public static RadicalSum Add(RadicalSum left, RadicalSum right)
{
var z = new Radical[left.Radicals.Length + right.Radicals.Length];
left.Radicals.CopyTo(z, 0);
right.Radicals.CopyTo(z, left.Radicals.Length);
return(new RadicalSum(z));
}
public static RadicalSum Negate(RadicalSum value)
{
Radical[] radicals = new Radical[value.Radicals.Length];
for (int i = 0; i < value.Radicals.Length; i++)
{
radicals[i] = -value.Radicals[i];
}
var result = new RadicalSum(radicals);
return(result);
}
public static RadicalSum Multiply(RadicalSum left, RadicalSum right)
{
var z = new Radical[left.Radicals.Length * right.Radicals.Length];
for (int i = 0; i < left.Radicals.Length; i++)
{
for (int j = 0; j < right.Radicals.Length; j++)
{
z[(i * right.Radicals.Length) + j] = left.Radicals[i] * right.Radicals[j];
}
}
return(new RadicalSum(z));
}
public static RadicalSum Divide(RadicalSum left, Radical right)
{
if (right == 0)
{
throw new DivideByZeroException("Cannot divide by zero");
}
var z = new Radical[left.Radicals.Length];
for (int i = 0; i < left.Radicals.Length; i++)
{
z[i] = left.Radicals[i] / right;
}
return(new RadicalSum(z));
}
public static Radical[] SimplifyRadicals(Radical[] radicals)
{
if (radicals == null)
{
return(null);
}
if (radicals.Length == 0)
{
return(new Radical[0]);
}
if (radicals.Length == 1)
{
return(radicals);
}
var uniqueRadicals = new Dictionary <Tuple <BigInteger, int>, Radical>();
for (int i = 0; i < radicals.Length; i++)
{
Radical b = radicals[i];
var key = new Tuple <BigInteger, int>(b.Radicand, b.Index);
if (uniqueRadicals.ContainsKey(key))
{
uniqueRadicals[key] = Radical.AddCompatible(uniqueRadicals[key], b);
}
else if (!b.IsZero)
{
uniqueRadicals[key] = b;
}
}
Radical[] result =
uniqueRadicals.Values
.Where(f => !f.IsZero)
.OrderBy(f => f.Index)
.ThenBy(f => f.Radicand)
.ToArray();
if (result.Length == 0)
{
result = new Radical[1] {
Radical.Zero
}
}
;
return(result);
}
public RadicalSumRatio(Radical radical)
: this(new RadicalSum(radical), RadicalSum.One)
{
}
/// <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 RadicalSum(Radical radical)
{
_radicals = new Radical[1] {
radical
};
}