protected override PrettyString VisitProduct(Product M)
{
Expression N = Product.Numerator(M);
Expression D = Product.Denominator(M);
bool negative = false;
if (N is Product && IsNegative(N))
{
negative = !negative;
N = -N;
}
if (D is Product && IsNegative(D))
{
negative = !negative;
D = -D;
}
if (!D.Equals(1))
{
return(PrettyString.ConcatColumns(negative ? "- " : "", VisitDivide(N, D)));
}
else if (N is Product)
{
return(PrettyString.ConcatColumns(negative ? "-" : "", UnSplit(Product.TermsOf(N), "*")));
}
else
{
return(PrettyString.ConcatColumns(negative ? "-" : "", Visit(N)));
}
}
/// <summary>
/// Expand a multiplication expression.
/// </summary>
/// <param name="f"></param>
/// <param name="x"></param>
/// <returns></returns>
private static Expression ExpandMultiply(Expression f, Expression x)
{
// If the denominator is multiplication, expand partial fractions.
if (!ReferenceEquals(x, null))
{
Expression d = Product.Denominator(f).Factor(x);
if (d is Product)
{
return(ExpandPartialFractions(Product.Numerator(f), (Product)d, x));
}
}
// If f contains an add expression, distribute it.
if (Product.TermsOf(f).Any(i => i is Sum))
{
Expression e = 1;
foreach (Expression i in Product.TermsOf(f))
{
e = Distribute(i.Expand(x), e);
}
return(e);
}
return(f);
}
protected override string VisitProduct(Product M)
{
int pr = Parser.Precedence(Operator.Multiply);
Expression N = Product.Numerator(M);
string minus = "";
if (IsNegative(N))
{
minus = "-";
N = -N;
}
string n = String.Join(" ", Product.TermsOf(N).Select(i => Visit(i, pr)));
string d = String.Join(" ", Product.TermsOf(Product.Denominator(M)).Select(i => Visit(i, pr)));
if (d != "1")
{
return(minus + Frac(n, d));
}
else
{
return(minus + n);
}
}
/// <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);
}
private static bool IsNegative(Expression x)
{
Constant C = Product.TermsOf(x).First() as Constant;
if (C != null)
{
return((Real)C < 0);
}
return(false);
}
protected static bool IsNegative(Expression x)
{
Constant C = Product.TermsOf(x).FirstOrDefault(i => i is Constant) as Constant;
if (C != null)
{
return((Real)C < 0);
}
return(false);
}
/// <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))));
}
private void AddTerm(IEnumerable <Expression> B, Expression t)
{
foreach (Expression b in B)
{
if (Product.TermsOf(t).Count(i => i.Equals(b)) == 1)
{
terms[b] += Product.New(Product.TermsOf(t).Except(b));
return;
}
}
terms[1] += t;
}
// Combine like terms and multiply constants.
protected override Expression VisitProduct(Product M)
{
// Map terms to exponents.
DefaultDictionary <Expression, Real> terms = new DefaultDictionary <Expression, Real>(0);
// Accumulate constants and sum exponent of each term.
Real C = 1;
foreach (Expression i in M.Terms.SelectMany(i => Product.TermsOf(Visit(i))))
{
if (i is Constant)
{
Real Ci = (Real)i;
// Early exit if 0.
if (Ci.EqualsZero())
{
return(0);
}
C *= Ci;
}
else
{
Power Pi = i as Power;
if (!ReferenceEquals(Pi, null) && Pi.Right is Constant)
{
terms[Pi.Left] += (Real)Pi.Right;
}
else
{
terms[i] += 1;
}
}
}
// Build a new expression with the accumulated terms.
if (!C.EqualsOne())
{
// Find a sum term that has a constant term to distribute into.
KeyValuePair <Expression, Real> A = terms.FirstOrDefault(i => Real.Abs(i.Value).EqualsOne() && i.Key is Sum);
if (!ReferenceEquals(A.Key, null))
{
terms.Remove(A.Key);
terms[ExpandExtension.Distribute(C ^ A.Value, A.Key)] += A.Value;
}
else
{
terms.Add(C, 1);
}
}
return(Product.New(terms
.Where(i => !i.Value.EqualsZero())
.Select(i => !i.Value.EqualsOne() ? Power.New(i.Key, Constant.New(i.Value)) : i.Key)));
}
// V((A*x)^n) = A^(1/n)*V(x^n)
protected override Expression VisitPower(Power P)
{
if (!IsConstant(P.Right))
{
return(base.VisitPower(P));
}
Expression L = P.Left.Factor();
IEnumerable <Expression> A = Product.TermsOf(L).Where(i => IsConstant(i));
if (A.Any())
{
return(Product.New(Power.New(Product.New(A), 1 / P.Right), Visit(Product.New(Product.TermsOf(L).Where(i => !IsConstant(i))))));
}
return(base.VisitPower(P));
}
// Enumerates x, splitting negative constants into a positive constant and -1.
private static IEnumerable <Expression> FactorsOf(Expression x)
{
foreach (Expression i in Product.TermsOf(x))
{
if (i is Constant && (Real)i < 0)
{
yield return(-1);
yield return(Real.Abs((Real)i));
}
else if (i is Power power)
{
yield return(i);
yield return(power.Left);
}
else
{
yield return(i);
}
}
}
public static Expression EvaluateSum(IEnumerable <Expression> Terms)
{
// Map terms to their coefficients.
DefaultDictionary <Expression, Real> terms = new DefaultDictionary <Expression, Real>(0);
// Accumulate constants and sum coefficient of each term.
Real C = 0;
foreach (Expression i in Terms)
{
if (i is Constant)
{
C += (Real)i;
}
else
{
// Find constant term.
Constant coeff = Product.TermsOf(i).OfType <Constant>().FirstOrDefault();
if (!(coeff is null))
{
terms[Product.New(Product.TermsOf(i).ExceptUnique(coeff, Expression.RefComparer))] += (Real)coeff;
}
public static Expression EvaluateSum(IEnumerable <Expression> Terms)
{
// Map terms to their coefficients.
DefaultDictionary <Expression, Real> terms = new DefaultDictionary <Expression, Real>(0);
// Accumulate constants and sum coefficient of each term.
Real C = 0;
foreach (Expression i in Terms)
{
if (i is Constant)
{
C += (Real)i;
}
else
{
// Find constant term.
Constant coeff = Product.TermsOf(i).OfType <Constant>().FirstOrDefault();
if (!ReferenceEquals(coeff, null))
{
terms[Product.New(Product.TermsOf(i).ExceptUnique(coeff, Expression.RefComparer))] += (Real)coeff;
}
else
{
terms[i] += 1;
}
}
}
// Build a new expression with the accumulated terms.
if (!C.EqualsZero())
{
terms.Add(Constant.New(C), (Real)1);
}
return(Sum.New(terms
.Where(i => !i.Value.EqualsZero())
.Select(i => !i.Value.EqualsOne() ? Product.New(i.Key, Constant.New(i.Value)) : i.Key)));
}
public static Expression Denominator(Expression x)
{
return(Product.New(Product.TermsOf(x).Where(i => IsInDenominator(i)).Select(i => (Expression)(i ^ -1))));
}
public static Expression Numerator(Expression x)
{
return(Product.New(Product.TermsOf(x).Where(i => !IsInDenominator(i))));
}