libgetfem++-dev 4.2.1~beta1~svn4482~dfsg-2build1 / usr / include / getfem / getfem_continuation.h

/* -*- c++ -*- (enables emacs c++ mode) */
/*===========================================================================
 
 Copyright (C) 2011-2012 Tomas Ligursky, Yves Renard
 
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/** @file getfem_continuation.h
    @author Tomas Ligursky <tomas.ligursky@gmail.com>
    @author Yves Renard <Yves.Renard@insa-lyon.fr>
    @date October 17, 2011.
    @brief inexact Moore-Penrose continuation method.
*/
#ifndef GETFEM_CONTINUATION_H__
#define GETFEM_CONTINUATION_H__

#include <getfem/getfem_model_solvers.h>

namespace getfem {


  //=========================================================================
  // Abstract Moore-Penrose continuation method
  //=========================================================================

  const double tau_init = 1.e6;
  enum build_data { BUILD_F = 1, BUILD_F_x = 2, BUILD_ALL = 3 };

  /* Compute a unit tangent at (x, gamma) that is accute to the incoming
     vector. */
  template <typename CONT_S, typename VECT>
  void compute_tangent(CONT_S &S, const VECT &x, double gamma,
		       VECT &t_x, double &t_gamma) {
    double r;
    VECT g(x), y(x);
    S.F_gamma(x, gamma, g);
    S.solve_grad(x, gamma, y, g);
    t_gamma = 1. / (t_gamma - S.w_sp(t_x, y));
    S.scale(y, -t_gamma); S.copy(y, t_x);
    
    double no = S.w_norm(t_x, t_gamma);
    S.scale(t_x, 1./no); t_gamma /= no;

    S.mult_grad(x, gamma, t_x, y); S.scaled_add(y, g, t_gamma, y);
    r = S.norm(y);
    if (r > 1.e-10)
      GMM_WARNING1("Tangent computed with the residual " << r);
  }

  /* Calculate a tangent vector at (x, gamma) + h * (T_x, T_gamma) and test
     whether it is close to (T_x, T_gamma). Informatively, compare it with
     (t_x, t_gamma), as well. */
  template <typename CONT_S, typename VECT>
  bool test_tangent(CONT_S &S, const VECT &x, double gamma,
		    const VECT &T_x, double T_gamma,
		    const VECT &t_x, double t_gamma, double h) {
    bool res = false;
    double Gamma, T_Gamma = T_gamma, cang;
    VECT X(x), T_X(T_x);
    
    S.scaled_add(x, T_x, h, X); Gamma = gamma + h * T_gamma;
    S.set_build(BUILD_ALL);
    compute_tangent(S, X, Gamma, T_X, T_Gamma);
    
    cang = S.cosang(T_X, T_x, T_Gamma, T_gamma);
    if (S.noisy() > 1)
      cout << "cos of the angle with the tested tangent " << cang << endl;
    if (cang >= S.mincos()) res = true;
    else {    
      cang = S.cosang(T_X, t_x, T_Gamma, t_gamma);
      if (S.noisy() > 1)
	cout << "cos of the angle with the initial tangent " << cang << endl;
    }
    return res;
  }

  /* Simple tangent switch. */
  template <typename CONT_S, typename VECT>
  bool switch_tangent(CONT_S &S, const VECT &x, double gamma,
		      VECT &t_x, double &t_gamma, double &h) {    
    bool accepted;
    double t_gamma0 = t_gamma, T_gamma = t_gamma, Gamma;
    VECT t_x0(t_x), T_x(t_x), X(x);

    if (S.noisy() > 0) cout  << "trying simple tangent switch" << endl;
    if (S.noisy() > 0) cout << "starting computing a new tangent" << endl;
    h *= 1.5;
    S.scaled_add(x, T_x, h, X); Gamma = gamma + h * T_gamma;
    S.set_build(BUILD_ALL);
    compute_tangent(S, X, Gamma, T_x, T_gamma);
    // One can test the cosine of the angle between (T_x, T_gamma) and
    // (t_x, t_gamma), for sure, and increase h_min if it were greater or
    // equal to S.mincos(). However, this seems to be superfluous.
    
    if (S.noisy() > 0)
      cout << "starting testing the computed tangent" << endl;
    double h_test = (-0.9) * S.h_min();
    do {
      h_test = -h_test
	+ pow(10., floor(log10(-h_test / S.h_min()))) * S.h_min();
      accepted = test_tangent(S, x, gamma, T_x, T_gamma,
			      t_x, t_gamma, h_test);
      if (!accepted) {
	h_test *= -1.;
	accepted = test_tangent(S, x, gamma, T_x, T_gamma,
				t_x, t_gamma, h_test);
      }
    } while (!accepted && (h_test > -S.h_max()));
    
    if (accepted) {
      S.copy(T_x, t_x); t_gamma = T_gamma;
      if (h_test < 0) { 
	S.scale(t_x, -1.); t_gamma *= -1.; h_test *= -1.;
      }
      if (S.noisy() > 0)
	cout << "tangent direction switched, "
	     << "starting computing a suitable step size" << endl;
      bool h_adapted = false; h = S.h_init();
      while (!h_adapted && (h > h_test)) {
	h_adapted = test_tangent(S, x, gamma, t_x, t_gamma,
				 t_x0, t_gamma0, h);
	h *= S.h_dec();
      }
      h = (h_adapted) ? h / S.h_dec() : h_test;
    } else
      if (S.noisy() > 0) cout << "simple tangent switch has failed" << endl;
    
    return accepted;
  }


  /* Test function for bifurcation points for a given matrix. The first part
     of the solution of the augmented system is passed in
     (v_x, v_gamma). */
  template <typename CONT_S, typename MAT, typename VECT>
  double test_function(CONT_S &S, const MAT &A, const VECT &g,
		       const VECT &t_x, double t_gamma,
		       VECT &v_x, double &v_gamma) {
    double q, r, tau;
    VECT y(g), z(g);

    S.solve(A, y, z, g, S.b_x());
    v_gamma = (S.b_gamma() - S.sp(t_x, z)) / (t_gamma - S.sp(t_x, y));
    S.scaled_add(z, y, -v_gamma, v_x);
    tau = 1. / (S.d() - S.sp(S.c_x(), v_x) - S.c_gamma() * v_gamma);
    S.scale(v_x, -tau); v_gamma *= -tau;

    // control of the norm of the residual
    S.mult(A, v_x, y);
    S.scaled_add(y, g, v_gamma, y); S.scaled_add(y, S.b_x(), tau, y); 
    r = S.sp(y, y);
    q = S.sp(t_x, v_x) + t_gamma * v_gamma + S.b_gamma() * tau; r += q * q;
    q = S.sp(S.c_x(), v_x) + S.c_gamma() * v_gamma + S.d() * tau - 1.;
    r += q * q; r = sqrt(r);
    if (r > 1.e-10)
      GMM_WARNING1("Test function evaluated with the residual " << r);

    return tau;
  }

  template <typename CONT_S, typename MAT, typename VECT>
  double test_function(CONT_S &S, const MAT &A, const VECT &g,
		       const VECT &t_x, double t_gamma) {
    VECT v_x(g); double v_gamma;
    return test_function(S, A, g, t_x, t_gamma, v_x, v_gamma);
  }

  /* Test function for bifurcation points for the gradient computed at
     (x, gamma). */
  template <typename CONT_S, typename VECT>
  double test_function(CONT_S &S, const VECT &x, double gamma,
		       const VECT &t_x, double t_gamma,
		       VECT &v_x, double &v_gamma) {
    typename CONT_S::MAT A; S.F_x(x, gamma, A);
    VECT g(x); S.F_gamma(x, gamma, g);
    return test_function(S, A, g, t_x, t_gamma, v_x, v_gamma);
  }

  template <typename CONT_S, typename VECT>
  double test_function(CONT_S &S, const VECT &x, double gamma,
		       const VECT &t_x, double t_gamma) {
    VECT v_x(x); double v_gamma;
    return test_function(S, x, gamma, t_x, t_gamma, v_x, v_gamma);
  }

  /* Test for smooth bifurcation points. */
  template <typename CONT_S, typename VECT>
  bool test_smooth_bifurcation(CONT_S &S, const VECT &x, double gamma,
			       const VECT &t_x, double t_gamma) {
    double tau0 = S.get_tau1(), tau1 = S.get_tau2(),
      tau2 = test_function(S, x, gamma, t_x, t_gamma);
    S.set_tau1(tau1); S.set_tau2(tau2);
    return (tau2 * tau1 < 0) && (S.abs(tau1) < S.abs(tau0));
  }

  /* Test for non-smooth bifurcation points. */
  template <typename CONT_S, typename VECT>
  bool test_nonsmooth_bifurcation (CONT_S &S, const VECT &x1, double gamma1,
				   const VECT &t_x1, double t_gamma1,
				   const VECT &x2, double gamma2,
				   const VECT &t_x2, double t_gamma2) {
    unsigned long nb_changes = 0;
    double alpha = 0., delta = S.delta_min(),
      tau0 = tau_init, tau1= S.get_tau2(),tau2, tau_var_ref, t_gamma;
    VECT g1(x1), g2(x1), g(x1), t_x(x1);

    // compute gradients at the two given points
    typename CONT_S::MAT A1, A2, A;
    S.F_x(x2, gamma2, A2); S.F_x(x2, gamma2, A); S.F_gamma(x2, gamma2, g2);
    S.F_x(x1, gamma1, A1); S.F_gamma(x1, gamma1, g1);
    S.init_tau_graph();
    tau2 = test_function(S, A2, g2, t_x2, t_gamma2);
    tau_var_ref = std::max(S.abs(tau2 - tau1),
			   (S.abs(tau1) + S.abs(tau2)) / 200);

    // monitor sign changes of the test function on the convex combination
    do {
      alpha = std::min(alpha + delta, 1.);
      S.scaled_add(A1, 1. - alpha, A2, alpha, A);
      S.scaled_add(g1, 1. - alpha, g2, alpha, g);
      S.scaled_add(t_x1, 1. - alpha, t_x2, alpha, t_x);
      t_gamma = (1. - alpha) * t_gamma1 + alpha * t_gamma2;
      
      tau2 = test_function(S, A, g, t_x, t_gamma);
      if ((tau2 * tau1 < 0) && (S.abs(tau1) < S.abs(tau0))) ++nb_changes;
      S.insert_tau_graph(alpha, tau2);

      if (S.abs(tau2 - tau1) < 0.5 * S.thrvar() * tau_var_ref)
	delta = std::min(2 * delta, S.delta_max());
      else if (S.abs(tau2 - tau1) > S.thrvar() * tau_var_ref) 
	delta = std::max(0.1 * delta, S.delta_min());
      tau0 = tau1; tau1 = tau2; 
    } while (alpha < 1.);
    
    S.set_tau1(tau_init); S.set_tau2(tau2);
    return nb_changes % 2;
  }
  
  /* Newton-type corrections for the couple ((X, Gamma), (T_x, T_gamma)).
     The current direction of (T_x, T_gamma) is informatively compared with
     (t_x, t_gamma). */
  template <typename CONT_S, typename VECT>
  bool newton_corr(CONT_S &S, VECT &X, double &Gamma, VECT &T_x,
		   double &T_gamma, const VECT &t_x, double t_gamma,
		   unsigned long &it) {
    bool converged = false;
    double Delta_Gamma, no, res, diff;
    VECT F(X), g(X), Delta_X(X), y(X);

    if (S.noisy() > 0) cout << "starting correction " << endl;
    it = 0;
    S.F(X, Gamma, F);
    
    do {
      S.F_gamma(X, Gamma, g);
      S.solve_grad(X, Gamma, Delta_X, y, F, g);
      
      Delta_Gamma = S.sp(T_x, Delta_X) / (S.sp(T_x, y) - T_gamma);
      S.scaled_add(Delta_X, y, -Delta_Gamma, Delta_X);
      S.scaled_add(X, Delta_X, -1., X); Gamma -= Delta_Gamma;
      S.set_build(BUILD_ALL);
      
      T_gamma = 1. / (T_gamma - S.w_sp(T_x, y));
      S.scale(y, -T_gamma); S.copy(y, T_x);
      no = S.w_norm(T_x, T_gamma);
      S.scale(T_x, 1./no); T_gamma /= no;

      S.F(X, Gamma, F); res = S.norm(F); 
      diff = S.w_norm(Delta_X, Delta_Gamma);
      if (S.noisy() > 1)
	cout << " iter " << it << " residual " << res
	     << " difference " << diff 
	     << " cosang " << S.cosang(T_x, t_x, T_gamma, t_gamma) << endl;

      if (res <= S.maxres() && diff <= S.maxdiff()) {
	converged = true;
	// recalculate the final tangent, for sure
	compute_tangent(S, X, Gamma, T_x, T_gamma);
	break;
      }

      it++;      
    } while (it < S.maxit() && res < 1.e8);
    return converged;
  }
  
  template <typename CONT_S, typename VECT>
  bool newton_corr(CONT_S &S, VECT &X, double &Gamma, VECT &T_x,
		   double &T_gamma, const VECT &t_x, double t_gamma) {
    unsigned long it;
    return newton_corr(S, X, Gamma, T_x, T_gamma, t_x, t_gamma, it);
  }

  /* Try to perform one predictor-corrector step starting from the couple
     ((x, gamma), (t_x, t_gamma)). Return the resulting couple in the case of
     convergence. */
  template <typename CONT_S, typename VECT>
  bool test_predict_dir(CONT_S &S, VECT &x, double &gamma,
			VECT &t_x, double &t_gamma) {
    bool converged = false;
    double h =  S.h_init(), Gamma, T_gamma;
    VECT X(x), T_x(x);
    do { //step control
      
      // prediction
      if (S.noisy() > 0) cout << "prediction with h = " << h << endl;
      S.scaled_add(x, t_x, h, X); Gamma = gamma + h * t_gamma;
      S.set_build(BUILD_ALL);
      S.copy(t_x, T_x); T_gamma = t_gamma;

      //correction
      converged = newton_corr(S, X, Gamma, T_x, T_gamma, t_x, t_gamma);
      
      if (converged) {
	// check the direction of the tangent found
	S.scaled_add(X, x, -1., t_x); t_gamma = Gamma - gamma;
	if (S.sp(T_x, t_x, T_gamma, t_gamma) < 0)
	  { S.scale(T_x, -1.); T_gamma *= -1.; }
	S.copy(X, x); gamma = Gamma;
	S.copy(T_x, t_x); t_gamma = T_gamma;
      }
      else if (h > S.h_min())
	  h = (0.199 * S.h_dec() * h > S.h_min()) ?
	    0.199 * S.h_dec() * h : S.h_min();
      else break;

    } while(!converged);
    return converged;
  }

  /* A tool for approximating a smooth bifurcation point close to (x, gamma)
     and locating the two branches emanating from there. */
  template <typename CONT_S, typename VECT>
  void treat_smooth_bif_point(CONT_S &S, const VECT &x, double gamma,
			      const VECT &t_x, double t_gamma, double h) {
    unsigned long i = 0;
    double tau0 = S.get_tau1(), tau1 = S.get_tau2(),
      gamma0 = gamma, Gamma, t_gamma0 = t_gamma, T_gamma = t_gamma, v_gamma;
    VECT x0(x), X(x), t_x0(t_x), T_x(t_x), v_x(t_x);
    
    if (S.noisy() > 0)
      cout  << "starting locating the bifurcation point" << endl;

    // predictor-corrector steps with a secant-type step-length adaptation
    h *= tau1 / (tau0 - tau1);
    while ((S.abs(h) >= S.h_min()) && i < 10) {
      if (S.noisy() > 0) cout << "prediction with h = " << h << endl;
      S.scaled_add(x0, t_x0, h, X); Gamma = gamma0 + h * t_gamma0;
      S.set_build(BUILD_ALL);
      if (newton_corr(S, X, Gamma, T_x, T_gamma, t_x0, t_gamma0)) {
	S.copy(X, x0); gamma0 = Gamma;
	if (S.cosang(T_x, t_x0, T_gamma, t_gamma0) >= S.mincos())
	  { S.copy(T_x, t_x0); t_gamma0 = T_gamma; }
	tau0 = tau1;
	tau1 = test_function(S, X, Gamma, t_x0, t_gamma0, v_x, v_gamma);
	h *= tau1 / (tau0 - tau1);
      }	else {
	S.scaled_add(x0, t_x0, h, x0); gamma0 += h * t_gamma0;
	test_function(S, x0, gamma0, t_x0, t_gamma0, v_x, v_gamma);
	break;
      }
      ++i;
    }
    S.set_sing_point(x0, gamma0);
    S.insert_tangent_sing(t_x0, t_gamma0);

    if (S.noisy() > 0)
      cout  << "starting searching for the second branch" << endl;
    double no = S.w_norm(v_x, v_gamma);
    S.scale(v_x, 1./no); v_gamma /= no;
    if (test_predict_dir(S, x0, gamma0, v_x, v_gamma)
	&& S.insert_tangent_sing(v_x, v_gamma))
      { if (S.noisy() > 0) cout << "second branch found" << endl; }
    else if (S.noisy() > 0) cout << "Second branch not found!" << endl;
  }
  
  /* A tool for approximating a non-smooth point close to (x, gamma) and
     locating (preferably) all smooth one-sided solution branches emanating
     from there. It is supposed that (x, gamma) is a point on the previously
     traversed smooth solution branch within the distance of S.h_min() from
     the end point of this branch and (t_x, t_gamma) is the corresponding
     tangent that is directed towards the end point. */
  template <typename CONT_S, typename VECT>
  void treat_nonsmooth_point(CONT_S &S, const VECT &x, double gamma,
			     const VECT &t_x, double t_gamma, int version) {
    double gamma_end = gamma, Gamma, t_gamma0 = t_gamma, T_gamma = t_gamma,
      h = S.h_min(), cang, mcos = S.mincos();
    VECT x_end(x), X(x), t_x0(t_x), T_x(t_x);

    // approximate the non-smooth point by a bisection-like algorithm
    if (S.noisy() > 0)
      cout  << "starting locating a non-smooth point" << endl;
    S.scaled_add(x, t_x, h, X); Gamma = gamma + h * t_gamma;
    S.set_build(BUILD_ALL);
    if (newton_corr(S, X, Gamma, T_x, T_gamma, t_x0, t_gamma0)) {
      cang = S.cosang(T_x, t_x0, T_gamma, t_gamma0);
      if (cang >= mcos) mcos = (cang + 1.) / 2.;
    }

    S.copy(t_x0, T_x); T_gamma = t_gamma0;
    h /= 2.;
    for (unsigned long i = 0; i < 15; i++) {
      if (S.noisy() > 0) cout << "prediction with h = " << h << endl;
      S.scaled_add(x_end, t_x0, h, X); Gamma = gamma_end + h * t_gamma0;
      S.set_build(BUILD_ALL);
      if (newton_corr(S, X, Gamma, T_x, T_gamma, t_x0, t_gamma0)
	  && (S.cosang(T_x, t_x, T_gamma, t_gamma) >= mcos)) {
	S.copy(X, x_end); gamma_end = Gamma;
	S.copy(T_x, t_x0); t_gamma0 = T_gamma;
      } else {
	S.copy(t_x0, T_x); T_gamma = t_gamma0;
      }
      h /= 2.;
    }
    S.scaled_add(x_end, t_x0, h, x_end); gamma_end += h * t_gamma0;
    S.set_sing_point(x_end, gamma_end);

    // take two vectors to span a subspace of perturbations for the
    // non-smooth point
    if (S.noisy() > 0)
      cout  << "starting a thorough search for other branches" << endl;
    double t_gamma1 = t_gamma0, t_gamma2 = t_gamma0;
    VECT t_x1(t_x0), t_x2(t_x0);
    S.scale(t_x1, -1.); t_gamma1 *= -1.;
    S.insert_tangent_sing(t_x1, t_gamma1);

    h = S.h_min();
    S.scaled_add(x_end, t_x0, h, X); Gamma = gamma_end + h * t_gamma0;
    S.set_build(BUILD_ALL);
    compute_tangent(S, X, Gamma, t_x2, t_gamma2);

    // perturb the non-smooth point systematically to find new tangent
    // predictions
    unsigned long i1 = 0, i2 = 0, ncomb = 0;
    double a, a1, a2, no;
    S.clear(t_x0); t_gamma0 = 0.;

    do {
      for (unsigned long i = 0; i < S.nbdir(); i++) {
	a = (2 * M_PI * double(i)) / double(S.nbdir());
	a1 = h * sin(a); a2 = h * cos(a);
	S.scaled_add(x_end, t_x1, a1, X); Gamma = gamma_end + a1 * t_gamma1;
	S.scaled_add(X, t_x2, a2, X); Gamma += a2 * t_gamma2;
	S.set_build(BUILD_ALL);
	compute_tangent(S, X, Gamma, T_x, T_gamma);

	if (S.abs(S.cosang(T_x, t_x0, T_gamma, t_gamma0)) < S.mincos()) {
	  S.copy(T_x, t_x0); t_gamma0 = T_gamma;
	  if (S.insert_tangent_predict(T_x, T_gamma)) {
	    if (S.noisy() > 0)
	      cout << "new potential tangent vector found, "
		   << "trying one predictor-corrector step" << endl;
	    S.copy(x_end, X); Gamma = gamma_end;
	    
	    if (test_predict_dir(S, X, Gamma, T_x, T_gamma)) {
	      if (S.insert_tangent_sing(T_x, T_gamma)) {
		if ((a == 0) && (ncomb == 0)
		    && (S.abs(S.cosang(T_x, t_x0, T_gamma, t_gamma0))
			>= S.mincos())) { i2 = 1; ncomb = 1; }
		if (version) S.set_next_point(X, Gamma);
	      }
	      S.copy(x_end, X); Gamma = gamma_end;
	      S.copy(t_x0, T_x); T_gamma = t_gamma0;
	    }
	    
	    S.scale(T_x, -1.); T_gamma *= -1.;
	    if (test_predict_dir(S, X, Gamma, T_x, T_gamma)
		&& S.insert_tangent_sing(T_x, T_gamma) && version)
	      S.set_next_point(X, Gamma);
	  }
	}
      }
      
      // heuristics for varying the spanning vectors
      bool index_changed;
      if (i1 + 1 < i2) { ++i1; index_changed = true; }
      else if(i2 + 1 < S.nb_tangent_sing())
	{ ++i2; i1 = 0; index_changed = true; }
      else index_changed = false;
      if (index_changed) {
	S.copy(S.get_t_x_sing(i1), t_x1); t_gamma1 = S.get_t_gamma_sing(i1);
	S.copy(S.get_t_x_sing(i2), t_x2); t_gamma2 = S.get_t_gamma_sing(i2);
      } else {
	S.fill_random(T_x); T_gamma = S.random();
	no = S.w_norm(T_x, T_gamma);
	S.scaled_add(t_x2, T_x, 0.1/no, t_x2);
	t_gamma2 += 0.1/no * T_gamma;
	S.scaled_add(x_end, t_x2, h, X); Gamma = gamma_end + h * t_gamma2;
	S.set_build(BUILD_ALL);
	compute_tangent(S, X, Gamma, t_x2, t_gamma2);
      }
    } while (++ncomb < S.nbcomb());

    if (S.noisy() > 0)
      cout << "located branches " << S.nb_tangent_sing() << endl;
  }


  template <typename CONT_S, typename VECT>
  void init_test_function(CONT_S &S, const VECT &x, double gamma,
			  const VECT &t_x, double t_gamma) {
    if (S.noisy() > 0) cout << "starting computing an initial value of a "
			    << "test function for bifurcations" << endl;
    S.set_build(BUILD_ALL);
    double tau = test_function(S, x, gamma, t_x, t_gamma); S.set_tau2(tau);
  }

  template <typename CONT_S, typename VECT>
  void init_Moore_Penrose_continuation(CONT_S &S, const VECT &x,
				       double gamma, VECT &t_x,
				       double &t_gamma, double &h) {
    S.set_build(BUILD_ALL);
    S.clear(t_x); t_gamma = (t_gamma >= 0) ? 1. : -1.;
    if (S.noisy() > 0)
      cout << "starting computing an initial tangent" << endl;
    compute_tangent(S, x, gamma, t_x, t_gamma);
    h = S.h_init();
    if (S.bifurcations()) init_test_function(S, x, gamma, t_x, t_gamma);
  }

  
  /* Perform one step of the (non-smooth) Moore-Penrose continuation.
     NOTE: The new point need not to be saved in the model in the end! */
  template <typename CONT_S, typename VECT>
    void Moore_Penrose_continuation(CONT_S &S, VECT &x, double &gamma,
				    VECT &t_x, double &t_gamma, double &h) {
    bool converged, new_point = false, tangent_switched = false;
    unsigned long it, step_dec = 0;
    double t_gamma0 = t_gamma, Gamma, T_gamma;
    VECT t_x0(t_x), X(x), T_x(x);

    S.clear_tau_currentstep(); S.clear_sing_data();

    do {
      // prediction
      if (S.noisy() > 0) cout << "prediction with h = " << h << endl;
      S.scaled_add(x, t_x, h, X); Gamma = gamma + h * t_gamma;
      S.set_build(BUILD_ALL);
      S.copy(t_x, T_x); T_gamma = t_gamma;
      
      // correction
      converged = newton_corr(S, X, Gamma, T_x, T_gamma, t_x, t_gamma, it);

      if (converged
	  && (S.cosang(T_x, t_x, T_gamma, t_gamma) >= S.mincos())) {
	new_point = true;
	if (S.bifurcations()) {
	  if (S.noisy() > 0)
	    cout << "new point found, starting computing a test function "
		 << "for bifurcations" << endl;
	  if (!tangent_switched) {
	    if(test_smooth_bifurcation(S, X, Gamma, T_x, T_gamma)) {
	      S.set_sing_label("smooth bifurcation point");
	      if (S.noisy() > 0)
		cout << "Smooth bifurcation point detected!" << endl;
	      treat_smooth_bif_point(S, X, Gamma, T_x, T_gamma, h);
	    }
	  } else if (test_nonsmooth_bifurcation(S, x, gamma, t_x0, t_gamma0,
						X, Gamma, T_x, T_gamma)) {
	    S.set_sing_label("non-smooth bifurcation point");
	    if (S.noisy() > 0)
	      cout << "Non-smooth bifurcation point detected!" << endl;
	    treat_nonsmooth_point(S, x, gamma, t_x0, t_gamma0, 0);
	  }
	}
	
	if (step_dec == 0 && it < S.thrit())
	  h = (S.h_inc() * h < S.h_max()) ? S.h_inc() * h : S.h_max();
      } else if (h > S.h_min()) {
	h = (S.h_dec() * h > S.h_min()) ? S.h_dec() * h : S.h_min();
	step_dec++;
      } else if (S.non_smooth() && !tangent_switched) {
	if (S.noisy() > 0)
	  cout << "classical continuation has failed" << endl;
	if (switch_tangent(S, x, gamma, t_x, t_gamma, h)) {
	  tangent_switched = true;
	  step_dec = (h >= S.h_init()) ? 0 : 1;
	  if (S.noisy() > 0)
	    cout << "restarting the classical continuation" << endl;
	} else break;
      } else break;
    } while (!new_point);

    if (new_point) {
      S.copy(X, x); gamma = Gamma;
      S.copy(T_x, t_x); t_gamma = T_gamma;
    } else if (S.non_smooth()) {
      treat_nonsmooth_point(S, x, gamma, t_x0, t_gamma0, 1);
      if (S.next_point()) {
	if (S.bifurcations()) {
	  if (S.noisy() > 0)
	    cout << "starting computing a test function for bifurcations"
		 << endl;
	  S.set_build(BUILD_ALL);
	  bool bifurcation_detected = (S.nb_tangent_sing() > 2);
	  if (bifurcation_detected) {
	    // update the stored values of the test function only
	    S.set_tau1(tau_init);
	    S.set_tau2(test_function(S, S.get_x_next(), S.get_gamma_next(),
				     S.get_t_x_sing(1),
				     S.get_t_gamma_sing(1)));
	  } else
	    bifurcation_detected
	      = test_nonsmooth_bifurcation(S, x, gamma, t_x, t_gamma,
					   S.get_x_next(),
					   S.get_gamma_next(),
					   S.get_t_x_sing(1),
					   S.get_t_gamma_sing(1));
	  if (bifurcation_detected) {
	    S.set_sing_label("non-smooth bifurcation point");
	    if (S.noisy() > 0)
	      cout << "Non-smooth bifurcation point detected!" << endl;
	  }
	}
	
	S.copy(S.get_x_next(), x); gamma = S.get_gamma_next();
	S.copy(S.get_t_x_sing(1), t_x); t_gamma = S.get_t_gamma_sing(1);
	h = S.h_init();
	new_point = true;
      }
    }
    
    if (!new_point) {
      cout << "Continuation has failed!" << endl;
      h = 0;
    }
  }
  

  //=========================================================================
  // Moore-Penrose continuation method for Getfem models
  //=========================================================================


#ifdef GETFEM_MODELS_H__
 
  struct cont_struct_getfem_model {
    
    typedef base_vector VECT;
    typedef model_real_sparse_matrix MAT;
    
  private:
    model *md;
    bool bifurcations_, nonsmooth;
    std::string parameter_name_;
    bool with_parametrised_data;
    std::string initdata_name_, finaldata_name_, currentdata_name_;
    double scfac_;
    rmodel_plsolver_type lsolver;
    double h_init_, h_max_, h_min_, h_inc_, h_dec_;
    unsigned long maxit_, thrit_;
    double maxres_, maxdiff_, mincos_, maxres_solve_, delta_max_, delta_min_,
      thrvar_;
    unsigned long nbdir_, nbcomb_;
    int noisy_;
    VECT b_x_, c_x_;
    double b_gamma_, c_gamma_, d_;
    double tau1, tau2;
    VECT alpha_hist, tau_hist;
    std::map<double, double> tau_graph;
    std::string sing_label;
    VECT x_sing, x_next;
    double gamma_sing, gamma_next;
    std::vector<VECT> t_x_sing, t_x_predict;
    std::vector<double> t_gamma_sing, t_gamma_predict;
    build_data build;

  public:
    void init_border(void) {
      srand(unsigned(time(NULL)));
      unsigned long nbdof = md->nb_dof();
      gmm::resize(b_x_, nbdof); gmm::fill_random(b_x_);
      gmm::resize(c_x_, nbdof); gmm::fill_random(c_x_);
      b_gamma_ = gmm::random(1.); c_gamma_ = gmm::random(1.);
      d_ = gmm::random(1.);
    }

    cont_struct_getfem_model
    (model &m, const std::string &pn, double sfac, rmodel_plsolver_type ls,
     bool bif = false, double hin = 1.e-2, double hmax = 1.e-1,
     double hmin = 1.e-5, double hinc = 1.3, double hdec = 0.5,
     unsigned long mit = 10, unsigned long tit = 4, double mres = 1.e-6,
     double mdiff = 1.e-6, double mcos = 0.9, double mress = 1.e-8,
     int noi = 0, bool nonsm = false, double dmax = 0.005,
     double dmin = 0.00012, double tvar = 0.02, unsigned long ndir = 40,
     unsigned long ncomb = 1)
      : md(&m), bifurcations_(bif), nonsmooth(nonsm), parameter_name_(pn),
	with_parametrised_data(false), scfac_(sfac), lsolver(ls),
	h_init_(hin), h_max_(hmax), h_min_(hmin), h_inc_(hinc), h_dec_(hdec),
	maxit_(mit), thrit_(tit), maxres_(mres), maxdiff_(mdiff),
	mincos_(mcos), maxres_solve_(mress), delta_max_(dmax),
	delta_min_(dmin), thrvar_(tvar), nbdir_(ndir), nbcomb_(ncomb),
	noisy_(noi), tau1(tau_init), tau2(tau_init), gamma_sing(0.),
	gamma_next(0.), build(BUILD_ALL)
    { GMM_ASSERT1(!md->is_complex(),
		  "Continuation has only a real version, sorry.");
      if (bifurcations_) init_border(); }
    
    cont_struct_getfem_model
    (model &m, const std::string &pn, const std::string &in,
     const std::string &fn, const std::string &cn, double sfac,
     rmodel_plsolver_type ls, bool bif = false, double hin = 1.e-2,
     double hmax = 1.e-1, double hmin = 1.e-5, double hinc = 1.3,
     double hdec = 0.5, unsigned long mit = 10, unsigned long tit = 4,
     double mres = 1.e-6, double mdiff = 1.e-6, double mcos = 0.9,
     double mress = 1.e-8, int noi = 0, bool nonsm = false,
     double dmax = 0.005, double dmin = 0.00012, double tvar = 0.02,
     unsigned long ndir = 40, unsigned long ncomb = 1)
      : md(&m), bifurcations_(bif), nonsmooth(nonsm), parameter_name_(pn),
	with_parametrised_data(true), initdata_name_(in),
	finaldata_name_(fn), currentdata_name_(cn), scfac_(sfac),
	lsolver(ls), h_init_(hin), h_max_(hmax), h_min_(hmin), h_inc_(hinc),
	h_dec_(hdec), maxit_(mit), thrit_(tit), maxres_(mres),
	maxdiff_(mdiff), mincos_(mcos), maxres_solve_(mress),
	delta_max_(dmax), delta_min_(dmin), thrvar_(tvar), nbdir_(ndir),
	nbcomb_(ncomb), noisy_(noi), tau1(tau_init), tau2(tau_init),
	gamma_sing(0.), gamma_next(0.), build(BUILD_ALL)
    { GMM_ASSERT1(!md->is_complex(),
		  "Continuation has only a real version, sorry.");
      if (bifurcations_) init_border(); }

    cont_struct_getfem_model(void) {}
    

    // Linear algebra functions
    double abs(double a) { return gmm::abs(a); }
    void clear(VECT &v) { gmm::clear(v); }
    void copy(const VECT &v1, VECT &v) { gmm::copy(v1, v); }
    void scale(VECT &v, double a) { gmm::scale(v, a); }
    void scaled_add(const VECT &v1, const VECT &v2, double a, VECT &v)
    { gmm::add(v1, gmm::scaled(v2, a), v); }
    void scaled_add(const VECT &v1, double a1,
		    const VECT &v2, double a2, VECT &v)
    { gmm::add(gmm::scaled(v1, a1), gmm::scaled(v2, a2), v); }
    void scaled_add(const MAT &M1, double a1,
		    const MAT &M2, double a2, MAT &M)
    { gmm::add(gmm::scaled(M1, a1), gmm::scaled(M2, a2), M); }    
    void mult(const MAT &A, const VECT &v1, VECT &v)
    { gmm::mult(A, v1, v); }

    double sp(const VECT &v1, const VECT &v2)
    { return gmm::vect_sp(v1, v2); }
    double norm(const VECT &v)
    { return gmm::vect_norm2(v); }
    double w_sp(const VECT &v1, const VECT &v2)
    { return scfac_ * gmm::vect_sp(v1, v2); }
    double sp(const VECT &v1, const VECT &v2, double w1, double w2)
    { return sp(v1, v2) + w1 * w2; }
    double w_norm(const VECT &v, double w)
    { return sqrt(w_sp(v, v) + w * w); }
    double cosang(const VECT &v1, const VECT &v2, double w1, double w2) {
      double no = sqrt(sp(v1, v1, w1, w1) * sp(v2, v2, w2, w2)); 
      return ((no == 0) ? 0. : sp(v1, v2, w1, w2) / no);
    }
    
    double random(void) { return gmm::random(1.); }
    void fill_random(VECT &v) { gmm::fill_random(v); }

    void solve(const MAT &A, VECT &g, const VECT &L) { /* A * g = L */
      if (noisy_ > 2) cout << "starting linear solver" << endl;
      gmm::iteration iter(maxres_solve_, (noisy_ >= 2) ? noisy_ - 2 : 0,
			  40000);
      (*lsolver)(A, g, L, iter);
      if (noisy_ > 2) cout << "linear solver done" << endl;
    }

    void solve(const MAT &A, VECT &g1, VECT &g2,
	       const VECT &L1, const VECT &L2) { /* A * (g1|g2) = (L1|L2) */
      if (noisy_ > 2) cout << "starting linear solver" << endl;
      gmm::iteration iter(maxres_solve_, (noisy_ >= 2) ? noisy_ - 2 : 0,
			  40000);
      (*lsolver)(A, g1, L1, iter);
      iter.init(); (*lsolver)(A, g2, L2, iter); // (can be optimised)
      if (noisy_ > 2) cout << "linear solver done" << endl;
    }


    // Evaluation of  ...
    void set_variables(const VECT &x, double gamma) {
      md->set_real_variable(parameter_name_)[0] = gamma;
      if (with_parametrised_data) {
	gmm::add(gmm::scaled(md->real_variable(initdata_name_), 1. - gamma),
		 gmm::scaled(md->real_variable(finaldata_name_), gamma),
		 md->set_real_variable(currentdata_name_));
      }
      md->to_variables(x);
    }

    // F(x, gamma) --> f
    void F(const VECT &x, double gamma, VECT &f) {
      if (build == BUILD_ALL) set_variables(x, gamma);
      if (build & BUILD_F) {
	md->assembly(model::BUILD_RHS);
	build = build_data(build ^ BUILD_F);
      }
      gmm::copy(gmm::scaled(md->real_rhs(), -1.), f);
    }
    
    // (F(x, gamma + eps) - F(x, gamma)) / eps --> g
    void F_gamma(const VECT &x, double gamma, VECT &g) {
      const double eps = 1.e-8;
      VECT F0(x), F1(x);
      F(x, gamma, F0);
      build = BUILD_ALL; F(x, gamma + eps, F1); build = BUILD_ALL;
      gmm::add(F1, gmm::scaled(F0, -1.), g);
      gmm::scale(g, 1./eps);
    }

    void update_matrix(const VECT &x, double gamma) {
      if (build == BUILD_ALL) set_variables(x, gamma);
      if (build & BUILD_F_x) {
	if (noisy_ > 2) cout << "starting computing tangent matrix" << endl;
	md->assembly(model::BUILD_MATRIX);
	build = build_data(build ^ BUILD_F_x);
      }
    }
    // F_x(x, gamma) --> A
    void F_x(const VECT &x, double gamma, MAT &A) {
      update_matrix(x, gamma);
      unsigned long nbdof = md->nb_dof();
      gmm::resize(A, nbdof, nbdof);
      gmm::copy(md->real_tangent_matrix(), A);
    }

    // solve F_x(x, gamma) * g = L
    void solve_grad(const VECT &x, double gamma,
		    VECT &g, const VECT &L) {
      update_matrix(x, gamma);
      solve(md->real_tangent_matrix(), g, L);
    }

    // solve F_x(x, gamma) * (g1|g2) = (L1|L2)
    void solve_grad(const VECT &x, double gamma, VECT &g1,VECT &g2,
		    const VECT &L1, const VECT &L2) {
      update_matrix(x, gamma);
      solve(md->real_tangent_matrix(), g1, g2, L1, L2);
    }

    // F_x(x, gamma) * w --> y
    void mult_grad(const VECT &x, double gamma,
			const VECT &w, VECT &y) {
      update_matrix(x, gamma);
      mult(md->real_tangent_matrix(), w, y);
    }

    
    // Misc. for accessing private data
    model &linked_model(void) { return *md; }
    bool bifurcations(void) { return bifurcations_; }
    bool non_smooth(void) { return nonsmooth; }
    std::string parameter_name(void) { return parameter_name_; }
    double scfac(void) { return scfac_; }
    double h_init(void) { return h_init_; }
    double h_min(void) { return h_min_; }
    double h_max(void) { return h_max_; }
    double h_dec(void) { return h_dec_; }
    double h_inc(void) { return h_inc_; }
    unsigned long maxit(void) { return maxit_; }
    unsigned long thrit(void) { return thrit_; }
    double maxres(void) { return maxres_; }
    double maxdiff(void) { return maxdiff_; }
    double mincos(void) { return mincos_; }
    double delta_max(void) { return delta_max_; }
    double delta_min(void) { return delta_min_; }
    double thrvar(void) { return thrvar_; }
    unsigned long nbdir(void) { return nbdir_; }
    unsigned long nbcomb(void) { return nbcomb_; }
    int noisy(void) { return noisy_; }
    VECT &b_x(void) { return b_x_; }
    VECT &c_x(void) { return c_x_; }
    double b_gamma(void) { return b_gamma_; }
    double c_gamma(void) { return c_gamma_; }
    double d(void) { return d_; }

    void set_tau1(double tau) { tau1 = tau; }
    double get_tau1(void) { return tau1; }
    void set_tau2(double tau) { tau2 = tau; }
    double get_tau2(void) { return tau2; }
    void clear_tau_currentstep(void) {
      tau_graph.clear();
      gmm::resize(alpha_hist, 0); gmm::resize(tau_hist, 0);
    }
    void init_tau_graph(void) { tau_graph[0.] = tau2; }
    void insert_tau_graph(double alpha, double tau) {
      tau_graph[alpha] = tau;
    }
    VECT &get_alpha_hist(void) {
      unsigned long i = 0;
      gmm::resize(alpha_hist, tau_graph.size());
      for (std::map<double, double>::iterator it = tau_graph.begin();
	   it != tau_graph.end(); it++) {
	alpha_hist[i] = (*it).first; i++;
      }	
      return alpha_hist;
    }
    VECT &get_tau_hist(void) {
      unsigned long i = 0;
      gmm::resize(tau_hist, tau_graph.size());
      for (std::map<double, double>::iterator it = tau_graph.begin();
	   it != tau_graph.end(); it++) {
	tau_hist[i] = (*it).second; i++; 
      }	
      return tau_hist;
    }

    void clear_sing_data(void) {
      sing_label = "";
      gmm::resize(x_sing, 0); gmm::resize(x_next, 0);
      t_x_sing.clear(); t_gamma_sing.clear();
      t_x_predict.clear(); t_gamma_predict.clear();
    }
    void set_sing_label(std::string label) { sing_label = label; }
    std::string get_sing_label(void) { return sing_label; }
    void set_sing_point(const VECT &x, double gamma) {
      gmm::resize(x_sing, gmm::vect_size(x)); gmm::copy(x, x_sing);
      gamma_sing = gamma;
    }
    VECT &get_x_sing(void) { return x_sing; }
    double get_gamma_sing(void) { return gamma_sing; }
    unsigned long nb_tangent_sing(void) { return t_x_sing.size(); }
    bool insert_tangent_sing(const VECT &t_x, double t_gamma){
      bool is_included = false;
      unsigned long i = 0;
      double cang;
      while ((i < t_x_sing.size()) && (!is_included)){
	cang = cosang(t_x_sing[i], t_x, t_gamma_sing[i], t_gamma);
	is_included = (cang >= mincos_);
	++i;
      }
      if (!is_included) {
	t_x_sing.push_back(t_x); t_gamma_sing.push_back(t_gamma);
      }
      return !is_included;
    }
    VECT &get_t_x_sing(unsigned long i) { return t_x_sing[i]; }
    double get_t_gamma_sing(unsigned long i) { return t_gamma_sing[i]; }
    std::vector<VECT> &get_t_x_sing(void) { return t_x_sing; }
    std::vector<double> &get_t_gamma_sing(void) { return t_gamma_sing; }

    bool next_point(void) { return gmm::vect_size(x_next) > 0; }
    void set_next_point(const VECT &x, double gamma) {
      if (gmm::vect_size(x_next) == 0) {
	gmm::resize(x_next, gmm::vect_size(x)); gmm::copy(x, x_next);
	gamma_next = gamma;
      }
    }
    VECT &get_x_next(void) { return x_next; }
    double get_gamma_next(void) { return gamma_next; }

    bool insert_tangent_predict(const VECT &t_x, double t_gamma){
      bool is_included = false;
      unsigned long i = 0;
      double cang;
      while ((i < t_x_predict.size()) && (!is_included)){
	cang = gmm::abs(cosang(t_x_predict[i], t_x,
			       t_gamma_predict[i], t_gamma));
	is_included = (cang >= mincos_);
	++i;
      }
      if (!is_included) {
	t_x_predict.push_back(t_x); t_gamma_predict.push_back(t_gamma);
      }
      return !is_included;
    }

    void set_build(build_data build_) { build = build_; }
  };

#endif


}  /* end of namespace getfem.                                             */


#endif /* GETFEM_CONTINUATION_H__ */