// Use homotopy method with newton's method to find a solution for F(x) = 0.
private static List <Arrow> NSolve(List <Expression> F, List <Arrow> x0, double Epsilon, int MaxIterations)
{
int M = F.Count;
int N = x0.Count;
// Compute JxF, the Jacobian of F.
List <Dictionary <Expression, Expression> > JxF = Jacobian(F, x0.Select(i => i.Left)).ToList();
// Define a function to evaluate JxH(x), where H = F(x) - s*F(x0).
CodeGen code = new CodeGen();
ParamExpr _JxH = code.Decl <double[, ]>(Scope.Parameter, "JxH");
ParamExpr _x0 = code.Decl <double[]>(Scope.Parameter, "x0");
ParamExpr _s = code.Decl <double>(Scope.Parameter, "s");
// Load x_j from the input array and add them to the map.
for (int j = 0; j < N; ++j)
{
code.DeclInit(x0[j].Left, LinqExpr.ArrayAccess(_x0, LinqExpr.Constant(j)));
}
LinqExpr error = code.Decl <double>("error");
// Compile the expressions to assign JxH
for (int i = 0; i < M; ++i)
{
LinqExpr _i = LinqExpr.Constant(i);
for (int j = 0; j < N; ++j)
{
code.Add(LinqExpr.Assign(
LinqExpr.ArrayAccess(_JxH, _i, LinqExpr.Constant(j)),
code.Compile(JxF[i][x0[j].Left])));
}
// e = F(x) - s*F(x0)
LinqExpr e = code.DeclInit <double>("e", LinqExpr.Subtract(code.Compile(F[i]), LinqExpr.Multiply(LinqExpr.Constant((double)F[i].Evaluate(x0)), _s)));
code.Add(LinqExpr.Assign(LinqExpr.ArrayAccess(_JxH, _i, LinqExpr.Constant(N)), e));
// error += e * e
code.Add(LinqExpr.AddAssign(error, LinqExpr.Multiply(e, e)));
}
// return error
code.Return(error);
Func <double[, ], double[], double, double> JxH = code.Build <Func <double[, ], double[], double, double> >().Compile();
double[] x = new double[N];
// Remember where we last succeeded/failed.
double s0 = 0.0;
double s1 = 1.0;
do
{
try
{
// H(F, s) = F - s*F0
NewtonsMethod(M, N, JxH, s0, x, Epsilon, MaxIterations);
// Success at this s!
s1 = s0;
for (int i = 0; i < N; ++i)
{
x0[i] = Arrow.New(x0[i].Left, x[i]);
}
// Go near the goal.
s0 = Lerp(s0, 0.0, 0.9);
}
catch (FailedToConvergeException)
{
// Go near the last success.
s0 = Lerp(s0, s1, 0.9);
for (int i = 0; i < N; ++i)
{
x[i] = (double)x0[i].Right;
}
}
} while (s0 > 0.0 && s1 >= s0 + 1e-6);
// Make sure the last solution is at F itself.
if (s0 != 0.0)
{
NewtonsMethod(M, N, JxH, 0.0, x, Epsilon, MaxIterations);
for (int i = 0; i < N; ++i)
{
x0[i] = Arrow.New(x0[i].Left, x[i]);
}
}
return(x0);
}