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Improved detection of numerical errors in cubic equation solver

This commit is contained in:
Chlumsky 2021-11-13 17:02:40 +01:00
parent f1797ada88
commit 0b633e75f7
1 changed files with 21 additions and 26 deletions
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47
core/equation-solver.cpp
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@ -4,17 +4,15 @@
#define _USE_MATH_DEFINES #define _USE_MATH_DEFINES
#include <cmath> #include <cmath>
#define TOO_LARGE_RATIO 1e12
namespace msdfgen { namespace msdfgen {
int solveQuadratic(double x[2], double a, double b, double c) { int solveQuadratic(double x[2], double a, double b, double c) {
// a = 0 -> linear equation // a == 0 -> linear equation
if (a == 0 || fabs(b)+fabs(c) > TOO_LARGE_RATIO*fabs(a)) { if (a == 0 || fabs(b) > 1e12*fabs(a)) {
// a, b = 0 -> no solution // a == 0, b == 0 -> no solution
if (b == 0 || fabs(c) > TOO_LARGE_RATIO*fabs(b)) { if (b == 0) {
if (c == 0) if (c == 0)
return -1; // 0 = 0 return -1; // 0 == 0
return 0; return 0;
} }
x[0] = -c/b; x[0] = -c/b;
@ -35,41 +33,38 @@ int solveQuadratic(double x[2], double a, double b, double c) {
static int solveCubicNormed(double x[3], double a, double b, double c) { static int solveCubicNormed(double x[3], double a, double b, double c) {
double a2 = a*a; double a2 = a*a;
double q = (a2 - 3*b)/9; double q = 1/9.*(a2-3*b);
double r = (a*(2*a2-9*b) + 27*c)/54; double r = 1/54.*(a*(2*a2-9*b)+27*c);
double r2 = r*r; double r2 = r*r;
double q3 = q*q*q; double q3 = q*q*q;
double A, B; a *= 1/3.;
if (r2 < q3) { if (r2 < q3) {
double t = r/sqrt(q3); double t = r/sqrt(q3);
if (t < -1) t = -1; if (t < -1) t = -1;
if (t > 1) t = 1; if (t > 1) t = 1;
t = acos(t); t = acos(t);
a /= 3; q = -2*sqrt(q); q = -2*sqrt(q);
x[0] = q*cos(t/3)-a; x[0] = q*cos(1/3.*t)-a;
x[1] = q*cos((t+2*M_PI)/3)-a; x[1] = q*cos(1/3.*(t+2*M_PI))-a;
x[2] = q*cos((t-2*M_PI)/3)-a; x[2] = q*cos(1/3.*(t-2*M_PI))-a;
return 3; return 3;
} else { } else {
A = -pow(fabs(r)+sqrt(r2-q3), 1/3.); double u = (r < 0 ? 1 : -1)*pow(fabs(r)+sqrt(r2-q3), 1/3.);
if (r < 0) A = -A; double v = u == 0 ? 0 : q/u;
B = A == 0 ? 0 : q/A; x[0] = (u+v)-a;
a /= 3; if (u == v || fabs(u-v) < 1e-12*fabs(u+v)) {
x[0] = (A+B)-a; x[1] = -.5*(u+v)-a;
x[1] = -0.5*(A+B)-a;
x[2] = 0.5*sqrt(3.)*(A-B);
if (fabs(x[2]) < 1e-14)
return 2; return 2;
}
return 1; return 1;
} }
} }
int solveCubic(double x[3], double a, double b, double c, double d) { int solveCubic(double x[3], double a, double b, double c, double d) {
if (a != 0) { if (a != 0) {
double bn = b/a, cn = c/a, dn = d/a; double bn = b/a;
// Check that a isn't "almost zero" if (fabs(bn) < 1e6) // Above this ratio, the numerical error gets larger than if we treated a as zero
if (fabs(bn) < TOO_LARGE_RATIO && fabs(cn) < TOO_LARGE_RATIO && fabs(dn) < TOO_LARGE_RATIO) return solveCubicNormed(x, bn, c/a, d/a);
return solveCubicNormed(x, bn, cn, dn);
} }
return solveQuadratic(x, b, c, d); return solveQuadratic(x, b, c, d);
} }