private static ISymbolicExpressionTreeNode MakeSum(ISymbolicExpressionTreeNode treeNode, double alpha)
{
if (alpha.IsAlmost(0.0))
{
return(treeNode);
}
else
{
var addition = new Addition();
var node = addition.CreateTreeNode();
var alphaConst = MakeConstant(alpha);
node.AddSubtree(treeNode);
node.AddSubtree(alphaConst);
return(node);
}
}
public static ISymbolicExpressionTreeNode Derive(ISymbolicExpressionTreeNode branch, string variableName)
{
if (branch.Symbol is Constant)
{
return(CreateConstant(0.0));
}
if (branch.Symbol is Variable)
{
var varNode = branch as VariableTreeNode;
if (varNode.VariableName == variableName)
{
return(CreateConstant(varNode.Weight));
}
else
{
return(CreateConstant(0.0));
}
}
if (branch.Symbol is Addition)
{
var sum = addSy.CreateTreeNode();
foreach (var subTree in branch.Subtrees)
{
sum.AddSubtree(Derive(subTree, variableName));
}
return(sum);
}
if (branch.Symbol is Subtraction)
{
var sum = subSy.CreateTreeNode();
foreach (var subTree in branch.Subtrees)
{
sum.AddSubtree(Derive(subTree, variableName));
}
return(sum);
}
if (branch.Symbol is Multiplication)
{
// (f * g)' = f'*g + f*g'
// for multiple factors: (f * g * h)' = ((f*g) * h)' = (f*g)' * h + (f*g) * h'
if (branch.SubtreeCount >= 2)
{
var f = (ISymbolicExpressionTreeNode)branch.GetSubtree(0).Clone();
var fprime = Derive(f, variableName);
for (int i = 1; i < branch.SubtreeCount; i++)
{
var g = (ISymbolicExpressionTreeNode)branch.GetSubtree(i).Clone();
var fg = Product((ISymbolicExpressionTreeNode)f.Clone(), (ISymbolicExpressionTreeNode)g.Clone());
var gPrime = Derive(g, variableName);
var fgPrime = Sum(Product(fprime, g), Product(gPrime, f));
// prepare for next iteration
f = fg;
fprime = fgPrime;
}
return(fprime);
}
else
{
// multiplication with only one argument has no effect -> derive the argument
return(Derive(branch.GetSubtree(0), variableName));
}
}
if (branch.Symbol is Division)
{
// (f/g)' = (f'g - g'f) / g²
if (branch.SubtreeCount == 1)
{
var g = (ISymbolicExpressionTreeNode)branch.GetSubtree(0).Clone();
var gPrime = Product(CreateConstant(-1.0), Derive(g, variableName));
var sqrNode = new Square().CreateTreeNode();
sqrNode.AddSubtree(g);
return(Div(gPrime, sqrNode));
}
else
{
// for two subtrees:
// (f/g)' = (f'g - fg')/g²
// if there are more than 2 subtrees
// div(x,y,z) is interpretered as (x/y)/z
// which is the same as x / (y*z)
// --> make a product of all but the first subtree and differentiate as for the 2-argument case above
var f = (ISymbolicExpressionTreeNode)branch.GetSubtree(0).Clone();
var g = Product(branch.Subtrees.Skip(1).Select(n => (ISymbolicExpressionTreeNode)n.Clone()));
var fprime = Derive(f, variableName);
var gprime = Derive(g, variableName);
var gSqr = Square(g);
return(Div(Subtract(Product(fprime, g), Product(f, gprime)), gSqr));
}
}
if (branch.Symbol is Logarithm)
{
var f = (ISymbolicExpressionTreeNode)branch.GetSubtree(0).Clone();
return(Product(Div(CreateConstant(1.0), f), Derive(f, variableName)));
}
if (branch.Symbol is Exponential)
{
var f = (ISymbolicExpressionTreeNode)branch.Clone();
return(Product(f, Derive(branch.GetSubtree(0), variableName)));
}
if (branch.Symbol is Square)
{
var f = (ISymbolicExpressionTreeNode)branch.GetSubtree(0).Clone();
return(Product(Product(CreateConstant(2.0), f), Derive(f, variableName)));
}
if (branch.Symbol is SquareRoot)
{
var f = (ISymbolicExpressionTreeNode)branch.Clone();
var u = (ISymbolicExpressionTreeNode)branch.GetSubtree(0).Clone();
return(Product(Div(CreateConstant(1.0), Product(CreateConstant(2.0), f)), Derive(u, variableName)));
}
if (branch.Symbol is CubeRoot)
{
var f = (ISymbolicExpressionTreeNode)branch.Clone();
var u = (ISymbolicExpressionTreeNode)branch.GetSubtree(0).Clone();
return(Product(Div(CreateConstant(1.0), Product(CreateConstant(3.0), Square(f))), Derive(u, variableName))); // 1/3 1/cbrt(f(x))^2 d/dx f(x)
}
if (branch.Symbol is Cube)
{
var f = (ISymbolicExpressionTreeNode)branch.GetSubtree(0).Clone();
return(Product(Product(CreateConstant(3.0), Square(f)), Derive(f, variableName)));
}
if (branch.Symbol is Absolute)
{
var f = (ISymbolicExpressionTreeNode)branch.GetSubtree(0).Clone();
var absf = Abs((ISymbolicExpressionTreeNode)f.Clone());
return(Product(Div(f, absf), Derive(f, variableName)));
}
if (branch.Symbol is AnalyticQuotient)
{
// aq(a(x), b(x)) = a(x) / sqrt(b(x)²+1)
var a = (ISymbolicExpressionTreeNode)branch.GetSubtree(0).Clone();
var b = (ISymbolicExpressionTreeNode)branch.GetSubtree(1).Clone();
var definition = Div(a, SquareRoot(Sum(Square(b), CreateConstant(1.0))));
return(Derive(definition, variableName));
}
if (branch.Symbol is Sine)
{
var u = (ISymbolicExpressionTreeNode)branch.GetSubtree(0).Clone();
var cos = (new Cosine()).CreateTreeNode();
cos.AddSubtree(u);
return(Product(cos, Derive(u, variableName)));
}
if (branch.Symbol is Cosine)
{
var u = (ISymbolicExpressionTreeNode)branch.GetSubtree(0).Clone();
var sin = (new Sine()).CreateTreeNode();
sin.AddSubtree(u);
return(Product(CreateConstant(-1.0), Product(sin, Derive(u, variableName))));
}
if (branch.Symbol is Tangent)
{
// tan(x)' = 1 / cos²(x)
var fxp = Derive(branch.GetSubtree(0), variableName);
var u = (ISymbolicExpressionTreeNode)branch.GetSubtree(0).Clone();
return(Div(fxp, Square(Cosine(u))));
}
if (branch.Symbol is HyperbolicTangent)
{
// tanh(f(x))' = f(x)'sech²(f(x)) = f(x)'(1 - tanh²(f(x)))
var fxp = Derive(branch.GetSubtree(0), variableName);
var tanh = (ISymbolicExpressionTreeNode)branch.Clone();
return(Product(fxp, Subtract(CreateConstant(1.0), Square(tanh))));
}
throw new NotSupportedException(string.Format("Symbol {0} is not supported.", branch.Symbol));
}
private static ISymbolicExpressionTreeNode MakeSum(ISymbolicExpressionTreeNode treeNode, double alpha) {
if (alpha.IsAlmost(0.0)) {
return treeNode;
} else {
var addition = new Addition();
var node = addition.CreateTreeNode();
var alphaConst = MakeConstant(alpha);
node.AddSubtree(treeNode);
node.AddSubtree(alphaConst);
return node;
}
}
