public WWComplex Evaluate(WWComplex x)
{
WWComplex yN = WWComplex.Add(WWComplex.Mul(x, numer[1]), numer[0]);
WWComplex yD = WWComplex.Add(WWComplex.Mul(x, denom[1]), denom[0]);
return(WWComplex.Div(yN, yD));
}
public WWComplex Evaluate(WWComplex x)
{
WWComplex n = WWComplex.Zero();
{
WWComplex xN = WWComplex.Unity();
for (int i = 0; i < numer.Length; ++i)
{
n = WWComplex.Add(n, WWComplex.Mul(numer[i], xN));
xN = WWComplex.Mul(xN, x);
}
}
WWComplex d = WWComplex.Zero();
{
WWComplex xN = WWComplex.Unity();
for (int i = 0; i < denom.Length; ++i)
{
d = WWComplex.Add(d, WWComplex.Mul(denom[i], xN));
xN = WWComplex.Mul(xN, x);
}
}
return(WWComplex.Div(n, d));
}
PartialFractionDecomposition(
WWComplex [] nCoeffs, WWComplex [] dRoots)
{
var result = new List <FirstOrderComplexRationalPolynomial>();
if (dRoots.Length == 1 && nCoeffs.Length == 1)
{
result.Add(new FirstOrderComplexRationalPolynomial(WWComplex.Zero(),
nCoeffs[0], WWComplex.Unity(), WWComplex.Minus(dRoots[0])));
return(result);
}
if (dRoots.Length < 2)
{
throw new ArgumentException("denomR");
}
if (dRoots.Length < nCoeffs.Length)
{
throw new ArgumentException("nCoeffs");
}
for (int k = 0; k < dRoots.Length; ++k)
{
// cn = ... + nCoeffs[2]s^2 + nCoeffs[1]s + nCoeffs[0]
// 係数c[0]は、s==denomR[0]としたときの、cn/(s-denomR[1])(s-denomR[2])(s-denomR[3])…(s-denomR[p-1])
// 係数c[1]は、s==denomR[1]としたときの、cn/(s-denomR[0])(s-denomR[2])(s-denomR[3])…(s-denomR[p-1])
// 係数c[2]は、s==denomR[2]としたときの、cn/(s-denomR[0])(s-denomR[1])(s-denomR[3])…(s-denomR[p-1])
// 分子の値c。
var c = WWComplex.Zero();
var s = WWComplex.Unity();
for (int j = 0; j < nCoeffs.Length; ++j)
{
c = WWComplex.Add(c, WWComplex.Mul(nCoeffs[j], s));
s = WWComplex.Mul(s, dRoots[k]);
}
for (int i = 0; i < dRoots.Length; ++i)
{
if (i == k)
{
continue;
}
c = WWComplex.Div(c, WWComplex.Sub(dRoots[k], dRoots[i]));
}
result.Add(new FirstOrderComplexRationalPolynomial(
WWComplex.Zero(), c,
WWComplex.Unity(), WWComplex.Minus(dRoots[k])));
}
return(result);
}
Reduction(WWComplex [] aNumerCoeffList, WWComplex [] aDenomRootList)
{
var rv = new PolynomialAndRationalPolynomial();
rv.coeffList = new WWComplex[0];
rv.numerCoeffList = new WWComplex[aNumerCoeffList.Length];
Array.Copy(aNumerCoeffList, rv.numerCoeffList, aNumerCoeffList.Length);
rv.denomRootList = new WWComplex[aDenomRootList.Length];
Array.Copy(aDenomRootList, rv.denomRootList, aDenomRootList.Length);
if (rv.numerCoeffList.Length - 1 < rv.denomRootList.Length)
{
// 既に分子の多項式の次数が分母の多項式の次数よりも低い。
return(rv);
}
// 分母の根のリスト⇒分母の多項式の係数のリストに変換。
var denomCoeffList = RootListToCoeffList(aDenomRootList, new WWComplex(1, 0));
var quotient = new List <WWComplex>();
while (denomCoeffList.Length <= rv.numerCoeffList.Length)
{
// denomの最も次数が高い項の係数がcdn、numerの最も次数が高い項の係数がcnnとすると
// denomiCoeffListを-cnn/cdn * s^(numerの次数-denomの次数)倍してnumerCoeffListと足し合わせてnumerの次数を1下げる。
// このときrv.coeffListにc == cnn/cdnを足す。
var c = WWComplex.Div(rv.numerCoeffList[rv.numerCoeffList.Length - 1],
denomCoeffList[denomCoeffList.Length - 1]);
quotient.Insert(0, c);
var denomMulX = new ComplexPolynomial(denomCoeffList);
while (denomMulX.Degree + 1 < rv.numerCoeffList.Length)
{
denomMulX = ComplexPolynomial.MultiplyX(denomMulX);
}
denomMulX = ComplexPolynomial.MultiplyC(denomMulX, WWComplex.Mul(c, -1));
// ここで引き算することで分子の多項式の次数が1減る。
int expectedOrder = rv.numerCoeffList.Length - 2;
rv.numerCoeffList = ComplexPolynomial.TrimPolynomialOrder(
ComplexPolynomial.Add(denomMulX, new ComplexPolynomial(rv.numerCoeffList)),
expectedOrder).ToArray();
// 引き算によって一挙に2以上次数が減った場合coeffListに0を足す。
int actualOrder = rv.numerCoeffList.Length - 1;
for (int i = 0; i < expectedOrder - actualOrder; ++i)
{
quotient.Insert(0, WWComplex.Zero());
}
}
rv.coeffList = quotient.ToArray();
return(rv);
}
/// <summary>
/// see this document: http://people.csail.mit.edu/bkph/articles/Quadratics.pdf
/// </summary>
private void FindRootQuadratic(double[] coeffs)
{
// 2次多項式 quadratic poly equation
System.Diagnostics.Debug.Assert(3 == coeffs.Length);
double a = coeffs[2];
double b = coeffs[1];
double c = coeffs[0];
double discriminant = b * b - 4 * a * c;
if (0 <= discriminant)
{
double x1;
double x2;
if (0 <= b)
{
x1 = 2 * c / (-b - Math.Sqrt(discriminant));
x2 = (-b - Math.Sqrt(discriminant)) / 2 / a;
}
else
{
x1 = (-b + Math.Sqrt(discriminant)) / 2 / a;
x2 = 2 * c / (-b + Math.Sqrt(discriminant));
}
mRoots.Add(new WWComplex(x1, 0));
mRoots.Add(new WWComplex(x2, 0));
}
else
{
WWComplex x1;
WWComplex x2;
if (0 <= b)
{
x1 = new WWComplex(-b / 2 / a, -Math.Sqrt(-discriminant) / 2 / a);
x2 = WWComplex.Div(new WWComplex(2 * c, 0), new WWComplex(-b, -Math.Sqrt(-discriminant)));
}
else
{
x1 = WWComplex.Div(new WWComplex(2 * c, 0), new WWComplex(-b, Math.Sqrt(-discriminant)));
x2 = new WWComplex(-b / 2 / a, Math.Sqrt(-discriminant) / 2 / a);
}
mRoots.Add(x1);
mRoots.Add(x2);
}
}
public WWComplex Evaluate(WWComplex x)
{
WWComplex n = WWComplex.Zero();
WWComplex xN = WWComplex.Unity();
for (int i = 0; i < numer.Length; ++i)
{
n = WWComplex.Add(n, WWComplex.Mul(xN, numer[i]));
xN = WWComplex.Mul(xN, x);
}
WWComplex d = WWComplex.Zero();
xN = WWComplex.Unity();
for (int i = 0; i < denom.Length; ++i)
{
d = WWComplex.Add(d, WWComplex.Mul(xN, denom[i]));
xN = WWComplex.Mul(xN, x);
}
WWComplex y = WWComplex.Div(n, d);
return(y);
}
private void FindRootCubic(double[] coeffs)
{
// 3次多項式 cubic poly equation
// https://en.wikipedia.org/wiki/Cubic_function#Algebraic_solution
System.Diagnostics.Debug.Assert(4 == coeffs.Length);
double a = coeffs[3];
double b = coeffs[2];
double c = coeffs[1];
double d = coeffs[0];
double Δ =
18 * a * b * c * d
- 4 * b * b * b * d
+ 1 * b * b * c * c
- 4 * a * c * c * c
- 27 * a * a * d * d;
double Δ0 = b * b - 3 * a * c;
// ↓ 雑な0かどうかの判定処理。
if (AlmostZero(Δ))
{
if (AlmostZero(Δ0))
{
// triple root
double root = -b / (3 * a);
mRoots.Add(new WWComplex(root, 0));
mRoots.Add(new WWComplex(root, 0));
mRoots.Add(new WWComplex(root, 0));
return;
}
// double root and a simple root
double root2 = (9 * a * d - b * c) / (2 * Δ0);
double root1 = (4 * a * b * c - 9 * a * a * d - b * b * b);
mRoots.Add(new WWComplex(root2, 0));
mRoots.Add(new WWComplex(root2, 0));
mRoots.Add(new WWComplex(root1, 0));
return;
}
{
double Δ1 =
+2 * b * b * b
- 9 * a * b * c
+ 27 * a * a * d;
WWComplex C;
{
WWComplex c1 = new WWComplex(Δ1 / 2, 0);
double c2Sqrt = Math.Sqrt(Math.Abs(-27 * a * a * Δ) / 4);
WWComplex c2;
if (Δ < 0)
{
//C2 is real number
c2 = new WWComplex(c2Sqrt, 0);
}
else
{
//C2 is imaginary number
c2 = new WWComplex(0, c2Sqrt);
}
WWComplex c1c2;
WWComplex c1Pc2 = WWComplex.Add(c1, c2);
WWComplex c1Mc2 = WWComplex.Sub(c1, c2);
if (c1Mc2.Magnitude() <= c1Pc2.Magnitude())
{
c1c2 = c1Pc2;
}
else
{
c1c2 = c1Mc2;
}
// 3乗根 = 大きさが3分の1乗で角度が3分の1．
double magnitude = c1c2.Magnitude();
double phase = c1c2.Phase();
double cMag = Math.Pow(magnitude, 1.0 / 3.0);
double cPhase = phase / 3;
C = new WWComplex(cMag * Math.Cos(cPhase), cMag * Math.Sin(cPhase));
}
var ζ = new WWComplex(-1.0 / 2.0, 1.0 / 2.0 * Math.Sqrt(3.0));
var ζ2 = new WWComplex(-1.0 / 2.0, -1.0 / 2.0 * Math.Sqrt(3.0));
var r3a = new WWComplex(-1.0 / 3.0 / a, 0);
WWComplex root0 = WWComplex.Mul(r3a, WWComplex.Add(
WWComplex.Add(new WWComplex(b, 0), C),
WWComplex.Div(new WWComplex(Δ0, 0), C)));
WWComplex root1 = WWComplex.Mul(r3a, WWComplex.Add(
WWComplex.Add(new WWComplex(b, 0), WWComplex.Mul(ζ, C)),
WWComplex.Div(new WWComplex(Δ0, 0), WWComplex.Mul(ζ, C))));
WWComplex root2 = WWComplex.Mul(r3a, WWComplex.Add(
WWComplex.Add(new WWComplex(b, 0), WWComplex.Mul(ζ2, C)),
WWComplex.Div(new WWComplex(Δ0, 0), WWComplex.Mul(ζ2, C))));
mRoots.Add(root0);
mRoots.Add(root1);
mRoots.Add(root2);
return;
}
}
