--Eigensystem module local ffi = require 'ffi' local gsl = require 'gsl' local sqrt, abs = math.sqrt, math.abs local format = string.format local check = require 'check' local gsl_check = require'gsl-check' local is_integer, is_real = check.is_integer, check.is_real local matrix = require 'matrix' ------------------------------------------------------------ --Eigensystem struct und function definitions ffi.cdef[[ typedef struct { size_t size; double * d; double * sd; } gsl_eigen_symm_workspace; gsl_eigen_symm_workspace * gsl_eigen_symm_alloc (const size_t n); void gsl_eigen_symm_free (gsl_eigen_symm_workspace * w); int gsl_eigen_symm (gsl_matrix * A, gsl_vector * eval, gsl_eigen_symm_workspace * w); typedef struct { size_t size; double * d; double * sd; double * gc; double * gs; } gsl_eigen_symmv_workspace; gsl_eigen_symmv_workspace * gsl_eigen_symmv_alloc (const size_t n); void gsl_eigen_symmv_free (gsl_eigen_symmv_workspace * w); int gsl_eigen_symmv (gsl_matrix * A, gsl_vector * eval, gsl_matrix * evec, gsl_eigen_symmv_workspace * w); typedef struct { size_t size; double * d; double * sd; double * tau; } gsl_eigen_herm_workspace; gsl_eigen_herm_workspace * gsl_eigen_herm_alloc (const size_t n); void gsl_eigen_herm_free (gsl_eigen_herm_workspace * w); int gsl_eigen_herm (gsl_matrix_complex * A, gsl_vector * eval, gsl_eigen_herm_workspace * w); typedef struct { size_t size; double * d; double * sd; double * tau; double * gc; double * gs; } gsl_eigen_hermv_workspace; gsl_eigen_hermv_workspace * gsl_eigen_hermv_alloc (const size_t n); void gsl_eigen_hermv_free (gsl_eigen_hermv_workspace * w); int gsl_eigen_hermv (gsl_matrix_complex * A, gsl_vector * eval, gsl_matrix_complex * evec, gsl_eigen_hermv_workspace * w); typedef struct { size_t size; /* matrix size */ size_t max_iterations; /* max iterations since last eigenvalue found */ size_t n_iter; /* number of iterations since last eigenvalue found */ size_t n_evals; /* number of eigenvalues found so far */ int compute_t; /* compute Schur form T = Z^t A Z */ gsl_matrix *H; /* pointer to Hessenberg matrix */ gsl_matrix *Z; /* pointer to Schur vector matrix */ } gsl_eigen_francis_workspace; gsl_eigen_francis_workspace * gsl_eigen_francis_alloc (void); void gsl_eigen_francis_free (gsl_eigen_francis_workspace * w); void gsl_eigen_francis_T (const int compute_t, gsl_eigen_francis_workspace * w); int gsl_eigen_francis (gsl_matrix * H, gsl_vector_complex * eval, gsl_eigen_francis_workspace * w); int gsl_eigen_francis_Z (gsl_matrix * H, gsl_vector_complex * eval, gsl_matrix * Z, gsl_eigen_francis_workspace * w); typedef struct { size_t size; /* size of matrices */ gsl_vector *diag; /* diagonal matrix elements from balancing */ gsl_vector *tau; /* Householder coefficients */ gsl_matrix *Z; /* pointer to Z matrix */ int do_balance; /* perform balancing transformation? */ size_t n_evals; /* number of eigenvalues found */ gsl_eigen_francis_workspace *francis_workspace_p; } gsl_eigen_nonsymm_workspace; gsl_eigen_nonsymm_workspace * gsl_eigen_nonsymm_alloc (const size_t n); void gsl_eigen_nonsymm_free (gsl_eigen_nonsymm_workspace * w); void gsl_eigen_nonsymm_params (const int compute_t, const int balance, gsl_eigen_nonsymm_workspace *w); int gsl_eigen_nonsymm (gsl_matrix * A, gsl_vector_complex * eval, gsl_eigen_nonsymm_workspace * w); int gsl_eigen_nonsymm_Z (gsl_matrix * A, gsl_vector_complex * eval, gsl_matrix * Z, gsl_eigen_nonsymm_workspace * w); typedef struct { size_t size; /* size of matrices */ gsl_vector *work; /* scratch workspace */ gsl_vector *work2; /* scratch workspace */ gsl_vector *work3; /* scratch workspace */ gsl_matrix *Z; /* pointer to Schur vectors */ gsl_eigen_nonsymm_workspace *nonsymm_workspace_p; } gsl_eigen_nonsymmv_workspace; gsl_eigen_nonsymmv_workspace * gsl_eigen_nonsymmv_alloc (const size_t n); void gsl_eigen_nonsymmv_free (gsl_eigen_nonsymmv_workspace * w); void gsl_eigen_nonsymmv_params (const int balance, gsl_eigen_nonsymmv_workspace *w); int gsl_eigen_nonsymmv (gsl_matrix * A, gsl_vector_complex * eval, gsl_matrix_complex * evec, gsl_eigen_nonsymmv_workspace * w); int gsl_eigen_nonsymmv_Z (gsl_matrix * A, gsl_vector_complex * eval, gsl_matrix_complex * evec, gsl_matrix * Z, gsl_eigen_nonsymmv_workspace * w); typedef struct { size_t size; /* size of matrices */ gsl_eigen_symm_workspace *symm_workspace_p; } gsl_eigen_gensymm_workspace; gsl_eigen_gensymm_workspace * gsl_eigen_gensymm_alloc (const size_t n); void gsl_eigen_gensymm_free (gsl_eigen_gensymm_workspace * w); int gsl_eigen_gensymm (gsl_matrix * A, gsl_matrix * B, gsl_vector * eval, gsl_eigen_gensymm_workspace * w); int gsl_eigen_gensymm_standardize (gsl_matrix * A, const gsl_matrix * B); typedef struct { size_t size; /* size of matrices */ gsl_eigen_symmv_workspace *symmv_workspace_p; } gsl_eigen_gensymmv_workspace; gsl_eigen_gensymmv_workspace * gsl_eigen_gensymmv_alloc (const size_t n); void gsl_eigen_gensymmv_free (gsl_eigen_gensymmv_workspace * w); int gsl_eigen_gensymmv (gsl_matrix * A, gsl_matrix * B, gsl_vector * eval, gsl_matrix * evec, gsl_eigen_gensymmv_workspace * w); typedef struct { size_t size; /* size of matrices */ gsl_eigen_herm_workspace *herm_workspace_p; } gsl_eigen_genherm_workspace; gsl_eigen_genherm_workspace * gsl_eigen_genherm_alloc (const size_t n); void gsl_eigen_genherm_free (gsl_eigen_genherm_workspace * w); int gsl_eigen_genherm (gsl_matrix_complex * A, gsl_matrix_complex * B, gsl_vector * eval, gsl_eigen_genherm_workspace * w); int gsl_eigen_genherm_standardize (gsl_matrix_complex * A, const gsl_matrix_complex * B); typedef struct { size_t size; /* size of matrices */ gsl_eigen_hermv_workspace *hermv_workspace_p; } gsl_eigen_genhermv_workspace; gsl_eigen_genhermv_workspace * gsl_eigen_genhermv_alloc (const size_t n); void gsl_eigen_genhermv_free (gsl_eigen_genhermv_workspace * w); int gsl_eigen_genhermv (gsl_matrix_complex * A, gsl_matrix_complex * B, gsl_vector * eval, gsl_matrix_complex * evec, gsl_eigen_genhermv_workspace * w); typedef struct { size_t size; /* size of matrices */ gsl_vector *work; /* scratch workspace */ size_t n_evals; /* number of eigenvalues found */ size_t max_iterations; /* maximum QZ iterations allowed */ size_t n_iter; /* number of iterations since last eigenvalue found */ double eshift; /* exceptional shift counter */ int needtop; /* need to compute top index? */ double atol; /* tolerance for splitting A matrix */ double btol; /* tolerance for splitting B matrix */ double ascale; /* scaling factor for shifts */ double bscale; /* scaling factor for shifts */ gsl_matrix *H; /* pointer to hessenberg matrix */ gsl_matrix *R; /* pointer to upper triangular matrix */ int compute_s; /* compute generalized Schur form S */ int compute_t; /* compute generalized Schur form T */ gsl_matrix *Q; /* pointer to left Schur vectors */ gsl_matrix *Z; /* pointer to right Schur vectors */ } gsl_eigen_gen_workspace; gsl_eigen_gen_workspace * gsl_eigen_gen_alloc (const size_t n); void gsl_eigen_gen_free (gsl_eigen_gen_workspace * w); void gsl_eigen_gen_params (const int compute_s, const int compute_t, const int balance, gsl_eigen_gen_workspace * w); int gsl_eigen_gen (gsl_matrix * A, gsl_matrix * B, gsl_vector_complex * alpha, gsl_vector * beta, gsl_eigen_gen_workspace * w); int gsl_eigen_gen_QZ (gsl_matrix * A, gsl_matrix * B, gsl_vector_complex * alpha, gsl_vector * beta, gsl_matrix * Q, gsl_matrix * Z, gsl_eigen_gen_workspace * w); typedef struct { size_t size; /* size of matrices */ gsl_vector *work1; /* 1-norm of columns of A */ gsl_vector *work2; /* 1-norm of columns of B */ gsl_vector *work3; /* real part of eigenvector */ gsl_vector *work4; /* imag part of eigenvector */ gsl_vector *work5; /* real part of back-transformed eigenvector */ gsl_vector *work6; /* imag part of back-transformed eigenvector */ gsl_matrix *Q; /* pointer to left Schur vectors */ gsl_matrix *Z; /* pointer to right Schur vectors */ gsl_eigen_gen_workspace *gen_workspace_p; } gsl_eigen_genv_workspace; gsl_eigen_genv_workspace * gsl_eigen_genv_alloc (const size_t n); void gsl_eigen_genv_free (gsl_eigen_genv_workspace * w); int gsl_eigen_genv (gsl_matrix * A, gsl_matrix * B, gsl_vector_complex * alpha, gsl_vector * beta, gsl_matrix_complex * evec, gsl_eigen_genv_workspace * w); int gsl_eigen_genv_QZ (gsl_matrix * A, gsl_matrix * B, gsl_vector_complex * alpha, gsl_vector * beta, gsl_matrix_complex * evec, gsl_matrix * Q, gsl_matrix * Z, gsl_eigen_genv_workspace * w); typedef enum { GSL_EIGEN_SORT_VAL_ASC, GSL_EIGEN_SORT_VAL_DESC, GSL_EIGEN_SORT_ABS_ASC, GSL_EIGEN_SORT_ABS_DESC, GSL_EIGEN_SORT_NONE } gsl_eigen_sort_t; /* Sort eigensystem results based on eigenvalues. * Sorts in order of increasing value or increasing * absolute value. * * exceptions: GSL_EBADLEN */ int gsl_eigen_symmv_sort(gsl_vector * eval, gsl_matrix * evec, gsl_eigen_sort_t sort_type); int gsl_eigen_hermv_sort(gsl_vector * eval, gsl_matrix_complex * evec, gsl_eigen_sort_t sort_type); int gsl_eigen_nonsymmv_sort(gsl_vector_complex * eval, gsl_matrix_complex * evec, gsl_eigen_sort_t sort_type); int gsl_eigen_gensymmv_sort (gsl_vector * eval, gsl_matrix * evec, gsl_eigen_sort_t sort_type); int gsl_eigen_genhermv_sort (gsl_vector * eval, gsl_matrix_complex * evec, gsl_eigen_sort_t sort_type); int gsl_eigen_genv_sort (gsl_vector_complex * alpha, gsl_vector * beta, gsl_matrix_complex * evec, gsl_eigen_sort_t sort_type); /* Prototypes for the schur module */ int gsl_schur_gen_eigvals(const gsl_matrix *A, const gsl_matrix *B, double *wr1, double *wr2, double *wi, double *scale1, double *scale2); int gsl_schur_solve_equation(double ca, const gsl_matrix *A, double z, double d1, double d2, const gsl_vector *b, gsl_vector *x, double *s, double *xnorm, double smin); int gsl_schur_solve_equation_z(double ca, const gsl_matrix *A, gsl_complex *z, double d1, double d2, const gsl_vector_complex *b, gsl_vector_complex *x, double *s, double *xnorm, double smin); /* The following functions are obsolete: */ /* Eigensolve by Jacobi Method * * The data in the matrix input is destroyed. * * exceptions: */ int gsl_eigen_jacobi(gsl_matrix * matrix, gsl_vector * eval, gsl_matrix * evec, unsigned int max_rot, unsigned int * nrot); /* Invert by Jacobi Method * * exceptions: */ int gsl_eigen_invert_jacobi(const gsl_matrix * matrix, gsl_matrix * ainv, unsigned int max_rot); ]] ------------------------------------------------------------------------------- eigen = {} order_lookup = { asc = gsl.GSL_EIGEN_SORT_VAL_ASC, desc = gsl.GSL_EIGEN_SORT_VAL_DESC, abs_asc = gsl.GSL_EIGEN_SORT_ABS_ASC, abs_desc = gsl.GSL_EIGEN_SORT_ABS_DESC, none = gsl.GSL_EIGEN_SORT_NONE } local SORT_NONE = gsl.GSL_EIGEN_SORT_NONE local function get_order(order) local sel = order and order_lookup[order] or gsl.GSL_EIGEN_SORT_VAL_DESC if not sel then error('invalid order specification: '..order, 3) end return sel end --Calculates the eigenvalues/eigenvectors of the symmetric matrix m --the order can be used to determine the sorting of the eigenvalues according to their value function eigen.symm(m, order) local size = m.size1 local A = matrix.copy(m) local eval = matrix.alloc(size, 1) local evec = matrix.alloc (size, size) local xeval = gsl.gsl_matrix_column(eval, 0) local order_sel = get_order(order) local w = gsl.gsl_eigen_symmv_alloc (size) gsl_check(gsl.gsl_eigen_symmv (A, xeval, evec, w)) gsl.gsl_eigen_symmv_free (w) if order_sel ~= SORT_NONE then gsl.gsl_eigen_symmv_sort (xeval, evec, order_sel) end return eval,evec end --Calculates the eigenvalues/eigenvectors of the real nonsymmetric matrix m --the order can be used to determine the sorting of the eigenvalues according to their value function eigen.non_symm(m, order) local size = m.size1 local A = matrix.copy(m) local eval = matrix.calloc(size, 1) local evec = matrix.calloc (size, size) local xeval = gsl.gsl_matrix_complex_column(eval, 0) local order_sel = get_order(order) local w = gsl.gsl_eigen_nonsymmv_alloc (size) gsl_check(gsl.gsl_eigen_nonsymmv (A, xeval, evec, w)) gsl.gsl_eigen_nonsymmv_free (w) if order_sel ~= SORT_NONE then gsl.gsl_eigen_nonsymmv_sort (xeval, evec, order_sel) end return eval,evec end function eigen.herm(m, order, eigenvalues_only) local size = m.size1 local A = matrix_complex.copy(m) local eval = matrix.alloc(size, 1) local xeval = gsl.gsl_matrix_column(eval, 0) local evec = matrix.calloc (size, size) local order_sel = get_order(order) local w = gsl.gsl_eigen_hermv_alloc (size) gsl_check(gsl.gsl_eigen_hermv(A, xeval, evec, w)) gsl.gsl_eigen_hermv_free (w) if order_sel ~= SORT_NONE then gsl.gsl_eigen_hermv_sort (xeval, evec, order_sel) end return eval,evec end function eigen.gensymm(a, b) local size = a.size1 local A = matrix.copy(a) local B = matrix.copy(b) local eval = matrix.alloc(size, 1) local xeval = gsl.gsl_matrix_column(eval, 0) local evec = matrix.alloc (size, size) local order_sel = get_order(order) local w = gsl.gsl_eigen_gensymmv_alloc (size) gsl_check(gsl.gsl_eigen_gensymmv(A,B, xeval, evec, w)) gsl.gsl_eigen_gensymmv_free (w) if order_sel ~= SORT_NONE then gsl.gsl_eigen_gensymmv_sort (xeval, evec, order_sel) end return eval,evec end function eigen.genherm(a, b) local size = a.size1 local A = matrix_complex.copy(a) local B = matrix_complex.copy(b) local eval = matrix.alloc(size, 1) local xeval = gsl.gsl_matrix_column(eval, 0) local evec = matrix.calloc (size, size) local order_sel = get_order(order) local w = gsl.gsl_eigen_genhermv_alloc (size) gsl_check(gsl.gsl_eigen_genhermv(A,B, xeval, evec, w)) gsl.gsl_eigen_genhermv_free (w) if order_sel ~= SORT_NONE then gsl.gsl_eigen_genhermv_sort (xeval, evec, order_sel) end return eval,evec end function eigen.gen(a, b) local size = a.size1 local A = matrix.copy(a) local B = matrix.copy(b) local alpha = matrix.calloc(size, 1) local alpha_vec = gsl.gsl_matrix_complex_column(alpha, 0) local beta = matrix.alloc(size,size) local beta_vec = gsl.gsl_matrix_column(beta, 0) local evec = matrix.calloc (size, size) local order_sel = get_order(order) local w = gsl.gsl_eigen_genv_alloc (size) gsl_check(gsl.gsl_eigen_genv(A,B, alpha_vec, beta_vec, evec, w)) gsl.gsl_eigen_genv_free (w) if order_sel ~= SORT_NONE then gsl.gsl_eigen_genv_sort (alpha_vec, beta_vec, evec, order_sel) end return alpha, beta, evec end

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