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DataPlayer/ChaosDataPlayer/GSL/include/gsl/gsl_linalg.h

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2022-07-30 11:50:01 +08:00
/* 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 <stdlib.h>
#include <gsl/gsl_mode.h>
#include <gsl/gsl_permutation.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_inline.h>
#include <gsl/gsl_blas.h>
#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__ */
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