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AngouriMath.Functions

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Classes within the AngouriMath.Functions namespace

• BaseConversion

• ExpressionNumerical

• Fraction

• InverseTrigonometricTableValues

  Summary

  The trigonometric tables read backwards: given a value, the angle in the principal
  range whose sine or tangent it is.
  https://github.com/asc-community/AngouriMath/issues/569
  https://github.com/asc-community/AngouriMath/issues/179
  

• MonomialOrder

  Summary

  Which monomial is the leading one.

• MultivariatePolynomial

  Summary

  A polynomial in several variables over the rationals, stored sparsely.
  

  Remarks

  The coefficient domain is deliberately only Q. A polynomial whose
  coefficients are arbitrary expressions is not one this can reason about: deciding
  whether sqrt(2) - a is zero is as hard as the problem that sent us here, and
  a divisor wrongly believed nonzero produces a wrong cancellation rather than a
  missing one. Anything outside Q is refused at the door by
  TryParse(AngouriMath.Entity,System.Collections.Generic.IReadOnlyDictionary{AngouriMath.Entity.Variable,System.Int32}).
  An exponent vector is packed into one UInt64, a byte to a variable,
  the first variable in the most significant byte. Comparing packed monomials is then
  exactly the lexicographic order on exponents that DivideExact(AngouriMath.Functions.MultivariatePolynomial) needs,
  and looking a monomial up costs one hash.
  This and its neighbours in Functions/Algebra/Polynomials are kernel algebra:
  the solvers, the evaluator and the simplifier all depend on them, and they depend on
  none of those. That direction is the point of the folder. PolynomialGcd in particular is reached from simplification, from Core/Transformations and
  from evaluation, which is why it lived under Functions/Simplification for as
  long as simplification was the only caller anybody had counted.
  Partial so that the operations only a Gröbner basis needs — monomial divisibility, an
  order other than lexicographic, reduction against a set — live beside the solver that
  wants them, in Functions/Algebra/Groebner, rather than here where every other
  caller would have to read past them. See MultivariatePolynomial.Groebner.cs.
  

• Patterns

• PolynomialFactoring

  Summary

  Factors a polynomial in one variable into linear factors with whole roots:
  x^2 + 2x + 1 becomes (x + 1)^2, and
  x^3 - 6x^2 + 11x - 6 becomes (x - 1)(x - 2)(x - 3).
  

  Remarks

  Deliberately narrow, on two counts.
  Rational roots only. Factoring through every root would answer
  (x - i)(x + i) for x^2 + 1 and (x - sqrt(2))(x + sqrt(2)) for
  x^2 - 2, which is not what anyone means by factoring those.
  And only when the polynomial splits completely into whole roots. A partly factored
  answer is not obviously better than the sum it came from, and fractional roots turn
  up mostly in the output of calculus, where the expanded form is the conventional
  one: the antiderivative of x^2 + x reads better as x^3/3 + x^2/2 than
  as x^2 * (x + 3/2) / 3.
  What comes out of here is a candidate, not a decision. The simplifier keeps it
  alongside the other forms it has found and picks between them by its complexity
  metric, which is why x^2 - 1 stays as it is while (x + 1)^2 wins.
  

• PolynomialGcd

  Summary

  The greatest common divisor of two polynomials in several variables over the
  rationals, and the cancellation of a quotient that it makes possible:
  (x^2 + 2xy + y^2) / (x^2 - y^2) is (x + y) / (x - y) (#55).
  

  Remarks

  A polynomial in several variables is a polynomial in one of them whose coefficients
  are polynomials in the rest, so the algorithm is the univariate one applied down a
  recursion on the variable count. What it cannot be is the plain Euclidean algorithm
  over that coefficient ring: pseudo-division multiplies through by the leading
  coefficient every step, and the coefficients of the remainder sequence then grow
  exponentially in the degree. The classic example is Knuth's, where two degree-8
  polynomials over the integers produce a remainder with a coefficient near 10^35.
  Two things keep that in hand, and both are needed. The greatest common divisor
  splits as the gcd of the contents times the gcd of the primitive parts, because the
  content carries everything free of the main variable and the primitive parts carry
  nothing of it; so the content is taken out first, recursively, in one variable
  fewer. And within the remainder sequence the subresultant coefficients are divided
  out at each step — that division is exact, since what is left is a subresultant, and
  it is what turns exponential growth into linear. Knuth, TAOCP vol. 2,
  §4.6.1, algorithms C and E; Geddes, Czapor and Labahn, Algorithms for Computer
  Algebra, §7.3.
  Nothing here trusts the result of that machinery. The divisor it computes is
  divided out of both sides and multiplied back, and the quotient is only used when
  both come out equal to what they started as. A divisor that is merely common and
  not greatest costs an incomplete cancellation; one that is not a divisor at all
  would be a wrong answer, and is the thing the check is there to stop.
  

• RootExtraction

  Summary

  Writes a rational raised to a rational power as a whole part times a radical that has
  nothing left to give up, so that sqrt(12) is 2 * sqrt(3) and sqrt(1/2) is sqrt(2) / 2.
  https://github.com/asc-community/AngouriMath/issues/281
  

• Simplificator

• TreeAnalyzer

• TrigonometricAngleExpansion

• TrigonometryTableValues