Modularize the libdimension codebase.

1 files changed, 443 insertions, 0 deletions

diff --git a/libdimension/math/polynomial.c b/libdimension/math/polynomial.c
new file mode 100644
index 0000000..09e9603
--- /dev/null
+++ b/libdimension/math/polynomial.c
@@ -0,0 +1,443 @@
+/*************************************************************************
+ * Copyright (C) 2010-2011 Tavian Barnes <tavianator@tavianator.com>     *
+ *                                                                       *
+ * This file is part of The Dimension Library.                           *
+ *                                                                       *
+ * The Dimension Library is free software; you can redistribute it and/  *
+ * or modify it under the terms of the GNU Lesser General Public License *
+ * as published by the Free Software Foundation; either version 3 of the *
+ * License, or (at your option) any later version.                       *
+ *                                                                       *
+ * The Dimension Library 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  *
+ * Lesser General Public License for more details.                       *
+ *                                                                       *
+ * You should have received a copy of the GNU Lesser General Public      *
+ * License along with this program.  If not, see                         *
+ * <http://www.gnu.org/licenses/>.                                       *
+ *************************************************************************/
+
+/**
+ * @file
+ * Real root isolation algorithm based on work by Vincent, Uspensky, Collins and
+ * Akritas, Johnson, Krandick, and Rouillier and Zimmerman.
+ */
+
+#include "internal.h"
+#include "internal/polynomial.h"
+#include "dimension/math.h"
+#include <math.h>
+
+/// Get the real degree of a polynomial, ignoring leading zeros.
+static inline size_t
+dmnsn_real_degree(const double poly[], size_t degree)
+{
+  for (size_t i = degree + 1; i-- > 0;) {
+    if (dmnsn_likely(fabs(poly[i]) >= dmnsn_epsilon)) {
+      return i;
+    }
+  }
+
+  return 0;
+}
+
+/// Divide each coefficient by the leading coefficient.
+static inline void
+dmnsn_polynomial_normalize(double poly[], size_t degree)
+{
+  for (size_t i = 0; i < degree; ++i) {
+    poly[i] /= poly[degree];
+  }
+  poly[degree] = 1.0;
+}
+
+/// Eliminate trivial zero roots from \p poly[].
+static inline void
+dmnsn_eliminate_zero_roots(double **poly, size_t *degree)
+{
+  size_t i;
+  for (i = 0; i <= *degree; ++i) {
+    if (dmnsn_likely(fabs((*poly)[i]) >= dmnsn_epsilon)) {
+      break;
+    }
+  }
+
+  *poly += i;
+  *degree -= i;
+}
+
+/// Calculate a finite upper bound on the roots of a normalized polynomial.
+static inline double
+dmnsn_root_bound(const double poly[], size_t degree)
+{
+  double bound = fabs(poly[0]);
+  for (size_t i = 1; i < degree; ++i) {
+    bound = dmnsn_max(bound, fabs(poly[i]));
+  }
+  bound += 1.0;
+  return bound;
+}
+
+/// Copy a polynomial.
+static inline void
+dmnsn_polynomial_copy(double dest[], const double src[], size_t degree)
+{
+  for (size_t i = 0; i <= degree; ++i) {
+    dest[i] = src[i];
+  }
+}
+
+/// Transform a polynomial by P'(x) = P(x + 1).
+static inline void
+dmnsn_polynomial_translate(double poly[], size_t degree)
+{
+  for (size_t i = 0; i <= degree; ++i) {
+    for (size_t j = degree - i; j <= degree - 1; ++j) {
+      poly[j] += poly[j + 1];
+    }
+  }
+}
+
+/// Transform a polynomial by P'(x) = P(c*x).
+static inline void
+dmnsn_polynomial_scale(double poly[], size_t degree, double c)
+{
+  double factor = c;
+  for (size_t i = 1; i <= degree; ++i) {
+    poly[i] *= factor;
+    factor *= c;
+  }
+}
+
+/// Returns the result of Descartes' rule on x^degree * poly(1/(x + 1)).
+static size_t
+dmnsn_descartes_bound(const double poly[], size_t degree)
+{
+  // Copy the polynomial so we can be destructive
+  double p[degree + 1];
+  dmnsn_polynomial_copy(p, poly, degree);
+
+  // Calculate poly(1/(1/x + 1)) which avoids reversal
+  for (size_t i = 1; i <= degree; ++i) {
+    for (size_t j = i; j >= 1; --j) {
+      p[j] += p[j - 1];
+    }
+  }
+
+  // Find the number of sign changes in p[]
+  size_t changes = 0;
+  int lastsign = dmnsn_sgn(p[0]);
+  for (size_t i = 1; changes <= 1 && i <= degree; ++i) {
+    int sign = dmnsn_sgn(p[i]);
+    if (sign != 0 && sign != lastsign) {
+      ++changes;
+      lastsign = sign;
+    }
+  }
+
+  return changes;
+}
+
+/// Depth-first search of possible isolating intervals.
+static size_t
+dmnsn_root_bounds_recursive(double poly[], size_t degree, double *c, double *k,
+                            double bounds[][2], size_t nbounds)
+{
+  size_t s = dmnsn_descartes_bound(poly, degree);
+  if (s >= 2) {
+    // Get the left child
+    dmnsn_polynomial_scale(poly, degree, 1.0/2.0);
+    *c *= 2.0;
+    *k /= 2.0;
+    double currc = *c, currk = *k;
+
+    // Test the left child
+    size_t n = dmnsn_root_bounds_recursive(poly, degree, c, k, bounds, nbounds);
+    if (nbounds == n) {
+      return n;
+    }
+    bounds += n;
+    nbounds -= n;
+
+    // Get the right child from the last tested polynomial
+    dmnsn_polynomial_translate(poly, degree);
+    dmnsn_polynomial_scale(poly, degree, currk/(*k));
+    *c = currc + 1.0;
+    *k = currk;
+
+    // Test the right child
+    n += dmnsn_root_bounds_recursive(poly, degree, c, k, bounds, nbounds);
+    return n;
+  } else if (s == 1) {
+    bounds[0][0] = (*c)*(*k);
+    bounds[0][1] = (*c + 1.0)*(*k);
+    return 1;
+  } else {
+    return 0;
+  }
+}
+
+/// Find ranges that contain a single root.
+static size_t
+dmnsn_root_bounds(const double poly[], size_t degree, double bounds[][2],
+                  size_t nbounds)
+{
+  // Copy the polynomial so we can be destructive
+  double p[degree + 1];
+  dmnsn_polynomial_copy(p, poly, degree);
+
+  // Scale the roots to within (0, 1]
+  double bound = dmnsn_root_bound(p, degree);
+  dmnsn_polynomial_scale(p, degree, bound);
+
+  // Bounding intervals are of the form (c*k, (c + 1)*k)
+  double c = 0.0, k = 1.0;
+
+  // Isolate the roots
+  size_t n = dmnsn_root_bounds_recursive(p, degree, &c, &k, bounds, nbounds);
+
+  // Scale the roots back to within (0, bound]
+  for (size_t i = 0; i < n; ++i) {
+    bounds[i][0] *= bound;
+    bounds[i][1] *= bound;
+  }
+
+  return n;
+}
+
+/// Maximum number of iterations in dmnsn_bisect_root() before bailout.
+#define DMNSN_BISECT_ITERATIONS 64
+
+/// Use the false position method to find a root in a range that contains
+/// exactly one root.
+static inline double
+dmnsn_bisect_root(const double poly[], size_t degree, double min, double max)
+{
+  double evmin = dmnsn_polynomial_evaluate(poly, degree, min);
+  double evmax = dmnsn_polynomial_evaluate(poly, degree, max);
+
+  // Handle equal bounds, and equal values at the bounds.
+  if (dmnsn_unlikely(fabs(evmax - evmin) < dmnsn_epsilon)) {
+    return (min + max)/2.0;
+  }
+
+  double evinitial = dmnsn_min(fabs(evmin), fabs(evmax));
+  double mid, evmid;
+  int lastsign = 0;
+
+  for (size_t i = 0; i < DMNSN_BISECT_ITERATIONS; ++i) {
+    mid = (min*evmax - max*evmin)/(evmax - evmin);
+    evmid = dmnsn_polynomial_evaluate(poly, degree, mid);
+    int sign = dmnsn_sgn(evmid);
+
+    if ((fabs(evmid) < fabs(mid)*dmnsn_epsilon
+         // This condition improves stability when one of the bounds is close to
+         // a different root than we are trying to find
+         && fabs(evmid) <= evinitial)
+        || max - min < fabs(mid)*dmnsn_epsilon)
+    {
+      break;
+    }
+
+    if (mid < min) {
+      // This can happen due to numerical instability in the root bounding
+      // algorithm, so behave like the normal secant method
+      max = min;
+      evmax = evmin;
+      min = mid;
+      evmin = evmid;
+    } else if (mid > max) {
+      min = max;
+      evmin = evmax;
+      max = mid;
+      evmax = evmid;
+    } else if (sign == dmnsn_sgn(evmax)) {
+      max = mid;
+      evmax = evmid;
+      if (sign == lastsign) {
+        // Don't allow the algorithm to keep the same endpoint for three
+        // iterations in a row; this ensures superlinear convergence
+        evmin /= 2.0;
+      }
+    } else {
+      min = mid;
+      evmin = evmid;
+      if (sign == lastsign) {
+        evmax /= 2.0;
+      }
+    }
+
+    lastsign = sign;
+  }
+
+  return mid;
+}
+
+/// Use synthetic division to eliminate the root \p r from \p poly[].
+static inline size_t
+dmnsn_eliminate_root(double poly[], size_t degree, double r)
+{
+  double rem = poly[degree];
+  for (size_t i = degree; i-- > 0;) {
+    double temp = poly[i];
+    poly[i] = rem;
+    rem = temp + r*rem;
+  }
+  return degree - 1;
+}
+
+/// Solve a normalized linear polynomial algebraically.
+static inline size_t
+dmnsn_solve_linear(const double poly[2], double x[1])
+{
+  x[0] = -poly[0];
+  if (x[0] >= dmnsn_epsilon)
+    return 1;
+  else
+    return 0;
+}
+
+/// Solve a normalized quadratic polynomial algebraically.
+static inline size_t
+dmnsn_solve_quadratic(const double poly[3], double x[2])
+{
+  double disc = poly[1]*poly[1] - 4.0*poly[0];
+  if (disc >= 0.0) {
+    double s = sqrt(disc);
+    x[0] = (-poly[1] + s)/2.0;
+    x[1] = (-poly[1] - s)/2.0;
+
+    if (x[1] >= dmnsn_epsilon)
+      return 2;
+    else if (x[0] >= dmnsn_epsilon)
+      return 1;
+    else
+      return 0;
+  } else {
+    return 0;
+  }
+}
+
+/// Solve a normalized cubic polynomial algebraically.
+static inline size_t
+dmnsn_solve_cubic(double poly[4], double x[3])
+{
+  // Reduce to a monic trinomial (t^3 + p*t + q, t = x + b/3)
+  double b2 = poly[2]*poly[2];
+  double p = poly[1] - b2/3.0;
+  double q = poly[0] - poly[2]*(9.0*poly[1] - 2.0*b2)/27.0;
+
+  double disc = 4.0*p*p*p + 27.0*q*q;
+  double bdiv3 = poly[2]/3.0;
+
+  if (disc < 0.0) {
+    // Three real roots -- this implies p < 0
+    double msqrtp3 = -sqrt(-p/3.0);
+    double theta = acos(3*q/(2*p*msqrtp3))/3.0;
+
+    // Store the roots in order from largest to smallest
+    x[2] = 2.0*msqrtp3*cos(theta) - bdiv3;
+    x[0] = -2.0*msqrtp3*cos(4.0*atan(1.0)/3.0 - theta) - bdiv3;
+    x[1] = -(x[0] + x[2] + poly[2]);
+
+    if (x[2] >= dmnsn_epsilon)
+      return 3;
+    else if (x[1] >= dmnsn_epsilon)
+      return 2;
+  } else if (disc > 0.0) {
+    // One real root
+    double cbrtdiscq = cbrt(sqrt(disc/108.0) + fabs(q)/2.0);
+    double abst = cbrtdiscq - p/(3.0*cbrtdiscq);
+
+    if (q >= 0) {
+      x[0] = -abst - bdiv3;
+    } else {
+      x[0] = abst - bdiv3;
+    }
+  } else if (fabs(p) < dmnsn_epsilon) {
+    // Equation is a perfect cube
+    x[0] = -bdiv3;
+  } else {
+    // Two real roots; one duplicate
+    double t1 = -(3.0*q)/(2.0*p), t2 = -2.0*t1;
+    x[0] = dmnsn_max(t1, t2) - bdiv3;
+    x[1] = dmnsn_min(t1, t2) - bdiv3;
+    if (x[1] >= dmnsn_epsilon)
+      return 2;
+  }
+
+  if (x[0] >= dmnsn_epsilon)
+    return 1;
+  else
+    return 0;
+}
+
+// Solve a polynomial
+DMNSN_HOT size_t
+dmnsn_polynomial_solve(const double poly[], size_t degree, double x[])
+{
+  // Copy the polynomial so we can be destructive
+  double copy[degree + 1], *p = copy;
+  dmnsn_polynomial_copy(p, poly, degree);
+
+  // Index into x[]
+  size_t i = 0;
+
+  // Account for leading zero coefficients
+  degree = dmnsn_real_degree(p, degree);
+  // Normalize the leading coefficient to 1.0
+  dmnsn_polynomial_normalize(p, degree);
+  // Eliminate simple zero roots
+  dmnsn_eliminate_zero_roots(&p, &degree);
+
+  static const size_t max_algebraic = 3;
+  if (degree > max_algebraic) {
+    // Find isolating intervals for (degree - max_algebraic) roots of p[]
+    double ranges[degree - max_algebraic][2];
+    size_t n = dmnsn_root_bounds(p, degree, ranges, degree - max_algebraic);
+
+    for (size_t j = 0; j < n; ++j) {
+      // Bisect within the found range
+      double r = dmnsn_bisect_root(p, degree, ranges[j][0], ranges[j][1]);
+
+      // Use synthetic division to eliminate the root `r'
+      degree = dmnsn_eliminate_root(p, degree, r);
+
+      // Store the found root
+      x[i] = r;
+      ++i;
+    }
+  }
+
+  switch (degree) {
+  case 1:
+    i += dmnsn_solve_linear(p, x + i);
+    break;
+  case 2:
+    i += dmnsn_solve_quadratic(p, x + i);
+    break;
+  case 3:
+    i += dmnsn_solve_cubic(p, x + i);
+    break;
+  }
+
+  return i;
+}
+
+// Print a polynomial
+void
+dmnsn_polynomial_print(FILE *file, const double poly[], size_t degree)
+{
+  for (size_t i = degree + 1; i-- > 0;) {
+    if (i < degree) {
+      fprintf(file, (poly[i] >= 0.0) ? " + " : " - ");
+    }
+    fprintf(file, "%.17g", fabs(poly[i]));
+    if (i >= 2) {
+      fprintf(file, "*x^%zu", i);
+    } else if (i == 1) {
+      fprintf(file, "*x");
+    }
+  }
+}