public lsfit(vector x, vector y, vector dy, Func <double, double>[] fs)
{
int n = x.size;
int m = fs.Length;
// We move the functions into the variable f
f = fs;
matrix A = new matrix(n, m);
vector b = new vector(n);
for (int i = 0; i < n; i++)
{
b[i] = y[i] / dy[i];
for (int j = 0; j < m; j++)
{
A[i, j] = f[j](x[i]) / dy[i];
}
}
// Solve the system of equations
var qrGS = new qrDecompositionGS(A);
c = qrGS.solve(b);
// At last we calculate the covariance matrix, A^T * A
matrix Ainv = qrGS.inverse();
sigma = Ainv * Ainv.T;
}
void covariance()
{
matrix sigmaInv = qr.r.transpose() * qr.r;
qrDecompositionGS cov = new qrDecompositionGS(sigmaInv);
sigma = cov.inverse();
}
static void Main()
{
var rand = new Random();
// We can pull out new random numbers between 0 and 1 with rand.NextDouble() and
// stuff it into a matrix
int n = 4;
matrix A = new matrix(n, n);
for (int i = 0; i < n; i++)
{
for (int j = 0; j < n; j++)
{
A[i, j] = 2 - 4 * rand.NextDouble();
}
}
WriteLine("Set up a random matrix, and try to find the inverse matrix via QR" +
" decomposition.\n");
WriteLine("Printing random matrix A:");
A.print();
// Factorize the matrix into two matrices Q and R, and calculate the inverse matrix B
var qrGS = new qrDecompositionGS(A);
matrix B = qrGS.inverse();
matrix Q = qrGS.Q;
matrix R = qrGS.R;
WriteLine("\nPrinting matrix R:");
R.print();
WriteLine("\nPrinting matrix Q:");
Q.print();
WriteLine("\nPrinting matrix B (inverse to A):");
B.print();
// Let's check if B is actually the inverse by calculating A*B which then should
// give the identity matrix
matrix AB = A * B;
WriteLine("\nPrinting matrix A*B:");
AB.print();
matrix I = new matrix(n, n);
I.set_identity();
bool approx = AB.approx(I);
if (approx == true)
{
WriteLine("A*B is approximately equal to the identity matrix.");
}
else
{
WriteLine("A*B is not approximately equal to the identity matrix.");
}
}
public static void Main()
{
Write("==== B ====\n");
Write("Testing inverse by Gram-Schmidt in random matrix A:\n");
matrix a = rndMat.randomMatrix(5, 5);
a.print("A = ");
var qr = new qrDecompositionGS(a);
Write("Inverse is found to be:\n");
(qr.inverse()).print("A^(-1) = ");
Write("Checking that this is indeed the desired inverse matrix by calculating I=A^-1 * A:\n");
(qr.inverse() * a).print("A^(-1)*A = ");
} //Main
public ols(Func <double, double>[] fs, vector x, vector y, vector dy)
{
int m = fs.Length;
int n = y.size;
matrix a = new matrix(n, m);
vector b = new vector(n);
for (int i = 0; i < n; i++)
{
b[i] = y[i] / dy[i];
for (int k = 0; k < m; k++)
{
a[i, k] = fs[k](x[i]) / dy[i];
}
} // matrices and vector for qr-demposition
qr = new qrDecompositionGS(a);
c = qr.solve(b);
} //constructor
public static vector newton(Func <vector, vector> f, vector x, double epsilon = 1e-3, double dx = 1e-7)
{
vector root = x;
vector step;
do
{
/* calculate jacobian */
vector fx = f(root);
int i = fx.size;
int k = root.size;
/* Add assertion of i!=k (only square J) */
matrix jacobian = new matrix(i, k);
for (int r = 0; r < i; r++)
{
for (int c = 0; c < k; c++)
{
vector deltax = root.copy();
deltax[c] += dx;
jacobian[r, c] = (f(deltax)[r] - f(root)[r]) / dx;
}
}
/* solve J * deltax = - f(x) with QR-factorization */
qrDecompositionGS qr = new qrDecompositionGS(jacobian);
step = qr.solve(-1 * fx);
/* Do line search and choose lambda when (8) is fulfilled. */
double lambda = 1;
while (f(root + lambda * step).norm() > (1 - lambda / 2) * fx.norm() && lambda > 1 / 64)
{
lambda /= 2;
}
root += lambda * step;
} while(f(x).norm() > epsilon && step.norm() > dx);
return(root);
}
public static void Main()
{
Write("==== A.1 ====\n");
Write("Testing on random tall matrix A:\n");
matrix a = rndMat.randomMatrix(5, 3);
a.print("A =");
qrDecompositionGS qr = new qrDecompositionGS(a);
Write("Q and R matrices found from QR-factorization:\n");
(qr.q).print("Q = ");;
(qr.r).print("R = ");;
Write("Testing that Q is orthogonal:\n");
(qr.q.transpose() * qr.q).print("Q^T*Q = ");
Write("Calculating the difference between A and QR, which should be 0:\n");
(a - qr.q * qr.r).print("A-QR = ");
Write("\n==== A.2 ====\n");
Write("Testing solver on random matrix A:\n");
a = rndMat.randomMatrix(5, 5);
a.print("A = ");
qr = new qrDecompositionGS(a);
Write("and random vector b:\n");
vector b = rndMat.randomVector(5);
b.print("b = ");
Write("Solution to Ax=b is found to be:\n");
vector x = qr.solve(b);
x.print("x = ");
Write("Checking that x is a solution by calculating 0=Ax-b:\n");
vector diff = a * x - b;
diff.print("A*x-b = ");
} //Main
static void Main()
{
// Part 1 of exercise A
WriteLine("-------Part 1 of exercise A:-------\n");
// We start out by creating a random matrix.
var rand = new Random();
// We can pull out new random numbers between 0 and 1 with rand.NextDouble() and
// stuff it into a matrix
int nA = 5;
int mA = 4;
matrix A = new matrix(nA, mA);
for (int i = 0; i < nA; i++)
{
for (int j = 0; j < mA; j++)
{
A[i, j] = 2 - 4 * rand.NextDouble();
}
}
// Print A
WriteLine("We will setup a random tall matrix and perform QR factorization on it.");
WriteLine("Printing random tall matrix A:");
A.print();
// We can now try to factorize the matrix A into the two matríces Q and R
var qrGSA = new qrDecompositionGS(A);
matrix Q = qrGSA.Q;
matrix R = qrGSA.R;
// Let's check that R is upper triangular
WriteLine("\nPrinting matrix R:");
R.print();
WriteLine("\nPrinting matrix Q:");
Q.print();
// Checking that Q^T * Q = 1
WriteLine("\nPrinting (Q^T)*Q:");
matrix QTQ = Q.transpose() * Q;
QTQ.print();
// Let's check if (Q^T)*Q is approximately the identity matrix
matrix I = new matrix(mA, mA);
I.set_identity();
bool approx = QTQ.approx(I);
if (approx == true)
{
WriteLine("(Q^T)*Q is approximately equal to the identity matrix.");
}
else
{
WriteLine("(Q^T)*Q is not approximately equal to the identity matrix.");
}
// Checking that Q*R=A, or in other words that A-Q*R = 0
WriteLine("\nPrinting out A-Q*R:");
matrix AmQR = A - Q * R;
AmQR.print();
// Let's check if Q*R is approximately equal to A, or that A-QR is approx. the zero
// matrix
matrix nullMatrix = new matrix(AmQR.size1, AmQR.size2);
nullMatrix.set_zero();
approx = AmQR.approx(nullMatrix);
if (approx == true)
{
WriteLine("QR is thus approximately equal to A.");
}
else
{
WriteLine("QR is thus not approximately equal to A.");
}
// Part 2 of exercise A
WriteLine("\n\n-------Part 2 of exercise A:-------\n");
// At first we create a random square matrix - let's call it C now to avoid confusion
// with part 1 above
WriteLine("Setting up system C*x = b.");
int nC = 4;
matrix C = new matrix(nC, nC);
for (int i = 0; i < nC; i++)
{
for (int j = 0; j < nC; j++)
{
C[i, j] = 2 - 4 * rand.NextDouble();
}
}
WriteLine("Printing matrix C:");
C.print();
// Next we generate a vector b of a length corresponding to the dimension of the
// square matrix
vector b = new vector(nC);
for (int i = 0; i < nC; i++)
{
b[i] = 3 * rand.NextDouble() - 1.5;
}
WriteLine("\nPrinting out b:");
b.print();
// Once again we factorize our matrix into two matrices Q and R
var qrGSC = new qrDecompositionGS(C);
Q = qrGSC.Q;
R = qrGSC.R;
WriteLine("\nSolving by QR decomposition and back-substitution.");
vector x = qrGSC.solve(b);
WriteLine("\nThe obtained x-vector from the routine is:");
x.print();
WriteLine("\nPrinting vector C*x :");
vector Cx = C * x;
Cx.print();
// Let's check if C*x is approximately equal to b
approx = Cx.approx(b);
if (approx == true)
{
WriteLine("Cx is approximately equal to b. The system has been solved.");
}
else
{
WriteLine("Cx is not approximately equal to b. The system has not been solved.");
}
}
public static vector newton(Func <vector, vector> f, vector x, double epsilon = 1e-3,
double dx = 1e-7)
{
int n = x.size;
vector fx = f(x);
vector x1;
vector fx1;
// Run a loop to find the roots
while (true)
{
// Create the Jacobian
matrix J = jacobian(f, x, fx);
// Run a QR-decomposition algorithm on J, and find it's inverse
var qr = new qrDecompositionGS(J);
matrix B = qr.inverse();
// Find the Newton stepsize
vector deltaX = -B * fx;
// Lambda factor to take a more conservative step
double lambda = 1;
// Backtracking linesearch algorithm
while (true)
{
// Attempt a step and check if we arrive at a point closer to a root
x1 = x + deltaX * lambda;
fx1 = f(x1);
if (fx1.norm() < fx.norm() * (1 - lambda / 2))
{
// This should be a good step, so we accept it by breaking
// the loop.
break;
}
if (lambda < 1.0 / 64)
{
// This is not a good step, but we surrender and accept it
// anyway, hoping that the next step will be better.
break;
}
// Reduce lambda by a factor of 2 before next attempt
lambda /= 2;
}
// Move us to the newfound better positon before attempting the next step
x = x1;
fx = fx1;
// Check if we have converged to a value closer to zero than the given
// accuracy epsilon. If yes, we are done, so we break the loop and return
// the position of the root, x.
if (fx.norm() < epsilon)
{
break;
}
}
return(x);
} // end newton
