private static void ToSimplestForm(PolynomialTerm[] terms_in, out PolynomialTerm[] terms_out)
{
var termDictionary = new Dictionary <int, PolynomialTerm>();
for (int i = 0; i < terms_in.Length; i++)
{
var degree = terms_in[i].Degree;
if (termDictionary.ContainsKey(degree))
{
var currentTerm = termDictionary[degree];
termDictionary[degree] = new PolynomialTerm(currentTerm.Coefficient + terms_in[i].Coefficient, degree);
}
else if (terms_in[i] != PolynomialTerm.Zero)
{
termDictionary.Add(terms_in[i].Degree, terms_in[i]);
}
}
var result =
termDictionary.Values
.Where(t => t != PolynomialTerm.Zero)
.OrderBy(t => t.Degree)
.ToArray();
if (result.Length == 0)
{
result = new PolynomialTerm[1] {
PolynomialTerm.Zero
}
}
;
terms_out = result;
}
public void ExtractTermsContainingNthRoot(
BigInteger radicand,
int index,
out Polynomial p_reduced,
out Polynomial p_extract)
{
p_reduced = Polynomial.Zero;
p_extract = Polynomial.Zero;
for (int i = 0; i < Terms.Length; i++)
{
var coeff = Terms[i].Coefficient;
RadicalSum coeff_reduced = coeff;
RadicalSum coeff_target = RadicalSum.Zero;
for (int j = 0; j < coeff.Radicals.Length; j++)
{
if (coeff.Radicals[j].Index == index)
{
if (coeff.Radicals[j].Radicand % radicand == 0)
{
coeff_target += coeff.Radicals[j];
coeff_reduced -= coeff.Radicals[j];
}
}
}
p_reduced += new PolynomialTerm(coeff_reduced, Terms[i].Degree);
p_extract += new PolynomialTerm(coeff_target, Terms[i].Degree);
}
}
public static Polynomial Divide(Polynomial left, Radical right)
{
var terms = new PolynomialTerm[left.Terms.Length];
for (int i = 0; i < left.Terms.Length; i++)
{
terms[i] = new PolynomialTerm(left.Terms[i].Coefficient / right, left.Terms[i].Degree);
}
return(new Polynomial(terms));
}
public static Polynomial Add(Polynomial left, Polynomial right)
{
var terms = new PolynomialTerm[left.Terms.Length + right.Terms.Length];
left.Terms.CopyTo(terms, 0);
right.Terms.CopyTo(terms, left.Terms.Length);
var result = new Polynomial(terms);
return(result);
}
public static Polynomial DivideByX(Polynomial value)
{
var terms = new PolynomialTerm[value.Terms.Length];
for (int i = 0; i < value.Terms.Length; i++)
{
terms[i] = new PolynomialTerm(value.Terms[i].Coefficient, value.Terms[i].Degree - 1);
}
var result = new Polynomial(terms);
return(result);
}
public static Polynomial Negate(Polynomial value)
{
var terms = new PolynomialTerm[value.Terms.Length];
for (int i = 0; i < value.Terms.Length; i++)
{
terms[i] = -value.Terms[i];
}
var result = new Polynomial(terms);
return(result);
}
public static Polynomial Multiply(Polynomial left, Polynomial right)
{
var terms = new PolynomialTerm[left.Terms.Length * right.Terms.Length];
for (int i = 0; i < left.Terms.Length; i++)
{
for (int j = 0; j < right.Terms.Length; j++)
{
terms[(i * right.Terms.Length) + j] = left.Terms[i] * right.Terms[j];
}
}
return(new Polynomial(terms));
}
/// <summary>
/// Returns multiplicative inverse, i.e., B such that this * B = 1
/// </summary>
/// <returns></returns>
public static RadicalSum GetRationalizer(RadicalSum value)
{
// Method derived from:
// Rationalizing Denominators
// Allan Berele and Stefan Catoiu
// https://www.jstor.org/stable/10.4169/mathmaga.88.issue-2
// See Example 9
if (value == 0)
{
throw new Exception("Cannot get rationalization of zero");
}
// Solve for the largest field extensions in turn
// S = r_1 + r_2 + ... + r_n
// r_n = S - r_1 - r_2 - ... - r_(n-1)
// r_n^2 = [S - r_1 - r_2 - ... - r_(n-1)]^2
// Create polynomial in S
// S - r_1 - r_2 - ... - r_n = 0
var term1 = new Polynomials.PolynomialTerm(1, 1);
var term0 = new Polynomials.PolynomialTerm(-value, 0);
var p = new Polynomials.Polynomial(new Polynomials.PolynomialTerm[2] {
term0, term1
});
// Get the set of unique indexes and radicands for each index
var uniquePrimeNthRoots = p.GetUniquePrimeNthRoots();
// Main loop
for (int i = uniquePrimeNthRoots.Length - 1; i >= 0; i--)
{
var currentPrimeNthRoot = uniquePrimeNthRoots[i];
if (currentPrimeNthRoot.Item1 < 2) // Index
{
continue;
}
if (currentPrimeNthRoot.Item2 < 2) // Radicand
{
continue;
}
// Solve for largest radicand
// p = p_reduced + p_extract = 0
// p_extract = root[currentIndex](currentRadicand) * p' for some p'
Polynomials.Polynomial p_reduced;
Polynomials.Polynomial p_extract;
p.ExtractTermsContainingNthRoot(
radicand: currentPrimeNthRoot.Item2,
index: currentPrimeNthRoot.Item1,
p_reduced: out p_reduced,
p_extract: out p_extract);
// Remove root[currentIndex](currentRadicand) to keep p' === p_extract_coefficient
var r = new Radical(1, currentPrimeNthRoot.Item2, currentPrimeNthRoot.Item1);
var p_extract_coefficient = p_extract /= r;
// p_reduced = -p_extract
// p_reduced^currentIndex = (-p_extract)^currentIndex
// p_reduced^currentIndex = (-p_extract_coefficient)^currentIndex * currentRadicand
// p => p_reduced^currentIndex - currentRadicand * (-p_extract_coefficient)^currentIndex = 0
var left = Polynomials.Polynomial.Pow(p_reduced, currentPrimeNthRoot.Item1);
var right = Polynomials.Polynomial.Pow(-p_extract, currentPrimeNthRoot.Item1) * currentPrimeNthRoot.Item2;
p = left - right;
}
// We now have a polynomial in S with all radicals removed:
// a_0 + a_1 * S + a_2 * S^2 + ... + a_n * S^n = 0
// Radicalizer then becomes:
// -a_0 / S = a_1 + a_2 * S + ... + a_n * S^(n-1)
//
// 1 a_1 + a_2 * S + ... + a_n * S^(n-1)
// - = -----------------------------------
// S -a_0
var constantTerm = p.GetConstantTerm();
var constantP = new Polynomials.Polynomial(constantTerm);
p -= constantP;
if (!p.IsDivisibleByX)
{
throw new Exception("All terms should have at least degree 1");
}
if (!constantTerm.IsRational)
{
throw new Exception("No radicals should remain in any terms");
}
p = Polynomials.Polynomial.DivideByX(p);
p /= (Rational)(-constantTerm);
var rationalizer = p.EvaluateAt(value);
return(rationalizer);
}
public Polynomial(PolynomialTerm term)
: this(new PolynomialTerm[1] {
term
})
{
}
public static Polynomial Pow(Polynomial value, int exponent)
{
// Naive implementation is SLOW
// Attempt to make more efficient using multinomial theorem:
// https://en.wikipedia.org/wiki/Multinomial_theorem
// Replace x1, x2, ..., xm with polynomial terms X^0, X^1, ..., X^m-1
// Need to partition exponent as a sum of at most value.Terms.Length === m integers
// Pre-calculate powers of each term's coefficient in the polynomial, and cache them
// Item1: term index
// Item2: power
var polynomialTermCoefficientPowers = new Dictionary <Tuple <int, int>, RadicalSum>();
for (int termIndex = 0; termIndex < value.Terms.Length; termIndex++)
{
for (int power = 0; power <= exponent; power++)
{
var key = new Tuple <int, int>(termIndex, power);
var termPower = RadicalSum.Pow(value.Terms[termIndex].Coefficient, power);
polynomialTermCoefficientPowers.Add(key, termPower);
}
}
// Get the unique partitions first, along with the multinomial coefficient associated with them
// These correspond to unique integers k1, k2, ..., km such that k1 + k2 + ... + km = n
var maxTerms = Math.Min(exponent, value.Terms.Length);
var degreePartitions = Combinatorics.IntegerPartition.GetIntegerPartitions(exponent, maxTerms);
var degreePartitionsWithMultinomialCoefficient =
degreePartitions
.Select(p => new Tuple <Combinatorics.IntegerPartition, BigInteger>(
p,
GetMultinomialCoefficient(p.Values, exponent)))
.ToList();
// From the unique partitions, get all permutations
// These correspond to all possible k1 + k2 + ... + km = n in the multinomial theorem
// For each permutation, calculate the degree in X; this will be used to help group terms later
// Item1: permutation
// Item2: multinomial coefficient
// Item3: degree in X (e.g., X^2, X^3, etc)
var termDescriptions = new List <Tuple <Combinatorics.Permutation, BigInteger, int> >();
foreach (Tuple <Combinatorics.IntegerPartition, BigInteger> partition in degreePartitionsWithMultinomialCoefficient)
{
var permutations =
Combinatorics.Permutation.ArrangeInSlots(partition.Item1.Values, value.Terms.Length, false);
foreach (Combinatorics.Permutation p in permutations)
{
var degreeX = GetPermutationDegree(p.Values, value);
var termDescription =
new Tuple <Combinatorics.Permutation, BigInteger, int>(
p,
partition.Item2,
degreeX);
termDescriptions.Add(termDescription);
}
}
// Start multithreading block
// Initialize dictionary of terms for each degree in X
var degreeRadicalSumDictionary = new Dictionary <int, List <RadicalSum> >();
int processCount = termDescriptions.Count;
var doneEvent = new ManualResetEvent(false);
for (int termDescriptionIndex = 0; termDescriptionIndex < termDescriptions.Count; termDescriptionIndex++)
{
ThreadPool.QueueUserWorkItem(new WaitCallback((obj) =>
{
var termDescription =
(Tuple <Combinatorics.Permutation, BigInteger, int>)obj;
// Calculate term for this permutation in multinomial formula
var coefficient = RadicalSum.One;
for (int i = 0; i < value.Terms.Length; i++)
{
// term index, coefficient degree
var key = new Tuple <int, int>(i, termDescription.Item1.Values[i]);
var polynomialTermCoefficientPower = polynomialTermCoefficientPowers[key];
coefficient *= polynomialTermCoefficientPower;
}
coefficient *= termDescription.Item2;
lock (degreeRadicalSumDictionary)
{
if (degreeRadicalSumDictionary.ContainsKey(termDescription.Item3))
{
var existingTermList = degreeRadicalSumDictionary[termDescription.Item3];
existingTermList.Add(coefficient);
}
else
{
var termList = new List <RadicalSum>();
termList.Add(coefficient);
degreeRadicalSumDictionary.Add(termDescription.Item3, termList);
}
}
if (Interlocked.Decrement(ref processCount) == 0)
{
doneEvent.Set();
}
}), termDescriptions[termDescriptionIndex]);
}
doneEvent.WaitOne();
// Combine terms for each degree in X
var polynomialTerms = new List <PolynomialTerm>();
doneEvent = new ManualResetEvent(false);
processCount = degreeRadicalSumDictionary.Count;
foreach (KeyValuePair <int, List <RadicalSum> > kvp in degreeRadicalSumDictionary)
{
ThreadPool.QueueUserWorkItem(new WaitCallback((obj) =>
{
var key =
(KeyValuePair <int, List <RadicalSum> >)obj;
// Combine terms
var coefficient = RadicalSum.Zero;
foreach (RadicalSum radicalSum in key.Value)
{
coefficient += radicalSum;
}
var term = new PolynomialTerm(coefficient, key.Key);
lock (polynomialTerms)
{
polynomialTerms.Add(term);
}
if (Interlocked.Decrement(ref processCount) == 0)
{
doneEvent.Set();
}
}), kvp);
}
doneEvent.WaitOne();
var result = new Polynomials.Polynomial(polynomialTerms.OrderBy(p => p.Degree).ToArray());
return(result);
}
