/// <summary>
/// Distribute products across sums.
/// </summary>
/// <param name="f"></param>
/// <param name="x"></param>
/// <returns></returns>
public static Expression Factor(this Expression f, Expression x)
{
// If f is a product, just factor its terms.
if (f is Product)
{
return(Product.New(((Product)f).Terms.Select(i => i.Factor(x))));
}
// If if is l^r, factor l and distribute r.
if (f is Power)
{
Expression l = ((Power)f).Left.Factor(x);
Expression r = ((Power)f).Right;
return(Product.New(Product.TermsOf(l).Select(i => Power.New(i, r))));
}
// If f is a polynomial of x, use polynomial factoring methods.
if (f is Polynomial && (((Polynomial)f).Variable.Equals(x) || ReferenceEquals(x, null)))
{
return(((Polynomial)f).Factor());
}
// Try interpreting f as a polynomial of x.
if (!ReferenceEquals(x, null))
{
// If f is a polynomial of x, factor it.
try
{
return(Polynomial.New(f, x).Factor());
}
catch (Exception) { }
}
// Just factor out common sub-expressions.
if (f is Sum)
{
Sum s = (Sum)f;
IEnumerable <Expression> terms = s.Terms.Select(i => i.Factor()).Buffer();
// All of the distinct factors.
IEnumerable <Expression> factors = terms.SelectMany(i => FactorsOf(i).Except(1, -1)).Distinct();
// Choose the most common factor to use.
Expression factor = factors.ArgMax(i => terms.Count(j => FactorsOf(j).Contains(i)));
// Find the terms that contain the factor.
IEnumerable <Expression> contains = terms.Where(i => FactorsOf(i).Contains(factor)).Buffer();
// If more than one term contains the factor, pull it out and factor the resulting expression (again).
if (contains.Count() > 1)
{
return(Sum.New(
Product.New(factor, Sum.New(contains.Select(i => Binary.Divide(i, factor))).Evaluate()),
Sum.New(terms.Except(contains, Expression.RefComparer))).Factor(null));
}
}
return(f);
}
/// <summary>
/// Distribute products across sums.
/// </summary>
/// <param name="f"></param>
/// <param name="x"></param>
/// <returns></returns>
public static Expression Factor(this Expression f, Expression x)
{
// If f is a product, just factor its terms.
if (f is Product product)
{
return(Product.New(product.Terms.Select(i => i.Factor(x))));
}
// If if is l^r, factor l and distribute r.
if (f is Power power)
{
Expression l = power.Left.Factor(x);
Expression r = power.Right;
return(Product.New(Product.TermsOf(l).Select(i => Power.New(i, r))));
}
// If f is a polynomial of x, use polynomial factoring methods.
if (f is Polynomial p && (p.Variable.Equals(x) || (x is null)))
{
return(p.Factor());
}
// Try interpreting f as a polynomial of x.
if (!(x is null))
{
// If f is a polynomial of x, factor it.
try
{
return(Polynomial.New(f, x).Factor());
}
catch (Exception) { }
}
// Just factor out common sub-expressions.
if (f is Sum s)
{
// Make a list of each terms' products.
List <List <Expression> > terms = s.Terms.Select(i => FactorsOf(i).ToList()).ToList();
// All of the distinct factors.
IEnumerable <Expression> factors = terms.SelectMany(i => i.Except(1, -1)).Distinct();
// Choose the most common factor to factor.
Expression factor = factors.ArgMax(i => terms.Count(j => j.Contains(i)));
// Find the terms that contain the factor.
List <List <Expression> > contains = terms.Where(i => i.Contains(factor)).ToList();
// If more than one term contains the factor, pull it out and factor the resulting expressions (again).
if (contains.Count() > 1)
{
Expression factored = Sum.New(contains.Select(i => Product.New(i.Except(factor))));
Expression not_factored = Sum.New(terms.Except(contains).Select(i => Product.New(i)));
return(Sum.New(Product.New(factor, factored), not_factored).Factor(null));
}
}
return(f);
}
// Expand N(x)/D(x) using partial fractions.
private static Expression ExpandPartialFractions(Expression N, Expression D, Expression x)
{
List <Expression> terms = new List <Expression>();
List <Variable> unknowns = new List <Variable>();
List <Expression> basis = new List <Expression>();
foreach (Expression i in Product.TermsOf(D))
{
// Get the multiplicity of this basis term.
Expression e = i;
int n = Power.IntegralExponentOf(e);
if (n != 1)
{
e = ((Power)i).Left;
}
// Convert to a polynomial.
Polynomial Pi = Polynomial.New(e, x);
// Add new terms for each multiplicity n.
for (int j = 1; j <= n; ++j)
{
// Expression for the unknown numerator of this term.
Expression unknown = 0;
for (int k = 0; k < Pi.Degree; ++k)
{
Variable Ai = Variable.New("_A" + unknowns.Count.ToString());
unknown += Ai * (x ^ k);
unknowns.Add(Ai);
}
terms.Add(Product.New(unknown, Power.New(e, -j)));
}
basis.Add(i);
}
// Equate the original expression with the decomposed expressions.
D = Sum.New(terms.Select(j => (Expression)(D * j))).Expand();
Polynomial l = Polynomial.New(N, x);
Polynomial r = Polynomial.New(D, x);
// Equate terms of equal degree and solve for the unknowns.
int degree = Math.Max(l.Degree, r.Degree);
List <Equal> eqs = new List <Equal>(degree + 1);
for (int i = 0; i <= degree; ++i)
{
eqs.Add(Equal.New(l[i], r[i]));
}
List <Arrow> A = eqs.Solve(unknowns);
// Substitute the now knowns.
return(Sum.New(terms.Select(i => i.Evaluate(A))));
}
