public static vector newton(Func<vector,vector> f, vector x, double eps=1e-3, double dx=1e-7){
int n=x.size;
vector fx=f(x);
vector z,fz;
while(true){
matrix J=jacobian(f,x,fx); // define the Jacobian matrix with partial derivatives
qrdecomposition qrJ=new qrdecomposition(J); // use QR-decomposition
matrix B=qrJ.inverse(); // construct the inverse of our QR-decomposition
vector Dx=-B*fx; // solve the Newton's algorithm in matrix form (using eq. 5)
double s=1;
while(true){ // backtracking linesearch
z=x+Dx*s; // creating left-hand side of eq. 8
fz=f(z); // creating left-hand side of eq. 8
if(fz.norm()<(1-s/2)*fx.norm()){ // stop if step is good
break;
}
if(s<1.0/32){ // stop if minimum step-size is reached
break;
}
s/=2; // backtrack using half the step-size
}
x=z;
fx=fz;
if(fx.norm()<eps)break; // until converged
}
return x;
}//newton
}//broyden
public static int broyden
(Func <vector, vector> f, ref vector x, double eps = 1e-3)
{
vector fx = f(x), z, fz;
matrix J = jacobian(f, x, fx);
var qrJ = new qrdecomposition(J);
matrix B = qrJ.inverse();
int nsteps = 0;
while (++nsteps < 999)
{
vector Dx = -B * fx;
double s = 1;
while (true)
{
z = x + Dx * s;
fz = f(z);
if (fz.norm() < (1 - s / 2) * fx.norm())
{
break;
}
if (s < 1.0 / 32)
{
J = jacobian(f, x, fx);
qrJ = new qrdecomposition(J);
B = qrJ.inverse();
break;
}
s /= 2;
}
vector dx = z - x;
vector df = fz - fx;
if (dx.dot(df) > 1e-9)
{
vector c = (dx - B * df) / dx.dot(df);
B.update(c, dx);
}
//vector c=(dx-B*df)/(df%df); B.update(c,df);
//vector c=(dx-B*df)/(dx%(B*df)); B.update(c,B*df);
x = z;
fx = fz;
if (fx.norm() < eps)
{
break;
}
}
return(nsteps);
}//broyden
static void Main()
{
int n = 7, m = n;
var A = new matrix(n, m);
var rnd = new System.Random();
for (int i = 0; i < A.size1; i++)
{
for (int j = 0; j < A.size2; j++)
{
A[i, j] = 2 * (rnd.NextDouble() - 1);
}
}
A.print("random matrix A:");
var b = new vector(m);
for (int i = 0; i < b.size; i++)
{
b[i] = rnd.NextDouble();
}
b.print("random vector b:\n");
var qra = new qrdecomposition(A);
var x = qra.solve(b);
x.print("solution x to system A*x=b:\n");
var Ax = A * x;
Ax.print("check: A*x (should be equal b):\n");
if (vector.approx(Ax, b))
{
WriteLine("test passed");
}
else
{
WriteLine("test failed");
}
var B = qra.inverse();
var C = (B * A);
C.print("A^-1*A=");
var D = (A * B);
D.print("A*A^-1=");
}
public static vector newton
(Func <vector, vector> f, vector x, double eps = 1e-3, double dx = 1e-7)
{
int n = x.size;
vector fx = f(x), z, fz;
while (true)
{
matrix J = jacobian(f, x, fx);
qrdecomposition qrJ = new qrdecomposition(J);
matrix B = qrJ.inverse();
vector Dx = -B * fx;
double s = 1;
while (true) // backtracking linesearch
{
z = x + Dx * s;
fz = f(z);
if (fz.norm() < (1 - s / 2) * fx.norm())
{
break;
}
if (s < 1.0 / 32)
{
break;
}
s /= 2;
}
x = z;
fx = fz;
if (fx.norm() < eps)
{
break;
}
}
return(x);
} //broyden