la.c

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/******************************************************************************
 * la.c
 * wrapper modules for linear algebra problems
 * linking to BLAS / LAPACK (and others?)
 
 * @Copyright David D.Gray <ddgray@armadce.demon.co.uk>
 * 26th. Sep. 2000
 * Last updated:
 * 2006-11-23
 * 2015-01-20
 
 * This file is part of GRASS GIS. It 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 2 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.
 
 ******************************************************************************/
 
#include <grass/config.h>
 
#if defined(HAVE_LIBBLAS) && defined(HAVE_LIBLAPACK)
 
#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
 
#include <lapacke.h>
#if defined(HAVE_CBLAS_ATLAS_H)
#include <cblas-atlas.h>
#else
#include <cblas.h>
#endif // HAVE_CBLAS_ATLAS_H
 
#include <grass/gis.h>
#include <grass/glocale.h>
#include <grass/la.h>
 
static int egcmp(const void *pa, const void *pb);
 
/*!
 * \fn mat_struct *G_matrix_init(int rows, int cols, int ldim)
 *
 * \brief Initialize a matrix structure
 *
 * Initialize a matrix structure
 *
 * \param rows
 * \param cols
 * \param ldim
 * \return mat_struct
 */
 
mat_struct *G_matrix_init(int rows, int cols, int ldim)
{
    mat_struct *tmp_arry;
 
    if (rows < 1 || cols < 1 || ldim < rows) {
        G_warning(_("Matrix dimensions out of range"));
        return NULL;
    }
 
    tmp_arry = (mat_struct *)G_malloc(sizeof(mat_struct));
    tmp_arry->rows = rows;
    tmp_arry->cols = cols;
    tmp_arry->ldim = ldim;
    tmp_arry->type = MATRIX_;
    tmp_arry->v_indx = -1;
 
    tmp_arry->vals = (double *)G_calloc(ldim * cols, sizeof(double));
    tmp_arry->is_init = 1;
 
    return tmp_arry;
}
 
/*!
 * \fn int G_matrix_zero (mat_struct *A)
 *
 * \brief Clears (or resets) the matrix values to 0
 *
 * \param A
 * \return 0 on error; 1 on success
 */
 
int G_matrix_zero(mat_struct *A)
{
    if (!A->vals)
        return 0;
 
    memset(A->vals, 0, (A->ldim * A->cols) * sizeof(double));
 
    return 1;
}
 
/*!
 * \fn int G_matrix_set(mat_struct *A, int rows, int cols, int ldim)
 *
 * \brief Set parameters for an initialized matrix
 *
 * Set parameters for matrix <b>A</b> that is allocated,
 * but not yet fully initialized.  Is an alternative to G_matrix_init().
 *
 * \param A
 * \param rows
 * \param cols
 * \param ldim
 * \return int
 */
 
int G_matrix_set(mat_struct *A, int rows, int cols, int ldim)
{
    if (rows < 1 || cols < 1 || ldim < 0) {
        G_warning(_("Matrix dimensions out of range"));
        return -1;
    }
 
    A->rows = rows;
    A->cols = cols;
    A->ldim = ldim;
    A->type = MATRIX_;
    A->v_indx = -1;
 
    A->vals = (double *)G_calloc(ldim * cols, sizeof(double));
    A->is_init = 1;
 
    return 0;
}
 
/*!
 * \fn mat_struct *G_matrix_copy (const mat_struct *A)
 *
 * \brief Copy a matrix
 *
 * Copy matrix <b>A</b> by exactly duplicating its contents.
 *
 * \param A
 * \return mat_struct
 */
 
mat_struct *G_matrix_copy(const mat_struct *A)
{
    mat_struct *B;
 
    if (!A->is_init) {
        G_warning(_("Matrix is not initialised fully."));
        return NULL;
    }
 
    if ((B = G_matrix_init(A->rows, A->cols, A->ldim)) == NULL) {
        G_warning(_("Unable to allocate space for matrix copy"));
        return NULL;
    }
 
    memcpy(&B->vals[0], &A->vals[0],
           (size_t)A->cols * A->ldim * sizeof(double));
 
    return B;
}
 
/*!
 * \fn mat_struct *G_matrix_add (mat_struct *mt1, mat_struct *mt2)
 *
 * \brief Adds two matricies
 *
 *  Adds two matricies <b>mt1</b> and <b>mt2</b> and returns a
 * resulting matrix. The return structure is automatically initialized.
 *
 * \param mt1
 * \param mt2
 * \return mat_struct
 */
 
mat_struct *G_matrix_add(mat_struct *mt1, mat_struct *mt2)
{
    return G__matrix_add(mt1, mt2, 1, 1);
}
 
/*!
 * \fn mat_struct *G_matrix_subtract (mat_struct *mt1, mat_struct *mt2)
 *
 * \brief Subtract two matricies
 *
 * Subtracts two matricies <b>mt1</b> and <b>mt2</b> and returns
 * a resulting matrix. The return matrix is automatically initialized.
 *
 * \param mt1
 * \param mt2
 * \return mat_struct
 */
 
mat_struct *G_matrix_subtract(mat_struct *mt1, mat_struct *mt2)
{
    return G__matrix_add(mt1, mt2, 1, -1);
}
 
/*!
 * \fn mat_struct *G_matrix_scalar_mul(double scalar, mat_struct *matrix,
 * mat_struct *out)
 *
 * \brief Calculates the scalar-matrix multiplication
 *
 * Calculates the scalar-matrix multiplication
 *
 * \param scalar
 * \param A
 * \return mat_struct
 */
 
mat_struct *G_matrix_scalar_mul(double scalar, mat_struct *matrix,
                                mat_struct *out)
{
    int m, n, i, j;
 
    if (matrix == NULL) {
        G_warning(_("Input matrix is uninitialized"));
        return NULL;
    }
 
    if (out == NULL)
        out = G_matrix_init(matrix->rows, matrix->cols, matrix->rows);
 
    if (out->rows != matrix->rows || out->cols != matrix->cols)
        out = G_matrix_resize(out, matrix->rows, matrix->cols);
 
    m = matrix->rows;
    n = matrix->cols;
 
    for (i = 0; i < m; i++) {
        for (j = 0; j < n; j++) {
            double value = scalar * G_matrix_get_element(matrix, i, j);
 
            G_matrix_set_element(out, i, j, value);
        }
    }
 
    return (out);
}
 
/*!
 * \fn mat_struct *G_matrix_scale (mat_struct *mt1, const double c)
 *
 * \brief Scale a matrix by a scalar value
 *
 * Scales matrix <b>mt1</b> by scalar value <b>c</b>. The
 * resulting matrix is automatically initialized.
 *
 * \param mt1
 * \param c
 * \return mat_struct
 */
 
mat_struct *G_matrix_scale(mat_struct *mt1, const double c)
{
    return G__matrix_add(mt1, NULL, c, 0);
}
 
/*!
 * \fn mat_struct *G__matrix_add (mat_struct *mt1, mat_struct *mt2, const double
 * c1, const double c2)
 *
 * \brief General add/subtract/scalar multiply routine
 *
 * General add/subtract/scalar multiply routine. <b>c2</b> may be
 * zero, but <b>c1</b> must be non-zero.
 *
 * \param mt1
 * \param mt2
 * \param c1
 * \param c2
 * \return mat_struct
 */
 
mat_struct *G__matrix_add(mat_struct *mt1, mat_struct *mt2, const double c1,
                          const double c2)
{
    mat_struct *mt3;
    int i, j; /* loop variables */
 
    if (c1 == 0) {
        G_warning(_("First scalar multiplier must be non-zero"));
        return NULL;
    }
 
    if (c2 == 0) {
        if (!mt1->is_init) {
            G_warning(_("One or both input matrices uninitialised"));
            return NULL;
        }
    }
 
    else {
 
        if (!((mt1->is_init) && (mt2->is_init))) {
            G_warning(_("One or both input matrices uninitialised"));
            return NULL;
        }
 
        if (mt1->rows != mt2->rows || mt1->cols != mt2->cols) {
            G_warning(_("Matrix order does not match"));
            return NULL;
        }
    }
 
    if ((mt3 = G_matrix_init(mt1->rows, mt1->cols, mt1->ldim)) == NULL) {
        G_warning(_("Unable to allocate space for matrix sum"));
        return NULL;
    }
 
    if (c2 == 0) {
 
        for (i = 0; i < mt3->rows; i++) {
            for (j = 0; j < mt3->cols; j++) {
                mt3->vals[i + mt3->ldim * j] =
                    c1 * mt1->vals[i + mt1->ldim * j];
            }
        }
    }
 
    else {
 
        for (i = 0; i < mt3->rows; i++) {
            for (j = 0; j < mt3->cols; j++) {
                mt3->vals[i + mt3->ldim * j] =
                    c1 * mt1->vals[i + mt1->ldim * j] +
                    c2 * mt2->vals[i + mt2->ldim * j];
            }
        }
    }
 
    return mt3;
}
 
/*!
 * \fn mat_struct *G_matrix_product (mat_struct *mt1, mat_struct *mt2)
 *
 * \brief Returns product of two matricies
 *
 *  Returns a matrix with the product of matrix <b>mt1</b> and
 * <b>mt2</b>. The return matrix is automatically initialized.
 *
 * \param mt1
 * \param mt2
 * \return mat_struct
 */
 
mat_struct *G_matrix_product(mat_struct *mt1, mat_struct *mt2)
{
    mat_struct *mt3;
    double unity = 1., zero = 0.;
    int rows, cols, interdim, lda, ldb;
 
    if (!((mt1->is_init) || (mt2->is_init))) {
        G_warning(_("One or both input matrices uninitialised"));
        return NULL;
    }
 
    if (mt1->cols != mt2->rows) {
        G_warning(_("Matrix order does not match"));
        return NULL;
    }
 
    if ((mt3 = G_matrix_init(mt1->rows, mt2->cols, mt1->ldim)) == NULL) {
        G_warning(_("Unable to allocate space for matrix product"));
        return NULL;
    }
 
    /* Call the driver */
 
    rows = (int)mt1->rows;
    interdim = (int)mt1->cols;
    cols = (int)mt2->cols;
 
    lda = (int)mt1->ldim;
    ldb = (int)mt2->ldim;
 
    cblas_dgemm(CblasColMajor, CblasNoTrans, CblasNoTrans, rows, cols, interdim,
                unity, mt1->vals, lda, mt2->vals, ldb, zero, mt3->vals, lda);
 
    return mt3;
}
 
/*!
 * \fn mat_struct *G_matrix_transpose (mat_struct *mt)
 *
 * \brief Transpose a matrix
 *
 * Transpose matrix <b>m1</b> by creating a new one and
 * populating with transposed elements. The return matrix is
 * automatically initialized.
 *
 * \param mt
 * \return mat_struct
 */
 
mat_struct *G_matrix_transpose(mat_struct *mt)
{
    mat_struct *mt1;
    int ldim, ldo;
    double *dbo, *dbt, *dbx, *dby;
    int cnt, cnt2;
 
    /* Word align the workspace blocks */
    if (mt->cols % 2 == 0)
        ldim = mt->cols;
    else
        ldim = mt->cols + 1;
 
    mt1 = G_matrix_init(mt->cols, mt->rows, ldim);
 
    /* Set initial values for reading arrays */
    dbo = &mt->vals[0];
    dbt = &mt1->vals[0];
    ldo = mt->ldim;
 
    for (cnt = 0; cnt < mt->cols; cnt++) {
        dbx = dbo;
        dby = dbt;
 
        for (cnt2 = 0; cnt2 < ldo - 1; cnt2++) {
            *dby = *dbx;
            dby += ldim;
            dbx++;
        }
 
        *dby = *dbx;
 
        if (cnt < mt->cols - 1) {
            dbo += ldo;
            dbt++;
        }
    }
 
    return mt1;
}
 
/*!
 * \fn int G_matrix_LU_solve (const mat_struct *mt1, mat_struct **xmat0,
 *                      const mat_struct *bmat, mat_type mtype)
 *
 * \brief Solve a general system A.X = B
 *
 * Solve a general system A.X = B, where A is a NxN matrix, X and B are
 * NxC matrices, and we are to solve for C arrays in X given B. Uses LU
 * decomposition.<br>
 * Links to LAPACK function dgesv_() and similar to perform the core routine.
 * (By default solves for a general non-symmetric matrix.)<br>
 * mtype is a flag to indicate what kind of matrix (real/complex, Hermitian,
 * symmetric, general etc.) is used (NONSYM, SYM, HERMITIAN).<br>
 * <b>Warning:</b> NOT YET COMPLETE: only some solutions' options
 * available. Now, only general real matrix is supported.
 *
 *  \param mt1
 *  \param xmat0
 *  \param bmat
 *  \param mtype
 *  \return int
 */
 
/*** NOT YET COMPLETE: only some solutions' options available ***/
 
int G_matrix_LU_solve(const mat_struct *mt1, mat_struct **xmat0,
                      const mat_struct *bmat, mat_type mtype)
{
    mat_struct *wmat, *xmat, *mtx;
 
    if (mt1->is_init == 0 || bmat->is_init == 0) {
        G_warning(_("Input: one or both data matrices uninitialised"));
        return -1;
    }
 
    if (mt1->rows != mt1->cols || mt1->rows < 1) {
        G_warning(_("Principal matrix is not properly dimensioned"));
        return -1;
    }
 
    if (bmat->cols < 1) {
        G_warning(_("Input: you must have at least one array to solve"));
        return -1;
    }
 
    /* Now create solution matrix by copying the original coefficient matrix */
    if ((xmat = G_matrix_copy(bmat)) == NULL) {
        G_warning(_("Could not allocate space for solution matrix"));
        return -1;
    }
 
    /* Create working matrix for the coefficient array */
    if ((mtx = G_matrix_copy(mt1)) == NULL) {
        G_warning(_("Could not allocate space for working matrix"));
        return -1;
    }
 
    /* Copy the contents of the data matrix, to preserve the
       original information
     */
    if ((wmat = G_matrix_copy(bmat)) == NULL) {
        G_warning(_("Could not allocate space for working matrix"));
        return -1;
    }
 
    /* Now call appropriate LA driver to solve equations */
    switch (mtype) {
 
    case NONSYM: {
        int *perm, res_info;
        int num_eqns, nrhs, lda, ldb;
 
        perm = (int *)G_malloc(wmat->rows * sizeof(int));
 
        /* Set fields to pass to fortran routine */
        num_eqns = (int)mt1->rows;
        nrhs = (int)wmat->cols;
        lda = (int)mt1->ldim;
        ldb = (int)wmat->ldim;
 
        /* Call LA driver */
        res_info = LAPACKE_dgesv(LAPACK_COL_MAJOR, num_eqns, nrhs, mtx->vals,
                                 lda, perm, wmat->vals, ldb);
 
        /* Copy the results from the modified data matrix, taking account
           of pivot permutations ???
         */
 
        /*
           for(indx1 = 0; indx1 < num_eqns; indx1++) {
           iperm = perm[indx1];
           ptin = &wmat->vals[0] + indx1;
           ptout = &xmat->vals[0] + iperm;
 
           for(indx2 = 0; indx2 < nrhs - 1; indx2++) {
           *ptout = *ptin;
           ptin += wmat->ldim;
           ptout += xmat->ldim;
           }
 
           *ptout = *ptin;
           }
         */
 
        memcpy(xmat->vals, wmat->vals,
               (size_t)wmat->cols * wmat->ldim * sizeof(double));
 
        /* Free temp arrays */
        G_free(perm);
        G_matrix_free(wmat);
        G_matrix_free(mtx);
 
        if (res_info > 0) {
            G_warning(
                _("Matrix (or submatrix is singular). Solution undetermined"));
            return 1;
        }
        else if (res_info < 0) {
            G_warning(_("Problem in LA routine."));
            return -1;
        }
        break;
    }
    default: {
        G_warning(_("Procedure not yet available for selected matrix type"));
        return -1;
    }
    } /* end switch */
 
    *xmat0 = xmat;
 
    return 0;
}
 
/*!
 * \fn mat_struct *G_matrix_inverse (mat_struct *mt)
 *
 * \brief Returns the matrix inverse
 *
 * Calls G_matrix_LU_solve() to obtain matrix inverse using LU
 * decomposition. Returns NULL on failure.
 *
 * \param mt
 * \return mat_struct
 */
 
mat_struct *G_matrix_inverse(mat_struct *mt)
{
    mat_struct *mt0, *res;
    int i, j, k; /* loop */
 
    if (mt->rows != mt->cols) {
        G_warning(_("Matrix is not square. Cannot determine inverse"));
        return NULL;
    }
 
    if ((mt0 = G_matrix_init(mt->rows, mt->rows, mt->ldim)) == NULL) {
        G_warning(_("Unable to allocate space for matrix"));
        return NULL;
    }
 
    /* Set `B' matrix to unit matrix */
    for (i = 0; i < mt0->rows - 1; i++) {
        mt0->vals[i + i * mt0->ldim] = 1.0;
 
        for (j = i + 1; j < mt0->cols; j++) {
            mt0->vals[i + j * mt0->ldim] = mt0->vals[j + i * mt0->ldim] = 0.0;
        }
    }
 
    mt0->vals[mt0->rows - 1 + (mt0->rows - 1) * mt0->ldim] = 1.0;
 
    /* Solve system */
    if ((k = G_matrix_LU_solve(mt, &res, mt0, NONSYM)) == 1) {
        G_warning(_("Matrix is singular"));
        G_matrix_free(mt0);
        return NULL;
    }
    else if (k < 0) {
        G_warning(_("Problem in LA procedure."));
        G_matrix_free(mt0);
        return NULL;
    }
    else {
        G_matrix_free(mt0);
        return res;
    }
}
 
/*!
 * \fn void G_matrix_free (mat_struct *mt)
 *
 * \brief Free up allocated matrix
 *
 * Free up allocated matrix.
 *
 * \param mt
 * \return void
 */
 
void G_matrix_free(mat_struct *mt)
{
    if (mt->is_init)
        G_free(mt->vals);
 
    G_free(mt);
}
 
/*!
 * \fn void G_matrix_print (mat_struct *mt)
 *
 * \brief Print out a matrix
 *
 * Print out a representation of the matrix to standard output.
 *
 *  \param mt
 *  \return void
 */
 
void G_matrix_print(mat_struct *mt)
{
    int i, j;
    char buf[2048], numbuf[64];
 
    for (i = 0; i < mt->rows; i++) {
        G_strlcpy(buf, "", sizeof(buf));
 
        for (j = 0; j < mt->cols; j++) {
            snprintf(numbuf, sizeof(numbuf), "%14.6f",
                     G_matrix_get_element(mt, i, j));
            strcat(buf, numbuf);
            if (j < mt->cols - 1)
                strcat(buf, ", ");
        }
 
        G_message("%s", buf);
    }
 
    fprintf(stderr, "\n");
}
 
/*!
 * \fn int G_matrix_set_element (mat_struct *mt, int rowval, int colval, double
 * val)
 *
 * \brief Set the value of the (i, j)th element
 *
 * Set the value of the (i, j)th element to a double value. Index values
 * are C-like ie. zero-based. The row number is given first as is
 * conventional. Returns -1 if the accessed cell is outside the bounds.
 *
 *  \param mt
 *  \param rowval
 *  \param colval
 *  \param val
 *  \return int
 */
 
int G_matrix_set_element(mat_struct *mt, int rowval, int colval, double val)
{
    if (!mt->is_init) {
        G_warning(_("Element array has not been allocated"));
        return -1;
    }
 
    if (rowval >= mt->rows || colval >= mt->cols || rowval < 0 || colval < 0) {
        G_warning(_("Specified element is outside array bounds"));
        return -1;
    }
 
    mt->vals[rowval + colval * mt->ldim] = (double)val;
 
    return 0;
}
 
/*!
 * \fn double G_matrix_get_element (mat_struct *mt, int rowval, int colval)
 *
 * \brief Retrieve value of the (i,j)th element
 *
 * Retrieve the value of the (i, j)th element to a double value. Index
 * values are C-like ie. zero-based.
 * <b>Note:</b> Does currently not set an error flag for bounds checking.
 *
 *  \param mt
 *  \param rowval
 *  \param colval
 *  \return double
 */
 
double G_matrix_get_element(mat_struct *mt, int rowval, int colval)
{
    double val;
 
    /* Should do some checks, but this would require an error control
       system: later? */
 
    return (val = (double)mt->vals[rowval + colval * mt->ldim]);
}
 
/*!
 * \fn vec_struct *G_matvect_get_column (mat_struct *mt, int col)
 *
 * \brief Retrieve a column of the matrix to a vector structure
 *
 * Retrieve a column of matrix <b>mt</b> to a returning vector structure
 *
 * \param mt
 * \param col
 * \return vec_struct
 */
 
vec_struct *G_matvect_get_column(mat_struct *mt, int col)
{
    int i; /* loop */
    vec_struct *vc1;
 
    if (col < 0 || col >= mt->cols) {
        G_warning(_("Specified matrix column index is outside range"));
        return NULL;
    }
 
    if (!mt->is_init) {
        G_warning(_("Matrix is not initialised"));
        return NULL;
    }
 
    if ((vc1 = G_vector_init(mt->rows, mt->ldim, CVEC)) == NULL) {
        G_warning(_("Could not allocate space for vector structure"));
        return NULL;
    }
 
    for (i = 0; i < mt->rows; i++)
        G_matrix_set_element((mat_struct *)vc1, i, 0,
                             G_matrix_get_element(mt, i, col));
 
    return vc1;
}
 
/*!
 * \fn vec_struct *G_matvect_get_row (mat_struct *mt, int row)
 *
 * \brief Retrieve a row of the matrix to a vector structure
 *
 * Retrieves a row from matrix <b>mt</b> and returns it in a vector
 * structure.
 *
 * \param mt
 * \param row
 * \return vec_struct
 */
 
vec_struct *G_matvect_get_row(mat_struct *mt, int row)
{
    int i; /* loop */
    vec_struct *vc1;
 
    if (row < 0 || row >= mt->cols) {
        G_warning(_("Specified matrix row index is outside range"));
        return NULL;
    }
 
    if (!mt->is_init) {
        G_warning(_("Matrix is not initialised"));
        return NULL;
    }
 
    if ((vc1 = G_vector_init(mt->cols, mt->ldim, RVEC)) == NULL) {
        G_warning(_("Could not allocate space for vector structure"));
        return NULL;
    }
 
    for (i = 0; i < mt->cols; i++)
        G_matrix_set_element((mat_struct *)vc1, 0, i,
                             G_matrix_get_element(mt, row, i));
 
    return vc1;
}
 
/*!
 * \fn int G_matvect_extract_vector (mat_struct *mt, vtype vt, int indx)
 *
 * \brief Convert matrix to vector
 *
 * Convert the matrix <b>mt</b> to a vector structure. The vtype,
 * <b>vt</b>, is RVEC or CVEC which specifies a row vector or column
 * vector. The index, <b>indx</b>, indicates the row/column number (zero based).
 *
 * \param mt
 * \param vt
 * \param indx
 * \return int
 */
 
int G_matvect_extract_vector(mat_struct *mt, vtype vt, int indx)
{
    if (vt == RVEC && indx >= mt->rows) {
        G_warning(_("Specified row index is outside range"));
        return -1;
    }
 
    else if (vt == CVEC && indx >= mt->cols) {
        G_warning(_("Specified column index is outside range"));
        return -1;
    }
 
    switch (vt) {
 
    case RVEC: {
        mt->type = ROWVEC_;
        mt->v_indx = indx;
        break;
    }
 
    case CVEC: {
        mt->type = COLVEC_;
        mt->v_indx = indx;
        break;
    }
 
    default: {
        G_warning(_("Unknown vector type."));
        return -1;
    }
    }
 
    return 0;
}
 
/*!
 * \fn int G_matvect_retrieve_matrix (vec_struct *vc)
 *
 * \brief Revert a vector to matrix
 *
 * Revert vector <b>vc</b> to a matrix.
 *
 *  \param vc
 *  \return int
 */
 
int G_matvect_retrieve_matrix(vec_struct *vc)
{
    /* We have to take the integrity of the vector structure
       largely on trust
     */
 
    vc->type = MATRIX_;
    vc->v_indx = -1;
 
    return 0;
}
 
/*!
 * \fn vec_struct *G_matvect_product(mat_struct *A, vec_struct *b, vec_struct
 * *out)
 *
 * \brief Calculates the matrix-vector product
 *
 * Calculates the product of a matrix and a vector
 *
 * \param A
 * \param b
 * \return vec_struct
 */
 
vec_struct *G_matvect_product(mat_struct *A, vec_struct *b, vec_struct *out)
{
    unsigned int i, m, n, j;
    register double sum;
 
    /* G_message("A=%d,%d,%d", A->cols, A->rows, A->ldim); */
    /* G_message("B=%d,%d,%d", b->cols, b->rows, b->ldim); */
    if (A->cols != b->cols) {
        G_warning(_("Input matrix and vector have differing dimensions1"));
 
        return NULL;
    }
    if (!out) {
        G_warning(_("Output vector is uninitialized"));
        return NULL;
    }
    /*  if (out->ldim != A->rows) { */
    /*    G_warning (_("Output vector has incorrect dimension")); */
    /*    exit(1); */
    /*    return NULL; */
    /*  } */
 
    m = A->rows;
    n = A->cols;
 
    for (i = 0; i < m; i++) {
        sum = 0.0;
        /* int width = A->rows; */
 
        for (j = 0; j < n; j++) {
 
            sum +=
                G_matrix_get_element(A, i, j) * G_matrix_get_element(b, 0, j);
            /*sum += A->vals[i + j * width] * b->vals[j]; */
            out->vals[i] = sum;
        }
    }
    return (out);
}
 
/*!
 * \fn vec_struct *G_vector_init (int cells, int ldim, vtype vt)
 *
 * \brief Initialize a vector structure
 *
 * Returns an initialized vector structure with <b>cell</b> cells,
 * of dimension <b>ldim</b>, and of type <b>vt</b>.
 *
 * \param cells
 * \param ldim
 * \param vt
 * \return vec_struct
 */
 
vec_struct *G_vector_init(int cells, int ldim, vtype vt)
{
    vec_struct *tmp_arry;
 
    if ((cells < 1) || (vt == RVEC && ldim < 1) ||
        (vt == CVEC && ldim < cells) || ldim < 0) {
        G_warning("Vector dimensions out of range.");
        return NULL;
    }
 
    tmp_arry = (vec_struct *)G_malloc(sizeof(vec_struct));
 
    if (vt == RVEC) {
        tmp_arry->rows = 1;
        tmp_arry->cols = cells;
        tmp_arry->ldim = ldim;
        tmp_arry->type = ROWVEC_;
    }
 
    else if (vt == CVEC) {
        tmp_arry->rows = cells;
        tmp_arry->cols = 1;
        tmp_arry->ldim = ldim;
        tmp_arry->type = COLVEC_;
    }
 
    tmp_arry->v_indx = 0;
 
    tmp_arry->vals = (double *)G_calloc(ldim * tmp_arry->cols, sizeof(double));
    tmp_arry->is_init = 1;
 
    return tmp_arry;
}
 
/*!
 * \fn void G_vector_free (vec_struct *v)
 *
 * \brief Free an allocated vector structure
 *
 * Free an allocated vector structure.
 *
 * \param v
 * \return void
 */
 
void G_vector_free(vec_struct *v)
{
    if (v->is_init)
        G_free(v->vals);
 
    G_free(v);
}
 
/*!
 * \fn vec_struct *G_vector_sub (vec_struct *v1, vec_struct *v2, vec_struct
 * *out)
 *
 * \brief Subtract two vectors
 *
 * Subtracts two vectors, <b>v1</b> and <b>v2</b>, and returns and
 * populates vector <b>out</b>.
 *
 * \param v1
 * \param v2
 * \param out
 * \return vec_struct
 */
 
vec_struct *G_vector_sub(vec_struct *v1, vec_struct *v2, vec_struct *out)
{
    int idx1, idx2, idx0;
    int i;
 
    if (!out->is_init) {
        G_warning(_("Output vector is uninitialized"));
        return NULL;
    }
 
    if (v1->type != v2->type) {
        G_warning(_("Vectors are not of the same type"));
        return NULL;
    }
 
    if (v1->type != out->type) {
        G_warning(_("Output vector is of incorrect type"));
        return NULL;
    }
 
    if (v1->type == MATRIX_) {
        G_warning(_("Matrices not allowed"));
        return NULL;
    }
 
    if ((v1->type == ROWVEC_ && v1->cols != v2->cols) ||
        (v1->type == COLVEC_ && v1->rows != v2->rows)) {
        G_warning(_("Vectors have differing dimensions"));
        return NULL;
    }
 
    if ((v1->type == ROWVEC_ && v1->cols != out->cols) ||
        (v1->type == COLVEC_ && v1->rows != out->rows)) {
        G_warning(_("Output vector has incorrect dimension"));
        return NULL;
    }
 
    idx1 = (v1->v_indx > 0) ? v1->v_indx : 0;
    idx2 = (v2->v_indx > 0) ? v2->v_indx : 0;
    idx0 = (out->v_indx > 0) ? out->v_indx : 0;
 
    if (v1->type == ROWVEC_) {
        for (i = 0; i < v1->cols; i++)
            G_matrix_set_element(out, idx0, i,
                                 G_matrix_get_element(v1, idx1, i) -
                                     G_matrix_get_element(v2, idx2, i));
    }
    else {
        for (i = 0; i < v1->rows; i++)
            G_matrix_set_element(out, i, idx0,
                                 G_matrix_get_element(v1, i, idx1) -
                                     G_matrix_get_element(v2, i, idx2));
    }
 
    return out;
}
 
/*!
 * \fn int G_vector_set (vec_struct *A, int cells, int ldim, vtype vt, int
 * vindx)
 *
 * \brief Set parameters for vector structure
 *
 * Set parameters for a vector structure that is
 * allocated but not yet initialised fully. The vtype is RVEC or
 * CVEC which specifies a row vector or column vector.
 *
 * \param A
 * \param cells
 * \param ldim
 * \param vt
 * \param vindx
 * \return int
 */
 
int G_vector_set(vec_struct *A, int cells, int ldim, vtype vt, int vindx)
{
    if ((cells < 1) || (vt == RVEC && ldim < 1) ||
        (vt == CVEC && ldim < cells) || ldim < 0) {
        G_warning(_("Vector dimensions out of range"));
        return -1;
    }
 
    if ((vt == RVEC && vindx >= A->cols) || (vt == CVEC && vindx >= A->rows)) {
        G_warning(_("Row/column out of range"));
        return -1;
    }
 
    if (vt == RVEC) {
        A->rows = 1;
        A->cols = cells;
        A->ldim = ldim;
        A->type = ROWVEC_;
    }
    else {
        A->rows = cells;
        A->cols = 1;
        A->ldim = ldim;
        A->type = COLVEC_;
    }
 
    if (vindx < 0)
        A->v_indx = 0;
    else
        A->v_indx = vindx;
 
    A->vals = (double *)G_calloc(ldim * A->cols, sizeof(double));
    A->is_init = 1;
 
    return 0;
}
 
/*!
 * \fn double G_vector_norm_euclid (vec_struct *vc)
 *
 * \brief Calculates euclidean norm
 *
 * Calculates the euclidean norm of a row or column vector, using BLAS
 * routine dnrm2_().
 *
 * \param vc
 * \return double
 */
 
double G_vector_norm_euclid(vec_struct *vc)
{
    int incr, Nval;
    double *startpt;
 
    if (!vc->is_init)
        G_fatal_error(_("Matrix is not initialised"));
 
    if (vc->type == ROWVEC_) {
        Nval = (int)vc->cols;
        incr = (int)vc->ldim;
        if (vc->v_indx < 0)
            startpt = vc->vals;
        else
            startpt = vc->vals + vc->v_indx;
    }
    else {
        Nval = (int)vc->rows;
        incr = 1;
        if (vc->v_indx < 0)
            startpt = vc->vals;
        else
            startpt = vc->vals + vc->v_indx * vc->ldim;
    }
 
    /* Call the BLAS routine dnrm2_() */
    return cblas_dnrm2(Nval, startpt, incr);
}
 
/*!
 * \fn double G_vector_norm_maxval (vec_struct *vc, int vflag)
 *
 * \brief Calculates maximum value
 *
 * Calculates the maximum value of a row or column vector.
 * The vflag setting defines which value to be calculated:
 * vflag:
 * 1 Indicates maximum value<br>
 * -1  Indicates minimum value<br>
 * 0 Indicates absolute value [???]
 *
 * \param vc
 * \param vflag
 * \return double
 */
 
double G_vector_norm_maxval(vec_struct *vc, int vflag)
{
    double xval, *startpt, *curpt;
    double cellval;
    int ncells, incr;
 
    if (!vc->is_init)
        G_fatal_error(_("Matrix is not initialised"));
 
    if (vc->type == ROWVEC_) {
        ncells = (int)vc->cols;
        incr = (int)vc->ldim;
        if (vc->v_indx < 0)
            startpt = vc->vals;
        else
            startpt = vc->vals + vc->v_indx;
    }
    else {
        ncells = (int)vc->rows;
        incr = 1;
        if (vc->v_indx < 0)
            startpt = vc->vals;
        else
            startpt = vc->vals + vc->v_indx * vc->ldim;
    }
 
    xval = *startpt;
    curpt = startpt;
 
    while (ncells > 0) {
        if (curpt != startpt) {
            switch (vflag) {
 
            case MAX_POS: {
                if (*curpt > xval)
                    xval = *curpt;
                break;
            }
 
            case MAX_NEG: {
                if (*curpt < xval)
                    xval = *curpt;
                break;
            }
 
            case MAX_ABS: {
                cellval = (double)(*curpt);
                if (hypot(cellval, cellval) > (double)xval)
                    xval = *curpt;
            }
            } /* switch */
        } /* if(curpt != startpt) */
 
        curpt += incr;
        ncells--;
    }
 
    return (double)xval;
}
 
/*!
 * \fn double G_vector_norm1 (vec_struct *vc)
 *
 * \brief Calculates the 1-norm of a vector
 *
 * Calculates the 1-norm of a vector
 *
 * \param vc
 * \return double
 */
 
double G_vector_norm1(vec_struct *vc)
{
    double result = 0.0;
    int idx;
    int i;
 
    if (!vc->is_init) {
        G_warning(_("Matrix is not initialised"));
        return NAN;
    }
 
    idx = (vc->v_indx > 0) ? vc->v_indx : 0;
 
    if (vc->type == ROWVEC_) {
        for (i = 0; i < vc->cols; i++)
            result += fabs(G_matrix_get_element(vc, idx, i));
    }
    else {
        for (i = 0; i < vc->rows; i++)
            result += fabs(G_matrix_get_element(vc, i, idx));
    }
 
    return result;
}
 
/*!
 * \fn vec_struct *G_vector_product (vec_struct *v1, vec_struct *v2, vec_struct
 * *out)
 *
 * \brief Calculates the vector product
 *
 * Calculates the vector product of two vectors
 *
 * \param v1
 * \param v2
 * \return vec_struct
 */
 
vec_struct *G_vector_product(vec_struct *v1, vec_struct *v2, vec_struct *out)
{
    int idx1, idx2, idx0;
    int i;
 
    if (!out->is_init) {
        G_warning(_("Output vector is uninitialized"));
        return NULL;
    }
 
    if (v1->type != v2->type) {
        G_warning(_("Vectors are not of the same type"));
        return NULL;
    }
 
    if (v1->type != out->type) {
        G_warning(_("Output vector is not the same type as others"));
        return NULL;
    }
 
    if (v1->type == MATRIX_) {
        G_warning(_("Matrices not allowed"));
        return NULL;
    }
 
    if ((v1->type == ROWVEC_ && v1->cols != v2->cols) ||
        (v1->type == COLVEC_ && v1->rows != v2->rows)) {
        G_warning(_("Vectors have differing dimensions"));
        return NULL;
    }
 
    if ((v1->type == ROWVEC_ && v1->cols != out->cols) ||
        (v1->type == COLVEC_ && v1->rows != out->rows)) {
        G_warning(_("Output vector has incorrect dimension"));
        return NULL;
    }
 
    idx1 = (v1->v_indx > 0) ? v1->v_indx : 0;
    idx2 = (v2->v_indx > 0) ? v2->v_indx : 0;
    idx0 = (out->v_indx > 0) ? out->v_indx : 0;
 
    if (v1->type == ROWVEC_) {
        for (i = 0; i < v1->cols; i++)
            G_matrix_set_element(out, idx0, i,
                                 G_matrix_get_element(v1, idx1, i) *
                                     G_matrix_get_element(v2, idx2, i));
    }
    else {
        for (i = 0; i < v1->rows; i++)
            G_matrix_set_element(out, i, idx0,
                                 G_matrix_get_element(v1, i, idx1) *
                                     G_matrix_get_element(v2, i, idx2));
    }
 
    return out;
}
 
/*!
 * \fn vec_struct *G_vector_copy (const vec_struct *vc1, int comp_flag)
 *
 * \brief Returns a vector copied from <b>vc1</b>. Underlying structure
 * is preserved unless DO_COMPACT flag.
 *
 * \param vc1
 * \param comp_flag
 * \return vec_struct
 */
 
vec_struct *G_vector_copy(const vec_struct *vc1, int comp_flag)
{
    vec_struct *tmp_arry;
    int incr1, incr2;
    double *startpt1, *startpt2, *curpt1, *curpt2;
    int cnt;
 
    if (!vc1->is_init) {
        G_warning(_("Vector structure is not initialised"));
        return NULL;
    }
 
    tmp_arry = (vec_struct *)G_malloc(sizeof(vec_struct));
 
    if (comp_flag == DO_COMPACT) {
        if (vc1->type == ROWVEC_) {
            tmp_arry->rows = 1;
            tmp_arry->cols = vc1->cols;
            tmp_arry->ldim = 1;
            tmp_arry->type = ROWVEC_;
            tmp_arry->v_indx = 0;
        }
        else if (vc1->type == COLVEC_) {
            tmp_arry->rows = vc1->rows;
            tmp_arry->cols = 1;
            tmp_arry->ldim = vc1->ldim;
            tmp_arry->type = COLVEC_;
            tmp_arry->v_indx = 0;
        }
        else {
            G_warning("Type is not vector.");
            return NULL;
        }
    }
    else if (comp_flag == NO_COMPACT) {
        tmp_arry->v_indx = vc1->v_indx;
        tmp_arry->rows = vc1->rows;
        tmp_arry->cols = vc1->cols;
        tmp_arry->ldim = vc1->ldim;
        tmp_arry->type = vc1->type;
    }
    else {
        G_warning("Copy method must be specified: [DO,NO]_COMPACT.\n");
        return NULL;
    }
 
    tmp_arry->vals =
        (double *)G_calloc(tmp_arry->ldim * tmp_arry->cols, sizeof(double));
    if (comp_flag == DO_COMPACT) {
        if (tmp_arry->type == ROWVEC_) {
            startpt1 = tmp_arry->vals;
            startpt2 = vc1->vals + vc1->v_indx;
            curpt1 = startpt1;
            curpt2 = startpt2;
            incr1 = 1;
            incr2 = vc1->ldim;
            cnt = vc1->cols;
        }
        else if (tmp_arry->type == COLVEC_) {
            startpt1 = tmp_arry->vals;
            startpt2 = vc1->vals + vc1->v_indx * vc1->ldim;
            curpt1 = startpt1;
            curpt2 = startpt2;
            incr1 = 1;
            incr2 = 1;
            cnt = vc1->rows;
        }
        else {
            G_warning("Structure type is not vector.");
            return NULL;
        }
    }
    else if (comp_flag == NO_COMPACT) {
        startpt1 = tmp_arry->vals;
        startpt2 = vc1->vals;
        curpt1 = startpt1;
        curpt2 = startpt2;
        incr1 = 1;
        incr2 = 1;
        cnt = vc1->ldim * vc1->cols;
    }
    else {
        G_warning("Copy method must be specified: [DO,NO]_COMPACT.\n");
        return NULL;
    }
 
    while (cnt > 0) {
        memcpy(curpt1, curpt2, sizeof(double));
        curpt1 += incr1;
        curpt2 += incr2;
        cnt--;
    }
 
    tmp_arry->is_init = 1;
 
    return tmp_arry;
}
 
/*!
 * \fn int G_matrix_read (FILE *fp, mat_struct *out)
 *
 * \brief Read a matrix from a file stream
 *
 * Populates matrix structure <b>out</b> with matrix read from file
 * stream <b>fp</b>. Matrix <b>out</b> is automatically initialized.
 * Returns -1 on error and 0 on success.
 *
 * \param fp
 * \param out
 * \return int
 */
 
int G_matrix_read(FILE *fp, mat_struct *out)
{
    char buff[100];
    int rows, cols;
    int i, j, row;
    double val;
 
    /* skip comments */
    for (;;) {
        if (!G_getl(buff, sizeof(buff), fp))
            return -1;
        if (buff[0] != '#')
            break;
    }
 
    if (sscanf(buff, "Matrix: %d by %d", &rows, &cols) != 2) {
        G_warning(_("Input format error"));
        return -1;
    }
 
    G_matrix_set(out, rows, cols, rows);
 
    for (i = 0; i < rows; i++) {
        if (fscanf(fp, "row%d:", &row) != 1 || row != i) {
            G_warning(_("Input format error"));
            return -1;
        }
        for (j = 0; j < cols; j++) {
            if (fscanf(fp, "%lf:", &val) != 1) {
                G_warning(_("Input format error"));
                return -1;
            }
 
            G_matrix_set_element(out, i, j, val);
        }
    }
 
    return 0;
}
 
/*!
 * \fn mat_struct *G_matrix_resize(mat_struct *in, int rows, int cols)
 *
 * \brief Resizes a matrix
 *
 * Resizes a matrix
 *
 * \param A
 * \param rows
 * \param cols
 * \return mat_struct
 */
 
mat_struct *G_matrix_resize(mat_struct *in, int rows, int cols)
{
    mat_struct *matrix;
 
    matrix = G_matrix_init(rows, cols, rows);
    int i, j, p /*, index = 0 */;
 
    for (i = 0; i < rows; i++)
        for (j = 0; j < cols; j++)
            G_matrix_set_element(matrix, i, j, G_matrix_get_element(in, i, j));
    /*    matrix->vals[index++] = in->vals[i + j * cols]; */
 
    int old_size = in->rows * in->cols;
    int new_size = rows * cols;
 
    if (new_size > old_size)
        for (p = old_size; p < new_size; p++)
            G_matrix_set_element(matrix, i, j, 0.0);
 
    return (matrix);
}
 
/*!
 * \fn int G_matrix_read_stdin (mat_struct *out)
 *
 * \brief Read a matrix from standard input
 *
 * Populates matrix <b>out</b> with matrix read from stdin. Matrix
 * <b>out</b> is automatically initialized. Returns -1 on failure or 0
 * on success.
 *
 * \param out
 * \return int
 */
 
int G_matrix_stdin(mat_struct *out)
{
    return G_matrix_read(stdin, out);
}
 
/*!
 * \fn int G_matrix_eigen_sort (vec_struct *d, mat_struct *m)
 *
 * \brief Sort eigenvectors according to eigenvalues
 *
 * Sort eigenvectors according to eigenvalues. Returns 0.
 *
 * \param d
 * \param m
 * \return int
 */
 
int G_matrix_eigen_sort(vec_struct *d, mat_struct *m)
{
    mat_struct tmp;
    int i, j;
    int idx;
 
    G_matrix_set(&tmp, m->rows + 1, m->cols, m->ldim + 1);
 
    idx = (d->v_indx > 0) ? d->v_indx : 0;
 
    /* concatenate (vertically) m and d into tmp */
    for (i = 0; i < m->cols; i++) {
        for (j = 0; j < m->rows; j++)
            G_matrix_set_element(&tmp, j + 1, i, G_matrix_get_element(m, j, i));
        if (d->type == ROWVEC_)
            G_matrix_set_element(&tmp, 0, i, G_matrix_get_element(d, idx, i));
        else
            G_matrix_set_element(&tmp, 0, i, G_matrix_get_element(d, i, idx));
    }
 
    /* sort the combined matrix */
    qsort(tmp.vals, tmp.cols, tmp.ldim * sizeof(double), egcmp);
 
    /* split tmp into m and d */
    for (i = 0; i < m->cols; i++) {
        for (j = 0; j < m->rows; j++)
            G_matrix_set_element(m, j, i, G_matrix_get_element(&tmp, j + 1, i));
        if (d->type == ROWVEC_)
            G_matrix_set_element(d, idx, i, G_matrix_get_element(&tmp, 0, i));
        else
            G_matrix_set_element(d, i, idx, G_matrix_get_element(&tmp, 0, i));
    }
 
    G_free(tmp.vals);
 
    return 0;
}
 
static int egcmp(const void *pa, const void *pb)
{
    double a = *(double *const)pa;
    double b = *(double *const)pb;
 
    if (a > b)
        return 1;
    if (a < b)
        return -1;
 
    return 0;
}
 
#endif // HAVE_LIBLAPACK HAVE_LIBBLAS
 
typedef int suppress_empty_translation_unit_compiler_warning;