static void Main()
{
int ncalls;
Func <vector, double> f;
ncalls = 0;
Func <vector, double> f1 = (z) => { ncalls++; return((1 - z[0]) * (1 - z[0]) + 100 * (z[1] - z[0] * z[0]) * (z[1] - z[0] * z[0])); };
double dx = 1e-3;
vector p = new vector("2 3");
Write("simplex.downhill: Rosenbrock's valley function\n");
p.print("start point:");
double psize = simplex.downhill(f1, ref p, dx: dx);
WriteLine($"simplex size= {(float)psize}");
WriteLine($"ncalls = {ncalls}");
p.print("minimum: ");
Write($"f(x_min) = {(float)f1(p)}\n");
ncalls = 0;
Func <vector, double> f2 = (z) => { ncalls++; return((z[0] * z[0] + z[1] - 11) * (z[0] * z[0] + z[1] - 11) + (z[0] + z[1] * z[1] - 7) * (z[0] + z[1] * z[1] - 7)); };
double dx2 = 1e-3;
vector p2 = new vector("0 1");
Write("\nsimplex.downhill: Himmelblau's function\n");
p2.print("start point:");
double psize2 = simplex.downhill(f2, ref p2, dx: dx2);
WriteLine($"simplex size= {(float)psize2}");
WriteLine($"ncalls = {ncalls}");
p2.print("minimum: ");
Write($"f(x_min) = {(float)f2(p2)}\n");
}
public static void Main()
{
Write("==== Finding the minimum of the Rosenbrock's Valley function with a=1 and b=100. ==== \n\n");
Write("Start guess of minimum:");
Func <vector, double> f = delegate(vector x) {
double val = (1 - x[0]) * (1 - x[0]) + 100 * (x[1] - x[0] * x[0]) * (x[1] - x[0] * x[0]);
return(val);
};
vector xstart = new vector(0.0, 0.0);
xstart.print();
vector xmin = minimizer.qnewton(f, xstart);
Write("\n The found minimum is:\n");
xmin.print();
Write("\n Minimum according to Wikipedia is:\n");
Write(" 1 1\n");
Write("\n \n ==== Finding the minimum of Himmelblau's function. ====\n\n");
f = delegate(vector x) {
double val = Pow(x[0] * x[0] + x[1] - 11, 2) + Pow(x[0] + x[1] * x[1] - 7, 2);
return(val);
};
Write("Start guess of minimum:\n");
xstart = new vector(0.0, 0.0);
xstart.print();
xmin = minimizer.qnewton(f, xstart);
Write("\n The found minimum is:\n");
xmin.print();
Write("\n One minimum according to Wikipedia is:\n");
Write(" 3.0 2.0\n");
}
public static void Main(string[] args)
{
// Rosenbrock's valley function
WriteLine("Rosenbrock's valley function:");
WriteLine($"Expected minimum: (a, a*a) = (1, 1)");
double eps = 1e-4;
Func <vector, double> f = delegate(vector z) {
double x = z[0], y = z[1];
return(Pow(1 - x, 2) + 100 * Pow(y - x * x, 2));
};
vector x0 = new vector(0, 0), min;
int n; // steps
x0.print("Starting point:");
(min, n) = mini.qnewton(f, x0, eps);
min.print($"Minimum reached in {n} steps:");
// Himmeblau's function
WriteLine("\nHimmelblau's function:");
WriteLine("Reference minimums: (3.0, 2.0), (-2.81, 3.13), (-3.78, -3.28), (3.58, -1.85)");
f = delegate(vector z) {
double x = z[0], y = z[1];
return(Pow(x * x + y - 11, 2) + Pow(x + y * y - 7, 2));
};
x0.print("\nStarting point:");
(min, n) = mini.qnewton(f, x0, eps);
min.print($"Minimum reached in {n} steps:");
x0 = new vector(-6, -6);
x0.print("\nStarting point:");
(min, n) = mini.qnewton(f, x0, eps);
min.print($"Minimum reached in {n} steps:");
// Higgs discovery
WriteLine("\nHiggs discovery");
var lines = File.ReadAllLines("higgs.txt").Select(line => line.Split(' ', StringSplitOptions.RemoveEmptyEntries)).Select(line => new { x = double.Parse(line[0]), y = double.Parse(line[1]), z = double.Parse(line[2]) }).ToList(); // maps every line in higgs.txt to a (x, y, z) double value. RemoveEmptyEntries is required for some reason - probably due to the format of the .txt file?
int l = lines.Count(); // number of lines
vector E = new vector(l), sig = new vector(l), err = new vector(l);
for (int i = 0; i < l; i++) // not really necessary, but it tidies up the rest of the code
{
E[i] = lines[i].x;
sig[i] = lines[i].y;
err[i] = lines[i].z;
}
Func <double, double, double, double, double> F = (e, m, g, a) => a / (Pow(e - m, 2) + g * g / 4); // Breit-Wigner
Func <vector, double> chi2 = delegate(vector z) {
double m = z[0], gamma = z[1], A = z[2];
double sum = 0;
for (int i = 0; i < l; i++)
{
sum += Pow(F(E[i], m, gamma, A) - sig[i], 2) / Pow(err[i], 2);
}
return(sum);
};
x0 = new vector(120, 2, 6); // i have no idea which starting point to use, so these are dmitris values
(min, n) = mini.qnewton(chi2, x0, eps);
WriteLine($"Fitted values in {n} steps: mass: {min[0]:F3}, gamma: {min[1]:F3}, sqrt(chi2/n): {chi2(min)/l:F3}");
}
public static void Main()
{
Func <vector, vector> f = delegate(vector z) {
double x = z[0];
return(new vector((2 - x) * (6 - x)));
};
vector x0 = new vector(1); x0[0] = 0; // start pos
vector root = roots.newton(f, x0);
WriteLine($"Testing with (2 - x)*(6 - x).");
root.print("The root is: ");
f(root).print("The function value at the root:");
f = delegate(vector z) {
double x = z[0], y = z[1];
return(new vector((4 - x) * (6 - x), (y - x) * (y - 3)));
};
x0 = new vector(0, 0);
root = roots.newton(f, x0);
WriteLine($"\nTesting with (4 - x)*(6 - x), (y - x)*(y - 3)");
root.print("The roots are: ");
f(root).print("The function values at these roots: ");
// Gradient of Rosenbrock
WriteLine("\nExtremums of Rosenbrock's valley function");
f = delegate(vector z) {
double x = z[0], y = z[1];
return(new vector(2 * (1 - x) + 2 * 100 * (y - x * x) * 2 * x, 2 * 100 * (y - x * x))); // the gradient
};
x0 = new vector(2, 2);
root = roots.newton(f, x0);
root.print("The roots are");
f(root).print("The function values at these roots: ");
// Hydrogen atom
WriteLine("\n# Hydrogen atom");
double rmax = 8;
Func <vector, vector> aux = delegate(vector z) {
double eps = z[0];
double frmax = Feps(eps, rmax);
return(new vector(frmax));
};
x0 = new vector(-1.0);
root = roots.newton(aux, x0);
WriteLine($"# rmax is {rmax}, energy is {root[0]}");
WriteLine(string.Format("# {0, -6} {1, -8} {2, -8}", "r", "Feps(r)", "exact"));
for (double r = 0; r <= rmax; r += rmax / 100)
{
WriteLine($"{r, -8:F3} {Feps(root[0], r), -8:F3} {r*Exp(-r), -8:F3}");
}
}
static void Main()
{
Func <vector, double> f1 = (z) => (1 - z[0]) * (1 - z[0]) + 100 * (z[1] - z[0] * z[0]) * (z[1] - z[0] * z[0]); // Rosenbrock's valley function (a=1,b=100)
double eps = 1e-4;
vector p = new vector("2 3"); // starting point
Write("SR1: Rosenbrock's valley function\n");
p.print("start point:");
int nsteps = qnewton.sr1(f1, ref p, eps);
WriteLine($"nsteps={nsteps}");
p.print("minimum: ");
Write($"f(x_min) = {(float)f1(p)}\n");
vector g = qnewton.gradient(f1, p);
Write($"|gradient| goal = {(float)eps}\n");
Write($"|gradient| actual = {(float)g.norm()}\n");
if (g.norm() < eps)
{
WriteLine("test passed");
}
else
{
WriteLine("test failed");
}
// Minimum at (1,1)
Func <vector, double> f2 = (z) => (z[0] * z[0] + z[1] - 11) * (z[0] * z[0] + z[1] - 11) + (z[0] + z[1] * z[1] - 7) * (z[0] + z[1] * z[1] - 7); // Himmelblau's function
eps = 1e-4;
p = new vector("0 1"); // starting point
Write("\nSR1: Himmelblau's function\n");
p.print("start point:");
nsteps = qnewton.sr1(f2, ref p, eps);
WriteLine($"nsteps={nsteps}");
p.print("minimum: ");
Write($"f(x_min) = {(float)f2(p)}\n");
g = qnewton.gradient(f, p);
Write($"|gradient| goal = {(float)eps}\n");
Write($"|gradient| actual = {(float)g.norm()}\n");
if (g.norm() < eps)
{
WriteLine("test passed");
}
else
{
WriteLine("test failed");
}
// Minima at (3,2), (-2.805118,3.131312), (-3.779310,-3.283186), and (3.584428,-1.848126)
}
static void Main()
{
/// Test on random matrix ///
WriteLine("/////////////// Part A - Test on random matrix ///////////////");
int n = 5;
var rand = new Random(1);
matrix A = new matrix(n, n);
vector e = new vector(n);
matrix V = new matrix(n, n);
for (int i = 0; i < n; i++)
{
for (int j = i; j < n; j++)
{
A[i, j] = rand.NextDouble();
A[j, i] = A[i, j];
}
}
A.print($"random {n}x{n} matrix"); WriteLine();
matrix B = A.copy();
int sweeps = jacobi.cyclic(B, e, V);
WriteLine($"Number of sweeps in jacobi={sweeps}"); WriteLine();
matrix D = (V.T * A * V);
(D).print("Should be a diagonal matrix V.T*B*V="); WriteLine();
e.print("Eigenvalues should equal the diagonal elements above");
vector D_diagonal = new vector(n);
for (int i = 0; i < n; i++)
{
D_diagonal[i] = D[i, i];
}
if (D_diagonal.approx(e))
{
WriteLine("Eigenvalues agree \tTest passed");
}
else
{
WriteLine("Test failed");
}
WriteLine();
matrix A2 = (V * D * V.T);
(A2).print("Check that V*D*V.T=A");
if (A2.approx(A))
{
WriteLine("V*D*V.T = A \tTest passed");
}
else
{
WriteLine("V*D*V.T != A \tTest failed");
}
}
public static void Main()
{
//now test highest EV
int n = 5;
matrix A = new matrix(n, n);
matrix V = new matrix(n, n);
vector e = new vector(n);
var rand = new Random(1);
for (int i = 0; i < n; i++)
{
for (int j = i; j < n; j++)
{
A[i, j] = 2 * (rand.NextDouble() - 0.5);
A[j, i] = A[i, j];
}
}
//2 highest EV
matrix AA2 = A.copy();
string order = "high";
int count = jacobiB.jacobirow(AA2, e, 2, V, order);
e.print("First 2 values should be highest EVs");
//full diag
matrix Vh = new matrix(n, n);
vector eh = new vector(n);
matrix AAh = A.copy();
int counth = jacobiB.jacobirow(AAh, eh, n, Vh);
eh.print("all eigenvalues");
}
public static void Main()
{
// First part of A
int n = 5; // Size of quadratic matrix M.
var rnd = new Random(1);
Console.WriteLine("Part A1");
matrix M = new matrix(n, n);
for (int i = 0; i < n; i++)
{
for (int j = 0; j < n; j++)
{
M[i, j] = 4 * rnd.NextDouble() - 1.5;
M[j, i] = M[i, j];
}
}
matrix M2 = M.copy();
M.print("We choose an arbitrary matrix M = ");
vector D = new vector(n);
matrix V = new matrix(n, n);
int sweeps = Jacobi_diagonalization.cyclic_sweep(M, V, D);
matrix temp2 = V.T * M2 * V;
Console.WriteLine($"Total number of sweeps done is = {sweeps}");
temp2.print("V.T*M*V = ");
D.print("Eigenvalues = ");
}
static void probC()
{
Write("Problem C:\n");
var rand = new System.Random();
int n = 5 + rand.Next(3);
matrix A = makeRandomMatrix(n, n);
A.print("Make a random square matrix of random size: A = ");
qr_givens decomp_givens = new qr_givens(A);
qr_gs decomp_gs = new qr_gs(A);
Write("\nDo the Givens decomposition and store the restult in the matrix G, which cotains the elements of the component R in the upper triangular part, and the angles for the Givens rotations in the relevant sub-diagonal entries. Compare with the R matrix found from the Gramm-Schmitt method:");
decomp_givens.G.print("Givens G: ");
decomp_gs.R.print("Gram-Schmitt R: ");
Write("The upper triangular parts are the same.\n");
vector b = makeRandomVector(n);
b.print("\nMake a random vector b with the same size as A: b = ");
Write("\nCompare using the Givens rotations to apply Q^(T) to b, and doing the explicit matrix multiplication with the Q matrix found by the Gramm-Schmitt method:\n");
decomp_givens.applyQT(b).print("Givens: Q^(T)*b = ");
(decomp_gs.Q.transpose() * b).print("GS: Q^(T)*b = ");
vector x = decomp_givens.solve(b);
x.print("\nSolution to A*x=b using Givens: ");
(A * x - b).print("\nCheck solution satisfies A*x = b: A*x-b = ");
Write("\nFind the inverse of A, and check that A^(-1)*A is the identity matrix:\n");
decomp_givens.inverse().print("A^(-1)");
(decomp_givens.inverse() * A).print("A^(-1)*A = ");
}
public static void Main()
{
double EPS = Pow(10, -6);
//
Func <vector, double> f = (z) => Pow(1 - z[0], 2) + 100 * Pow(z[1] - Pow(z[0], 2), 2);
vector x01 = new vector(0, 0);
int StepCounts = Quasi.qnewton(f, ref x01);
WriteLine("");
x01.print("Starting point: ");
WriteLine($"steps = {StepCounts}");
WriteLine($"Found minimum = ({x01[0]:f5}, {x01[1]:f5},)");
WriteLine($"Exact minimum = (1,1)");
WriteLine($"Tolerance for gradient = {EPS}");
//
Func <vector, double> f2 = (z) => Pow((Pow(z[0], 2) + z[1] - 11), 2) + Pow(z[0] + Pow(z[1], 2) - 7, 2);
vector x02 = new vector(-2, 3);
int StepCounts2 = Quasi.qnewton(f2, ref x02);
x02.print("Starting point: ");
WriteLine($"steps = {StepCounts2}");
WriteLine($"The minimum was found at minimum = ({x02[0]:f5}, {x02[1]:f5})");
WriteLine($"Where the analytical minimum = (-2.805118, 3.131312)");
WriteLine($"The tolorence for gradient is = {EPS}");
}
static void Main()
{
int ncalls = 0;
double eps = 1e-7;
vector p = new vector(2);
p[0] = 0.5; p[1] = 1.7;
Func <vector, vector> f = delegate(vector z){
ncalls++;
vector r = new vector(2);
double x = z[0], y = z[1], b = 100;
// Rosenbrock's valley function: f(x,y) = (1-x)^2+100*(y-x^2)^2
// The gradient of Rosenbrock's valley function:
r[0] = 2 * (1 - x) + 2 * 100 * (y - x * x) * (-2 * x); // d/dx
r[1] = b * 2 * (y - x * x); // d/dy
return(r);
};
vector root = roots.newton(f, p, eps);
WriteLine($"start values={p[0]} {p[1]}");
WriteLine($"ncalls={ncalls}");
root.print("root=");
f(root).print("f(root)=");
WriteLine($"eps = {eps}");
WriteLine($"f(root).norm() = {f(root).norm()}");
if (f(root).norm() < eps)
{
WriteLine("test passed");
}
else
{
WriteLine("test failed");
}
}
static void probA2()
{
Write("\nProblem A2:\n");
// Need to make the tests for A2
var rand = new System.Random();
// Remember, square matrix:
Write("\nMake a random square matrix, A, with random dimmensions, and a random vector, b, of the same dimmension:\n");
int n = 2 + rand.Next(10);
matrix A = makeRandomMatrix(n, n);
vector b = makeRandomVector(n);
A.print("A = ");
b.print("b = ");
qr_gs decomposer = new qr_gs(A);
Write("\nSolve the system A*x = b for the vector x:\n");
//decomposer.Q.print("Decomposition Q: ");
//decomposer.R.print("Decomposition R: ");
//(decomposer.Q.transpose()*b).print("Q^(T)*b = ");
vector x = decomposer.solve(b);
x.print("Solution: x = ");
Write("\nCheck that the vector x satisfies A*x=b:\n");
(A * x - b).print("A*x-b = ");
}
// public double eval(double x, vector weights)
// {
// vector hold = x*weights;
// // hold.print();
// hold = relu(hold);
// double res = 0;
// for(int i = 0; i<hold.size; i++)
// {
// res+=hold[i];
// }
// return res;
// }
// static Func<vector, double> trainingFunc = delegate(vector x)
// {
// double res = 0;
// for(int i = 0; i<data.size;i++)
// {
// res+=MSE(eval(data[i], x),labels[i]);
// }
// return res;
// };
public void fit(double[] data, double[] labels)
{
parameters.print();
Func <vector, double> trainingFunc = (x) =>
{
parameters = x;
double res = 0;
for (int i = 0; i < data.Length; i++)
{
// parameters.fprint(Console.Error);
// Error.Write($"{data[i]}\n");
// Error.Write($"{eval(data[i])}\n");
res += MSE(eval(data[i]), labels[i]);
}
// Error.Write($"current error is {res/data.Length} \n");
return(res);
};
// var guesses = new vector(parameters.size);
// for(int i = 0;i<guesses.size;i++)
// {
// guesses[i]=1;
// }
parameters = qnewton.QNewton(trainingFunc, parameters, 0.01);
parameters.print();
}
public static void Main(string[] args)
{
int n = 5; //size of real symmetric matrix
matrix A = new matrix(n, n);
var rnd = new Random(1);
for (int i = 0; i < n; i++) /* Construction of a random vector */
{
for (int j = 0; j < n; j++)
{
A[i, j] = (rnd.NextDouble() * 10);
A[j, i] = A[i, j]; /* We make sure that the matrix is symmetric */
}
}
A.print("Random symmetric matrix, A:");
matrix V = new matrix(n, n);
vector e = new vector(n);
matrix B = A.copy(); /* We make a copy of A */
int sweeps = jacobi.cyclic(A, e, V); /* We use the jacobi matlib for the implementation of the cyclic sweeps: public static int cyclic(matrix A, vector e, matrix V=null){...} */
WriteLine($"\nNumber of sweeps: {sweeps}");
matrix D = (V.T * B * V);
// D.print("\nEigenvalue-decomposition, V^T*A*V (should be diagonal):");
D.printfloat("\nEigenvalue-decomposition, V^T*A*V (should be diagonal):"); /* using my print2 to get 0's instead of values with e.g. "e-16"*/
e.print("\nEigenvalues:\n");
}
public static void Main()
{
int n = 3;
matrix A = new matrix(n, n);
matrix V = new matrix(n, n);
vector e = new vector(n);
var rand = new Random();
for (int i = 0; i < n; i++)
{
for (int j = i; j < n; j++)
{
A[i, j] = rand.NextDouble() * 10;
A[j, i] = A[i, j];
}
}
A.print("Symmetric matrix A:");
matrix Ao = new matrix(n, n);
Ao = A.copy();
int count = jacobi.cycsweep(A, e, V);
WriteLine($"Number of sweeps: {count}");
V.print("Eigenvectors");
A.print("new A");
e.print("Eigenvalues");
(V.T * Ao * V).print("V^T*A*V diagonal, test");
}
static void Main()
{
int n = 6;
WriteLine($"Generate symmetric {n}x{n} matrix A");
matrix A = randomMatrix(n, n);
restoreUpperTriang(A);
A.print("A: ");
matrix V = new matrix(n, n);
vector e = new vector(n);
int sweeps;
// Cyclic
sweeps = Jacobi.cyclic(A, V, e);
WriteLine($"Cyclic Jacobi done in {sweeps} sweeps");
e.print("Cyclic eigenvalues: ");
// Lowest k
restoreUpperTriang(A);
V = new matrix(n, n);
e = new vector(n);
int k = n / 3;
vector v = Jacobi.lowestK(A, V, e, k);
v.print($"Lowest {k} values:");
A.print($"Test that {k} rows are cleared");
// Highest k
restoreUpperTriang(A);
V = new matrix(n, n);
e = new vector(n);
k = n / 3;
v = Jacobi.highestK(A, V, e, k);
v.print($"Highest {k} values:");
A.print($"Test that {k} rows are cleared");
restoreUpperTriang(A);
V = new matrix(n, n);
e = new vector(n);
int rots = Jacobi.classic(A, V, e);
e.print($"Classical done! rotations: {rots}");
A.print($"Test that all rows are cleared");
}
static void Main()
{
int n = 6;
int m = 4;
// Problem A1:
WriteLine($"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:
}
static void Main()
{
int callCount = 0;
Func <vector, vector> f = (x) => { callCount++; return(new vector(x[0] * x[0] - 4)); };
double epsilon = 1e-3;
vector x0 = new vector(1.0);
vector root = newton(f, x0, epsilon);
vector exact = new vector(2.0);
WriteLine("\n__________________________________________________________________________________________________________");
WriteLine("Question A\n Newton's method with numerical Jacobian and back-tracking linesearch");
WriteLine("Test function f(x)=x^2-4, root = +-2 (only looking for 2)");
root.print("Numericaly found root : ");
exact.print("Exact root : ");
f(root).print("Function value at root : ");
WriteLine("Error goal : {0}", epsilon);
WriteLine("Actual error (norm at rooot) : {0}", f(root).norm());
WriteLine("Number of func call : {0}", callCount);
WriteLine("");
callCount = 0;
// Gradient
// Func<vector,vector> f = (x) => {callCount++;return new vector(pow(1-x,2)+100,);};
f = (x) => {
callCount++;
double dx = -2 * (1 - x[0]) - 400 * x[0] * (x[1] - x[0] * x[0]);
double dy = 200 * (x[1] - x[0] * x[0]);
return(new vector(dx, dy));
};
epsilon = 1e-3;
x0 = new vector(1.1, 0.5);
root = newton(f, x0, epsilon);
exact = new vector(1.0, 1.0);
WriteLine("Test function extreem of Rosenbrock's valley function");
root.print("Numericaly found root : ");
exact.print("Exact root : ");
f(root).print("Function value at root : ");
WriteLine("Error goal : {0}", epsilon);
WriteLine("Actual error (norm at rooot) : {0}", f(root).norm());
WriteLine("Number of func call : {0}", callCount);
WriteLine("__________________________________________________________________________________________________________\n");
callCount = 0;
}
static void Main()
{
WriteLine("Testing to find roots of f(x,y) = (x*x - 9, 81 - y*y)");
Func <vector, vector> f = (x) => new vector(x[0] * x[0] - 9, 81 - x[1] * x[1]);
vector p = new vector(10, 20);
vector roots = newton(f, p);
roots.print("Found solution: ");
p = new vector(-10, -20);
roots = newton(f, p);
roots.print("Found second solution: ");
p = new vector(-13, 27);
roots = newton(f, p);
roots.print("Found third solution: ");
p = new vector(42, -65);
roots = newton(f, p);
roots.print("Found fourth solution: ");
}
static void Main()
{
var rnd = new Random();
var A = new matrix(3, 3);
for (int i = 0; i < A.size1; i++)
{
for (int j = 0; j < A.size2; j++)
{
A[i, j] = 10 * (rnd.NextDouble() - 0.5);
}
}
var b = new vector(3);
for (int i = 0; i < 3; i++)
{
b[i] = 10 * (rnd.NextDouble() - 0.5);
}
WriteLine("\n__________________________________________________________________________________________________________");
WriteLine("Question C: \nTest QR-decomposition by Givens rotations:");
var QR = A.copy();
A.print("A = ");
qr_givens.qr_givens_QR(QR);
WriteLine("QR-decomposition");
QR.print("QR = ");
WriteLine("Random vector");
b.print("b = ");
qr_givens.qr_givens_solve(QR, b);
WriteLine("Solve Ax = b:");
b.print("x = ");
WriteLine("Check result:");
(A * b).print("Ax= ");
var B = qr_givens.qr_givens_inverse(QR);
WriteLine("Inverse of A:");
B.print("A^-1=B=");
WriteLine("Product should give identity");
(A * B).print("I=AB=");
WriteLine("__________________________________________________________________________________________________________\n");
}
static void Main()
{
WriteLine("--- Part A: Search for extremums of the Rosenbrock Valley Function ---");
Func <vector, vector> f = (x) =>
new vector(2 * (200 * x[0] * x[0] * x[0] - 200 * x[0] * x[1] + x[0] - 1),
200 * (x[1] - x[0] * x[0]));
vector p = new vector(10, 20);
p.print("Starting point: ");
vector roots = newton(f, p);
roots.print("Found solution: ");
p = new vector(-123, 123);
p.print("Starting point: ");
roots = newton(f, p);
roots.print("Found solution: ");
p = new vector(-123, 123);
p.print("Starting point: ");
roots = newton(f, p);
roots.print("Found solution: ");
p = new vector(-100, 0);
p.print("Starting point: ");
roots = newton(f, p);
roots.print("Found solution: ");
p = new vector(100, -123);
p.print("Starting point: ");
roots = newton(f, p);
roots.print("Found solution: ");
WriteLine("\n\n--- Part B ---");
double rmax = 8;
Func <vector, vector> M = (x) => new vector(f_E(x[0], rmax));
p = new vector(-2.0);
roots = newton(M, p);
WriteLine($"For rmax = {rmax} we get an energy estimate of {roots[0]},\ncompared to the analytical result of -1/2");
using (StreamWriter sw = new StreamWriter("data.txt")){
for (double r = 0; r <= rmax; r += 1.0 / 64)
{
sw.WriteLine($"{r}\t{f_E(roots[0],r)}");
}
}
}
static void Main()
{
// Testing that the jacobiAlgorithm works:
int n = 5;
vector e = new vector(n);
matrix V = new matrix(n, n);
matrix A = generateRandommatrix(n, n);
matrix B = A.copy();
int sweeps = jacobi.jacobi_cyclic(A, e, V);
WriteLine("---------Problem A1----------");
WriteLine("\nTesting that the Jacobi algorithm works as intended:");
A.print("\nRandom square matrix A:");
WriteLine("\nEigenvalue decomposition of A=V*D*V^T:");
(V.transpose() * B * V).print("\nD = V^T*A*V = ");
e.print("\nListed eigenvalues:\t");
WriteLine($"\nNumber of sweeps required to diagonalize A: {sweeps}");
// Particle in a box
n = 50;
matrix H = Hamiltonian.boxHamilton(n);
B = H.copy();
V = new matrix(n, n);
e = new vector(n);
sweeps = jacobi.jacobi_cyclic(H, e, V);
WriteLine("\n---------Problem A2----------");
WriteLine("\nNow solving for a particle in a box.");
WriteLine($"\nThe dimensions of the soon to be diagonalized Hamiltonian is {n}x{n}");
WriteLine("\nComparison of the calculated eigenvalues and the exact:\n");
WriteLine("n Calculated Exact:");
for (int k = 0; k < n / 3; k++)
{
double exact = PI * PI * (k + 1) * (k + 1);
double calculated = e[k];
WriteLine($"{k} {calculated:f4}\t {exact:f4}");
}
// plotting time
StreamWriter solutions = new StreamWriter("SolutionsA.txt");
for (int i = 0; i < n; i++)
{
solutions.Write($"{i*(1.0/n)}");
for (int k = 0; k < 5; k++)
{
solutions.Write($" {V[i,k]}");
}
solutions.Write("\n");
}
solutions.Close();
}
public static void Main()
{
vector guesses = new vector(new double[] { 0 });
vector roots = Newton.newton(F, guesses);
roots.print();
vector guesses_RosenBrock = new vector(new double[] { 0, 0 });
vector roots_RosenBrock = Newton.newton(DRosenbrock, guesses_RosenBrock);
roots_RosenBrock.print();
}
static int Main()
{
int n = 3;
vector v = new vector(n);
vector u = new vector(n);
for (int i = 0; i < n; i++)
{
v[i] = i;
u[i] = -2 * i;
}
v.print("v=");
v[0] = 5;
v.print("v=");
vector w = u + v;
w.print("w=");
return(0);
}
} // end Main function
public static void printResults(vector estimate, vector a, vector b, int N, double expected)
{
Write("Lower integration limits: ");
a.print();
Write("Upper integration limits: ");
b.print();
WriteLine("Analytic value: \t\t{0}", expected);
WriteLine("Estimate of integral: \t\t{0}", estimate[0]);
WriteLine("Estimate of error: \t\t{0}", estimate[1]);
WriteLine("Actual error: \t\t\t{0}", Abs(expected - estimate[0]));
WriteLine("The function was sampled in {0} points.", N);
} // end printResults
public static int Main()
{
double epsilon = 1e-5;
Func <vector, double> f_rosen = (x) =>
{
return((1 - x[0]) * (1 - x[0]) + 100 * (x[1] - x[0] * x[0]) * (x[1] - x[0] * x[0]));
};
WriteLine("The Rosenbruck Valley function: f(x,y)=(1-x)^2+100(y-x^2)^2");
vector start = new vector(new double [] { 1.3, 0.2 });
start.print("starting point : ");
var min_rosen = minimization.qnewton(f_rosen, ref start, epsilon);
vector x_rosen = new vector(new double[] { 1.0, 1.0 });
start.print("Quasi newton minimum : ");
x_rosen.print($"The exact minimum : ");
WriteLine($"Number of steps : {min_rosen} steps.");
WriteLine("");
Func <vector, double> f_himmel = (x) =>
{
return((x[0] * x[0] + x[1] - 11) * (x[0] * x[0] + x[1] - 11) + (x[0] + x[1] * x[1] - 7) * (x[0] + x[1] * x[1] - 7));
};
epsilon = 1e-4;
WriteLine("The Himmelblau function: f(x,y)=(x^2+y-11)^2+(x+y^2-7)^2.");
vector start_h = new vector(new double [] { 1, 1 });
start_h.print("starting point : ");
vector x_himmel = new vector(new double [] { 3.0, 2.0 });
var min_himmel = minimization.qnewton(f_himmel, ref start_h, epsilon);
start_h.print("Quasi newton minimum : ");
x_himmel.print($"The exact minimum : ");
WriteLine($"Number of steps : {min_himmel} steps.");
return(0);
}
static int Main()
{
// Load data (not from file...)
vector t = new vector(new double[] { 1, 2, 3, 4, 6, 9, 10, 13, 15 });
vector A = new vector(new double[] { 117, 100, 88, 72, 53, 29.5, 25.2, 15.2, 11.1 });
vector dA = new vector(A.size); // uncertainty of A
for (int i = 0; i < A.size; i++)
{
dA[i] = 0.05; // 5% uncertainty
}
// Transform data from y = a*exp(-lambda*t) --> ln(y) = ln(a) - lambda*t
for (int i = 0; i < A.size; i++)
{
dA[i] = dA[i] / A[i]; // delta ln(y) --> (delta y )/ y
A[i] = Log(A[i]);
}
// Make function array
Func <double, double>[] funcs = new Func <double, double>[] { x => 1, x => x };
// t -> 1 is for ln(a), t -> t is for lambda
// Make fit
Fit FitResult = qr_stuff.qrFitter(t, A, dA, funcs);
vector cErrors = FitResult.getParamErrors();
double a = Exp(FitResult.c[0]);
double aErr = cErrors[0];
double lambda = -FitResult.c[1];
double lambdaErr = cErrors[1];
// Generate Answer to answer questions:
FitResult.c.print("Fit result parameters: ");
WriteLine($"This means that a: {a} and lambda: {lambda}");
WriteLine($"This gives a half life of {Log(2)/lambda} days");
WriteLine("The half life of Ra224 is actually around 3.6319 days");
WriteLine("");
FitResult.cov.print("The Covariance matrix is estimated to");
cErrors.print("This give the following errors of the parameters");
WriteLine($"Thus, the half of ThX is estimated to be between {Log(2)/lambda} +- {Log(2)/lambda/lambda*lambdaErr} days");
// Generate plotting data to plot with
TextWriter DataWriter = Error;
//DataWriter.WriteLine("This is in the data stream");
for (double x = 0; x < 16; x += 1.0 / 16)
{
DataWriter.WriteLine($"{x} {Exp(FitResult.evaluate(x))} {Exp(FitResult.upper(x))} {Exp(FitResult.lower(x))}");
}
return(0);
}
static int Main()
{
Func <double, double>[] lnexp = new Func <double, double> [2];
lnexp[0] = (x) => 1;
lnexp[1] = (x) => - x;
double[] t = { 1, 2, 3, 4, 6, 9, 10, 13 };
double[] yt = { 117, 100, 88, 72, 53, 29.5, 25.2, 15.2 };
vector xs = new vector(t);
vector ys = new vector(yt);
vector lny = new vector(yt.Length);
vector yerr = new vector(yt.Length);
vector lnerr = new vector(yt.Length);
for (int i = 0; i < yt.Length; i++)
{
lny[i] = Log(yt[i]);
yerr[i] = yt[i] / 20;
lnerr[i] = yerr[i] / yt[i];
}
lsfit fit1 = new lsfit(xs, lny, lnerr, lnexp);
vector c = fit1.C;
WriteLine("\n Data:");
xs.print("time in days = ");
ys.print("y = ");
WriteLine("\n Attempting to make fit y=a*Exp(-lambda*t) by using ln(y)=ln(a)-lambda*t");
c.print("c = ln(a) lambda = ");
WriteLine("a = {0:f6}", Exp(c[0]));
WriteLine("half life = {0:f6}\n", Log(2) * 1 / c[1]);
Func <double, double> fit1fun = x => c[0] * lnexp[0](x) + c[1] * lnexp[1](x);
var fitwriter = new System.IO.StreamWriter("out.fit.txt");
for (double x = 0; x < 20; x += 0.1)
{
fitwriter.Write("{0:f16} {1:f16}\n", x, Exp(fit1fun(x)));
}
fitwriter.Close();
var datwriter = new System.IO.StreamWriter("out.data.txt");
for (int i = 0; i < xs.size; i++)
{
datwriter.Write("{0:f16} {1:f16} {2:f16}\n", xs[i], yt[i], yerr[i]);
}
datwriter.Close();
return(0);
}
static void Main()
{
Write("==== Finding the extrenum of the Rosenbrock's Valley function with a=1 and b=100. ==== \n\n");
Write("Start guess of root:\n");
Func <vector, vector> f = x => new vector(-2 * (1 - x[0]) - 200 * x[0] * (x[1] - x[0] * x[0]), 200 * (x[1] - x[0] * x[0]));
vector startx = new vector(0, 0);
startx.print();
vector x0 = root.newton(f, startx);
Write("\n After running newtons method, the found extremum is:\n");
x0.print("");
}
static void Main()
{
int callCount = 0;
Func <vector, double> f = (X) => Pow(1 - X[0], 2) + 100 * Pow(X[1] - X[0] * X[0], 2);
double epsilon = 1e-4;
vector x = new vector(1.5, 0.5);
int num_of_steps = qnewton(f, ref x, epsilon);
vector x_exact = new vector(1.0, 1.0);
WriteLine("\n__________________________________________________________________________________________________________");
WriteLine("Question A\n Quasi-Newton method with numerical gradient, back-tracking linesearch, and rank-1 update");
WriteLine("Test function f(x,y)=(1-x)^2+100(y-x^2)^2, min at 1,1");
x.print("Numericaly found min x : ");
x.print("Exact min x : ");
WriteLine("Function value at min : {0}", f(x));
WriteLine("Gradient tollerence : {0}", epsilon);
WriteLine("Actual gradient : {0}", gradient(f, x).norm());
WriteLine("Actual error : {0}", Abs(f(x) - f(x_exact)));
WriteLine("Number of steps : {0}", num_of_steps);
WriteLine("");
f = (X) => Pow(X[0] * X[0] + X[1] - 11, 2) + Pow(X[0] + X[1] * X[1] - 7, 2);
epsilon = 1e-4;
x = new vector(2.4, 2.4);
num_of_steps = qnewton(f, ref x, epsilon);
x_exact = new vector(3, 2);
WriteLine("");
WriteLine("Test function f(x,y)=(x^2+y-11)^2+(x+y^2-7)^2, min at 3,2");
x.print("Numericaly found min x : ");
x.print("Exact min x : ");
WriteLine("Function value at min : {0}", f(x));
WriteLine("Gradient tollerence : {0}", epsilon);
WriteLine("Actual gradient : {0}", gradient(f, x).norm());
WriteLine("Actual error : {0}", Abs(f(x) - f(x_exact)));
WriteLine("Number of steps : {0}", num_of_steps);
WriteLine("__________________________________________________________________________________________________________\n");
}