public static Tuple <vector, matrix> box(int n)
{
double s = 1.0 / (n + 1);
matrix H = new matrix(n, n);
for (int i = 0; i < n - 1; i++)
{
matrix.set(H, i, i, -2);
matrix.set(H, i, i + 1, 1);
matrix.set(H, i + 1, i, 1);
}
matrix.set(H, n - 1, n - 1, -2);
H = -1 / s / s * H;
var res = new jacobi_diagonalization(H);
vector e = res.get_eigenvalues();
matrix V = res.get_eigenvectors();
return(new Tuple <vector, matrix>(e, V));
}
public static int Main()
{
int dim = 30;
double tau = 1e-6; double eps = 1e-6;
int updates = 999; int n_max = 999; // If updates=999, no Rayleigh updates are performed
var rnd = new Random(); int i = rnd.Next(dim - 1);
double deviation = 1.05;
matrix A = misc.gen_matrix(dim);
matrix Ac = A.copy();
var jacobi = new jacobi_diagonalization(A);
vector e = jacobi.get_eigenvalues();
matrix V = jacobi.get_eigenvectors();
double s_0 = e[i] * deviation;
double s_1 = e[i] * deviation;
vector v_0 = misc.gen_vector(dim); // Random vector
vector v_1 = V[i] / V[i].norm(); // Jacobi eigenvector
for (int j = 0; j < v_1.size; j++)
{
v_1[j] = v_1[j] * deviation;
}
double n_0 = power_method.inverse_iteration(Ac, ref s_0, ref v_0, tau, eps, n_max, updates); // Random
double n_1 = power_method.inverse_iteration(Ac, ref s_1, ref v_1, tau, eps, n_max, updates); // Jacobi
var outfile = new System.IO.StreamWriter($"../test_out.txt", append: false);
outfile.WriteLine($"--------------------------------------------");
outfile.WriteLine($"Inverse iteration method (Jacobi comparison)");
outfile.WriteLine($"--------------------------------------------");
outfile.WriteLine($"Matrix dimension: {dim}");
outfile.WriteLine($"Random eigenvalue index: {i}");
outfile.WriteLine($"Maximum iterations: {n_max}\n");
outfile.WriteLine($"Jacobi eigenvalues:");
outfile.WriteLine($"e_(i-1): {e[i-1]}");
outfile.WriteLine($"e_(i): {e[i]}");
outfile.WriteLine($"e_(i+1): {e[i+1]}\n");
outfile.WriteLine($"Inverse iteration method:\n");
outfile.WriteLine($"Initial deviation: {deviation}");
outfile.WriteLine($"Initial eigenvalue: {e[i]*deviation}");
outfile.WriteLine($"Absolute accuracy: {tau}");
outfile.WriteLine($"Relative accuracy: {eps}\n");
outfile.WriteLine($"Inverse iteration method with random initial eigenvector:");
outfile.WriteLine($"Algorithm result: {s_0}");
outfile.WriteLine($"v^(T)*A*v: {v_0.dot(Ac*v_0)}");
outfile.WriteLine($"Iterations: {n_0}");
outfile.WriteLine($"Errors compared to Jacobi eigenvalues:");
outfile.WriteLine($"Abs(e_(i-1) - s): {Abs(e[i-1]-s_0)}");
outfile.WriteLine($"Abs(e_(i) - s): {Abs(e[i]-s_0)}");
outfile.WriteLine($"Abs(e_(i+1) - s): {Abs(e[i+1]-s_0)}\n");
outfile.WriteLine($"Inverse iteration method with deviated Jacobi eigenvector:");
outfile.WriteLine($"Algorithm result: {s_1}");
outfile.WriteLine($"v^(T)*A*v: {v_1.dot(Ac*v_1)}");
outfile.WriteLine($"Iterations: {n_1}");
outfile.WriteLine($"Errors compared to Jacobi eigenvalues:");
outfile.WriteLine($"Abs(e_(i-1) - s): {Abs(e[i-1]-s_1)}");
outfile.WriteLine($"Abs(e_(i) - s): {Abs(e[i]-s_1)}");
outfile.WriteLine($"Abs(e_(i+1) - s): {Abs(e[i+1]-s_1)}\n");
outfile.Close();
return(0);
}
public static int Main()
{
// A: Jacobi diagonalization with cyclic sweeps and quantum particle in a box
matrix A = misc.gen_matrix(5); // Generate random symmetric matrix A
matrix Ac = A.copy();
var res = new jacobi_diagonalization(A); // Create an instance of jacobi diagonalization for matrix A
vector e = res.get_eigenvalues(); // Retrieve eigenvalues
matrix V = res.get_eigenvectors(); // Retrieves eigenvectors
matrix D = new matrix(A.size1, A.size1); // Create D matrix
for (int i = 0; i < A.size1; i++)
{
D[i][i] = e[i];
}
matrix VTAV = V * Ac * V.transpose();
Tuple <vector, matrix> boxres = box(20);
var outfile = new System.IO.StreamWriter("../out_A.txt", append: false);
outfile.WriteLine($"-----------------------------------------");
outfile.WriteLine($"Jacobi diagonalization with cyclic sweeps");
outfile.WriteLine($"-----------------------------------------");
outfile.WriteLine("Random real symmetric matrix A:");
for (int ir = 0; ir < Ac.size1; ir++)
{
for (int ic = 0; ic < Ac.size2; ic++)
{
outfile.Write("{0,10:g3} ", Ac[ir, ic]);
}
outfile.WriteLine("");
}
outfile.WriteLine("");
outfile.WriteLine("Eigenvalues of A:");
for (int ir = 0; ir < e.size; ir++)
{
outfile.Write("{0,10:g3} ", e[ir]);
}
outfile.WriteLine("");
outfile.WriteLine("");
outfile.WriteLine("Eigenvectors of A (matrix V):");
for (int ir = 0; ir < V.size1; ir++)
{
for (int ic = 0; ic < V.size2; ic++)
{
outfile.Write("{0,10:g3} ", V[ir, ic]);
}
outfile.WriteLine("");
}
outfile.WriteLine("");
outfile.WriteLine("V*A*V^T:");
for (int ir = 0; ir < VTAV.size1; ir++)
{
for (int ic = 0; ic < VTAV.size2; ic++)
{
outfile.Write("{0,10:g3} ", VTAV[ir, ic]);
}
outfile.WriteLine("");
}
outfile.WriteLine("");
outfile.WriteLine("Matrix D:");
for (int ir = 0; ir < D.size1; ir++)
{
for (int ic = 0; ic < D.size2; ic++)
{
outfile.Write("{0,10:g3} ", D[ir, ic]);
}
outfile.WriteLine("");
}
outfile.WriteLine("");
outfile.WriteLine($"-----------------------------------------");
outfile.WriteLine($"Quantum particle in a box");
outfile.WriteLine($"-----------------------------------------");
int n = 20;
outfile.WriteLine("Eigenvalues of the Hamiltonian (k, calculated, exact, error):");
for (int k = 0; k < n / 3; k++)
{
double exact = PI * PI * (k + 1) * (k + 1);
double calculated = boxres.Item1[k];
outfile.WriteLine($"{k} {calculated} {exact} {exact-calculated}");
}
for (int k = 0; k < 3; k++)
{
var eigenvectors = new System.IO.StreamWriter($"./plot_files/eigenvectors{k}.txt", append: false);
eigenvectors.WriteLine($"{0} {0} {0}");
for (int i = 0; i < n; i++)
{
eigenvectors.WriteLine($"{(i+1.0)/(n+1)} {boxres.Item2[k,i]} {Psi(k+1,(i+1.0)/(n+1))}");
}
eigenvectors.WriteLine($"{1} {0} {0}");
eigenvectors.Close();
}
outfile.Close();
return(0);
}