Dgees Dgees_r Dgeesx Dgeesx_r Dgeev Dgeevx

Dgeev() Sub Dgeev ( Jobvl As  String, Jobvr As  String, N As  Long, A() As  Double, Wr() As  Double, Wi() As  Double, Vl() As  Double, Vr() As  Double, Info As  Long  ) (Simple driver) Eigenvalues and left and/or right eigenvectors of a general matrix Purpose This routine computes for an n x n real nonsymmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors. The right eigenvector v(j) of A satisfies A * v(j) = λ(j) * v(j) where λ(j) is its eigenvalue. The left eigenvector u(j) of A satisfies u(j)^H * A = λ(j) * u(j)^H where u(j)^H denotes the conjugate transpose of u(j). The computed eigenvectors are normalized to have Euclidean norm equal to 1 and largest component real. Parameters [in] JobVl = "N": Left eigenvectors of A are not computed. = "V": Left eigenvectors of A are computed. [in] JobVr = "N": Right eigenvectors of A are not computed. = "V": Right eigenvectors of A are computed. [in] N Order of the matrix A. (N >= 0) (If N = 0, returns without computation) [in,out] A() Array A(LA1 - 1, LA2 - 1) (LA1 >= N, LA2 >= N) [in] N x N matrix A. [out] A() has been overwritten. [out] Wr() Array Wr(LWr - 1) (LWr >= N) [out] Wi() Array Wi(LWi - 1) (LWi >= N) 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. [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)-st eigenvalues form a complex conjugate pair, then u(j) = (j-th column of Vl()) + i*((j+1)-th column of Vl()) and u(j+1) = (j-th column of Vl()) - i*((j+1)-st column of Vl()). 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()) + i*((j+1)-st column of Vr()) and v(j+1) = (j-th column of Vr()) - i*((j+1)-st column of Vr()). 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 Wr() is invalid. = -6: The argument Wi() is invalid. = -7: The argument Vl() is invalid. = -8: The argument Vr() is invalid. = i > 0: The QR algorithm failed to compute all the eigenvalues, and no eigenvectors have been computed; elements i to N-1 of Wr() and Wi() contain eigenvalues which have converged. Reference LAPACK Example Program Compute all eigenvalues and eigenvectors of the general matrix A, where ( 0.20 -0.11 -0.93 ) A = ( -0.32 0.81 0.37 ) ( -0.80 -0.92 -0.29 ) Sub Ex_Dgeev() Const N = 3 Dim A(N - 1, N - 1) As Double, Wr(N - 1) As Double, Wi(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 Call Dgeev("V", "V", N, A(), Wr(), Wi(), Vl(), Vr(), Info) Debug.Print "Eigenvalues (r) =", Wr(0), Wr(1), Wr(2) Debug.Print "Eigenvalues (i) =", Wi(0), Wi(1), Wi(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) = -0.904130023851345 0.812065011925673 0.812065011925673 Eigenvalues (i) = 0 0.48915757543818 -0.48915757543818 Eigenvectors (L) = -0.610555216308549 6.20242550062375E-02 -0.532191939813679 -0.404592057902723 0.7174496715502 0 -0.680828608770563 -9.02631541142068E-03 0.445094625232521 Eigenvectors (R) = -0.640873419668578 -0.22876069887917 -0.504184181286349 4.57634889263765E-02 0.671832713345003 0 -0.766281255834689 -0.194097516422216 0.45214357607639 Info = 0