*error in Schur decomposition of a non-symmetric matrix ?@ 2010-03-23 15:33 Francesco Abbate0 siblings, 0 replies; only message in thread From: Francesco Abbate @ 2010-03-23 15:33 UTC (permalink / raw) To: gsl-discuss Hi all, it seems that I've found an error in GSL manual in the section about eigenvalues/vectors determination of non-symmetric real matrices. The manual says: 14.3 Real Nonsymmetric Matrices =============================== The solution of the real nonsymmetric eigensystem problem for a matrix A involves computing the Schur decomposition A = Z T Z^T where Z is an orthogonal matrix of Schur vectors and T, the Schur form, is quasi upper triangular with diagonal 1-by-1 blocks which are real eigenvalues of A, and diagonal 2-by-2 blocks whose eigenvalues are complex conjugate eigenvalues of A. The algorithm used is the double-shift Francis method. ----------------------- and after about the function gsl_eigen_nonsymmv and gsl_eigen_nonsymmv_Z : -- Function: int gsl_eigen_nonsymmv (gsl_matrix * A, gsl_vector_complex * EVAL, gsl_matrix_complex * EVEC, gsl_eigen_nonsymmv_workspace * W) This function computes eigenvalues and right eigenvectors of the N-by-N real nonsymmetric matrix A. It first calls `gsl_eigen_nonsymm' to compute the eigenvalues, Schur form T, and Schur vectors. Then it finds eigenvectors of T and backtransforms them using the Schur vectors. The Schur vectors are destroyed in the process, but can be saved by using `gsl_eigen_nonsymmv_Z'. The computed eigenvectors are normalized to have unit magnitude. On output, the upper portion of A contains the Schur form T. If `gsl_eigen_nonsymm' fails, no eigenvectors are computed, and an error code is returned. -- Function: int gsl_eigen_nonsymmv_Z (gsl_matrix * A, gsl_vector_complex * EVAL, gsl_matrix_complex * EVEC, gsl_matrix * Z, gsl_eigen_nonsymmv_workspace * W) This function is identical to `gsl_eigen_nonsymmv' except that it also saves the Schur vectors into Z. ------------------------------ The problems seems to be about the Schur decomposition formula. What it is true is that: A = Z T Z^(-1) and not that: A = Z T Z^T as asserted in the documentation. I've found that empirically. Otherwise I'm perplexed about the statement in the gsl_eigen_nonsymmv: "On output, the upper portion of A contains the Schur form T". I don't understand this statement because before it was stated that: "T, the Schur form, is quasi upper triangular with diagonal 1-by-1 blocks which are real eigenvalues of A, and diagonal 2-by-2 blocks whose eigenvalues are complex conjugate eigenvalues of A". So the T matrix is not really upper triangular. I've made some tests and it seems that the function gsl_eigen_nonsymmv set the matrix A to the Schur form T and not only the upper part. It would be nice if someone could check because what I've seen is based on empirical tests using the functions. Best regards, Francesco ^ permalink raw reply [flat|nested] only message in thread

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