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+/* glpmat.h (linear algebra routines) */
+
+/***********************************************************************
+*  This code is part of GLPK (GNU Linear Programming Kit).
+*
+*  Copyright (C) 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008,
+*  2009, 2010, 2011, 2013 Andrew Makhorin, Department for Applied
+*  Informatics, Moscow Aviation Institute, Moscow, Russia. All rights
+*  reserved. E-mail: <mao@gnu.org>.
+*
+*  GLPK 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.
+*
+*  GLPK 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 GLPK. If not, see <http://www.gnu.org/licenses/>.
+***********************************************************************/
+
+#ifndef GLPMAT_H
+#define GLPMAT_H
+
+/***********************************************************************
+*  FULL-VECTOR STORAGE
+*
+*  For a sparse vector x having n elements, ne of which are non-zero,
+*  the full-vector storage format uses two arrays x_ind and x_vec, which
+*  are set up as follows:
+*
+*  x_ind is an integer array of length [1+ne]. Location x_ind[0] is
+*  not used, and locations x_ind[1], ..., x_ind[ne] contain indices of
+*  non-zero elements in vector x.
+*
+*  x_vec is a floating-point array of length [1+n]. Location x_vec[0]
+*  is not used, and locations x_vec[1], ..., x_vec[n] contain numeric
+*  values of ALL elements in vector x, including its zero elements.
+*
+*  Let, for example, the following sparse vector x be given:
+*
+*     (0, 1, 0, 0, 2, 3, 0, 4)
+*
+*  Then the arrays are:
+*
+*     x_ind = { X; 2, 5, 6, 8 }
+*
+*     x_vec = { X; 0, 1, 0, 0, 2, 3, 0, 4 }
+*
+*  COMPRESSED-VECTOR STORAGE
+*
+*  For a sparse vector x having n elements, ne of which are non-zero,
+*  the compressed-vector storage format uses two arrays x_ind and x_vec,
+*  which are set up as follows:
+*
+*  x_ind is an integer array of length [1+ne]. Location x_ind[0] is
+*  not used, and locations x_ind[1], ..., x_ind[ne] contain indices of
+*  non-zero elements in vector x.
+*
+*  x_vec is a floating-point array of length [1+ne]. Location x_vec[0]
+*  is not used, and locations x_vec[1], ..., x_vec[ne] contain numeric
+*  values of corresponding non-zero elements in vector x.
+*
+*  Let, for example, the following sparse vector x be given:
+*
+*     (0, 1, 0, 0, 2, 3, 0, 4)
+*
+*  Then the arrays are:
+*
+*     x_ind = { X; 2, 5, 6, 8 }
+*
+*     x_vec = { X; 1, 2, 3, 4 }
+*
+*  STORAGE-BY-ROWS
+*
+*  For a sparse matrix A, which has m rows, n columns, and ne non-zero
+*  elements the storage-by-rows format uses three arrays A_ptr, A_ind,
+*  and A_val, which are set up as follows:
+*
+*  A_ptr is an integer array of length [1+m+1] also called "row pointer
+*  array". It contains the relative starting positions of each row of A
+*  in the arrays A_ind and A_val, i.e. element A_ptr[i], 1 <= i <= m,
+*  indicates where row i begins in the arrays A_ind and A_val. If all
+*  elements in row i are zero, then A_ptr[i] = A_ptr[i+1]. Location
+*  A_ptr[0] is not used, location A_ptr[1] must contain 1, and location
+*  A_ptr[m+1] must contain ne+1 that indicates the position after the
+*  last element in the arrays A_ind and A_val.
+*
+*  A_ind is an integer array of length [1+ne]. Location A_ind[0] is not
+*  used, and locations A_ind[1], ..., A_ind[ne] contain column indices
+*  of (non-zero) elements in matrix A.
+*
+*  A_val is a floating-point array of length [1+ne]. Location A_val[0]
+*  is not used, and locations A_val[1], ..., A_val[ne] contain numeric
+*  values of non-zero elements in matrix A.
+*
+*  Non-zero elements of matrix A are stored contiguously, and the rows
+*  of matrix A are stored consecutively from 1 to m in the arrays A_ind
+*  and A_val. The elements in each row of A may be stored in any order
+*  in A_ind and A_val. Note that elements with duplicate column indices
+*  are not allowed.
+*
+*  Let, for example, the following sparse matrix A be given:
+*
+*     | 11  . 13  .  .  . |
+*     | 21 22  . 24  .  . |
+*     |  . 32 33  .  .  . |
+*     |  .  . 43 44  . 46 |
+*     |  .  .  .  .  .  . |
+*     | 61 62  .  .  . 66 |
+*
+*  Then the arrays are:
+*
+*     A_ptr = { X; 1, 3, 6, 8, 11, 11; 14 }
+*
+*     A_ind = { X;  1,  3;  4,  2,  1;  2,  3;  4,  3,  6;  1,  2,  6 }
+*
+*     A_val = { X; 11, 13; 24, 22, 21; 32, 33; 44, 43, 46; 61, 62, 66 }
+*
+*  PERMUTATION MATRICES
+*
+*  Let P be a permutation matrix of the order n. It is represented as
+*  an integer array P_per of length [1+n+n] as follows: if p[i,j] = 1,
+*  then P_per[i] = j and P_per[n+j] = i. Location P_per[0] is not used.
+*
+*  Let A' = P*A. If i-th row of A corresponds to i'-th row of A', then
+*  P_per[i'] = i and P_per[n+i] = i'.
+*
+*  References:
+*
+*  1. Gustavson F.G. Some basic techniques for solving sparse systems of
+*     linear equations. In Rose and Willoughby (1972), pp. 41-52.
+*
+*  2. Basic Linear Algebra Subprograms Technical (BLAST) Forum Standard.
+*     University of Tennessee (2001). */
+
+#define check_fvs _glp_mat_check_fvs
+int check_fvs(int n, int nnz, int ind[], double vec[]);
+/* check sparse vector in full-vector storage format */
+
+#define check_pattern _glp_mat_check_pattern
+int check_pattern(int m, int n, int A_ptr[], int A_ind[]);
+/* check pattern of sparse matrix */
+
+#define transpose _glp_mat_transpose
+void transpose(int m, int n, int A_ptr[], int A_ind[], double A_val[],
+      int AT_ptr[], int AT_ind[], double AT_val[]);
+/* transpose sparse matrix */
+
+#define adat_symbolic _glp_mat_adat_symbolic
+int *adat_symbolic(int m, int n, int P_per[], int A_ptr[], int A_ind[],
+      int S_ptr[]);
+/* compute S = P*A*D*A'*P' (symbolic phase) */
+
+#define adat_numeric _glp_mat_adat_numeric
+void adat_numeric(int m, int n, int P_per[],
+      int A_ptr[], int A_ind[], double A_val[], double D_diag[],
+      int S_ptr[], int S_ind[], double S_val[], double S_diag[]);
+/* compute S = P*A*D*A'*P' (numeric phase) */
+
+#define min_degree _glp_mat_min_degree
+void min_degree(int n, int A_ptr[], int A_ind[], int P_per[]);
+/* minimum degree ordering */
+
+#define amd_order1 _glp_mat_amd_order1
+void amd_order1(int n, int A_ptr[], int A_ind[], int P_per[]);
+/* approximate minimum degree ordering (AMD) */
+
+#define symamd_ord _glp_mat_symamd_ord
+void symamd_ord(int n, int A_ptr[], int A_ind[], int P_per[]);
+/* approximate minimum degree ordering (SYMAMD) */
+
+#define chol_symbolic _glp_mat_chol_symbolic
+int *chol_symbolic(int n, int A_ptr[], int A_ind[], int U_ptr[]);
+/* compute Cholesky factorization (symbolic phase) */
+
+#define chol_numeric _glp_mat_chol_numeric
+int chol_numeric(int n,
+      int A_ptr[], int A_ind[], double A_val[], double A_diag[],
+      int U_ptr[], int U_ind[], double U_val[], double U_diag[]);
+/* compute Cholesky factorization (numeric phase) */
+
+#define u_solve _glp_mat_u_solve
+void u_solve(int n, int U_ptr[], int U_ind[], double U_val[],
+      double U_diag[], double x[]);
+/* solve upper triangular system U*x = b */
+
+#define ut_solve _glp_mat_ut_solve
+void ut_solve(int n, int U_ptr[], int U_ind[], double U_val[],
+      double U_diag[], double x[]);
+/* solve lower triangular system U'*x = b */
+
+#endif
+
+/* eof */