Domaine_Poly_tools.h

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#ifndef Domaine_Poly_tools_included
#define Domaine_Poly_tools_included
 
#include <TRUSTTrav.h>
#include <Lapack.h>
#include <math.h>
#include <vector>
 
/* produit matricel et transposee de DoubleTab */
static inline DoubleTab prod(DoubleTab a, DoubleTab b)
{
  int i, j, k, m = a.dimension(0), n = a.dimension(1), p = b.dimension(1);
  assert(n == b.dimension(0));
  DoubleTab r(m, p);
  for (i = 0; i < m; i++)
    for (j = 0; j < n; j++)
      for (k = 0; k < p; k++) r(i, k) += a(i, j) * b(j, k);
  return r;
}
static inline DoubleTab transp(DoubleTab a)
{
  int i, j, m = a.dimension(0), n = a.dimension(1);
  DoubleTab r(n, m);
  for (i = 0; i < m; i++)
    for (j = 0; j < n; j++) r(j, i) = a(i, j);
  return r;
}
 
/*! @brief Solves the least squares problem ||M.x - b||_2, placing the kernel of M into P and returning the residual.
 *
 * This function uses Singular Value Decomposition (SVD) to solve the least squares problem for the equation M.x = b.
 * It decomposes the matrix M into its constituent parts using the LAPACK routine DGESVD, which is a Fortran subroutine
 * for computing the SVD of a general rectangular matrix. The kernel of M (null space) is placed into matrix P.
 *
 * @param M The input matrix of the linear system.
 * @param b The right-hand side vector of the linear system.
 * @param eps The threshold for considering singular values as zero.
 * @param P Pointer to a matrix where the kernel of M will be stored.
 * @param x The solution vector that minimizes the least squares problem.
 * @param S Vector to store the singular values of M.
 *
 * @return The residual of the least squares solution, i.e., the Euclidean norm of (M.x - b).
 */
static inline double kersol(const DoubleTab& M, DoubleTab& b, double eps, DoubleTab *P, DoubleTab& x, DoubleTab& S)
{
  int i, nk, m = M.dimension(0), n = M.dimension(1), k = std::min(m, n), l = std::max(m, n), w = 5 * l, info=-1, iP, jP;
  double res2 = 0;
  char a = 'A';
  //lapack en mode Fortran -> on decompose en fait Mt!!
  DoubleTab A = M, U(m, m), Vt(n, n), W(w), iS(n, m);
  S.resize(k);
  F77NAME(dgesvd)(&a, &a, &n, &m, A.addr(), &n, S.addr(), Vt.addr(), &n, U.addr(), &m, W.addr(), &w, &info);
  for (i = 0, nk = n; i < k && S(i) > eps * S(0); i++) nk--;
  if (P) P->resize(n, nk);
  for (i = 0, jP = -1; i < n; i++)
    if (i < k && S(i) > eps * S(0)) iS(i, i) = 1 / S(i); //terme diagonal de iS
    else if (P)
      for (iP = 0, jP++; iP < n; iP++) (*P)(iP, jP) = Vt(i, iP); //colonne de V -> colonne de P
  x = prod(transp(Vt), prod(iS, prod(transp(U), b)));
  DoubleTab res = prod(M, x);
  for (i = 0; i < m; i++) res2 += std::pow(res(i, 0) - b(i, 0), 2);
  return sqrt(res2);
}
 
/*! @def CRIMP(a)
 * @brief Compacts a multi-dimensional array by resizing it.
 *
 * This macro adjusts the size of a multi-dimensional array `a` by incrementing and then decrementing its first dimension.
 * It handles arrays with 1, 2, or 3 dimensions.
 *
 * @param a The array to be compacted.
 */
#define CRIMP(a) a.nb_dim() > 2 ? a.resize(a.dimension(0) + 1, a.dimension(1), a.dimension(2)) : a.nb_dim() > 1 ? a.resize(a.dimension(0) + 1, a.dimension(1)) : a.resize(a.dimension(0) + 1), \
        a.nb_dim() > 2 ? a.resize(a.dimension(0) - 1, a.dimension(1), a.dimension(2)) : a.nb_dim() > 1 ? a.resize(a.dimension(0) - 1, a.dimension(1)) : a.resize(a.dimension(0) - 1)
#endif