// The following method performs the inverse iteration method on a real symmetric matrix A
public static int inverse_iteration(matrix A, ref double s, ref vector v, double tau = 1e-6, double eps = 1e-6, int n_max = 999, int updates = 999)
{
int n = 0; int m = 0;
matrix As; matrix I = new matrix(A.size1, A.size1); I.set_identity();
v = v / v.norm();
As = A - s * I;
qr As_QR = new qr(As);
double abs = 0; double rel = 0;
while (converge(v, A, s, tau, eps, ref abs, ref rel) && n < n_max)
{
v = As_QR.solve(v);
v = v / v.norm();
s = v.dot(A * v);
if (m > updates) // Update QR decomposition if Rayleigh updates are used (if updates<999)
{
m = 0;
s = v.dot(A * v) / (v.dot(v));
As = A - s * I;
As_QR = new qr(As);
}
n++; m++;
}
s = v.dot(A * v) / (v.norm() * v.norm());
v = v / v.norm();
return(n);
}
public static void generate_errors(qr As_QR, ref matrix A, ref matrix I, ref List <double> errors, ref List <double> errors_s, double s, int updates, double e_J, vector v_0, double tau = 1e-6, double eps = 1e-6, double n_max = 999)
{
int n = 0; int m = 0;
matrix As;
vector u; vector v;
u = v_0 / v_0.norm();
double abs = 0; double rel = 0;
while (converge(u, A, s, tau, eps, ref abs, ref rel) && n < n_max)
{
v = As_QR.solve(u);
u = v / v.norm();
s = u.dot(A * u);
if (m > updates)
{
m = 0;
s = u.dot(A * u) / (u.dot(u));
As = A - s * I;
As_QR = new qr(As);
}
n++; m++; errors.Add(rel); errors_s.Add(Abs(s - e_J));
}
errors.Add(rel);
errors_s.Add(Abs(s - e_J));
s = u.dot(A * u) / (u.norm() * u.norm());
}
public static vector newton(Func <vector, vector> f, vector x, double eps = 1e-3, double dx = 1e-7)
{
matrix J = jacobian(f, x, dx);
qr Jqr = new qr(J);
vector deltax = Jqr.solve(-1.0 * f(x));
double a = 1;
// while((f(x) + a*deltax).norm() < (1-a/2)*f(x).norm() && a>1/64){a = a/2;}
while ((f(x + a * deltax).norm() > (1 - a / 2.0) * f(x).norm()) && a > 1.0 / 64)
{
a = a / 2.0;
}
x += a * deltax;
if (deltax.norm() < dx)
{
return(x);
}
else if (f(x).norm() < eps)
{
return(x);
}
else
{
return(newton(f, x, eps, dx));
}
}
public lsq_qr(double[] x, double[] y, double[] dy, Func <double, double>[] F)
{
A = new matrix(x.Length, F.Length);
for (int i = 0; i < F.Length; i++)
{
for (int j = 0; j < x.Length; j++)
{
A[i][j] = F[i](x[j]) / dy[j]; // Row/column convention interchanged
}
}
vector b = new vector(x.Length);
for (int i = 0; i < x.Length; i++)
{
b[i] = y[i] / dy[i];
}
qr AQR = new qr(A);
c = AQR.solve(b);
}
public static int Main()
{
int n = 5; int m = 3; // Row and column dimensions
matrix A = misc.random_matrix(n, m);
var data = new qr(A); // Instance of qr decomposition class of matrix A
matrix Q = data.Q;
matrix R = data.R;
matrix QTQ = Q.transpose() * Q;
matrix QR = Q * R;
// Output
var outfile = new System.IO.StreamWriter("../out_A.txt", append: false);
outfile.WriteLine($"--------------------------------");
outfile.WriteLine($"1: QR decomposition");
outfile.WriteLine($"--------------------------------");
outfile.WriteLine($"Random tall matrix A:");
for (int ir = 0; ir < A.size1; ir++)
{
for (int ic = 0; ic < A.size2; ic++)
{
outfile.Write("{0,10:g3} ", A[ir, ic]);
}
outfile.WriteLine("");
}
outfile.WriteLine("");
outfile.WriteLine($"Upper triangular matrix R:");
for (int ir = 0; ir < R.size1; ir++)
{
for (int ic = 0; ic < R.size2; ic++)
{
outfile.Write("{0,10:g3} ", R[ir, ic]);
}
outfile.WriteLine("");
}
outfile.WriteLine("");
outfile.WriteLine($"Q.transpose()*Q:");
for (int ir = 0; ir < QTQ.size1; ir++)
{
for (int ic = 0; ic < QTQ.size2; ic++)
{
outfile.Write("{0,10:g3} ", QTQ[ir, ic]);
}
outfile.WriteLine("");
}
outfile.WriteLine("");
outfile.WriteLine($"Q*R:");
for (int ir = 0; ir < QR.size1; ir++)
{
for (int ic = 0; ic < QR.size2; ic++)
{
outfile.Write("{0,10:g3} ", QR[ir, ic]);
}
outfile.WriteLine("");
}
outfile.WriteLine("");
n = 5; m = 5;
A = misc.random_matrix(n, m);
vector b = misc.gen_vector(n);
data = new qr(A);
vector x = new vector(n);
x = data.solve(b);
vector B = A * x;
outfile.WriteLine($"--------------------------------");
outfile.WriteLine($"2: Linear equations");
outfile.WriteLine($"--------------------------------");
outfile.WriteLine($"Random square matrix A:");
for (int ir = 0; ir < QR.size1; ir++)
{
for (int ic = 0; ic < QR.size2; ic++)
{
outfile.Write("{0,10:g3} ", QR[ir, ic]);
}
outfile.WriteLine("");
}
outfile.WriteLine($"Random vector b:");
for (int ir = 0; ir < b.size; ir++)
{
outfile.Write("{0,10:g3} ", b[ir]);
}
outfile.WriteLine("");
outfile.WriteLine($"A*x:");
for (int ir = 0; ir < B.size; ir++)
{
outfile.Write("{0,10:g3} ", B[ir]);
}
outfile.WriteLine("");
outfile.Close();
return(0);
}