/// <summary> /// Considers expr > 0 /// </summary> internal static Set Solve(Entity expr, Variable x) { { if (MathS.Utils.TryGetPolyLinear(expr, x, out var a, out var b)) { a = a.InnerSimplified; b = b.InnerSimplified; var root = PolynomialSolver.SolveLinear(a, b).First(); if (root is Complex and not Real) { return(Empty); } if (a is Real { IsNegative : true })
internal static void Solve(Entity expr, VariableEntity x, Set dst, bool compensateSolving) { if (expr == x) { dst.Add(0); return; } // Applies an attempt to downcast roots void DestinationAddRange(Set toAdd) { toAdd.FiniteApply(ent => TryDowncast(expr, x, ent)); dst.AddRange(toAdd); } var polyexpr = expr.DeepCopy(); Set res = PolynomialSolver.SolveAsPolynomial(polyexpr, x); if (res != null) { res.FiniteApply(e => e.InnerSimplify()); DestinationAddRange(res); return; } if (expr.entType == Entity.EntType.OPERATOR) { switch (expr.Name) { case "mulf": Solve(expr.Children[0], x, dst); Solve(expr.Children[1], x, dst); return; case "divf": bool IsSetNumeric(Set a) => a.Select(piece => piece.LowerBound().Item1).All(MathS.CanBeEvaluated); var zeroNumerators = new Set(); Solve(expr.Children[0], x, zeroNumerators); if (!IsSetNumeric(zeroNumerators)) { dst.AddRange(zeroNumerators); return; } var zeroDenominators = new Set(); Solve(expr.Children[1], x, zeroDenominators); if (!IsSetNumeric(zeroDenominators)) { dst.AddRange(zeroNumerators); return; } dst.AddRange((zeroNumerators & !zeroDenominators) as Set); return; case "powf": Solve(expr.Children[0], x, dst); return; case "minusf": if (expr.Children[1].FindSubtree(x) == null && compensateSolving) { if (expr.Children[0] == x) { dst.Add(expr.Children[1]); return; } var subs = 0; Entity lastChild = null; foreach (var child in expr.Children[0].Children) { if (child.FindSubtree(x) != null) { subs += 1; lastChild = child; } } if (subs != 1) { break; } var resInverted = TreeAnalyzer.FindInvertExpression(expr.Children[0], expr.Children[1], lastChild); foreach (var result in resInverted.FiniteSet()) { Solve(lastChild - result, x, dst, compensateSolving: true); } return; } break; } } else if (expr.entType == Entity.EntType.FUNCTION) { DestinationAddRange(TreeAnalyzer.InvertFunctionEntity(expr as FunctionEntity, 0, x)); return; } // Here we generate a unique variable name var uniqVars = MathS.Utils.GetUniqueVariables(expr); uniqVars.Pieces.Sort((a, b) => ((Entity)b).Name.Length.CompareTo(((Entity)a).Name.Length)); VariableEntity newVar = ((Entity)uniqVars.Pieces[0]).Name + "quack"; // // // // Here we find all possible replacements var replacements = new List <Tuple <Entity, Entity> >(); replacements.Add(new Tuple <Entity, Entity>(TreeAnalyzer.GetMinimumSubtree(expr, x), expr)); foreach (var alt in expr.Alternate(4).FiniteSet()) { if ((alt).FindSubtree(x) == null) { return; // in this case there is either 0 or +oo solutions } replacements.Add(new Tuple <Entity, Entity>(TreeAnalyzer.GetMinimumSubtree(alt, x), alt)); } // // // // Here we find one that has at least one solution foreach (var replacement in replacements) { Set solutions = null; if (replacement.Item1 == x) { continue; } var newExpr = replacement.Item2.DeepCopy(); TreeAnalyzer.FindAndReplace(ref newExpr, replacement.Item1, newVar); solutions = newExpr.SolveEquation(newVar); if (!solutions.IsEmpty()) { var bestReplacement = replacement.Item1; // Here we are trying to solve for this replacement Set newDst = new Set(); foreach (var solution in solutions.FiniteSet()) { var str = bestReplacement.ToString(); // TODO: make a smarter comparison than just comparison of complexities of two expressions // The idea is // similarToPrevious = ((bestReplacement - solution) - expr).Simplify() == 0 // But Simplify costs us too much time var similarToPrevious = (bestReplacement - solution).Complexity() >= expr.Complexity(); if (!compensateSolving || !similarToPrevious) { Solve(bestReplacement - solution, x, newDst, compensateSolving: true); } } DestinationAddRange(newDst); if (!dst.IsEmpty()) { break; } // // // } } // // // // if no replacement worked, try trigonometry solver if (dst.IsEmpty()) { var trigexpr = expr.DeepCopy(); res = TrigonometricSolver.SolveLinear(trigexpr, x); if (res != null) { DestinationAddRange(res); return; } } // // // // if nothing has been found so far if (dst.IsEmpty() && MathS.Settings.AllowNewton) { Set allVars = new Set(); TreeAnalyzer._GetUniqueVariables(expr, allVars); if (allVars.Count == 1) { DestinationAddRange(expr.SolveNt(x)); } } }