public static vector newton(Func<vector, vector> f, vector x, double eps = 1e-3, double dx = 1e-7){
int n = x.size;
matrix J = new matrix(n,n);
vector root = x.copy();
vector f_r = f(root);
vector z,fz;
while(f_r.norm()>eps){
for(int i = 0; i<n; i++){
root[i] = root[i]+dx;
vector df = f(root)-f_r;
for(int j=0; j<n; j++){
J[i,j] = df[j]/dx;
} // for
root[i] -= dx;
} // for
QRdecompositionGS qr = new QRdecompositionGS(J);
vector dr = qr.solve(-1*f_r);
double lambda = 1.0;
z = root+lambda*dr;
fz = f(z);
while(fz.norm()>(1-lambda/2)*f_r.norm()){
double lambda_min = 1/64.0;
if(lambda>lambda_min){
lambda = lambda/2;
fz = f(root+lambda*dr);
} // if
else{Error.WriteLine($"Bad step: lambda = {lambda_min}");
break;
} // else
} // while
root += lambda*dr;
f_r = f(root);
} // while
return root;
} // newton
static void Main()
{
int n = 6;
int m = 4;
// Problem A1:
WriteLine($"Problem A1: QR-factorization on a random {n}x{m} matrix A: \n");
matrix A = generatematrix(n, m);
// Printing our random matrix A
A.print("Matrix A =");
QRdecompositionGS decomp = new QRdecompositionGS(A);
// Printing the orthogonal matrix Q
matrix Q = decomp.Q;
Q.print("\nDecomposed component Q = ");
// Printing the upper triangle matrix R
matrix R = decomp.R;
R.print("\nDecomposed component R = ");
// Checking that Q^T*Q is equal to the identity
(Q.transpose() * Q).print("\nQ^(T)*Q = ");
// Checking that QR=A by substracting A from QR to yield the null matrix
(Q * R - A).print("\n Q*R-A = ");
n = 3;
// Problem A2:
WriteLine($"\nProblem A2: QR-factorization on a random {n}x{n} matrix A solving Ax=b: \n");
vector b = generatevector(n);
A = generatematrix(n, n);
decomp = new QRdecompositionGS(A);
b.print("Vector b = ");
A.print("\nSquare matrix A = ");
vector x = decomp.solve(b);
// Printing the solution to Ax=b
x.print("\nVector x = ");
WriteLine("\nChecking that x is indeed the solution: ");
(A * x - b).print("\nA*x-b = ");
// Problem B:
A = generatematrix(n, n);
decomp = new QRdecompositionGS(A);
WriteLine($"\nProblem B: Matrix inverse by Gram-Schmidt QR factorization \n");
matrix B = decomp.inverse();
B.print("A^(-1) = ");
(A * B).print("\n A*A^(-1) = ");
// Problem C:
}
public OrdLeastSquares(vector x, vector y, vector dy, Func <double, double>[] fs)
{
int n = y.size;
int m = fs.Length;
matrix A = new matrix(n, m);
vector b = new vector(n); // Creating the matrix and vector going into QRdecomp.
for (int i = 0; i < n; i = i + 1)
{
b[i] = y[i] / dy[i];
for (int j = 0; j < m; j = j + 1)
{
A[i, j] = fs[j](x[i]) / dy[i];
}
} // Matrix and vector ready for QRdecomp
QRdecompositionGS QRdecomp = new QRdecompositionGS(A);
c = QRdecomp.solve(b);
matrix sigmainv = QRdecomp.R.transpose() * QRdecomp.R;
QRdecompositionGS cov = new QRdecompositionGS(sigmainv);
sigma = cov.inverse();
}