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+/*************************************************************************
+ * 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");
+ }
+ }
+}
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