public lsfit(vector x, vector y, vector yerr, Func <double, double>[] fs)
{
int dsize = x.size;
int nf = fs.Length;
matrix A = new matrix(dsize, nf);
vector b = new vector(dsize);
fun = fs;
for (int i = 0; i < dsize; i++)
{
b[i] = y[i] / yerr[i];
for (int k = 0; k < nf; k++)
{
A[i, k] = fs[k](x[i]) / yerr[i];
}
}
qr_decomp_GS decomp = new qr_decomp_GS(A);
q = decomp.Q;
r = decomp.R;
c = decomp.solve(b);
qr_decomp_GS decomp_covar = new qr_decomp_GS(A.T * A);
covar = decomp_covar.inverse();
cerr = new vector(covar.size1);
for (int i = 0; i < covar.size1; i++)
{
cerr[i] = Sqrt(covar[i, i]);
}
}
} // newton complex
public static (vector, int) newton(
Func <vector, vector> f, // func takes vector(x1,...,xn) returns vector(f1(..),..,fn(..))
vector x, // starting values of vector(x1,...,xn)
double eps = 1e-3, // accuracy goal ||f(x)||<eps
double dx = 1e-7 // finite difference used in numerical eval of jacobian
)
{ // overload that implements a multiple-dimensional newton rootfinder for real functions of real variables
int n = x.size;
int nsteps = 0;
vector fx = f(x);
double lam;
do
{
nsteps++;
// create jacobian
matrix J = jacobian(f, x, fx, dx);
// find jacobian inverse by QR-decomposition
qr_decomp_GS qrd = new qr_decomp_GS(J);
matrix J_inv = qrd.inverse();
// solve J*Dx = - fx for newton stepsize Dx
vector Dx = -J_inv * fx;
// lambda factor
lam = 1.0;
// backtracking linesearch algorithm (first failing is good, second is step out try again)
while (f(x + lam * Dx).norm2() > fx.norm2() * (1 - lam / 2) & lam > 1.0 / 64)
{
lam /= 2;
}
// norm2 is an euclidean vector norm by its definition, with over/underflow potential
// Update values
x = x + lam * Dx;
fx = f(x);
}while (fx.norm2() > eps);
Error.WriteLine($"rootf.newton returning x, a condition is satisfied");
Error.WriteLine($"f(x).norm2() {fx.norm2()}");
Error.WriteLine($"eps {eps}");
Error.WriteLine($"lam {lam}");
Error.WriteLine($"dx {dx}\n");
return(x, nsteps);
} // qnewton real vector
static void B()
{
WriteLine("Problem B");
var rand = new System.Random();
int m = 2 + rand.Next(6);
int n = m;
WriteLine("Random square matrix A");
matrix A = make_random_matrix(n, m);
A.print("A = ");
qr_decomp_GS decomposition = new qr_decomp_GS(A);
matrix B = decomposition.inverse();
WriteLine("B inverse of Q*R.");
B.print("B = ");
matrix I = A * B;
I.print("A * B = ");
}