/// <summary>
/// Expand a power expression.
/// </summary>
/// <param name="f"></param>
/// <param name="x"></param>
/// <returns></returns>
private static Expression ExpandPower(Power f, Expression x)
{
// Get integral exponent of f.
int n = Power.IntegralExponentOf(f);
// If this is an an integral constant negative exponent, attempt to use partial fractions.
if (n < 0 && !ReferenceEquals(x, null))
{
Expression b = f.Left.Factor(x);
if (n != -1)
{
b = Power.New(b, Math.Abs(n));
}
return(ExpandPartialFractions(1, b, x));
}
// If f is an add expression, expand it as if it were multiplication.
if (n > 1 && f.Left is Sum)
{
Expression e = f.Left;
for (int i = 1; i < n; ++i)
{
e = Distribute(f.Left, e);
}
return(e);
}
return(f);
}
protected override Expression VisitPower(Power P)
{
Expression L = Visit(P.Left);
Expression R = Visit(P.Right);
if (R.IsInteger())
{
// Transform (x*y)^z => x^z*y^z if z is an integer.
Product M = L as Product;
if (!ReferenceEquals(M, null))
{
return(Visit(Product.New(M.Terms.Select(i => Power.New(i, R)))));
}
}
// Transform (x^y)^z => x^(y*z) if z is an integer.
Power LP = L as Power;
if (!ReferenceEquals(LP, null))
{
L = LP.Left;
R = Visit(Product.New(P.Right, LP.Right));
}
// Handle identities.
Real?LR = AsReal(L);
if (EqualsZero(LR))
{
return(0);
}
if (EqualsOne(LR))
{
return(1);
}
Real?RR = AsReal(R);
if (EqualsZero(RR))
{
return(1);
}
if (EqualsOne(RR))
{
return(L);
}
// Evaluate result.
if (LR != null && RR != null)
{
return(Constant.New(LR.Value ^ RR.Value));
}
else
{
return(Power.New(L, R));
}
}
/// <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))));
}
// 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));
}
protected override Expression VisitPower(Power P)
{
Expression f = P.Left;
Expression g = P.Right;
if (g.DependsOn(x))
{
// f(x)^g(x)
return(Product.New(P,
Sum.New(
Product.New(Visit(f), Binary.Divide(g, f)),
Product.New(Visit(g), Call.Ln(f)))).Evaluate());
}
else
{
// f(x)^g
return(Product.New(
g,
Power.New(f, Binary.Subtract(g, 1)),
Visit(f)).Evaluate());
}
}
/// <summary>
/// Construct a polynomial of x from f(x).
/// </summary>
/// <param name="f"></param>
/// <param name="x"></param>
/// <returns></returns>
public static Polynomial New(Expression f, Expression x)
{
// Match each term to A*x^N where A is constant with respect to x, and N is an integer.
Variable A = PatternVariable.New("A", i => !i.DependsOn(x));
Variable N = PatternVariable.New("N", i => i.IsInteger());
Expression TermPattern = Product.New(A, Power.New(x, N));
DefaultDictionary <int, Expression> P = new DefaultDictionary <int, Expression>(0);
foreach (Expression i in Sum.TermsOf(f))
{
MatchContext Matched = TermPattern.Matches(i, Arrow.New(x, x));
if (Matched == null)
{
throw new ArgumentException("f is not a polynomial of x.");
}
int n = (int)Matched[N];
P[n] += Matched[A];
}
return(new Polynomial(P, x));
}
public static LazyExpression operator /(Expression L, Expression R)
{
return(new LazyExpression(Product.New(L, Power.New(R, -1))));
}
public static LazyExpression operator /(LazyExpression L, LazyExpression R)
{
return(new LazyExpression(Product.New(L.value, Power.New(R.value, -1))));
}
