices (A, B), the generalized eigenvalues, the generalized Schur form (S, T), and the left and right matrices of Schur vectors, 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_Zgges() 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, Sdim As Long Dim Vsl(N - 1, N - 1) As Complex, Vsr(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 Zgges("V", "V", "S", AddressOf Selctg, N, A(), B(), Sdim, Alpha(), Beta(), Vsl(), Vsr(), 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 "Schur form S =" Debug.Print Creal(A(0, 0)), Cimag(A(0, 0)), Creal(A(0, 1)), Cimag(A(0, 1)) Debug.Print Creal(A(1, 0)), Cimag(A(1, 0)), Creal(A(1, 1)), Cimag(A(1, 1)) Debug.Print Creal(A(2, 0)), Cimag(A(2, 0)), Creal(A(2, 1)), Cimag(A(2, 1)) Debug.Print Creal(A(0, 2)), Cimag(A(0, 2)) Debug.Print Creal(A(1, 2)), Cimag(A(1, 2)) Debug.Print Creal(A(2, 2)), Cimag(A(2, 2)) Debug.Print "Schur form T =" Debug.Print Creal(B(0, 0)), Cimag(B(0, 0)), Creal(B(0, 1)), Cimag(B(0, 1)) Debug.Print Creal(B(1, 0)), Cimag(B(1, 0)), Creal(B(1, 1)), Cimag(B(1, 1)) Debug.Print Creal(B(2, 0)), Cimag(B(2, 0)), Creal(B(2, 1)), Cimag(B(2, 1)) Debug.Print Creal(B(0, 2)), Cimag(B(0, 2)) Debug.Print Creal(B(1, 2)), Cimag(B(1, 2)) Debug.Print Creal(B(2, 2)), Cimag(B(2, 2)) Debug.Print "Left Schur vectors =" Debug.Print Creal(Vsl(0, 0)), Cimag(Vsl(0, 0)), Creal(Vsl(0, 1)), Cimag(Vsl(0, 1)) Debug.Print Creal(Vsl(1, 0)), Cimag(Vsl(1, 0)), Creal(Vsl(1, 1)), Cimag(Vsl(1, 1)) Debug.Print Creal(Vsl(2, 0)), Cimag(Vsl(2, 0)), Creal(Vsl(2, 1)), Cimag(Vsl(2, 1)) Debug.Print Creal(Vsl(0, 2)), Cimag(Vsl(0, 2)) Debug.Print Creal(Vsl(1, 2)), Cimag(Vsl(1, 2)) Debug.Print Creal(Vsl(2, 2)), Cimag(Vsl(2, 2)) Debug.Print "Right Schur vectors =" Debug.Print Creal(Vsr(0, 0)), Cimag(Vsr(0, 0)), Creal(Vsr(0, 1)), Cimag(Vsr(0, 1)) Debug.Print Creal(Vsr(1, 0)), Cimag(Vsr(1, 0)), Creal(Vsr(1, 1)), Cimag(Vsr(1, 1)) Debug.Print Creal(Vsr(2, 0)), Cimag(Vsr(2, 0)), Creal(Vsr(2, 1)), Cimag(Vsr(2, 1)) Debug.Print Creal(Vsr(0, 2)), Cimag(Vsr(0, 2)) Debug.Print Creal(Vsr(1, 2)), Cimag(Vsr(1, 2)) Debug.Print Creal(Vsr(2, 2)), Cimag(Vsr(2, 2)) Debug.Print "Sdim =", Sdim, "Info =", Info End Sub Function Selctg(Alpha As Complex, Beta As Complex) As Long Selctg = 0 If Cimag(Alpha) <> 0 Then Selctg = 1 End Function Example Results Eigenvalues = -0.4784814787767 -0.640760182519056 1.62433774044009 1.10642263894432 -0.149178317709927 5.63446258373312 Schur form S = -0.72797081435173 -0.974864717993083 -0.273608952637649 0.236841131861002 0 0 1.50621819846239 1.02596515027546 0 0 0 0 1.07441516363149E-02 4.41155734877746E-02 -0.689834642053671 -0.367063708571792 -2.75225804680911E-02 1.03952740743988 Schur form T = 1.5214190029108 0 -0.330233470430214 -1.05504057337832 0 0 0.927281415042602 0 0 0 0 0 0.359567173199973 1.92492599356072 -0.911905206658519 -8.20406969051636E-02 0.184494508924602 0 Left Schur vectors = -0.129055100819468 0.380797390644917 0.459273707935736 -0.209026938555656 -0.693189782060827 0.176375779993619 -0.368487931998788 -0.564073489521078 0.149492866673543 0.551696946993847 0.512426331896747 -0.16980105101744 0.759605142163285 8.19362946843842E-02 -0.150340091122171 0.108697330557172 -0.614629176696597 6.40327579983544E-02 Right Schur vectors = -0.609843105250231 -0.453993250889564 0.531526779080892 0.119433675168335 -0.264988296049601 -9.90695388688886E-02 -0.740613399363287 -0.14310787368587 0.289272174294141 -0.508202276391652 -5.56309308836406E-02 -0.362121539068532 -0.265832927214418 -0.233514989432741 -0.344926967779475 -0.481667671217571 -0.51570410282413 0.507813473877466 Sdim = 3 Info = 0