WZggev WZggev2 WZhbgv WZhbgv2 WZhegv WZhegv2

WZggev() Function WZggev ( JobVl As  String, JobVr As  String, N As  Long, A As  Variant, B As  Variant  ) Generalized eigenvalue problem of complex matrices (complex number representation in Excel format) Purpose WZggev 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). To represent complex numbers in Excel cells, complex number format in Excel (e.g. 2.5+1i) is used. Worksheet function Complex can be used to input complex numbers into cells. Returns N+1 x 2 (JobVl = "N", JobVr = "N") Column 1 Column 2 Rows 1 to N Alpha(j) (j = 1 to N) Beta(j) (j = 1 to N). (Alphar(j)/Beta(j) (j = 1 to N) are the generalized eigenvalues. Row N+1 Return code 0 N+1 x N+2 (JobVl = "V", JobVr = "N") Column 1 Column 2 Columns 3 to N+2 Rows 1 to N Alpha(j) (j = 1 to N) Beta(j) (j = 1 to N). (Alpha(j)/Beta(j) (j = 1 to N) are the generalized eigenvalues. Left eigenvectors are stored one after another in the same order as their eigenvalues (Each eigenvector is scaled so the largest component has abs(real part) + abs(imaginary part) = 1) Row N+1 Return code 0 0 N+1 x N+2 (JobVl = "N", JobVr = "V") Column 1 Column 2 Columns 3 to N+2 Rows 1 to N Alpha(j) (j = 1 to N) Beta(j) (j = 1 to N). (Alphar(j)/Beta(j) (j = 1 to N) are the generalized eigenvalues. Right eigenvectors are stored one after another in the same order as their eigenvalues (Each eigenvector is scaled so the largest component has abs(real part) + abs(imaginary part) = 1) Row N+1 Return code 0 0 N+1 x 2N+2 (JobVl = "V", JobVr = "N") Column 1 Column 2 Columns 3 to N+2 Columns N+3 to 2N+2 Rows 1 to N Alpha(j) (j = 1 to N) Beta(j) (j = 1 to N). (Alpha(j)/Beta(j) (j = 1 to N) are the generalized eigenvalues. Left eigenvectors are stored one after another in the same order as their eigenvalues (Each eigenvector is scaled so the largest component has abs(real part) + abs(imaginary part) = 1) Right eigenvectors are stored one after another in the same order as their eigenvalues (Each eigenvector is scaled so the largest component has abs(real part) + abs(imaginary part) = 1) Row N+1 Return code 0 0 0 Return code. = 0: Successful exit. = i (0 < i <= N: Failed to converge for first to i-th eigenvalues, and no eigenvectors have been computed. = N+1, N+2: Failed to converge due to other errors. 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 matrices A and B. (N >= 1) [in] A (N x N) N x N complex matrix A. [in] B (N x N) N x N complex matrix B. Reference LAPACK Example 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 )