public static Q Brent(FunctionOfOneVariable f, Q left, Q right)
{
// extra info that callers may not always want
int iterationsUsed;
Q errorEstimate;
return Brent(f, left, right, new Q(1, 100000000), Q.Zero, out iterationsUsed, out errorEstimate);
}
private static IEnumerable <Range <Frac> > ApproximateRootsHelper(this Polynomial <XTerm> polynomial, Frac epsilon)
{
if (epsilon <= 0)
{
throw new ArgumentOutOfRangeException("epsilon");
}
if (!polynomial.Coefficients.Any())
{
throw new InvalidOperationException("Everything is a root...");
}
var degree = polynomial.Degree();
if (degree == 0)
{
return(new Range <Frac> [0]);
}
if (degree == 1)
{
return new Range <Frac>[] { -polynomial.Coefficient(XToThe(0)) / polynomial.Coefficient(XToThe(1)) }
}
;
var criticalRegions =
polynomial
.Derivative()
.ApproximateRoots(epsilon)
.ToArray();
if (criticalRegions.Length == 0)
{
criticalRegions = new Range <Frac>[] { Frac.Zero }
}
;
var lowerRoot = polynomial.BisectLower(criticalRegions.First().Min, epsilon);
var upperRoot = polynomial.BisectUpper(criticalRegions.Last().Max, epsilon);
var r0 = new[] { lowerRoot, upperRoot }.WhereHasValue();
var r1 = criticalRegions.Where(r => Frac.Abs(polynomial.EvaluateAt((r.Min + r.Max) / 2)) < epsilon);
var interRegions =
criticalRegions
.Window(2)
.Where(e => e.Count() == 2)
.Select(e => new Range <Frac>(e.First().Max, e.Last().Min, false, false))
.ToArray();
var r2 = interRegions.Select(e => polynomial.Bisect(e.Min, e.Max, epsilon)).WhereHasValue();
return(r0.Concat(r1).Concat(r2));
}
public static Q Bisect(FunctionOfOneVariable f, Q left, Q right, Q tolerance, Q target, out int iterationsUsed, out Q errorEstimate)
{
if (tolerance <= Q.Zero)
{
string msg = string.Format("Tolerance must be positive. Recieved {0}.", tolerance);
throw new ArgumentOutOfRangeException(msg);
}
iterationsUsed = 0;
errorEstimate = tolerance * 2;
// Standardize the problem. To solve f(x) = target,
// solve g(x) = 0 where g(x) = f(x) - target.
FunctionOfOneVariable g = delegate (Q x) { return f(x) - target; };
Q g_left = g(left); // evaluation of f at left end of interval
Q g_right = g(right);
Q mid;
Q g_mid;
if (g_left * g_right >= Q.Zero)
{
string str = "Invalid starting bracket. Function must be above target on one end and below target on other end.";
string msg = string.Format("{0} Target: {1}. f(left) = {2}. f(right) = {3}", str, g_left + target, g_right + target);
throw new ArgumentException(msg);
}
Q intervalWidth = right - left;
for (iterationsUsed = 0; iterationsUsed < maxIterations && intervalWidth > tolerance; iterationsUsed++)
{
intervalWidth *= new Q(1, 2);
mid = left + intervalWidth;
if ((g_mid = g(mid)) == Q.Zero)
{
errorEstimate = Q.Zero;
return mid;
}
if (g_left * g_mid < Q.Zero) // g changes sign in (left, mid)
g_right = g(right = mid);
else // g changes sign in (mid, right)
g_left = g(left = mid);
}
errorEstimate = right - left;
return left;
}
public static Q Brent(FunctionOfOneVariable g, Q left, Q right, Q tolerance, Q target, out int iterationsUsed, out Q errorEstimate)
{
if (tolerance <= Q.Zero)
{
string msg = string.Format("Tolerance must be positive. Recieved {0}.", tolerance);
throw new ArgumentOutOfRangeException(msg);
}
errorEstimate = tolerance * 2;
// Standardize the problem. To solve g(x) = target,
// solve f(x) = 0 where f(x) = g(x) - target.
FunctionOfOneVariable f = delegate (Q x) { return g(x) - target; };
// Implementation and notation based on Chapter 4 in
// "Algorithms for Minimization without Derivatives"
// by Richard Brent.
Q c, d, e, fa, fb, fc, tol, m, p, q, r, s;
// set up aliases to match Brent's notation
Q a = left; Q b = right; Q t = tolerance;
iterationsUsed = 0;
fa = f(a);
fb = f(b);
if (fa * fb > Q.Zero)
{
string str = "Invalid starting bracket. Function must be above target on one end and below target on other end.";
string msg = string.Format("{0} Target: {1}. f(left) = {2}. f(right) = {3}", str, target, fa + target, fb + target);
throw new ArgumentException(msg);
}
label_int:
c = a; fc = fa; d = e = b - a;
label_ext:
if (Q.Abs(fc) < Q.Abs(fb))
{
a = b; b = c; c = a;
fa = fb; fb = fc; fc = fa;
}
iterationsUsed++;
tol = 2.0 * t * Q.Abs(b) + t;
errorEstimate = m = 0.5 * (c - b);
if (Q.Abs(m) > tol && fb != Q.Zero) // exact comparison with 0 is OK here
{
// See if bisection is forced
if (Q.Abs(e) < tol || Q.Abs(fa) <= Q.Abs(fb))
{
d = e = m;
}
else
{
s = fb / fa;
if (a == c)
{
// linear interpolation
p = 2.0 * m * s; q = 1.0 - s;
}
else
{
// Inverse quadratic interpolation
q = fa / fc; r = fb / fc;
p = s * (2.0 * m * q * (q - r) - (b - a) * (r - 1.0));
q = (q - 1.0) * (r - 1.0) * (s - 1.0);
}
if (p > 0.0)
q = -q;
else
p = -p;
s = e; e = d;
if (new Q(2, 1) * p < new Q(3, 1) * m * q - Q.Abs(tol * q) && p < Q.Abs(new Q(1, 2) * s * q))
d = p / q;
else
d = e = m;
}
a = b; fa = fb;
if (Q.Abs(d) > tol)
b += d;
else if (m > Q.Zero)
b += tol;
else
b -= tol;
if (iterationsUsed == maxIterations)
return b;
fb = f(b);
if ((fb > Q.Zero && fc > Q.Zero) || (fb <= Q.Zero && fc <= Q.Zero))
goto label_int;
else
goto label_ext;
}
else
return b;
}
public static Q Newton(FunctionOfOneVariable f, FunctionOfOneVariable fprime, Q guess, Q tolerance, Q target, out int iterationsUsed, out Q errorEstimate)
{
if (tolerance <= Q.Zero)
{
string msg = string.Format("Tolerance must be positive. Recieved {0}.", tolerance);
throw new ArgumentOutOfRangeException(msg);
}
iterationsUsed = 0;
errorEstimate = tolerance * 2;
// Standardize the problem. To solve f(x) = target,
// solve g(x) = 0 where g(x) = f(x) - target.
// Note that f(x) and g(x) have the same derivative.
FunctionOfOneVariable g = delegate (Q x) { return f(x) - target; };
Q oldX, newX = guess;
for (iterationsUsed = 0; iterationsUsed < maxIterations && errorEstimate > tolerance; iterationsUsed++)
{
oldX = newX;
Q gx = g(oldX);
Q gprimex = fprime(oldX);
Q absgprimex = Q.Abs(gprimex);
newX = oldX - gx / gprimex;
errorEstimate = Q.Abs(newX - oldX);
}
return newX;
}
public static Q Newton(FunctionOfOneVariable f, FunctionOfOneVariable fprime, Q guess)
{
// extra info that callers may not always want
int iterationsUsed;
Q errorEstimate;
return Newton(f, fprime, guess, new Q(1, 100000000), Q.Zero, out iterationsUsed, out errorEstimate);
}
private static May <Range <Frac> > Bisect(this Polynomial <XTerm> polynomial, Frac minX, Frac maxX, Frac epsilon)
{
var minS = polynomial.EvaluateAt(minX).Sign;
var maxS = polynomial.EvaluateAt(maxX).Sign;
if (minS == 0)
{
return((Range <Frac>)minX);
}
if (maxS == 0)
{
return((Range <Frac>)maxX);
}
if (minS == maxS)
{
return(May.NoValue);
}
while (maxX - minX > epsilon)
{
var x = (minX + maxX) / 2;
var y = polynomial.EvaluateAt(x);
if (y == 0)
{
return((Range <Frac>)x);
}
if (y.Sign == minS)
{
minX = x;
}
else
{
maxX = x;
}
}
return(new Range <Frac>(minX, maxX, false, false));
}
private static May <Range <Frac> > BisectUpper(this Polynomial <XTerm> polynomial, Frac minX, Frac epsilon)
{
var increasingLimitSign = polynomial.Coefficients.MaxBy(e => e.Key.XPower).Value.Sign;
var minS = polynomial.EvaluateAt(minX).Sign;
if (minS == increasingLimitSign)
{
return(May.NoValue);
}
if (minS == 0)
{
return((Range <Frac>)minX);
}
var d = Frac.One;
while (true)
{
d *= 2;
var maxX = minX + d;
if (polynomial.EvaluateAt(maxX).Sign != increasingLimitSign)
{
continue;
}
return(polynomial.Bisect(minX, maxX, epsilon));
}
}
private static May <Range <Frac> > BisectLower(this Polynomial <XTerm> polynomial, Frac maxX, Frac epsilon)
{
var maxCoef = polynomial.Coefficients.MaxBy(e => e.Key.XPower);
var decreasingLimitSign = maxCoef.Value.Sign * (maxCoef.Key.XPower % 2 == 0 ? +1 : -1);
var maxS = polynomial.EvaluateAt(maxX).Sign;
if (maxS == decreasingLimitSign)
{
return(May.NoValue);
}
if (maxS == 0)
{
return((Range <Frac>)maxX);
}
var d = Frac.One;
while (true)
{
d *= 2;
var minX = maxX - d;
if (polynomial.EvaluateAt(minX).Sign != decreasingLimitSign)
{
continue;
}
return(polynomial.Bisect(minX, maxX, epsilon));
}
}
public static IEnumerable <Range <Frac> > ApproximateRoots(this Polynomial <XTerm> polynomial, Frac epsilon)
{
return(ApproximateRootsHelper(polynomial, epsilon).OrderBy(e => e.Min).Distinct());
}
public static Frac EvaluateAt(this Polynomial <XYTerm> polynomial, Frac x, Frac y)
{
return(polynomial.Coefficients.Select(e => Frac.Pow(x, (int)e.Key.XPower) * Frac.Pow(y, (int)e.Key.YPower) * e.Value).Sum());
}