/* linalg/gsl_linalg.h * * Copyright (C) 1996, 1997, 1998, 1999, 2000, 2006, 2007, 2019 Gerard Jungman, Brian Gough, Patrick Alken * * This program is free software; you can redistribute it and/or modify * it under the terms of the GNU General Public License as published by * the Free Software Foundation; either version 3 of the License, or (at * your option) any later version. * * This program is distributed in the hope that it will be useful, but * WITHOUT ANY WARRANTY; without even the implied warranty of * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU * General Public License for more details. * * You should have received a copy of the GNU General Public License * along with this program; if not, write to the Free Software * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA. */ #ifndef __GSL_LINALG_H__ #define __GSL_LINALG_H__ #include #include #include #include #include #include #include #include #undef __BEGIN_DECLS #undef __END_DECLS #ifdef __cplusplus #define __BEGIN_DECLS extern "C" { #define __END_DECLS } #else #define __BEGIN_DECLS /* empty */ #define __END_DECLS /* empty */ #endif __BEGIN_DECLS typedef enum { GSL_LINALG_MOD_NONE = 0, GSL_LINALG_MOD_TRANSPOSE = 1, GSL_LINALG_MOD_CONJUGATE = 2 } gsl_linalg_matrix_mod_t; /* Note: You can now use the gsl_blas_dgemm function instead of matmult */ /* Simple implementation of matrix multiply. * Calculates C = A.B * * exceptions: GSL_EBADLEN */ int gsl_linalg_matmult (const gsl_matrix * A, const gsl_matrix * B, gsl_matrix * C); /* Simple implementation of matrix multiply. * Allows transposition of either matrix, so it * can compute A.B or Trans(A).B or A.Trans(B) or Trans(A).Trans(B) * * exceptions: GSL_EBADLEN */ int gsl_linalg_matmult_mod (const gsl_matrix * A, gsl_linalg_matrix_mod_t modA, const gsl_matrix * B, gsl_linalg_matrix_mod_t modB, gsl_matrix * C); /* Calculate the matrix exponential by the scaling and * squaring method described in Moler + Van Loan, * SIAM Rev 20, 801 (1978). The mode argument allows * choosing an optimal strategy, from the table * given in the paper, for a given precision. * * exceptions: GSL_ENOTSQR, GSL_EBADLEN */ int gsl_linalg_exponential_ss( const gsl_matrix * A, gsl_matrix * eA, gsl_mode_t mode ); /* Householder Transformations */ double gsl_linalg_householder_transform (gsl_vector * v); double gsl_linalg_householder_transform2 (double * alpha, gsl_vector * v); gsl_complex gsl_linalg_complex_householder_transform (gsl_vector_complex * v); int gsl_linalg_householder_hm (double tau, const gsl_vector * v, gsl_matrix * A); int gsl_linalg_householder_mh (double tau, const gsl_vector * v, gsl_matrix * A); int gsl_linalg_householder_hv (double tau, const gsl_vector * v, gsl_vector * w); int gsl_linalg_householder_left(const double tau, const gsl_vector * v, gsl_matrix * A, gsl_vector * work); int gsl_linalg_householder_right(const double tau, const gsl_vector * v, gsl_matrix * A, gsl_vector * work); int gsl_linalg_householder_hm1 (double tau, gsl_matrix * A); int gsl_linalg_complex_householder_hm (gsl_complex tau, const gsl_vector_complex * v, gsl_matrix_complex * A); int gsl_linalg_complex_householder_mh (gsl_complex tau, const gsl_vector_complex * v, gsl_matrix_complex * A); int gsl_linalg_complex_householder_hv (gsl_complex tau, const gsl_vector_complex * v, gsl_vector_complex * w); int gsl_linalg_complex_householder_left (const gsl_complex tau, const gsl_vector_complex * v, gsl_matrix_complex * A, gsl_vector_complex * work); /* Hessenberg reduction */ int gsl_linalg_hessenberg_decomp(gsl_matrix *A, gsl_vector *tau); int gsl_linalg_hessenberg_unpack(gsl_matrix * H, gsl_vector * tau, gsl_matrix * U); int gsl_linalg_hessenberg_unpack_accum(gsl_matrix * H, gsl_vector * tau, gsl_matrix * U); int gsl_linalg_hessenberg_set_zero(gsl_matrix * H); int gsl_linalg_hessenberg_submatrix(gsl_matrix *M, gsl_matrix *A, size_t top, gsl_vector *tau); /* Hessenberg-Triangular reduction */ int gsl_linalg_hesstri_decomp(gsl_matrix * A, gsl_matrix * B, gsl_matrix * U, gsl_matrix * V, gsl_vector * work); /* Singular Value Decomposition * exceptions: */ int gsl_linalg_SV_decomp (gsl_matrix * A, gsl_matrix * V, gsl_vector * S, gsl_vector * work); int gsl_linalg_SV_decomp_mod (gsl_matrix * A, gsl_matrix * X, gsl_matrix * V, gsl_vector * S, gsl_vector * work); int gsl_linalg_SV_decomp_jacobi (gsl_matrix * A, gsl_matrix * Q, gsl_vector * S); int gsl_linalg_SV_solve (const gsl_matrix * U, const gsl_matrix * Q, const gsl_vector * S, const gsl_vector * b, gsl_vector * x); int gsl_linalg_SV_leverage(const gsl_matrix *U, gsl_vector *h); /* LU Decomposition, Gaussian elimination with partial pivoting */ int gsl_linalg_LU_decomp (gsl_matrix * A, gsl_permutation * p, int *signum); int gsl_linalg_LU_solve (const gsl_matrix * LU, const gsl_permutation * p, const gsl_vector * b, gsl_vector * x); int gsl_linalg_LU_svx (const gsl_matrix * LU, const gsl_permutation * p, gsl_vector * x); int gsl_linalg_LU_refine (const gsl_matrix * A, const gsl_matrix * LU, const gsl_permutation * p, const gsl_vector * b, gsl_vector * x, gsl_vector * work); int gsl_linalg_LU_invert (const gsl_matrix * LU, const gsl_permutation * p, gsl_matrix * inverse); int gsl_linalg_LU_invx (gsl_matrix * LU, const gsl_permutation * p); double gsl_linalg_LU_det (gsl_matrix * LU, int signum); double gsl_linalg_LU_lndet (gsl_matrix * LU); int gsl_linalg_LU_sgndet (gsl_matrix * lu, int signum); /* Banded LU decomposition */ int gsl_linalg_LU_band_decomp (const size_t M, const size_t lb, const size_t ub, gsl_matrix * AB, gsl_vector_uint * piv); int gsl_linalg_LU_band_solve (const size_t lb, const size_t ub, const gsl_matrix * LUB, const gsl_vector_uint * piv, const gsl_vector * b, gsl_vector * x); int gsl_linalg_LU_band_svx (const size_t lb, const size_t ub, const gsl_matrix * LUB, const gsl_vector_uint * piv, gsl_vector * x); int gsl_linalg_LU_band_unpack (const size_t M, const size_t lb, const size_t ub, const gsl_matrix * LUB, const gsl_vector_uint * piv, gsl_matrix * L, gsl_matrix * U); /* Complex LU Decomposition */ int gsl_linalg_complex_LU_decomp (gsl_matrix_complex * A, gsl_permutation * p, int *signum); int gsl_linalg_complex_LU_solve (const gsl_matrix_complex * LU, const gsl_permutation * p, const gsl_vector_complex * b, gsl_vector_complex * x); int gsl_linalg_complex_LU_svx (const gsl_matrix_complex * LU, const gsl_permutation * p, gsl_vector_complex * x); int gsl_linalg_complex_LU_refine (const gsl_matrix_complex * A, const gsl_matrix_complex * LU, const gsl_permutation * p, const gsl_vector_complex * b, gsl_vector_complex * x, gsl_vector_complex * work); int gsl_linalg_complex_LU_invert (const gsl_matrix_complex * LU, const gsl_permutation * p, gsl_matrix_complex * inverse); int gsl_linalg_complex_LU_invx (gsl_matrix_complex * LU, const gsl_permutation * p); gsl_complex gsl_linalg_complex_LU_det (gsl_matrix_complex * LU, int signum); double gsl_linalg_complex_LU_lndet (gsl_matrix_complex * LU); gsl_complex gsl_linalg_complex_LU_sgndet (gsl_matrix_complex * LU, int signum); /* QR decomposition */ int gsl_linalg_QR_decomp (gsl_matrix * A, gsl_vector * tau); int gsl_linalg_QR_decomp_old (gsl_matrix * A, gsl_vector * tau); int gsl_linalg_QR_decomp_r (gsl_matrix * A, gsl_matrix * T); int gsl_linalg_QR_solve (const gsl_matrix * QR, const gsl_vector * tau, const gsl_vector * b, gsl_vector * x); int gsl_linalg_QR_solve_r (const gsl_matrix * QR, const gsl_matrix * T, const gsl_vector * b, gsl_vector * x); int gsl_linalg_QR_svx (const gsl_matrix * QR, const gsl_vector * tau, gsl_vector * x); int gsl_linalg_QR_lssolve (const gsl_matrix * QR, const gsl_vector * tau, const gsl_vector * b, gsl_vector * x, gsl_vector * residual); int gsl_linalg_QR_lssolve_r (const gsl_matrix * QR, const gsl_matrix * T, const gsl_vector * b, gsl_vector * x, gsl_vector * work); int gsl_linalg_QR_QRsolve (gsl_matrix * Q, gsl_matrix * R, const gsl_vector * b, gsl_vector * x); int gsl_linalg_QR_Rsolve (const gsl_matrix * QR, const gsl_vector * b, gsl_vector * x); int gsl_linalg_QR_Rsvx (const gsl_matrix * QR, gsl_vector * x); int gsl_linalg_QR_update (gsl_matrix * Q, gsl_matrix * R, gsl_vector * w, const gsl_vector * v); int gsl_linalg_QR_QTvec (const gsl_matrix * QR, const gsl_vector * tau, gsl_vector * v); int gsl_linalg_QR_QTvec_r(const gsl_matrix * QR, const gsl_matrix * T, gsl_vector * b, gsl_vector * work); int gsl_linalg_QR_Qvec (const gsl_matrix * QR, const gsl_vector * tau, gsl_vector * v); int gsl_linalg_QR_QTmat (const gsl_matrix * QR, const gsl_vector * tau, gsl_matrix * A); int gsl_linalg_QR_QTmat_r(const gsl_matrix * QR, const gsl_matrix * T, gsl_matrix * B, gsl_matrix * work); int gsl_linalg_QR_matQ (const gsl_matrix * QR, const gsl_vector * tau, gsl_matrix * A); int gsl_linalg_QR_unpack (const gsl_matrix * QR, const gsl_vector * tau, gsl_matrix * Q, gsl_matrix * R); int gsl_linalg_QR_unpack_r(const gsl_matrix * QR, const gsl_matrix * T, gsl_matrix * Q, gsl_matrix * R); int gsl_linalg_R_solve (const gsl_matrix * R, const gsl_vector * b, gsl_vector * x); int gsl_linalg_R_svx (const gsl_matrix * R, gsl_vector * x); int gsl_linalg_QR_rcond(const gsl_matrix * QR, double * rcond, gsl_vector * work); /* complex QR decomposition */ int gsl_linalg_complex_QR_decomp (gsl_matrix_complex * A, gsl_vector_complex * tau); int gsl_linalg_complex_QR_decomp_r (gsl_matrix_complex * A, gsl_matrix_complex * T); int gsl_linalg_complex_QR_solve (const gsl_matrix_complex * QR, const gsl_vector_complex * tau, const gsl_vector_complex * b, gsl_vector_complex * x); int gsl_linalg_complex_QR_solve_r (const gsl_matrix_complex * QR, const gsl_matrix_complex * T, const gsl_vector_complex * b, gsl_vector_complex * x); int gsl_linalg_complex_QR_svx (const gsl_matrix_complex * QR, const gsl_vector_complex * tau, gsl_vector_complex * x); int gsl_linalg_complex_QR_lssolve (const gsl_matrix_complex * QR, const gsl_vector_complex * tau, const gsl_vector_complex * b, gsl_vector_complex * x, gsl_vector_complex * residual); int gsl_linalg_complex_QR_lssolve_r (const gsl_matrix_complex * QR, const gsl_matrix_complex * T, const gsl_vector_complex * b, gsl_vector_complex * x, gsl_vector_complex * work); int gsl_linalg_complex_QR_QHvec (const gsl_matrix_complex * QR, const gsl_vector_complex * tau, gsl_vector_complex * v); int gsl_linalg_complex_QR_QHvec_r(const gsl_matrix_complex * QR, const gsl_matrix_complex * T, gsl_vector_complex * b, gsl_vector_complex * work); int gsl_linalg_complex_QR_Qvec (const gsl_matrix_complex * QR, const gsl_vector_complex * tau, gsl_vector_complex * v); int gsl_linalg_complex_QR_unpack (const gsl_matrix_complex * QR, const gsl_vector_complex * tau, gsl_matrix_complex * Q, gsl_matrix_complex * R); int gsl_linalg_complex_QR_unpack_r(const gsl_matrix_complex * QR, const gsl_matrix_complex * T, gsl_matrix_complex * Q, gsl_matrix_complex * R); /* banded QR decomposition */ int gsl_linalg_QR_band_decomp_L2 (const size_t M, const size_t p, const size_t q, gsl_matrix * AB, gsl_vector * tau); int gsl_linalg_QR_band_unpack_L2 (const size_t p, const size_t q, const gsl_matrix * QRB, const gsl_vector * tau, gsl_matrix * Q, gsl_matrix * R); /* Q R P^T decomposition */ int gsl_linalg_QRPT_decomp (gsl_matrix * A, gsl_vector * tau, gsl_permutation * p, int *signum, gsl_vector * norm); int gsl_linalg_QRPT_decomp2 (const gsl_matrix * A, gsl_matrix * q, gsl_matrix * r, gsl_vector * tau, gsl_permutation * p, int *signum, gsl_vector * norm); int gsl_linalg_QRPT_solve (const gsl_matrix * QR, const gsl_vector * tau, const gsl_permutation * p, const gsl_vector * b, gsl_vector * x); int gsl_linalg_QRPT_lssolve (const gsl_matrix * QR, const gsl_vector * tau, const gsl_permutation * p, const gsl_vector * b, gsl_vector * x, gsl_vector * residual); int gsl_linalg_QRPT_lssolve2 (const gsl_matrix * QR, const gsl_vector * tau, const gsl_permutation * p, const gsl_vector * b, const size_t rank, gsl_vector * x, gsl_vector * residual); int gsl_linalg_QRPT_svx (const gsl_matrix * QR, const gsl_vector * tau, const gsl_permutation * p, gsl_vector * x); int gsl_linalg_QRPT_QRsolve (const gsl_matrix * Q, const gsl_matrix * R, const gsl_permutation * p, const gsl_vector * b, gsl_vector * x); int gsl_linalg_QRPT_Rsolve (const gsl_matrix * QR, const gsl_permutation * p, const gsl_vector * b, gsl_vector * x); int gsl_linalg_QRPT_Rsvx (const gsl_matrix * QR, const gsl_permutation * p, gsl_vector * x); int gsl_linalg_QRPT_update (gsl_matrix * Q, gsl_matrix * R, const gsl_permutation * p, gsl_vector * u, const gsl_vector * v); size_t gsl_linalg_QRPT_rank (const gsl_matrix * QR, const double tol); int gsl_linalg_QRPT_rcond(const gsl_matrix * QR, double * rcond, gsl_vector * work); /* triangle on top of diagonal QR decomposition */ int gsl_linalg_QR_UD_decomp (gsl_matrix * U, const gsl_vector * D, gsl_matrix * Y, gsl_matrix * T); int gsl_linalg_QR_UD_lssolve (const gsl_matrix * R, const gsl_matrix * Y, const gsl_matrix * T, const gsl_vector * b, gsl_vector * x, gsl_vector * work); /* triangle on top of rectangle QR decomposition */ int gsl_linalg_QR_UR_decomp (gsl_matrix * S, gsl_matrix * A, gsl_matrix * T); /* triangle on top of triangle QR decomposition */ int gsl_linalg_QR_UU_decomp (gsl_matrix * U, gsl_matrix * S, gsl_matrix * T); int gsl_linalg_QR_UU_lssolve (const gsl_matrix * R, const gsl_matrix * Y, const gsl_matrix * T, const gsl_vector * b, gsl_vector * x, gsl_vector * work); int gsl_linalg_QR_UU_QTvec(const gsl_matrix * Y, const gsl_matrix * T, gsl_vector * b, gsl_vector * work); /* triangle on top of trapezoidal QR decomposition */ int gsl_linalg_QR_UZ_decomp (gsl_matrix * S, gsl_matrix * A, gsl_matrix * T); /* QL decomposition */ int gsl_linalg_QL_decomp (gsl_matrix * A, gsl_vector * tau); int gsl_linalg_QL_unpack (const gsl_matrix * QL, const gsl_vector * tau, gsl_matrix * Q, gsl_matrix * L); /* COD decomposition */ int gsl_linalg_COD_decomp(gsl_matrix * A, gsl_vector * tau_Q, gsl_vector * tau_Z, gsl_permutation * p, size_t * rank, gsl_vector * work); int gsl_linalg_COD_decomp_e(gsl_matrix * A, gsl_vector * tau_Q, gsl_vector * tau_Z, gsl_permutation * p, double tol, size_t * rank, gsl_vector * work); int gsl_linalg_COD_lssolve (const gsl_matrix * QRZT, const gsl_vector * tau_Q, const gsl_vector * tau_Z, const gsl_permutation * perm, const size_t rank, const gsl_vector * b, gsl_vector * x, gsl_vector * residual); int gsl_linalg_COD_lssolve2 (const double lambda, const gsl_matrix * QRZT, const gsl_vector * tau_Q, const gsl_vector * tau_Z, const gsl_permutation * perm, const size_t rank, const gsl_vector * b, gsl_vector * x, gsl_vector * residual, gsl_matrix * S, gsl_vector * work); int gsl_linalg_COD_unpack(const gsl_matrix * QRZT, const gsl_vector * tau_Q, const gsl_vector * tau_Z, const size_t rank, gsl_matrix * Q, gsl_matrix * R, gsl_matrix * Z); int gsl_linalg_COD_matZ(const gsl_matrix * QRZT, const gsl_vector * tau_Z, const size_t rank, gsl_matrix * A, gsl_vector * work); /* LQ decomposition */ int gsl_linalg_LQ_decomp (gsl_matrix * A, gsl_vector * tau); int gsl_linalg_LQ_lssolve(const gsl_matrix * LQ, const gsl_vector * tau, const gsl_vector * b, gsl_vector * x, gsl_vector * residual); int gsl_linalg_LQ_QTvec(const gsl_matrix * LQ, const gsl_vector * tau, gsl_vector * v); int gsl_linalg_LQ_solve_T (const gsl_matrix * LQ, const gsl_vector * tau, const gsl_vector * b, gsl_vector * x); int gsl_linalg_LQ_svx_T (const gsl_matrix * LQ, const gsl_vector * tau, gsl_vector * x); int gsl_linalg_LQ_lssolve_T (const gsl_matrix * LQ, const gsl_vector * tau, const gsl_vector * b, gsl_vector * x, gsl_vector * residual); int gsl_linalg_LQ_Lsolve_T (const gsl_matrix * LQ, const gsl_vector * b, gsl_vector * x); int gsl_linalg_LQ_Lsvx_T (const gsl_matrix * LQ, gsl_vector * x); int gsl_linalg_L_solve_T (const gsl_matrix * L, const gsl_vector * b, gsl_vector * x); int gsl_linalg_LQ_vecQ (const gsl_matrix * LQ, const gsl_vector * tau, gsl_vector * v); int gsl_linalg_LQ_vecQT (const gsl_matrix * LQ, const gsl_vector * tau, gsl_vector * v); int gsl_linalg_LQ_unpack (const gsl_matrix * LQ, const gsl_vector * tau, gsl_matrix * Q, gsl_matrix * L); int gsl_linalg_LQ_update (gsl_matrix * Q, gsl_matrix * R, const gsl_vector * v, gsl_vector * w); int gsl_linalg_LQ_LQsolve (gsl_matrix * Q, gsl_matrix * L, const gsl_vector * b, gsl_vector * x); /* P^T L Q decomposition */ int gsl_linalg_PTLQ_decomp (gsl_matrix * A, gsl_vector * tau, gsl_permutation * p, int *signum, gsl_vector * norm); int gsl_linalg_PTLQ_decomp2 (const gsl_matrix * A, gsl_matrix * q, gsl_matrix * r, gsl_vector * tau, gsl_permutation * p, int *signum, gsl_vector * norm); int gsl_linalg_PTLQ_solve_T (const gsl_matrix * QR, const gsl_vector * tau, const gsl_permutation * p, const gsl_vector * b, gsl_vector * x); int gsl_linalg_PTLQ_svx_T (const gsl_matrix * LQ, const gsl_vector * tau, const gsl_permutation * p, gsl_vector * x); int gsl_linalg_PTLQ_LQsolve_T (const gsl_matrix * Q, const gsl_matrix * L, const gsl_permutation * p, const gsl_vector * b, gsl_vector * x); int gsl_linalg_PTLQ_Lsolve_T (const gsl_matrix * LQ, const gsl_permutation * p, const gsl_vector * b, gsl_vector * x); int gsl_linalg_PTLQ_Lsvx_T (const gsl_matrix * LQ, const gsl_permutation * p, gsl_vector * x); int gsl_linalg_PTLQ_update (gsl_matrix * Q, gsl_matrix * L, const gsl_permutation * p, const gsl_vector * v, gsl_vector * w); /* Cholesky Decomposition */ int gsl_linalg_cholesky_decomp (gsl_matrix * A); int gsl_linalg_cholesky_decomp1 (gsl_matrix * A); int gsl_linalg_cholesky_solve (const gsl_matrix * cholesky, const gsl_vector * b, gsl_vector * x); int gsl_linalg_cholesky_solve_mat (const gsl_matrix * cholesky, const gsl_matrix * B, gsl_matrix * X); int gsl_linalg_cholesky_svx (const gsl_matrix * cholesky, gsl_vector * x); int gsl_linalg_cholesky_svx_mat (const gsl_matrix * cholesky, gsl_matrix * X); int gsl_linalg_cholesky_invert(gsl_matrix * cholesky); /* Cholesky decomposition with unit-diagonal triangular parts. * A = L D L^T, where diag(L) = (1,1,...,1). * Upon exit, A contains L and L^T as for Cholesky, and * the diagonal of A is (1,1,...,1). The vector Dis set * to the diagonal elements of the diagonal matrix D. */ int gsl_linalg_cholesky_decomp_unit(gsl_matrix * A, gsl_vector * D); int gsl_linalg_cholesky_scale(const gsl_matrix * A, gsl_vector * S); int gsl_linalg_cholesky_scale_apply(gsl_matrix * A, const gsl_vector * S); int gsl_linalg_cholesky_decomp2(gsl_matrix * A, gsl_vector * S); int gsl_linalg_cholesky_svx2 (const gsl_matrix * LLT, const gsl_vector * S, gsl_vector * x); int gsl_linalg_cholesky_solve2 (const gsl_matrix * LLT, const gsl_vector * S, const gsl_vector * b, gsl_vector * x); int gsl_linalg_cholesky_rcond (const gsl_matrix * LLT, double * rcond, gsl_vector * work); /* Complex Cholesky Decomposition */ int gsl_linalg_complex_cholesky_decomp (gsl_matrix_complex * A); int gsl_linalg_complex_cholesky_solve (const gsl_matrix_complex * cholesky, const gsl_vector_complex * b, gsl_vector_complex * x); int gsl_linalg_complex_cholesky_svx (const gsl_matrix_complex * cholesky, gsl_vector_complex * x); int gsl_linalg_complex_cholesky_invert(gsl_matrix_complex * cholesky); /* Pivoted Cholesky LDLT decomposition */ int gsl_linalg_pcholesky_decomp (gsl_matrix * A, gsl_permutation * p); int gsl_linalg_pcholesky_solve(const gsl_matrix * LDLT, const gsl_permutation * p, const gsl_vector * b, gsl_vector * x); int gsl_linalg_pcholesky_svx(const gsl_matrix * LDLT, const gsl_permutation * p, gsl_vector * x); int gsl_linalg_pcholesky_decomp2(gsl_matrix * A, gsl_permutation * p, gsl_vector * S); int gsl_linalg_pcholesky_solve2(const gsl_matrix * LDLT, const gsl_permutation * p, const gsl_vector * S, const gsl_vector * b, gsl_vector * x); int gsl_linalg_pcholesky_svx2(const gsl_matrix * LDLT, const gsl_permutation * p, const gsl_vector * S, gsl_vector * x); int gsl_linalg_pcholesky_invert(const gsl_matrix * LDLT, const gsl_permutation * p, gsl_matrix * Ainv); int gsl_linalg_pcholesky_rcond (const gsl_matrix * LDLT, const gsl_permutation * p, double * rcond, gsl_vector * work); /* Modified Cholesky decomposition */ int gsl_linalg_mcholesky_decomp (gsl_matrix * A, gsl_permutation * p, gsl_vector * E); int gsl_linalg_mcholesky_solve(const gsl_matrix * LDLT, const gsl_permutation * p, const gsl_vector * b, gsl_vector * x); int gsl_linalg_mcholesky_svx(const gsl_matrix * LDLT, const gsl_permutation * p, gsl_vector * x); int gsl_linalg_mcholesky_rcond (const gsl_matrix * LDLT, const gsl_permutation * p, double * rcond, gsl_vector * work); int gsl_linalg_mcholesky_invert(const gsl_matrix * LDLT, const gsl_permutation * p, gsl_matrix * Ainv); /* Banded Cholesky decomposition */ int gsl_linalg_cholesky_band_decomp(gsl_matrix * A); int gsl_linalg_cholesky_band_solve (const gsl_matrix * LLT, const gsl_vector * b, gsl_vector * x); int gsl_linalg_cholesky_band_svx (const gsl_matrix * LLT, gsl_vector * x); int gsl_linalg_cholesky_band_solvem (const gsl_matrix * LLT, const gsl_matrix * B, gsl_matrix * X); int gsl_linalg_cholesky_band_svxm (const gsl_matrix * LLT, gsl_matrix * X); int gsl_linalg_cholesky_band_invert (const gsl_matrix * LLT, gsl_matrix * Ainv); int gsl_linalg_cholesky_band_unpack (const gsl_matrix * LLT, gsl_matrix * L); int gsl_linalg_cholesky_band_scale(const gsl_matrix * A, gsl_vector * S); int gsl_linalg_cholesky_band_scale_apply(gsl_matrix * A, const gsl_vector * S); int gsl_linalg_cholesky_band_rcond (const gsl_matrix * LLT, double * rcond, gsl_vector * work); /* L D L^T decomposition */ int gsl_linalg_ldlt_decomp (gsl_matrix * A); int gsl_linalg_ldlt_solve (const gsl_matrix * LDLT, const gsl_vector * b, gsl_vector * x); int gsl_linalg_ldlt_svx (const gsl_matrix * LDLT, gsl_vector * x); int gsl_linalg_ldlt_rcond (const gsl_matrix * LDLT, double * rcond, gsl_vector * work); /* Banded L D L^T decomposition */ int gsl_linalg_ldlt_band_decomp (gsl_matrix * A); int gsl_linalg_ldlt_band_solve (const gsl_matrix * LDLT, const gsl_vector * b, gsl_vector * x); int gsl_linalg_ldlt_band_svx (const gsl_matrix * LDLT, gsl_vector * x); int gsl_linalg_ldlt_band_unpack (const gsl_matrix * LDLT, gsl_matrix * L, gsl_vector * D); int gsl_linalg_ldlt_band_rcond (const gsl_matrix * LDLT, double * rcond, gsl_vector * work); /* Symmetric to symmetric tridiagonal decomposition */ int gsl_linalg_symmtd_decomp (gsl_matrix * A, gsl_vector * tau); int gsl_linalg_symmtd_unpack (const gsl_matrix * A, const gsl_vector * tau, gsl_matrix * Q, gsl_vector * diag, gsl_vector * subdiag); int gsl_linalg_symmtd_unpack_T (const gsl_matrix * A, gsl_vector * diag, gsl_vector * subdiag); /* Hermitian to symmetric tridiagonal decomposition */ int gsl_linalg_hermtd_decomp (gsl_matrix_complex * A, gsl_vector_complex * tau); int gsl_linalg_hermtd_unpack (const gsl_matrix_complex * A, const gsl_vector_complex * tau, gsl_matrix_complex * U, gsl_vector * diag, gsl_vector * sudiag); int gsl_linalg_hermtd_unpack_T (const gsl_matrix_complex * A, gsl_vector * diag, gsl_vector * subdiag); /* Linear Solve Using Householder Transformations * exceptions: */ int gsl_linalg_HH_solve (gsl_matrix * A, const gsl_vector * b, gsl_vector * x); int gsl_linalg_HH_svx (gsl_matrix * A, gsl_vector * x); /* Linear solve for a symmetric tridiagonal system. * The input vectors represent the NxN matrix as follows: * * diag[0] offdiag[0] 0 ... * offdiag[0] diag[1] offdiag[1] ... * 0 offdiag[1] diag[2] ... * 0 0 offdiag[2] ... * ... ... ... ... */ int gsl_linalg_solve_symm_tridiag (const gsl_vector * diag, const gsl_vector * offdiag, const gsl_vector * b, gsl_vector * x); /* Linear solve for a nonsymmetric tridiagonal system. * The input vectors represent the NxN matrix as follows: * * diag[0] abovediag[0] 0 ... * belowdiag[0] diag[1] abovediag[1] ... * 0 belowdiag[1] diag[2] ... * 0 0 belowdiag[2] ... * ... ... ... ... */ int gsl_linalg_solve_tridiag (const gsl_vector * diag, const gsl_vector * abovediag, const gsl_vector * belowdiag, const gsl_vector * b, gsl_vector * x); /* Linear solve for a symmetric cyclic tridiagonal system. * The input vectors represent the NxN matrix as follows: * * diag[0] offdiag[0] 0 ..... offdiag[N-1] * offdiag[0] diag[1] offdiag[1] ..... * 0 offdiag[1] diag[2] ..... * 0 0 offdiag[2] ..... * ... ... * offdiag[N-1] ... */ int gsl_linalg_solve_symm_cyc_tridiag (const gsl_vector * diag, const gsl_vector * offdiag, const gsl_vector * b, gsl_vector * x); /* Linear solve for a nonsymmetric cyclic tridiagonal system. * The input vectors represent the NxN matrix as follows: * * diag[0] abovediag[0] 0 ..... belowdiag[N-1] * belowdiag[0] diag[1] abovediag[1] ..... * 0 belowdiag[1] diag[2] * 0 0 belowdiag[2] ..... * ... ... * abovediag[N-1] ... */ int gsl_linalg_solve_cyc_tridiag (const gsl_vector * diag, const gsl_vector * abovediag, const gsl_vector * belowdiag, const gsl_vector * b, gsl_vector * x); /* Bidiagonal decomposition */ int gsl_linalg_bidiag_decomp (gsl_matrix * A, gsl_vector * tau_U, gsl_vector * tau_V); int gsl_linalg_bidiag_unpack (const gsl_matrix * A, const gsl_vector * tau_U, gsl_matrix * U, const gsl_vector * tau_V, gsl_matrix * V, gsl_vector * diag, gsl_vector * superdiag); int gsl_linalg_bidiag_unpack2 (gsl_matrix * A, gsl_vector * tau_U, gsl_vector * tau_V, gsl_matrix * V); int gsl_linalg_bidiag_unpack_B (const gsl_matrix * A, gsl_vector * diag, gsl_vector * superdiag); /* Balancing */ int gsl_linalg_balance_matrix (gsl_matrix * A, gsl_vector * D); int gsl_linalg_balance_accum (gsl_matrix * A, gsl_vector * D); int gsl_linalg_balance_columns (gsl_matrix * A, gsl_vector * D); /* condition estimation */ int gsl_linalg_tri_rcond(CBLAS_UPLO_t Uplo, const gsl_matrix * A, double * rcond, gsl_vector * work); int gsl_linalg_tri_upper_rcond(const gsl_matrix * A, double * rcond, gsl_vector * work); int gsl_linalg_tri_lower_rcond(const gsl_matrix * A, double * rcond, gsl_vector * work); int gsl_linalg_invnorm1(const size_t N, int (* Ainvx)(CBLAS_TRANSPOSE_t TransA, gsl_vector * x, void * params), void * params, double * Ainvnorm, gsl_vector * work); /* triangular matrices */ int gsl_linalg_tri_upper_invert(gsl_matrix * T); int gsl_linalg_tri_lower_invert(gsl_matrix * T); int gsl_linalg_tri_upper_unit_invert(gsl_matrix * T); int gsl_linalg_tri_lower_unit_invert(gsl_matrix * T); int gsl_linalg_tri_invert(CBLAS_UPLO_t Uplo, CBLAS_DIAG_t Diag, gsl_matrix * T); int gsl_linalg_complex_tri_invert(CBLAS_UPLO_t Uplo, CBLAS_DIAG_t Diag, gsl_matrix_complex * T); int gsl_linalg_tri_LTL(gsl_matrix * L); int gsl_linalg_tri_UL(gsl_matrix * LU); int gsl_linalg_complex_tri_LHL(gsl_matrix_complex * L); int gsl_linalg_complex_tri_UL(gsl_matrix_complex * LU); INLINE_DECL void gsl_linalg_givens (const double a, const double b, double *c, double *s); INLINE_DECL void gsl_linalg_givens_gv (gsl_vector * v, const size_t i, const size_t j, const double c, const double s); #ifdef HAVE_INLINE /* Generate a Givens rotation (cos,sin) which takes v=(x,y) to (|v|,0) From Golub and Van Loan, "Matrix Computations", Section 5.1.8 */ INLINE_FUN void gsl_linalg_givens (const double a, const double b, double *c, double *s) { if (b == 0) { *c = 1; *s = 0; } else if (fabs (b) > fabs (a)) { double t = -a / b; double s1 = 1.0 / sqrt (1 + t * t); *s = s1; *c = s1 * t; } else { double t = -b / a; double c1 = 1.0 / sqrt (1 + t * t); *c = c1; *s = c1 * t; } } /* gsl_linalg_givens() */ INLINE_FUN void gsl_linalg_givens_gv (gsl_vector * v, const size_t i, const size_t j, const double c, const double s) { /* Apply rotation to vector v' = G^T v */ double vi = gsl_vector_get (v, i); double vj = gsl_vector_get (v, j); gsl_vector_set (v, i, c * vi - s * vj); gsl_vector_set (v, j, s * vi + c * vj); } #endif /* HAVE_INLINE */ __END_DECLS #endif /* __GSL_LINALG_H__ */