/// <summary>
/// Method for calculatin (<paramref name="r"/>, <paramref name="s"/>) Hasse derivative value for bivariate polynomial <paramref name="polynomial"/> at point the (<paramref name="xValue"/>, <paramref name="yValue"/>)
/// </summary>
/// <param name="polynomial">Polynomial which Hasse derivative'll be calculated</param>
/// <param name="r">Hasse derivative order by x variable</param>
/// <param name="s">Hasse derivative order by y variable</param>
/// <param name="xValue">Value for x variable</param>
/// <param name="yValue">Value fro y variable</param>
/// <param name="combinationsCountCalculator">Combinations calculator contract implementation</param>
/// <param name="combinationsCache">Cache for storing calculated numbers of combinations</param>
/// <returns>Hasse derivative value</returns>
public static FieldElement CalculateHasseDerivative(this BiVariablePolynomial polynomial,
int r, int s, FieldElement xValue, FieldElement yValue,
ICombinationsCountCalculator combinationsCountCalculator, FieldElement[][] combinationsCache = null)
{
var field = polynomial.Field;
var derivativeValue = 0;
foreach (var coefficient in polynomial)
{
if (coefficient.Key.Item1 < r || coefficient.Key.Item2 < s)
{
continue;
}
var currentAddition = combinationsCountCalculator.Calculate(field, coefficient.Key.Item1, r, combinationsCache).Representation;
currentAddition = field.Multiply(currentAddition,
combinationsCountCalculator.Calculate(field, coefficient.Key.Item2, s, combinationsCache).Representation);
currentAddition = field.Multiply(currentAddition, coefficient.Value.Representation);
currentAddition = field.Multiply(currentAddition, field.Pow(xValue.Representation, coefficient.Key.Item1 - r));
currentAddition = field.Multiply(currentAddition, field.Pow(yValue.Representation, coefficient.Key.Item2 - s));
derivativeValue = field.Add(derivativeValue, currentAddition);
}
return(new FieldElement(field, derivativeValue));
}
/// <summary>
/// Method for creating linear equations system for finding interpolation polynomial
/// </summary>
/// <param name="field">Finite field from which system coefficients'll be taken</param>
/// <param name="degreeWeight">Weight of bivariate monomials degree</param>
/// <param name="maxWeightedDegree">Maximum value of bivariate monomial degree</param>
/// <param name="maxXDegree">Maximum value of bivariate monomial x variable degree</param>
/// <param name="roots">Roots of the interpolation polynomial</param>
/// <param name="rootsMultiplicity">Multiplicity of bivariate polynomial's roots</param>
/// <param name="variableIndexByMonomial">Mapping for bivariate monomials to their numbers</param>
/// <param name="monomialByVariableIndex">Mapping from numbers to bivariate monomials</param>
/// <returns>Linear system's representation</returns>
private List <List <Tuple <int, FieldElement> > > BuildEquationsSystem(
GaloisField field,
Tuple <int, int> degreeWeight,
int maxWeightedDegree,
int maxXDegree,
IEnumerable <Tuple <FieldElement, FieldElement> > roots,
int rootsMultiplicity,
IDictionary <Tuple <int, int>, int> variableIndexByMonomial,
IDictionary <int, Tuple <int, int> > monomialByVariableIndex)
{
var equations = new List <List <Tuple <int, FieldElement> > >();
var combinationsCache = new FieldElement[Math.Max(maxWeightedDegree / degreeWeight.Item1, maxWeightedDegree / degreeWeight.Item2) + 1][]
.MakeSquare();
foreach (var root in roots)
{
for (var r = 0; r < rootsMultiplicity; r++)
{
for (var s = 0; r + s < rootsMultiplicity; s++)
{
var equation = new List <Tuple <int, FieldElement> >();
equations.Add(equation);
for (var i = r; i <= maxXDegree; i++)
{
for (var j = s; i *degreeWeight.Item1 + j *degreeWeight.Item2 <= maxWeightedDegree; j++)
{
var variableIndex = GetVariableIndexByMonomial(variableIndexByMonomial, monomialByVariableIndex, i, j);
var coefficient = _combinationsCountCalculator.Calculate(field, i, r, combinationsCache)
* _combinationsCountCalculator.Calculate(field, j, s, combinationsCache)
* FieldElement.Pow(root.Item1, i - r)
* FieldElement.Pow(root.Item2, j - s);
equation.Add(new Tuple <int, FieldElement>(variableIndex, coefficient));
}
}
}
}
}
return(equations);
}