public static vector newton(Func <vector, vector> f, vector x, double eps = 1e-3, double dx = 1e-7)
{
int n = x.size;
vector fx = f(x), z, fz;
while (true)
{
matrix J = jacobi(f, fx, x);
matrix R = new matrix(n, n);
QR.qr_gs_decomp(J, R);
matrix B = QR.qr_gs_inverse(J, R);
//B.print("inverse B");
vector del_x = -B * fx;
double lambda = 1;
while (true)
{
z = x + lambda * del_x;
fz = f(z);
if (fz.norm() < (1 - lambda / 2) * fx.norm())
{
break;
}
else if (lambda < 1.0 / 64)
{
break;
}
else
{
lambda /= 2;
}
}
x = z;
fx = fz;
if (fx.norm() < eps)
{
break;
}
else if (x.norm() < dx)
{
break;
}
}
return(x);
}
static int Main()
{
//public Func<double,double>[] f;
int n = 8;
int m = 2;
matrix R = new matrix(m, m);
matrix Q = new matrix(n, m);
vector c = new vector(m);
vector y = new vector(n);
matrix A = new matrix(n, m);
matrix B = new matrix(m, n);
matrix sigma = new matrix(m, m);
Func <double, double>[] f = { t => 1, t => t };
vector b = new vector(new double [] { 117, 100, 88, 72, 53, 29.5, 25.2, 15.2, 11.1 });
var dy = 0.05 * b;
vector x = new vector(new double [] { 1, 2, 3, 4, 6, 9, 10, 13, 15 });
for (int i = 0; i < n; i++)
{
y[i] = Log(b[i]);
y[i] = y[i] / 0.05;
dy[i] = dy[i] / b[i];
}
for (int i = 0; i < n; i++)
{
for (int k = 0; k < m; k++)
{
A[i, k] = f[k](x[i]) / dy[i];
Q[i, k] = A[i, k];
}
}
//A.print($"matrix A: ");
QR.qr_gs_decomp(Q, R);
c = QR.qr_qs_solve(Q, R, y);//coeff
B = QR.inverse(Q, R);
sigma = B * B.T;
vector delta = new vector(m);//delta are uncertainties of coeff
for (int i = 0; i < m; i++)
{
delta[i] = Sqrt(sigma[i, i]);
}
for (double i = 0; i < 20; i = i + 0.1)
{
WriteLine($"{i} {Exp(c[0])*Exp(c[1]*i)}");
}
c.print($"solution x");
Error.WriteLine($"N0={Exp(c[0])}");
Error.WriteLine($"lambda={c[1]}");
Error.WriteLine($"The half life time calculated via the fitting function is ln(2)/lambda= {Log(2)/-c[1]:f1} days");
Error.WriteLine("The actual half life time is 3.6 days.");
//(A*c).print("check Ax~b");
//B
double unc = Log(2) / (c[1] * c[1]) * delta[1];
Error.WriteLine($"Uncertainty for estimated halflife: {unc:f1} days");
Error.WriteLine($"The estimated value doesn't agree with the modern value");
return(0);
}
