text: /*
   * 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 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.
   *
   * 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.
   *
   * The Original Code is Copyright (C) 2015 by Blender Foundation.
   * All rights reserved.
   * */
  
  /** \file
   * \ingroup bli
   */
  
  #include "MEM_guardedalloc.h"
  
  #include "BLI_math.h"
  #include "BLI_utildefines.h"
  
  #include "BLI_strict_flags.h"
  
  #include "eigen_capi.h"
  
  /********************************** Eigen Solvers *********************************/
  
  /**
   * \brief Compute the eigen values and/or vectors of given 3D symmetric (aka adjoint) matrix.
   *
   * \param m3: the 3D symmetric matrix.
   * \return r_eigen_values the computed eigen values (NULL if not needed).
   * \return r_eigen_vectors the computed eigen vectors (NULL if not needed).
   */
  bool BLI_eigen_solve_selfadjoint_m3(const float m3[3][3],
                                      float r_eigen_values[3],
                                      float r_eigen_vectors[3][3])
  {
  #ifndef NDEBUG
    /* We must assert given matrix is self-adjoint (i.e. symmetric) */
    if ((m3[0][1] != m3[1][0]) || (m3[0][2] != m3[2][0]) || (m3[1][2] != m3[2][1])) {
      BLI_assert(0);
    }
  #endif
  
    return EIG_self_adjoint_eigen_solve(
        3, (const float *)m3, r_eigen_values, (float *)r_eigen_vectors);
  }
  
  /**
   * \brief Compute the SVD (Singular Values Decomposition) of given 3D matrix (m3 = USV*).
   *
   * \param m3: the matrix to decompose.
   * \return r_U the computed left singular vector of \a m3 (NULL if not needed).
   * \return r_S the computed singular values of \a m3 (NULL if not needed).
   * \return r_V the computed right singular vector of \a m3 (NULL if not needed).
   */
  void BLI_svd_m3(const float m3[3][3], float r_U[3][3], float r_S[3], float r_V[3][3])
  {
    EIG_svd_square_matrix(3, (const float *)m3, (float *)r_U, (float *)r_S, (float *)r_V);
  }
  
  /***************************** Simple Solvers ************************************/
  
  /**
   * \brief Solve a tridiagonal system of equations:
   *
   * a[i] * r_x[i-1] + b[i] * r_x[i] + c[i] * r_x[i+1] = d[i]
   *
   * Ignores a[0] and c[count-1]. Uses the Thomas algorithm, e.g. see wiki.
   *
   * \param r_x: output vector, may be shared with any of the input ones
   * \return true if success
   */
  bool BLI_tridiagonal_solve(
      const float *a, const float *b, const float *c, const float *d, float *r_x, const int count)
  {
    if (count < 1) {
      return false;
    }
  
    size_t bytes = sizeof(double) * (unsigned)count;
    double *c1 = (double *)MEM_mallocN(bytes * 2, "tridiagonal_c1d1");
    double *d1 = c1 + count;
  
    if (!c1) {
      return false;
    }
  
    int i;
    double c_prev, d_prev, x_prev;
  
    /* forward pass */
  
    c1[0] = c_prev = ((double)c[0]) / b[0];
    d1[0] = d_prev = ((double)d[0]) / b[0];
  
    for (i = 1; i < count; i++) {
      double denum = b[i] - a[i] * c_prev;
  
      c1[i] = c_prev = c[i] / denum;
      d1[i] = d_prev = (d[i] - a[i] * d_prev) / denum;
    }
  
    /* back pass */
  
    x_prev = d_prev;
    r_x[--i] = ((float)x_prev);
  
    while (--i >= 0) {
      x_prev = d1[i] - c1[i] * x_prev;
      r_x[i] = ((float)x_prev);
    }
  
    MEM_freeN(c1);
  
    return isfinite(x_prev);
  }
  
  /**
   * \brief Solve a possibly cyclic tridiagonal system using the Sherman-Morrison formula.
   *
   * \param r_x: output vector, may be shared with any of the input ones
   * \return true if success
   */
  bool BLI_tridiagonal_solve_cyclic(
      const float *a, const float *b, const float *c, const float *d, float *r_x, const int count)
  {
    if (count < 1) {
      return false;
    }
  
    float a0 = a[0], cN = c[count - 1];
  
    /* if not really cyclic, fall back to the simple solver */
    if (a0 == 0.0f && cN == 0.0f) {
      return BLI_tridiagonal_solve(a, b, c, d, r_x, count);
    }
  
    size_t bytes = sizeof(float) * (unsigned)count;
    float *tmp = (float *)MEM_mallocN(bytes * 2, "tridiagonal_ex");
    float *b2 = tmp + count;
  
    if (!tmp) {
      return false;
    }
  
    /* prepare the noncyclic system; relies on tridiagonal_solve ignoring values */
    memcpy(b2, b, bytes);
    b2[0] -= a0;
    b2[count - 1] -= cN;
  
    memset(tmp, 0, bytes);
    tmp[0] = a0;
    tmp[count - 1] = cN;
  
    /* solve for partial solution and adjustment vector */
    bool success = BLI_tridiagonal_solve(a, b2, c, tmp, tmp, count) &&
                   BLI_tridiagonal_solve(a, b2, c, d, r_x, count);
  
    /* apply adjustment */
    if (success) {
      float coeff = (r_x[0] + r_x[count - 1]) / (1.0f + tmp[0] + tmp[count - 1]);
  
      for (int i = 0; i < count; i++) {
        r_x[i] -= coeff * tmp[i];
      }
    }
  
    MEM_freeN(tmp);
  
    return success;
  }
  
  /**
   * \brief Solve a generic f(x) = 0 equation using Newton's method.
   *
   * \param func_delta: Callback computing the value of f(x).
   * \param func_jacobian: Callback computing the Jacobian matrix of the function at x.
   * \param func_correction: Callback for forcing the search into an arbitrary custom domain.
   * May be NULL.
   * \param userdata: Data for the callbacks.
   * \param epsilon: Desired precision.
   * \param max_iterations: Limit on the iterations.
   * \param trace: Enables logging to console.
   * \param x_init: Initial solution vector.
   * \param result: Final result.
   * \return true if success
   */
  bool BLI_newton3d_solve(Newton3D_DeltaFunc func_delta,
                          Newton3D_JacobianFunc func_jacobian,
                          Newton3D_CorrectionFunc func_correction,
                          void *userdata,
                          float epsilon,
                          int max_iterations,
                          bool trace,
                          const float x_init[3],
                          float result[3])
  {
    float fdelta[3], fdeltav, next_fdeltav;
    float jacobian[3][3], step[3], x[3], x_next[3];
  
    epsilon *= epsilon;
  
    copy_v3_v3(x, x_init);
  
    func_delta(userdata, x, fdelta);
    fdeltav = len_squared_v3(fdelta);
  
    if (trace) {
      printf("START (%g, %g, %g) %g %g\n", x[0], x[1], x[2], fdeltav, epsilon);
    }
  
    for (int i = 0; i == 0 || (i < max_iterations && fdeltav > epsilon); i++) {
      /* Newton's method step. */
      func_jacobian(userdata, x, jacobian);
  
      if (!invert_m3(jacobian)) {
        return false;
      }
  
      mul_v3_m3v3(step, jacobian, fdelta);
      sub_v3_v3v3(x_next, x, step);
  
      /* Custom out-of-bounds value correction. */
      if (func_correction) {
        if (trace) {
          printf("%3d * (%g, %g, %g)\n", i, x_next[0], x_next[1], x_next[2]);
        }
  
        if (!func_correction(userdata, x, step, x_next)) {
          return false;
        }
      }
  
      func_delta(userdata, x_next, fdelta);
      next_fdeltav = len_squared_v3(fdelta);
  
      if (trace) {
        printf("%3d ? (%g, %g, %g) %g\n", i, x_next[0], x_next[1], x_next[2], next_fdeltav);
      }
  
      /* Line search correction. */
      while (next_fdeltav > fdeltav && next_fdeltav > epsilon) {
        float g0 = sqrtf(fdeltav), g1 = sqrtf(next_fdeltav);
        float g01 = -g0 / len_v3(step);
        float det = 2.0f * (g1 - g0 - g01);
        float l = (det == 0.0f) ? 0.1f : -g01 / det;
        CLAMP_MIN(l, 0.1f);
  
        mul_v3_fl(step, l);
        sub_v3_v3v3(x_next, x, step);
  
        func_delta(userdata, x_next, fdelta);
        next_fdeltav = len_squared_v3(fdelta);
  
        if (trace) {
          printf("%3d . (%g, %g, %g) %g\n", i, x_next[0], x_next[1], x_next[2], next_fdeltav);
        }
      }
  
      copy_v3_v3(x, x_next);
      fdeltav = next_fdeltav;
    }
  
    bool success = (fdeltav <= epsilon);
  
    if (trace) {
      printf("%s  (%g, %g, %g) %g\n", success ? "OK  " : "FAIL", x[0], x[1], x[2], fdeltav);
    }
  
    copy_v3_v3(result, x);
    return success;
  }
  