/// <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)))); }