//method to calculate the root for a system of functions given an initial guess/vector
//Gradient Descent with Fixed Alpha
public double[] GradientAlpha(FunctionMatrix f, double[] x, double toler)
{
int counter = 0;
int dim = x.GetLength(0);
int r = f.Rows;
double[,] Jmat = new double[r, dim];
Matrix MJmat = new Matrix(Jmat);
Vector VFg1 = new Vector(r);
Vector Vz = new Vector(dim);
Vector VFg3 = new Vector(r);
Vector VFg2 = new Vector(r);
Vector VFgc = new Vector(r);
Vector VFgnew = new Vector(r);
Vector Vxnew = new Vector(dim);
double alpha2 = 0;
double alpha3 = 0;
double[] xold = x;
double[] xnew = new double[dim];
double g1 = 0;
double g2 = 0;
double g3 = 0;
double gc = 0;
double gnew = 0;
double normz = 0;
double h1 = 0;
double h2 = 0;
double h3 = 0;
double ac = 0;
do
{
VFg1 = f.EvaluateVector(xold);
g1 = VFg1.Magnitude();
//calculate the Jacobian
Jmat = jacobi(f, xold, 0.001);
//now update the Matrix object which represents the Jacobian
for (int i = 0; i < r; i++)
for (int j = 0; j < dim; j++)
MJmat[i, j] = Jmat[i, j];
Vz = 2 * (MJmat * VFg1);
normz = Math.Sqrt(Vz.Magnitude());
if (normz > toler)
{
Vz = (1 / normz) * Vz;
alpha3 = 1;
VFg3 = f.EvaluateVector((double[])((Vector)xold - (alpha3 * Vz)));
g3 = VFg3.Magnitude();
while (Math.Abs(g3 - g1) > toler)
{
alpha3 = 0.5 * alpha3;
VFg3 = f.EvaluateVector((double[])((Vector)xold - (alpha3 * Vz)));
g3 = VFg3.Magnitude();
if (Math.Abs(alpha3) < 0.5 * toler)
break;
}
alpha2 = 0.5 * alpha3;
VFg2 = f.EvaluateVector((double[])((Vector)xold - (alpha2 * Vz)));
g2 = VFg2.Magnitude();
//perform interpolation
h1 = (g2 - g1) / alpha2;
h2 = (g3 - g2) / (alpha3 - alpha2);
h3 = (h2 - h1) / alpha3;
ac = 0.5 * (alpha2 - (h1 / h3));
VFgc = f.EvaluateVector((double[])((Vector)xold - (ac * Vz)));
gc = VFgc.Magnitude();
if (gc < g3)
{
Vxnew = ((Vector)xold) - (ac * Vz);
VFgnew = f.EvaluateVector((double[])Vxnew);
gnew = VFgnew.Magnitude();
}
else
{
Vxnew = ((Vector)xold) - (alpha3 * Vz);
VFgnew = f.EvaluateVector((double[])Vxnew);
gnew = VFgnew.Magnitude();
}
//test for convergence
if (Math.Abs(gnew - g1) < toler)
break;
else
{
xold = (double[])Vxnew;
counter++;
}
//end if
}
} while (counter < 100);
return (double[])Vxnew;
//end of method
}
//method to calculate the root for a system of functions given an initial guess/vector
public double[] newtonraphson(FunctionMatrix f, double[] x, double toler)
{
int maxiter = 1000;
double err = 10000000;
int counter = 0;
int dim = x.GetLength(0);
int r = f.Rows;
double[] xn = new double[dim];
//populating vector xn
for (int i = 0; i < dim; i++)
xn[i] = x[i];
double[] xnp1 = new double[dim];
while ((counter < maxiter) && (err > toler))
{
Vector Fn = f.EvaluateVector(xn);
//converting to Vector Objects
Vector Vxn = new Vector(xn);
//Need to calculate the Jacobi and convert it to a matrix
double[,] j = jacobi(f, xn, 0.001);
double[,] jinv = s.find_inverse(j, j.GetLength(0));
Matrix Jinv = new Matrix(jinv);
//this is just F(xn)/F'(xn)
Vector tmp = Jinv * Fn;
//xnp1 = xn - F(xn)/F'(xn)
Vector Vxnp1 = Vxn - tmp;
//what is the size of the err i.e. xnp1-xn
Vector Diff = Vxnp1 - Vxn;
err = Diff.maxNorm();
double[] t = new double[dim];
xnp1 = (double[])Vxnp1;
xn = xnp1;
counter++;
}
return xn;
//end of method
}
//method to calculate the inverse of a matrix using the Sherman Morrison formula
//this method will be utilised in the Broyden method
public double[,] BroydenAinv(double[,] Ainvold, double[] deltax, double[] deltaF)
{
Matrix MAinvold = new Matrix(Ainvold);
Vector Vdeltax = new Vector(deltax);
Vector VdeltaF = new Vector(deltaF);
//placeholder vector, won't be used for anything apart from calling the outer product and dot product Vector methods
Vector interim = new Vector(1);
Vector temp = new Vector(interim.dot(MAinvold, VdeltaF));
Vector u = new Vector(Vdeltax - temp);
Vector v = new Vector(interim.dot(Vdeltax, MAinvold));
Matrix numer = new Matrix(interim.outer(u, v));
double denom = interim.dot(Vdeltax, temp);
Matrix update = new Matrix(1 / denom * numer);
Matrix MAinvnew = new Matrix(MAinvold + update);
double[,] Ainvnew = (double[,])MAinvnew;
return Ainvnew;
}
//method to calculate the root for a system of functions given an initial guess/vector
public double[] BroydenShermanMorrison(FunctionMatrix f, double[] x, double toler)
{
double err = 10000000;
int counter = 0;
int dim = x.GetLength(0);
int r = f.Rows;
double[] xn = new double[dim];
//populating vector xn
for (int i = 0; i < dim; i++)
xn[i] = x[i];
double[] xnp1 = new double[dim];
//first iteration i.e. use Newton Raphson for one iteration, the Inverse of the Jacobi will be used as the Ainv for Broyden
Vector Fn = f.EvaluateVector(xn);
Vector Vxn = new Vector(xn);
//Need to calculate the Jacobi and convert it to a matrix
double[,] j = jacobi(f, xn, 0.001);
double[,] jinv = s.find_inverse(j, j.GetLength(0));
Matrix Jinv = new Matrix(jinv);
//this is just F(xn)/F'(xn)
Vector tmp = Jinv * Fn;
//xnp1 = xn - F(xn)/F'(xn)
Vector Vxnp1 = Vxn - tmp;
//what is the size of the err i.e. xnp1-xn
Vector Vdeltax = Vxnp1 - Vxn;
err = Vdeltax.maxNorm();
xnp1 = (double[])Vxnp1;
Vector placeholder = new Vector(1);
//we'll use these later
Vector VFnold = f.EvaluateVector(xn);
Vector VFnew = f.EvaluateVector(xnp1);
Vector VdeltaF = VFnew - VFnold;
double[,] Aold = new double[dim, dim];
double[,] Anew = new double[dim, dim];
//now we'll start on the Broyden iteration element
if (err < toler)
return xn;
else
{
while ((err > toler) && (counter < 1000))
{
if (counter == 0)
{
Aold = jinv;
Anew = BroydenAinv(Aold, (double[])Vdeltax, (double[])VdeltaF);
}
else
{
Anew = BroydenAinv(Aold, (double[])Vdeltax, (double[])VdeltaF);
}
//now update all the variables
Vxnp1 = Vxn - placeholder.dot(new Matrix(Anew), VFnew);
Vdeltax = Vxnp1 - Vxn;
err = Vdeltax.maxNorm();
Aold = Anew;
VFnold = f.EvaluateVector((double[])Vxn);
VFnew = f.EvaluateVector((double[])Vxnp1);
VdeltaF = VFnew - VFnold;
Vxn = Vxnp1;
xn = (double[])Vxn;
counter++;
}
return xn;
}
//end of method
}
//dot product of a vector and a Matrix
public Vector dot(Vector v, Matrix m)
{
int mrows = m.Rows;
int mcols = m.Cols;
int vrows = v.Rows;
if (mcols == vrows)
{
double[] temp = new double[mrows];
for (int i = 0; i < mrows; i++)
{
double sum = 0;
for (int j = 0; j < vrows; j++)
{
double c = m[i, j] * v[j];
sum = sum + m[i, j] * v[j];
}
temp[i] = sum;
}
Vector ans = new Vector(temp);
return ans;
}
else
return v;
}
//outer productover of two vectors
//dot product of two vectors
public Matrix outer(Vector v1, Vector v2)
{
int r1 = v1.Rows;
int r2 = v2.Rows;
double[,] temp = new double[v1.rows, v2.rows];
for (int i = 0; i < r1; i++)
{
for (int j = 0; j < r2; j++)
{
temp[i, j] = v1.data[i] * v2.data[j];
}
}
Matrix m = new Matrix(temp);
return m;
}
public static Vector operator -(Vector a)
{
Vector tmp = new Vector(a.rows);
for (int i = 0; i < a.rows; i++)
tmp.data[i] = -a.data[i];
return tmp;
}
//dot product of two vectors
public double dot(Vector v1, Vector v2)
{
int r1 = v1.Rows;
int r2 = v2.Rows;
if (r1 == r2)
{
double sum = 0;
for (int i = 0; i < r1; i++)
{
sum = sum + v1.data[i] * v2.data[i];
}
return sum;
}
else return 0;
}
public static Vector operator +(Vector a)
{
Vector tmp = new Vector(a);
return tmp;
}
public static Vector operator -(Vector a, Vector b)
{
if (a.rows != b.rows)
return null;
Vector tmp = new Vector(a.rows);
for (int i = 0; i < a.rows; i++)
tmp.data[i] = a.data[i] - b.data[i];
return tmp;
}
//Product of Matrix and vector
public static Vector operator *(Matrix a, Vector b)
{
if (a.Cols != b.rows)
return null;
int i, j;
double sum = 0;
Vector tmp = new Vector(a.Rows);
for (i = 0; i < a.Rows; i++)
{
sum = 0;
for (j = 0; j < a.Cols; j++)
sum += a[i, j] * b[j];
tmp[i] = sum;
}
return tmp;
}
//scalar product
public static Vector operator *(double a, Vector b)
{
Vector tmp = new Vector(b.rows);
for (int i = 0; i < b.rows; i++)
tmp[i] = a * b[i];
return tmp;
}
//scalar product
public static double operator *(Vector a, Vector b)
{
if (a.rows != b.rows)
return 0;
Vector tmp = new Vector(a.rows);
double sum = 0;
for (int i = 0; i < a.rows; i++)
sum += a[i] * b[i];
return sum;
}
public Vector(Vector val)
{
rows = val.rows;
data = new double[rows];
for (int i = 0; i < rows; i++)
data[i] = val.data[i];
}
public Vector EvaluateVector(double[] val)
{
Vector tmp = new Vector(rows);
int i;
for (i = 0; i < rows; i++)
tmp[i] = f[i, 0](val);
return tmp;
}