C#，码海拾贝（49）——求“复系数代数方程“全部根的“牛顿下山法“之C#源代码
using System;
namespace Zhou.CSharp.Algorithm
{
public delegate double delFunction_x(double x);
public delegate double delFunction_xa(double[] x);
public delegate double delFunction_x_y(double x, double y);
public delegate double delFunction_x_ya(double x, double[] y);
public delegate double delFunction_xa_ya(double[] x, double[] y);
///
/// 求解非线性方程组的类 NLEquations
/// 原作 周长发
/// 改编 深度混淆
///
public static partial class NLEquations
{
///
/// 求复系数代数方程全部根的牛顿下山法
///
/// 多项式方程的次数
/// 一维数组，长度为n+1，按降幂次序依次存放n次多项式方程的 n+1个系数的实部
/// 一维数组，长度为n+1，按降幂次序依次存放n次多项式方程的 n+1个系数的虚部
/// 一维数组，长度为n，返回n个根的实部
/// 一维数组，长度为n，返回n个根的虚部
///
public static bool GetRootNewtonDownHill(int n, double[] ar, double[] ai, double[] xr, double[] xi)
{
int m = 0, i = 0, jt = 0, k = 0, nis = 0, it = 0;
double t = 0, x = 0, y = 0, x1 = 0, y1 = 0, dx = 0, dy = 0;
double p = 0, q = 0, w = 0, dd = 0, dc = 0, c = 0;
double g = 0, u = 0, v = 0, pq = 0, g1 = 0, u1 = 0, v1 = 0;
// 初始判断
m = n;
p = Math.Sqrt(ar[m] * ar[m] + ai[m] * ai[m]);
while ((m > 0) && (p + 1.0 == 1.0))
{
m = m - 1;
p = Math.Sqrt(ar[m] * ar[m] + ai[m] * ai[m]);
}
// 求解失败
if (m <= 0)
{
return false;
}
for (i = 0; i <= m; i++)
{
ar[i] = ar[i] / p;
ai[i] = ai[i] / p;
}
for (i = 0; i <= m / 2; i++)
{
w = ar[i];
ar[i] = ar[m - i];
ar[m - i] = w;
w = ai[i];
ai[i] = ai[m - i];
ai[m - i] = w;
}
// 迭代求解
k = m;
nis = 0;
w = 1.0;
jt = 1;
while (jt == 1)
{
pq = Math.Sqrt(ar[k] * ar[k] + ai[k] * ai[k]);
while (pq < 1.0e-12)
{
xr[k - 1] = 0.0;
xi[k - 1] = 0.0;
k = k - 1;
if (k == 1)
{
p = ar[0] * ar[0] + ai[0] * ai[0];
xr[0] = -w * (ar[0] * ar[1] + ai[0] * ai[1]) / p;
xi[0] = w * (ar[1] * ai[0] - ar[0] * ai[1]) / p;
return true;
}
pq = Math.Sqrt(ar[k] * ar[k] + ai[k] * ai[k]);
}
q = Math.Log(pq);
q = q / (1.0 * k);
q = Math.Exp(q);
p = q;
w = w * p;
for (i = 1; i <= k; i++)
{
ar[i] = ar[i] / q;
ai[i] = ai[i] / q;
q = q * p;
}
x = 0.0001;
x1 = x;
y = 0.2;
y1 = y;
dx = 1.0;
g = 1.0e+37;
while (true)
{
u = ar[0];
v = ai[0];
for (i = 1; i <= k; i++)
{
p = u * x1;
q = v * y1;
pq = (u + v) * (x1 + y1);
u = p - q + ar[i];
v = pq - p - q + ai[i];
}
g1 = u * u + v * v;
if (g1 >= g)
{
if (nis != 0)
{
it = 1;
g65c(ref x, ref y, ref x1, ref y1, ref dx, ref dy, ref dd, ref dc, ref c, ref k, ref nis, ref it);
if (it == 0)
{
continue;
}
}
else
{
g60c(ref t, ref x, ref y, ref x1, ref y1, ref dx, ref dy, ref p, ref q, ref k, ref it);
if (t >= 1.0e-03)
{
continue;
}
if (g > 1.0e-18)
{
it = 0;
g65c(ref x, ref y, ref x1, ref y1, ref dx, ref dy, ref dd, ref dc, ref c, ref k, ref nis, ref it);
if (it == 0)
{
continue;
}
}
}
g90c(xr, xi, ar, ai, ref x, ref y, ref p, ref w, ref k);
break;
}
else
{
g = g1;
x = x1;
y = y1;
nis = 0;
if (g <= 1.0e-22)
{
g90c(xr, xi, ar, ai, ref x, ref y, ref p, ref w, ref k);
}
else
{
u1 = k * ar[0];
v1 = ai[0];
for (i = 2; i <= k; i++)
{
p = u1 * x;
q = v1 * y;
pq = (u1 + v1) * (x + y);
u1 = p - q + (k - i + 1) * ar[i - 1];
v1 = pq - p - q + (k - i + 1) * ai[i - 1];
}
p = u1 * u1 + v1 * v1;
if (p <= 1.0e-20)
{
it = 0;
g65c(ref x, ref y, ref x1, ref y1, ref dx, ref dy, ref dd, ref dc, ref c, ref k, ref nis, ref it);
if (it == 0)
{
continue;
}
g90c(xr, xi, ar, ai, ref x, ref y, ref p, ref w, ref k);
}
else
{
dx = (u * u1 + v * v1) / p;
dy = (u1 * v - v1 * u) / p;
t = 1.0 + 4.0 / k;
g60c(ref t, ref x, ref y, ref x1, ref y1, ref dx, ref dy, ref p, ref q, ref k, ref it);
if (t >= 1.0e-03)
{
continue;
}
if (g > 1.0e-18)
{
it = 0;
g65c(ref x, ref y, ref x1, ref y1, ref dx, ref dy, ref dd, ref dc, ref c, ref k, ref nis, ref it);
if (it == 0)
{
continue;
}
}
g90c(xr, xi, ar, ai, ref x, ref y, ref p, ref w, ref k);
}
}
break;
}
}
if (k == 1)
{
jt = 0;
}
else
{
jt = 1;
}
}
return true;
}
///
/// 内部函数
///
private static void g60c(ref double t, ref double x, ref double y, ref double x1, ref double y1, ref double dx, ref double dy, ref double p, ref double q, ref int k, ref int it)
{
it = 1;
while (it == 1)
{
t = t / 1.67;
it = 0;
x1 = x - t * (dx);
y1 = y - t * (dy);
if (k >= 30)
{
p = Math.Sqrt(x1 * (x1) + y1 * (y1));
q = Math.Exp(75.0 / (k));
if (p >= q)
it = 1;
}
}
}
///
/// 内部函数
///
private static void g90c(double[] xr, double[] xi, double[] ar, double[] ai, ref double x, ref double y, ref double p, ref double w, ref int k)
{
for (int i = 1; i <= k; i++)
{
ar[i] = ar[i] + ar[i - 1] * (x) - ai[i - 1] * (y);
ai[i] = ai[i] + ar[i - 1] * (y) + ai[i - 1] * (x);
}
xr[k - 1] = x * (w);
xi[k - 1] = y * (w);
k = k - 1;
if (k == 1)
{
p = ar[0] * ar[0] + ai[0] * ai[0];
xr[0] = -w * (ar[0] * ar[1] + ai[0] * ai[1]) / (p);
xi[0] = w * (ar[1] * ai[0] - ar[0] * ai[1]) / (p);
}
}
///
/// 内部函数
///
private static void g65c(ref double x, ref double y, ref double x1, ref double y1, ref double dx, ref double dy, ref double dd, ref double dc, ref double c, ref int k, ref int nis, ref int it)
{
if (it == 0)
{
nis = 1;
dd = Math.Sqrt(dx * (dx) + dy * (dy));
if (dd > 1.0)
dd = 1.0;
dc = 6.28 / (4.5 * (k));
c = 0.0;
}
while (true)
{
c = c + dc;
dx = dd * Math.Cos(c);
dy = dd * Math.Sin(c);
x1 = x + dx;
y1 = y + dy;
if (c <= 6.29)
{
it = 0;
return;
}
dd = dd / 1.67;
if (dd <= 1.0e-07)
{
it = 1;
return;
}
c = 0.0;
}
}
}
}