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// from http://tog.acm.org/resources/GraphicsGems/gems/Roots3And4.c
/*
* Roots3And4.c
*
* Utility functions to find cubic and quartic roots,
* coefficients are passed like this:
*
* c[0] + c[1]*x + c[2]*x^2 + c[3]*x^3 + c[4]*x^4 = 0
*
* The functions return the number of non-complex roots and
* put the values into the s array.
*
* Author: Jochen Schwarze (schwarze@isa.de)
*
* Jan 26, 1990 Version for Graphics Gems
* Oct 11, 1990 Fixed sign problem for negative q's in SolveQuartic
* (reported by Mark Podlipec),
* Old-style function definitions,
* IsZero() as a macro
* Nov 23, 1990 Some systems do not declare acos() and cbrt() in
* <math.h>, though the functions exist in the library.
* If large coefficients are used, EQN_EPS should be
* reduced considerably (e.g. to 1E-30), results will be
* correct but multiple roots might be reported more
* than once.
*/
#include <math.h>
#include "CubicUtilities.h"
#include "QuadraticUtilities.h"
#include "QuarticRoot.h"
int reducedQuarticRoots(const double t4, const double t3, const double t2, const double t1,
const double t0, const bool oneHint, double roots[4]) {
#if SK_DEBUG
// create a string mathematica understands
// GDB set print repe 15 # if repeated digits is a bother
// set print elements 400 # if line doesn't fit
char str[1024];
bzero(str, sizeof(str));
sprintf(str, "Solve[%1.19g x^4 + %1.19g x^3 + %1.19g x^2 + %1.19g x + %1.19g == 0, x]",
t4, t3, t2, t1, t0);
#endif
if (approximately_zero_when_compared_to(t4, t0) // 0 is one root
&& approximately_zero_when_compared_to(t4, t1)
&& approximately_zero_when_compared_to(t4, t2)
&& approximately_zero_when_compared_to(t4, t3)) {
if (approximately_zero_when_compared_to(t3, t0)
&& approximately_zero_when_compared_to(t3, t1)
&& approximately_zero_when_compared_to(t3, t2)) {
return quadraticRootsReal(t2, t1, t0, roots);
}
return cubicRootsReal(t3, t2, t1, t0, roots);
}
if (approximately_zero_when_compared_to(t0, t1) // 0 is one root
&& approximately_zero_when_compared_to(t0, t2)
&& approximately_zero_when_compared_to(t0, t3)
&& approximately_zero_when_compared_to(t0, t4)) {
int num = cubicRootsReal(t4, t3, t2, t1, roots);
for (int i = 0; i < num; ++i) {
if (approximately_zero(roots[i])) {
return num;
}
}
roots[num++] = 0;
return num;
}
if (oneHint) {
SkASSERT(approximately_zero(t4 + t3 + t2 + t1 + t0)); // 1 is one root
int num = cubicRootsReal(t4, t4 + t3, -(t1 + t0), -t0, roots); // note that -C==A+B+D+E
for (int i = 0; i < num; ++i) {
if (approximately_equal(roots[i], 1)) {
return num;
}
}
roots[num++] = 1;
return num;
}
return -1;
}
int quarticRootsReal(const double A, const double B, const double C, const double D,
const double E, double s[4]) {
double u, v;
/* normal form: x^4 + Ax^3 + Bx^2 + Cx + D = 0 */
const double invA = 1 / A;
const double a = B * invA;
const double b = C * invA;
const double c = D * invA;
const double d = E * invA;
/* substitute x = y - a/4 to eliminate cubic term:
x^4 + px^2 + qx + r = 0 */
const double a2 = a * a;
const double p = -3 * a2 / 8 + b;
const double q = a2 * a / 8 - a * b / 2 + c;
const double r = -3 * a2 * a2 / 256 + a2 * b / 16 - a * c / 4 + d;
int num;
if (approximately_zero(r)) {
/* no absolute term: y(y^3 + py + q) = 0 */
num = cubicRootsReal(1, 0, p, q, s);
s[num++] = 0;
} else {
/* solve the resolvent cubic ... */
double cubicRoots[3];
int roots = cubicRootsReal(1, -p / 2, -r, r * p / 2 - q * q / 8, cubicRoots);
int index;
#if 0 && SK_DEBUG // enable to verify that any cubic root is as good as any other
double tries[3][4];
int nums[3];
for (index = 0; index < roots; ++index) {
/* ... and take one real solution ... */
const double z = cubicRoots[index];
/* ... to build two quadric equations */
u = z * z - r;
v = 2 * z - p;
if (approximately_zero_squared(u)) {
u = 0;
} else if (u > 0) {
u = sqrt(u);
} else {
SkDebugf("%s u=%1.9g <0\n", __FUNCTION__, u);
continue;
}
if (approximately_zero_squared(v)) {
v = 0;
} else if (v > 0) {
v = sqrt(v);
} else {
SkDebugf("%s v=%1.9g <0\n", __FUNCTION__, v);
continue;
}
nums[index] = quadraticRootsReal(1, q < 0 ? -v : v, z - u, tries[index]);
nums[index] += quadraticRootsReal(1, q < 0 ? v : -v, z + u, tries[index] + nums[index]);
/* resubstitute */
const double sub = a / 4;
for (int i = 0; i < nums[index]; ++i) {
tries[index][i] -= sub;
}
}
for (index = 0; index < roots; ++index) {
SkDebugf("%s", __FUNCTION__);
for (int idx2 = 0; idx2 < nums[index]; ++idx2) {
SkDebugf(" %1.9g", tries[index][idx2]);
}
SkDebugf("\n");
}
#endif
/* ... and take one real solution ... */
double z;
num = 0;
int num2 = 0;
for (index = 0; index < roots; ++index) {
z = cubicRoots[index];
/* ... to build two quadric equations */
u = z * z - r;
v = 2 * z - p;
if (approximately_zero_squared(u)) {
u = 0;
} else if (u > 0) {
u = sqrt(u);
} else {
continue;
}
if (approximately_zero_squared(v)) {
v = 0;
} else if (v > 0) {
v = sqrt(v);
} else {
continue;
}
num = quadraticRootsReal(1, q < 0 ? -v : v, z - u, s);
num2 = quadraticRootsReal(1, q < 0 ? v : -v, z + u, s + num);
if (!((num | num2) & 1)) {
break; // prefer solutions without single quad roots
}
}
num += num2;
if (!num) {
return 0; // no valid cubic root
}
}
/* resubstitute */
const double sub = a / 4;
for (int i = 0; i < num; ++i) {
s[i] -= sub;
}
// eliminate duplicates
for (int i = 0; i < num - 1; ++i) {
for (int j = i + 1; j < num; ) {
if (AlmostEqualUlps(s[i], s[j])) {
if (j < --num) {
s[j] = s[num];
}
} else {
++j;
}
}
}
return num;
}
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