public static void Test()
{
var p = new RealPolynomial(new double[] { -1, 0, 1 });
double x = FindRoot(p, 2, 1e-8, 20);
Console.WriteLine("one of the root of p^2-1=0 exists near {0}", x);
}
AlgebraicLongDivision(RealPolynomial dividend, RealPolynomial divisor)
{
if (dividend.Degree < divisor.Degree)
{
throw new ArgumentException("dividendCoef degree is smaller than divisorCoef");
}
AllRealDivisionResult r = new AllRealDivisionResult();
var quotient = new double [dividend.Degree - divisor.Degree + 1];
var divRemain = new double[dividend.Degree + 1];
for (int i = 0; i < divRemain.Length; ++i)
{
divRemain[i] = dividend.C(i);
}
for (int i = 0; i < quotient.Length; ++i)
{
double q = divRemain[dividend.Degree - i]
/ divisor.C(divisor.Degree);
quotient[quotient.Length - 1 - i] = q;
for (int j = 0; j <= divisor.Degree; ++j)
{
divRemain[dividend.Degree - i - j] -=
q * divisor.C(divisor.Degree - j);
}
}
r.quotient = new RealPolynomial(quotient);
r.remainder = new RealRationalPolynomial(
new RealPolynomial(divRemain).RemoveLeadingZeros(),
divisor.RemoveLeadingZeros());
return(r);
}
public string ToString(string variableSymbol)
{
string n = new RealPolynomial(numer).ToString(variableSymbol);
string d = new RealPolynomial(denom).ToString(variableSymbol);
return(string.Format("{{ {0} }} / {{ {1} }}", n, d));
}
public RealRationalPolynomial(RealPolynomial aNumer, RealPolynomial aDenom)
{
numer = new double[aNumer.Degree + 1];
denom = new double[aDenom.Degree + 1];
for (int i = 0; i < numer.Length; ++i)
{
numer[i] = aNumer.C(i);
}
for (int i = 0; i < denom.Length; ++i)
{
denom[i] = aDenom.C(i);
}
}
public static RealPolynomial Add(RealPolynomial l, RealPolynomial r)
{
int degree = (l.Degree < r.Degree) ? r.Degree : l.Degree;
var c = new double[degree + 1];
for (int i = 0; i <= degree; ++i)
{
c[i] = 0;
if (i <= l.Degree)
{
c[i] += l.C(i);
}
if (i <= r.Degree)
{
c[i] += r.C(i);
}
}
return(new RealPolynomial(c));
}
/// <summary>
/// 実係数n次多項式p(x)の定積分 ∫_a^b{p(x)dx}を求める。
/// </summary>
/// <param name="p">実係数n次多項式p(x)。</param>
/// <param name="a">区間a。</param>
/// <param name="b">区間b。</param>
/// <returns>∫_a^b{p(x)dx}</returns>
public double Calc(RealPolynomial p, double a, double b)
{
// xの積分区間 [a b]
// ξの積分区間 [-1 +1]
double flipInterval = 1;
if (b < a)
{
// swap a and b
double t = a;
a = b;
b = t;
flipInterval = -1;
}
int np = NumberOfEvalPoints(p);
var ξwList = GetξwList(np);
// 定積分結果値r。
double r = 0;
for (int k = 0; k < np; ++k)
{
var ξ = ξwList[k].ξ;
var w = ξwList[k].w;
double x = ((a + b) + ξ * (b - a)) / 2.0;
// 地点xでpを評価：p(x)を求める。
double px = p.Evaluate(x);
r += w * px;
}
// 積分変数をx→ξに置き換えたことによるスケーリング。
r *= (b - a) / 2.0;
r *= flipInterval;
return(r);
}
Mul(RealPolynomial lhs, RealPolynomial rhs)
{
// 1次 p 2次 = 3次
// 2次 p 2次 = 4次
// 2次 p 3次 = 5次
// (p^2+2x+3) * (4x+5)
var c = new double[lhs.Degree + rhs.Degree + 1];
// C#では、cの要素は0で初期化される。
for (int iL = 0; iL <= lhs.Degree; ++iL)
{
for (int iR = 0; iR <= rhs.Degree; ++iR)
{
c[iL + iR] += lhs.C(iL) * rhs.C(iR);
}
}
return(new RealPolynomial(c));
}
/// <summary>
/// Find one real root of specified real poly
/// </summary>
/// <param name="coeffs">poly coeffs. coef[0]: constant, coef[1] 1st degree coeff</param>
/// <param name="initialX">initial estimate of the root position</param>
/// <returns>p</returns>
public static double FindRoot(RealPolynomial rpoly, double initialX, double kEpsilon, int loopCount)
{
var deriv = rpoly.Derivative();
double prev = initialX;
double x = initialX;
for (int i = 0; i < loopCount; ++i)
{
double f = rpoly.Evaluate(x);
double fPrime = deriv.Evaluate(x);
x = x - f / fPrime;
if (Math.Abs(prev - x) < kEpsilon)
{
break;
}
prev = x;
}
return(x);
}
/// <summary>
/// 多項式を評価する点の数。
/// </summary>
public static int NumberOfEvalPoints(RealPolynomial p)
{
return(p.Degree / 2 + 1);
}
public static void Test()
{
{
/*
* 部分分数分解のテスト。
* s + 3
* X(s) = -------------
* (s+1)s(s-2)
*
* 部分分数分解すると
*
* 2/3 -3/2 5/6
* X(s) = ------- + ------ + -------
* s + 1 s s - 2
*/
var numerCoeffs = new WWComplex [] {
new WWComplex(3, 0),
new WWComplex(1, 0)
};
var dRoots = new WWComplex [] {
new WWComplex(-1, 0),
WWComplex.Zero(),
new WWComplex(2, 0)
};
var p = WWPolynomial.PartialFractionDecomposition(numerCoeffs, dRoots);
for (int i = 0; i < p.Count; ++i)
{
Console.WriteLine(p[i].ToString("s"));
if (i != p.Count - 1)
{
Console.WriteLine(" + ");
}
}
Console.WriteLine("");
}
{
// 約分のテスト
// p-1 (p+1) -(p+1) + (p-1) -2
// ───── ⇒ ────────────────────── ⇒ 1 + ─────
// p+1 p+1 p+1
var numerC = new List <WWComplex>();
numerC.Add(new WWComplex(-1, 0)); // 定数項。
numerC.Add(WWComplex.Unity()); // 1乗の項。
var denomR = new List <WWComplex>();
denomR.Add(new WWComplex(-1, 0));
var r = Reduction(numerC.ToArray(), denomR.ToArray());
r.Print("p");
}
{
// 約分のテスト
// p^2+3x+2 (p^2+3x+2) -(p^2+7x+12) -4x-10
// ──────────── ⇒ 1+ ───────────────────────── ⇒ 1 + ────────────
// (p+3)(p+4) p^2+7x+12 (p+3)(p+4)
var numerC = new WWComplex [] {
new WWComplex(2, 0), // 定数項。
new WWComplex(3, 0), // 1乗の項。
WWComplex.Unity()
}; // 2乗の項。
var denomR = new WWComplex [] {
new WWComplex(-3, 0),
new WWComplex(-4, 0)
};
var r = Reduction(numerC, denomR);
r.Print("p");
}
{
// 部分分数分解。
// -4x-10 2 -6
// ──────────── ⇒ ───── + ─────
// (p+3)(p+4) p+3 p+4
var numerC = new WWComplex [] {
new WWComplex(-10, 0),
new WWComplex(-4, 0)
};
var denomR = new WWComplex [] {
new WWComplex(-3, 0),
new WWComplex(-4, 0)
};
var p = WWPolynomial.PartialFractionDecomposition(numerC, denomR);
for (int i = 0; i < p.Count; ++i)
{
Console.WriteLine(p[i].ToString("s"));
if (i != p.Count - 1)
{
Console.WriteLine(" + ");
}
}
Console.WriteLine("");
}
{
var deriv = new RealPolynomial(new double[] { 1, 1, 1, 1 }).Derivative();
Console.WriteLine("derivative of p^3+p^2+p+1={0}",
deriv.ToString("p"));
}
{
var r2 = AlgebraicLongDivision(new RealPolynomial(new double[] { 6, 3, 1 }),
new RealPolynomial(new double[] { 1, 1 }));
Console.WriteLine("(p^2+3x+6)/(p+1) = {0} r {1}",
r2.quotient.ToString(), r2.remainder.ToString());
}
}