/// <summary>
/// Evaluates the Jacobian of a multivariate function f at vector x.
/// </summary>
/// <remarks>
/// This function assumes that the length of vector x consistent with the argument count of f.
/// </remarks>
/// <param name="f">Multivariate function handle.</param>
/// <param name="x">Points at which to evaluate Jacobian.</param>
/// <returns>Jacobian vector.</returns>
public double[] Evaluate(Func <double[], double> f, double[] x)
{
var jacobian = new double[x.Length];
for (var i = 0; i < jacobian.Length; i++)
{
jacobian[i] = _df.EvaluatePartialDerivative(f, x, i, 1);
}
return(jacobian);
}
/// <summary>
/// Evaluates the Hessian of a multivariate function f at points x.
/// </summary>
/// <remarks>
/// This method of computing the Hessian is only valid for Lipschitz continuous functions.
/// The function mirrors the Hessian along the diagonal since d2f/dxdy = d2f/dydx for continuously differentiable functions.
/// </remarks>
/// <param name="f">Multivariate function handle.></param>
/// <param name="x">Points at which to evaluate Hessian.></param>
/// <returns>Hessian tensor.</returns>
public double[,] Evaluate(Func <double[], double> f, double[] x)
{
var hessian = new double[x.Length, x.Length];
// Compute diagonal elements
for (var row = 0; row < x.Length; row++)
{
hessian[row, row] = _df.EvaluatePartialDerivative(f, x, row, 2);
}
// Compute non-diagonal elements
for (var row = 0; row < x.Length; row++)
{
for (var col = 0; col < row; col++)
{
var mixedPartial = _df.EvaluateMixedPartialDerivative(f, x, new[] { row, col }, 2);
hessian[row, col] = mixedPartial;
hessian[col, row] = mixedPartial;
}
}
return(hessian);
}
public void VectorFunction1PartialDerivativeTest()
{
Func<double[], double>[] f =
{
(x) => Math.Pow(x[0],2) - 3*x[1],
(x) => x[1]*x[1] + 2*x[0]*x[1]
};
var x0 = new double[] { 2, 2 };
var g = new double[] { 4, 4 };
var df = new NumericalDerivative();
Assert.AreEqual(g, df.EvaluatePartialDerivative(f, x0, 0, 1));
Assert.AreEqual(new double[] { 2, 0 }, df.EvaluatePartialDerivative(f, x0, 0, 2));
Assert.AreEqual(new double[] { -3, 8 }, df.EvaluatePartialDerivative(f, x0, 1, 1));
}
public void RosenbrockFunctionMixedDerivativeOneVariableSecondOrderTest()
{
Func<double[], double> f = x => Math.Pow(1 - x[0], 2) + 100 * Math.Pow(x[1] - Math.Pow(x[0], 2), 2);
var df = new NumericalDerivative();
var x0 = new double[] { 2, 2 };
var parameterindex = new int[] { 0, 0 };
Assert.AreEqual(1602, df.EvaluatePartialDerivative(f, x0, 0, 1), 1e-6);
Assert.AreEqual(4002, df.EvaluateMixedPartialDerivative(f, x0, parameterindex, 2));
}
public void ExponentialFunctionPartialDerivativeCurrentValueTest()
{
//Test Function
Func<double[], double> f = (x) => Math.Sin(x[0] * x[1]) + Math.Exp(-x[0] / 2) + x[1] / x[0];
//Analytical partial dfdx
Func<double[], double> dfdx =
(x) => Math.Cos(x[0] * x[1]) * x[1] - Math.Exp(-x[0] / 2) / 2 - x[1] / Math.Pow(x[0], 2);
//Analytical partial dfdy
Func<double[], double> dfdy = (x) => Math.Cos(x[0] * x[1]) * x[0] + 1 / x[0];
// Current value
var x1 = new double[] { 3, 3 };
var current = f(x1);
var df = new NumericalDerivative(5, 2);
Assert.AreEqual(dfdx(x1), df.EvaluatePartialDerivative(f, x1, 0, 1, current), 1e-8);
Assert.AreEqual(dfdy(x1), df.EvaluatePartialDerivative(f, x1, 1, 1, current), 1e-8);
}
public void ExponentialFunctionPartialDerivativeTest()
{
//Test Function
Func<double[], double> f = (x) => Math.Sin(x[0] * x[1]) + Math.Exp(-x[0] / 2) + x[1] / x[0];
//Analytical partial dfdx
Func<double[], double> dfdx =
(x) => Math.Cos(x[0] * x[1]) * x[1] - Math.Exp(-x[0] / 2) / 2 - x[1] / Math.Pow(x[0], 2);
//Analytical partial dfdy
Func<double[], double> dfdy = (x) => Math.Cos(x[0] * x[1]) * x[0] + 1 / x[0];
var df = new NumericalDerivative(3, 1);
var x1 = new double[] { 3, 3 };
Assert.AreEqual(dfdx(x1), df.EvaluatePartialDerivative(f, x1, 0, 1), 1e-8);
Assert.AreEqual(dfdy(x1), df.EvaluatePartialDerivative(f, x1, 1, 1), 1e-8);
var x2 = new double[] { 300, -50 };
df.StepType = StepType.Absolute;
Assert.AreEqual(dfdx(x2), df.EvaluatePartialDerivative(f, x2, 0, 1), 1e-5);
Assert.AreEqual(dfdy(x2), df.EvaluatePartialDerivative(f, x2, 1, 1), 1e-2);
}
