/// <summary>
/// Returns the gradient of the specified parametric function
/// at the given argument.
/// </summary>
/// <param name="function">
/// The function to be differentiated.
/// </param>
/// <param name="argument">
/// The argument at which the gradient must be evaluated.
/// </param>
/// <param name="parameter">
/// The function parameter.
/// </param>
/// <typeparam name="TFunctionParameter">
/// The type of the function parameter.
/// </typeparam>
/// <returns>
/// The gradient of the specified function evaluated
/// at the given argument and parameter.
/// </returns>
/// <exception cref="ArgumentNullException">
/// <paramref name="function"/> is <b>null</b>.<br/>
/// -or-<br/>
/// <paramref name="argument"/> is <b>null</b>.
/// </exception>
public static DoubleMatrix Gradient <TFunctionParameter>(
Func <DoubleMatrix, TFunctionParameter, double> function,
DoubleMatrix argument,
TFunctionParameter parameter)
{
if (function is null)
{
throw new ArgumentNullException(nameof(function));
}
if (argument is null)
{
throw new ArgumentNullException(nameof(argument));
}
int numberOfArguments = argument.Count;
DoubleMatrix gradient = DoubleMatrix.Dense(
numberOfArguments, 1);
double h, h_by_2, arg_i, partialDerivative;
DoubleMatrix x = (DoubleMatrix)argument.Clone();
// Using central differences for an accuracy of order 4
// See also http://en.wikipedia.org/wiki/Finite_difference_coefficients
//
// Points: -2 -1 0 1 2
// Coefficients: 1/12 −2/3 0 2/3 −1/12
for (int i = 0; i < numberOfArguments; i++)
{
arg_i = argument[i];
h = NumericalDifferentiation.GetFirstOrderDelta(arg_i);
h_by_2 = h * 2.0;
partialDerivative = 0.0;
// argument - 2*h
x[i] -= h_by_2;
partialDerivative += function(x, parameter);
x[i] = arg_i;
// argument - h
x[i] -= h;
partialDerivative -= 8.0 * function(x, parameter);
x[i] = arg_i;
// argument + h
x[i] += h;
partialDerivative += 8.0 * function(x, parameter);
x[i] = arg_i;
// argument + 2*h
x[i] += h_by_2;
partialDerivative -= function(x, parameter);
x[i] = arg_i;
partialDerivative /= (12.0 * h);
gradient[i] = partialDerivative;
}
return(gradient);
}
/// <summary>
/// Returns the Hessian matrix of the specified parametric function
/// at the given argument.
/// </summary>
/// <param name="function">
/// The function to be differentiated.
/// </param>
/// <param name="argument">
/// The argument at which the Hessian must be evaluated.
/// </param>
/// <param name="parameter">
/// The function parameter.
/// </param>
/// <typeparam name="TFunctionParameter">
/// The type of the function parameter.
/// </typeparam>
/// <returns>
/// The Hessian matrix of the specified function evaluated
/// at the given argument.
/// </returns>
/// <exception cref="ArgumentNullException">
/// <paramref name="function"/> is <b>null</b>.<br/>
/// -or-<br/>
/// <paramref name="argument"/> is <b>null</b>.
/// </exception>
public static DoubleMatrix Hessian <TFunctionParameter>(
Func <DoubleMatrix, TFunctionParameter, double> function,
DoubleMatrix argument,
TFunctionParameter parameter)
{
if (function is null)
{
throw new ArgumentNullException(nameof(function));
}
if (argument is null)
{
throw new ArgumentNullException(nameof(argument));
}
int numberOfArguments = argument.Count;
DoubleMatrix hessian = DoubleMatrix.Dense(
numberOfArguments, numberOfArguments);
double h, arg_i;
DoubleMatrix x = (DoubleMatrix)argument.Clone();
double secondPartialDerivative;
double h2;
for (int i = 0; i < numberOfArguments; i++)
{
arg_i = argument[i];
h = NumericalDifferentiation.GetSecondOrderDelta(arg_i);
h2 = h * h;
secondPartialDerivative = 0.0;
// argument - h
x[i] -= h;
secondPartialDerivative += function(x, parameter);
x[i] = arg_i;
// argument
secondPartialDerivative -= 2.0 * function(x, parameter);
// argument + h
x[i] += h;
secondPartialDerivative += function(x, parameter);
x[i] = arg_i;
secondPartialDerivative /= h2;
hessian[i, i] = secondPartialDerivative;
}
// Mixed partial derivatives
// Points: -1,-1 -1,1 0 1,-1 1,1
// Coefficients: 1/4 -1/4 0 -1/4 1/4
double mixedPartialDerivative, k, arg_j, denominator;
for (int i = 0; i < numberOfArguments; i++)
{
arg_i = argument[i];
h = NumericalDifferentiation.GetSecondOrderDelta(arg_i);
for (int j = i + 1; j < numberOfArguments; j++)
{
arg_j = argument[j];
k = NumericalDifferentiation.GetSecondOrderDelta(arg_j);
denominator = h * k;
mixedPartialDerivative = 0.0;
// -1, -1
x[i] -= h;
x[j] -= k;
mixedPartialDerivative += .25 * function(x, parameter);
x[i] = arg_i;
x[j] = arg_j;
// Points: -1,-1 -1,1 0 1,-1 1,1
// Coefficients: 1/4 -1/4 0 -1/4 1/4
// -1, 1
x[i] -= h;
x[j] += k;
mixedPartialDerivative -= .25 * function(x, parameter);
x[i] = arg_i;
x[j] = arg_j;
// 1,-1
x[i] += h;
x[j] -= k;
mixedPartialDerivative -= .25 * function(x, parameter);
x[i] = arg_i;
x[j] = arg_j;
// 1, 1
x[i] += h;
x[j] += k;
mixedPartialDerivative += .25 * function(x, parameter);
x[i] = arg_i;
x[j] = arg_j;
mixedPartialDerivative /= denominator;
hessian[i, j] = mixedPartialDerivative;
hessian[j, i] = mixedPartialDerivative;
}
}
return(hessian);
}