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+/* linalg/gsl_linalg.h
+ *
+ * Copyright (C) 1996, 1997, 1998, 1999, 2000, 2006, 2007 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>
+
+#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);
+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_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);
+
+/* 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);
+
+/* To support gsl-1.9 interface: DEPRECATED */
+int gsl_linalg_hessenberg(gsl_matrix *A, 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 * residual);
+
+int gsl_linalg_LU_invert (const gsl_matrix * LU,
+ const gsl_permutation * p,
+ gsl_matrix * inverse);
+
+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);
+
+/* 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 * residual);
+
+int gsl_linalg_complex_LU_invert (const gsl_matrix_complex * LU,
+ const gsl_permutation * p,
+ gsl_matrix_complex * inverse);
+
+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_solve (const gsl_matrix * QR,
+ const gsl_vector * tau,
+ 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_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_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_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_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);
+
+
+/* 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_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);
+
+/* LQ decomposition */
+
+int gsl_linalg_LQ_decomp (gsl_matrix * A, gsl_vector * tau);
+
+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_solve (const gsl_matrix * cholesky,
+ const gsl_vector * b,
+ gsl_vector * x);
+
+int gsl_linalg_cholesky_svx (const gsl_matrix * cholesky,
+ gsl_vector * 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);
+
+/* 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);
+
+
+/* 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);
+
+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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