WZgeev WZgeev2 WZhbev WZhbev2 WZheev WZheev2

WZgeev2() Function WZgeev2 ( JobVl As  String, JobVr As  String, N As  Long, A As  Variant  ) Eigenvalues and left and/or right eigenvectors of a complex matrix (complex numbers in pairs of cells) Purpose WZgeev2 computes for an N x N complex 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. To represent complex numbers, a real part and an imaginary part are stored in a pair of adjacent cells (a real part in a left cell, and an imaginary part in a right cell). The computed results are stored in the same way. Returns N+1 x 2 (JobVl = "N", JobVr = "N") Column 1 Column 2 Rows 1 to N Real part of the computed eigenvalues (Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having the positive imaginary part first) Imaginary part of the computed eigenvalues (Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having the positive imaginary part first) Row N+1 Return code 0 N+1 x 2N+2 (JobVl = "V", JobVr = "N") Column 1 Column 2 Column 3 to 2N+2 Rows 1 to N Real part of the computed eigenvalues (Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having the positive imaginary part first) Imaginary part of the computed eigenvalues (Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having the positive imaginary part first) Left eigenvectors are stored one after another in the same order as their eigenvalues. Row N+1 Return code 0 0 N+1 x 2N+2 (JobVl = "N", JobVr = "V") Column 1 Column 2 Column 3 to 2N+2 Rows 1 to N Real part of the computed eigenvalues (Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having the positive imaginary part first) Imaginary part of the computed eigenvalues (Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having the positive imaginary part first) Right eigenvectors are stored one after another in the same order as their eigenvalues. Row N+1 Return code 0 0 N+1 x 4N+2 (JobVl = "V", JobVr = "V") Column 1 Column 2 Column 3 to 2N+2 Column 2N+3 to 4N+2 Rows 1 to N Real part of the computed eigenvalues (Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having the positive imaginary part first) Imaginary part of the computed eigenvalues (Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having the positive imaginary part first) Left eigenvectors are stored one after another in the same order as their eigenvalues. Right eigenvectors are stored one after another in the same order as their eigenvalues. Row N+1 Return code 0 0 0 Return code. = 0: Successful exit. = i > 0: Failed to converge for first to i-th eigenvalues, and no eigenvectors have been computed. 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 >= 1) [in] A (N x 2N) N x N complex matrix A. Reference LAPACK Example Compute all eigenvalues and eigenvectors of the general matrix A, where ( 0.20-0.11i -0.93-0.32i 0.81+0.37i ) A = ( -0.80-0.92i -0.29+0.86i 0.64+0.51i ) ( 0.71+0.59i -0.15+0.19i 0.20+0.94i )