public static IEnumerable <Tuple <Variable, Formula> > Solve(Polynomial[] equations)
{
var knowns = new List <Tuple <Variable, Formula> >();
var id = "Start";
while (true)
{
GeoUt.GenerateHtml(equations, knowns, id, null, 0, null);
if (equations.Length == 0)
{
return(knowns);
}
var step = SolveOneStep(equations).First();
var done = step.IfType(
(SolvedForVariable ss) =>
{
var nextId = "{0}›{1}".Fmt(id, ss.SolvedFor.Name);
GeoUt.GenerateHtml(equations, knowns, id, nextId, ss.EquationIndex.Value, "$ {0} = {1} $".Fmt(ss.SolvedFor, ss.Solution));
id = nextId;
equations = equations.Where((eq, i) => i != ss.EquationIndex).Select(eq => eq.SubstituteAsEquation(ss.SolvedFor, ss.Solution)).ToArray();
knowns = knowns.Select(kn => Tuple.Create(kn.Item1, kn.Item2.Substitute(ss.SolvedFor, ss.Solution))).ToList();
knowns.Add(Tuple.Create(ss.SolvedFor, ss.Solution));
return(false);
},
(RedundantEquation ss) =>
{
var nextId = "{0}ǀ{1}".Fmt(id, ss.EquationIndex);
GeoUt.GenerateHtml(equations, knowns, id, nextId, ss.EquationIndex.Value, "Redundant equation");
id = nextId;
equations = equations.Where((eq, i) => i != ss.EquationIndex).ToArray();
return(false);
},
(NewEquations ss) =>
{
var nextId = "{0}˟".Fmt(id);
GeoUt.GenerateHtml(equations, knowns, id, nextId, 0, "XL Algorithm");
id = nextId;
equations = ss.Equations;
return(false);
},
ss =>
{
throw new NotImplementedException();
}
);
if (done)
{
return(knowns);
}
}
}
public static void GenerateHtml(this Polynomial[] equations, List <Tuple <Variable, Formula> > knowns, string id, string nextId, int eqIndex, string inferred)
{
if (inferred != null)
{
_inferences.AddSafe(id, eqIndex, Tuple.Create(nextId, inferred));
}
else if (_inferences.ContainsKey(id))
{
return;
}
Dictionary <int, List <Tuple <string, string> > > tuples = null;
_inferences.TryGetValue(id, out tuples);
var filename = @"D:\Daten\Upload\Geocos\{0}.htm".Fmt(id);
Console.WriteLine("Writing {0}".Fmt(filename));
GeoUt.WriteHtml(
@"<h2>Knowns</h2>{0}<h2>Equations</h2><ul>{1}</ul><h2>Maxima</h2><div>solve ( [ {2} ], [ {3} ] );</div><h2>Sage</h2><div>{3} = var('{3}')</div><div>solve([{4}], {3})</div>".Fmt(
knowns.OrderBy(v => v.Item1.Name).Select(v => "$ {0} = {1} $".Fmt(v.Item1, v.Item2)).JoinString(", "),
equations.Select((eq, i) =>
{
List <Tuple <string, string> > theseTuples = null;
if (tuples != null)
{
tuples.TryGetValue(i, out theseTuples);
}
return(@"<li class='{0}'>$ {1} = 0 $ <small>{2}</small>{3}</li>".Fmt(
theseTuples != null ? "possible" : null,
eq,
eq.ToMaple(),
@"<blockquote><ol>{0}</ol>{1}</blockquote>".Fmt(
eq.Variables.Select((v, vi) =>
"<li>$ = {0} $</li>".Fmt(
eq.CollectTerms(vi)
.Select((ct, ctp) => new { Term = ct, Power = ctp })
.Where(inf => inf.Term != 0)
.Select(inf => (inf.Term == 1 ? (inf.Power == 0 ? "1" : inf.Power == 1 ? "{1}" : "{1}^{2}") : inf.Power == 0 ? "({0})" : inf.Power == 1 ? "({0}) {1}" : "({0}) {1}^{2}").Fmt(inf.Term, v, inf.Power)).JoinString(" + "))
).JoinString(),
theseTuples.NullOr(ts => ts.Select(t => @"<div>⇒ {0}{1}</div>".Fmt(t.Item2, t.Item1.NullOr(link => @" (<a href='{0}.htm'>go</a>)".Fmt(link)))).JoinString())
)
));
}).JoinString(),
equations.Select(eq => "{0} = 0".Fmt(eq.ToMaple())).JoinString(", "),
equations.SelectMany(eq => eq.Variables).Distinct().JoinString(", "),
equations.Select(eq => "{0} == 0".Fmt(eq.ToMaple())).JoinString(", ")
),
filename
);
}
public static IEnumerable <SolveStep> SolveOneStep(Polynomial[] equations)
{
if (equations.Length == 0)
{
yield break;
}
var redundantEquationIndex = equations.IndexOf(eq => eq == 0);
if (redundantEquationIndex != -1)
{
yield return(new RedundantEquation(redundantEquationIndex));
yield break;
}
var contradictionIndex = equations.IndexOf(eq => eq.Degree == 0);
if (contradictionIndex != -1)
{
yield return(new NoSolution(contradictionIndex));
yield break;
}
var extremelySimpleEquationIndex = equations.IndexOf(eq => eq.Degree == 1 && eq.Variables.Length < 2);
if (extremelySimpleEquationIndex != -1)
{
var extremelySimpleEquation = equations[extremelySimpleEquationIndex];
var terms = extremelySimpleEquation.CollectTerms(0);
yield return(new SolvedForVariable(extremelySimpleEquationIndex, extremelySimpleEquation.Variables[0], Fraction.Create(-terms[0], terms[1])));
yield break;
}
//var simpleQuadraticIndex = equations.IndexOf(eq => eq.Degree == 2 && eq.Variables.Length < 2);
//if (simpleQuadraticIndex != -1)
//{
// var simpleQuadratic = equations[simpleQuadraticIndex];
// var x = simpleQuadratic.Variables[0];
// var a = simpleQuadratic.Coefficients[2];
// var b = simpleQuadratic.Coefficients[1];
// var c = simpleQuadratic.Coefficients[0];
// var det = b * b - 4 * a * c;
// if (det < 0)
// {
// // No solution if 𝑏² − 4 𝑎 𝑐 is negative
// yield return new NoSolution(simpleQuadraticIndex);
// yield break;
// }
// var sols = new[] { (-b + Math.Sqrt(det)) / (2 * a), (-b - Math.Sqrt(det)) / (2 * a) }.Distinct();
// if (x.NonNegative)
// sols = sols.Where(sol => sol >= 0);
// var solArr = sols.ToArray();
// if (solArr.Length == 0)
// {
// yield return new NoSolution(simpleQuadraticIndex);
// yield break;
// }
// Array.Sort(solArr, (one, two) => Math.Abs(one - x.SuggestedValue).CompareTo(Math.Abs(two - x.SuggestedValue)));
// //if (x.Name == "x_J")
// // Ut.Swap(ref solArr[0], ref solArr[1]);
// foreach (var sol in solArr)
// yield return new SolvedForVariable(simpleQuadraticIndex, x, sol);
// yield break;
//}
// Find an equation we can simplify by dividing by one of its variables
for (int eqIndex = 0; eqIndex < equations.Length; eqIndex++)
{
var eq = equations[eqIndex];
// If the constant term is 0, we may be able to do this
Tuple <Polynomial, Variable> stuff;
if ((eq.Coefficients[0] == 0) && (stuff = eq.Decimate()) != null)
{
var simplerEquation = stuff.Item1;
var variable = stuff.Item2;
// Solution 1: variable = 0
yield return(new SolvedForVariable(eqIndex, variable, 0));
// Solution 2: the same set of equations, except with this one simplified
// (only exists if there are any other variables left in the equation;
// otherwise the remaining equation is just “1 = 0” which obviously has no solutions)
if (simplerEquation.Variables.Length > 0)
{
yield return(new EquationSimplified(eqIndex, simplerEquation));
}
yield break;
}
}
//// Find a single-variable equation and try Newton approximation on it
//var singleVariableIndex = equations.IndexOf(eq => eq.Variables.Length == 1);
//if (singleVariableIndex != -1)
//{
// var maxIterations = 1000;
// var threshold = 0.0000000001;
// var equation = equations[singleVariableIndex];
// var variable = equation.Variables[0];
// var currentBestGuess = variable.SuggestedValue;
// for (; maxIterations > 0; maxIterations--)
// {
// var curValue = equation.Evaluate(v => currentBestGuess);
// if (Math.Abs(curValue) < threshold)
// {
// yield return new SolvedForVariable(singleVariableIndex, variable, currentBestGuess);
// yield break;
// }
// var curDerivative = equation.EvaluateDerivative(currentBestGuess);
// currentBestGuess -= curValue / curDerivative;
// }
//}
Array.Sort(equations, (e1, e2) =>
{
var c = e1.Degree.CompareTo(e2.Degree);
if (c != 0)
{
return(c);
}
c = e1.Variables.Length.CompareTo(e2.Variables.Length);
if (c != 0)
{
return(c);
}
c = (e1.Coefficients[0] != 0).CompareTo(e2.Coefficients[0] != 0);
if (c != 0)
{
return(c);
}
// TODO: Come up with some more comparisons
return(0);
});
for (int eqIndex = 0; eqIndex < equations.Length; eqIndex++)
{
var eq = equations[eqIndex];
var varDegrees = eq.GetVariableDegrees();
var solveForIndex = Enumerable.Range(0, varDegrees.Length).MinElement(i => varDegrees[i]);
var solveFor = eq.Variables[solveForIndex];
if (varDegrees[solveForIndex] == 1)
{
// Any equation that is linear in one variable.
// 𝐄𝐱𝐚𝐦𝐩𝐥𝐞:
// 𝑥 𝑦 + 2 𝑥 + 3 𝑦 + 4 = 0
// ⇒ 𝑥 (𝑦 + 2) = −3 𝑦 − 4
// ⇒ 𝑥 = (−3 𝑦 − 4) / (𝑦 + 2)
var terms = eq.CollectTerms(solveForIndex);
yield return(new SolvedForVariable(eqIndex, solveFor, Fraction.Create(-terms[0], terms[1])));
// If only one other variable, it’s pretty safe to say that we don’t need to explore other avenues
if (eq.Variables.Length < 3)
{
yield break;
}
}
//else if (varDegrees[solveForIndex] == 2)
//{
// // Quadratic polynomial
// // 𝑎 𝑥² + 𝑏 𝑥 + 𝑐 = 0
// // ⇒ 𝑥 = (−𝑏 ± √(𝑏² − 4 𝑎 𝑐)) / 2 𝑎
// var xTerms = eq.CollectTerms(solveForIndex);
// var aTerm = xTerms[2];
// var bTerm = xTerms[1];
// var cTerm = xTerms[0]; // 𝑐 can’t be 0 because if it were 0 this equation would have been handled earlier
// if (aTerm.Degree == 0 && bTerm == 0)
// {
// // ⇒ 𝑥 = ± √(−4 𝑎 𝑐) / 2 𝑎
// var a = aTerm.Coefficients[0];
// // Easier case: if 𝑐 is of the form −(𝑠 𝑦 ± 𝑡)² = −𝑠² 𝑦² ∓ 2 𝑠 𝑡 𝑦 − 𝑡², then 𝑥 = (𝑠 𝑦 ± 𝑡) / √𝑎 or 𝑥 = −(𝑠 𝑦 ± 𝑡) / √𝑎
// if (cTerm.Variables.Length == 1 && cTerm.Coefficients[0] <= 0 && cTerm.Coefficients[2] <= 0)
// {
// if (a < 0)
// yield break;
// var sqrta = Math.Sqrt(a);
// var s = Math.Sqrt(-cTerm.Coefficients[2]);
// var t = Math.Sqrt(-cTerm.Coefficients[0]);
// if (cTerm.Coefficients[1] == 2 * s * t)
// {
// var xSolution = s / sqrta * cTerm.Variables[0] - t / sqrta;
// yield return new SolvedForVariable(eqIndex, solveFor, xSolution);
// yield return new SolvedForVariable(eqIndex, solveFor, -xSolution);
// yield break;
// }
// else if (cTerm.Coefficients[1] == -2 * s * t)
// {
// var xSolution = s / sqrta * cTerm.Variables[0] + t / sqrta;
// yield return new SolvedForVariable(eqIndex, solveFor, xSolution);
// yield return new SolvedForVariable(eqIndex, solveFor, -xSolution);
// yield break;
// }
// }
// // Now 𝑐 cannot be a constant because that would make this a simple quadratic in 𝑥, which would have been handled earlier.
// // It also cannot be higher degree than 2. So it fits the pattern:
// // 𝑎 𝑥² + 𝑐 = 0
// // ⇒ 𝑥² = 𝑐 / (−𝑎)
// var xsq = cTerm / (-a);
// yield return new SolvedForVariable2(eqIndex, solveFor, new Root(xsq, false), new Root(xsq, true));
// yield break;
// }
// else if (aTerm.Degree == 0)
// {
// var surdTerm = bTerm.Square() - 4 * aTerm * cTerm;
// if (surdTerm.Degree == 2 && surdTerm.Variables.Length == 1 && surdTerm.Coefficients[0] >= 0 && surdTerm.Coefficients[2] >= 0)
// {
// var s = Math.Sqrt(surdTerm.Coefficients[2]);
// var t = Math.Sqrt(surdTerm.Coefficients[0]);
// Polynomial[] solArr;
// if (surdTerm.Coefficients[1] == 2 * s * t)
// {
// // ⇒ 𝑥 = (−𝑏 ± (𝑠 𝑦 + 𝑡)) / 2 𝑎
// solArr = Ut.NewArray(
// (-bTerm + (s * surdTerm.Variables[0] + t)) / (2 * aTerm.Coefficients[0]),
// (-bTerm - (s * surdTerm.Variables[0] + t)) / (2 * aTerm.Coefficients[0])
// );
// }
// else if (surdTerm.Coefficients[1] == -2 * s * t)
// {
// // ⇒ 𝑥 = (−𝑏 ± (𝑠 𝑦 − 𝑡)) / 2 𝑎
// solArr = Ut.NewArray(
// (-bTerm + (s * surdTerm.Variables[0] - t)) / (2 * aTerm.Coefficients[0]),
// (-bTerm - (s * surdTerm.Variables[0] - t)) / (2 * aTerm.Coefficients[0])
// );
// }
// else
// {
// continue;
// }
// foreach (var sol in solArr)
// yield return new SolvedForVariable(eqIndex, solveFor, sol);
// yield break;
// }
// }
//}
}
//** BEGIN XL ALGORITHM
var xlEquations = equations;
for (int outerIteration = 0; outerIteration < 20; outerIteration++)
{
var degree = xlEquations.Max(eq => eq.Degree) + 2;
var variables = xlEquations.SelectMany(eq => eq.Variables).Distinct().ToArray();
var numCoeff = (degree + 1).Pow(variables.Length);
// Populate ‘cells’
var cells = new double[xlEquations.Length][];
for (int xlIndex = 0; xlIndex < xlEquations.Length; xlIndex++)
{
var eq = xlEquations[xlIndex];
var row = new double[numCoeff];
var varIndex = eq.Variables.Select(v => variables.IndexOf(v)).ToArray();
for (int i = 0; i < eq.Coefficients.Length; i++)
{
if (eq.Coefficients[i] == 0)
{
continue;
}
int index = 0, si = i;
for (int v = 0; v < eq.Variables.Length; v++)
{
var power = si % (eq.Degree + 1);
if (power != 0)
{
index += power * (degree + 1).Pow(varIndex[v]);
}
si /= (eq.Degree + 1);
}
row[index] = eq.Coefficients[i];
}
cells[xlIndex] = row;
}
// Calculate ‘totalDegree’ (e.g. 𝑥² 𝑦² has total degree 4) and ‘maxDegree’ (e.g. 𝑥² 𝑦² has max degree 2)
var totalDegree = new int[numCoeff];
var maxDegree = new int[numCoeff];
for (int i = 0; i < numCoeff; i++)
{
int si = i, maxDeg = 0, totalDeg = 0;
for (int v = 0; v < variables.Length; v++)
{
var power = si % (degree + 1);
maxDeg = Math.Max(maxDeg, power);
totalDeg += power;
si /= (degree + 1);
}
maxDegree[i] = maxDeg;
totalDegree[i] = totalDeg;
}
// Find a good ordering of the coefficients based on ‘totalDegree’ and ‘maxDegree’
var sortKey = maxDegree;
for (int i = 0; i < numCoeff; i++)
{
sortKey[i] += (degree + 1) * totalDegree[i];
}
var coeffMap = Ut.NewArray(numCoeff, index => index);
Array.Sort(sortKey, coeffMap);
Array.Reverse(coeffMap);
int col = 0;
for (int row = 0, rows = cells.Length, innerIteration = 0; ; col++)
{
var outputPath = @"D:\Daten\Upload\Geocos\XL-{0}-{1}.htm".Fmt(outerIteration, innerIteration);
var nextOutputFile = @"XL-{0}-{1}.htm".Fmt(outerIteration, innerIteration + 1);
Console.WriteLine("Writing: {0}".Fmt(outputPath));
GeoUt.WriteHtml(
@"
<ul>{0}</ul>
<p><a href='{1}'>Next</a></p>
".Fmt(
cells.Select(rw => "<li>$ {0} = 0 $</li>".Fmt(new Polynomial(degree, variables, rw))).JoinString(),
nextOutputFile
),
outputPath);
GeoUt.OutputTable(
coeffMap.Select(ci => "$ {0} $".Fmt(new Polynomial(degree, variables, Ut.NewArray(numCoeff, ci2 => ci == ci2 ? 1d : 0d)))).ToArray(),
//cells.Select(rw => "$ {0} = 0 $".Fmt(new Polynomial(degree, variables, rw))).ToArray(),
cells.Select(rw => "").ToArray(),
cells.Select(rw => rw.Select((d, i) => rw[coeffMap[i]].ToString("#.##")).ToArray()).ToArray(),
null,
@"D:\Daten\Upload\Geocos\XL-{0}-{1}-table.htm".Fmt(outerIteration, innerIteration)
);
Ut.Assert(col < numCoeff);
if (row >= rows)
{
break;
}
var equationToUse = Enumerable.Range(row, rows - row).FirstOrDefault(rw => cells[rw][coeffMap[col]] != 0, -1);
if (equationToUse == -1)
{
continue;
}
if (equationToUse != row)
{
var tmp = cells[equationToUse];
cells[equationToUse] = cells[row];
cells[row] = tmp;
}
var divideBy = cells[row][coeffMap[col]];
if (divideBy != 1)
{
for (int cl = col; cl < numCoeff; cl++)
{
cells[row][coeffMap[cl]] /= divideBy;
}
}
for (int r = row + 1; r < rows; r++)
{
var multiplier = cells[r][coeffMap[col]];
if (multiplier == 0)
{
continue;
}
for (int cl = col; cl < numCoeff; cl++)
{
cells[r][coeffMap[cl]] -= cells[row][coeffMap[cl]] * multiplier;
}
}
innerIteration++;
row++;
}
if (totalDegree[coeffMap[col]] <= 2)
{
yield return(new NewEquations(
Enumerable.Range(cells.Length - equations.Length, equations.Length).Select(
ri => new Polynomial(degree, variables, cells[ri]).Simplify()
).Reverse().ToArray()
));
yield break;
}
xlEquations = xlEquations
.Concat(variables.SelectMany(v => xlEquations.Select(equ => equ * v))).ToArray();
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
System.Diagnostics.Debugger.Break();
throw new NotImplementedException();
}