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AngouriMath.Functions.Algebra.AnalyticalSolving

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Classes within the AngouriMath.Functions.Algebra.AnalyticalSolving namespace

• AnalyticalEquationSolver

• AnalyticalInequalitySolver

• CommonDenominatorSolver

• ExponentialSolver

• LinearSystemSolver

  Summary

  A linear system with fewer equations than unknowns, answered as the family of all its
  solutions: the unknowns a row reduction leaves free become parameters, and the rest are
  written in terms of them.
  #212

  Remarks

  2x - 4y = 12 for x and y is the issue's own example. It has
  infinitely many solutions and one degree of freedom, so the answer is a single row
  [12/2 + 2t, t] rather than a list of them, and the t in it is a variable
  the caller did not name.
  The answer type did not need changing, which is why this is small. A solution is a
  row of Matrix whose i-th entry is the i-th unknown's value, and nothing
  says those entries may not mention a variable. The same device already carries the
  constant of integration out of the ODE solver, and
  InSolveSystem(System.Collections.Generic.List{AngouriMath.Entity},System.ReadOnlySpan{AngouriMath.Entity.Variable},AngouriMath.Entity,AngouriMath.Core.Budgets.BudgetLedger) already mints one t for the single case
  where its recursion bottoms out unconstrained
  (#550).
  Reached only where the count is short, which is the one shape that had no answer
  at all — SolveSystem threw WrongNumberOfArgumentsException for it before
  reaching any elimination. A square system that is rank-deficient is left to the
  eliminator, which already answers it; taking those here as well would change answers that
  are not wrong.
  Rational coefficients on the unknowns, and that is a soundness requirement rather than
  a convenience. Row reduction has to decide whether a pivot is zero, and the general
  test available here is structural — Matrix.ReducedRowEchelonForm asks
  a == 0, which an expression that is zero everywhere without being written as
  0 fails. Choosing such a pivot divides by zero and produces a wrong family rather
  than no answer. Over the rationals the question is decidable, so it is asked there. The
  constant term is under no such restriction: it is never a pivot, so
  2x - 4y = k is answered with k symbolic.
  

• ModulusSolver

  Summary

  Equations in moduli: where one is real for every complex x, and how one in moduli of
  linear functions is solved over the reals.
  

  Remarks

  Over the complex numbers |x - 2| = |x - 3| holds on the whole line
  Re x = 5/2: both sides are real whatever x is, so the equation is one real
  condition on the two real unknowns x is made of, and what solves it is a curve. A
  numerical search finds points of it and would answer them as the set. Over the reals the
  same equation is x = 5/2, found by cases: between two neighbouring kinks every
  modulus has one sign, and the equation there has no moduli.
  https://github.com/asc-community/AngouriMath/issues/1573
  

• PolynomialSignTable

  Summary

  p(x) > 0 for a univariate polynomial over Q of any degree, answered by
  the sign of p between consecutive real roots.
  

  Remarks

  A polynomial has one sign on each open interval between consecutive real roots, so the
  answer is the union of those intervals where the sign is positive. Everything hard is
  in the word *consecutive*: the intervals are only the answer if the list of real roots
  is complete, and a missed root merges two intervals of opposite sign into one
  and reports the wrong half of it as the solution. So this refuses wherever completeness
  cannot be established, rather than answering from whatever roots were found.
  Completeness is established algebraically and in two steps. First
  FactorPrimitive(AngouriMath.Functions.IntegerPolynomial) writes the polynomial as a
  product of powers of irreducibles over Q, and it verifies that the factors
  multiply back to what it was given — so every real root of the whole is a real root of
  exactly one factor, an irreducible being square-free and two distinct irreducibles
  being coprime. Second, the number of real roots of each factor is read off its
  Discriminant(AngouriMath.Functions.MultivariatePolynomial,System.Int32,System.Collections.Generic.IReadOnlyList{System.Int32}): a factor of degree two has two real
  roots where its discriminant is positive and none where it is negative, and one of
  degree three has three and one respectively. A factor of degree four needs two more
  quantities alongside the discriminant, and there is no such criterion at five — nor a
  formula for the roots — which is where this stops. That count is what makes a root
  list a complete root list rather than a list of the roots that happened to come back.
  The roots themselves come from the solver reached through SolveEquation,
  which is exact, and their order is decided numerically, which is not. So the
  ordering is checked rather than trusted: the sign of the polynomial at an exact
  rational point in each interval is computed in exact integer arithmetic, and the
  resulting sequence has to change sign at every root of odd multiplicity and keep it at
  every root of even multiplicity. A misordered or merged root shows up as a sequence
  that does not, and is refused. The two outermost sample points are placed beyond
  Cauchy's bound, so no root can lie outside the sampled range.
  Part of the polynomial layer of
  #746, item 43 —
  and the consumer that the resultant was waiting for.
  

• PolynomialSolver

  Summary

  Solves all forms of Polynomials that are trivially solved

• StatementSolver

• ThresholdSearch

  Summary

  { n in ZZ : 2^n > n^2 }, an inequality with an exponential or a factorial over
  the whole numbers, is searched over a window from the least member and its tails are
  proved: the members are the points of the window where it holds, and the whole numbers
  from the start of the final run where the quantifier decides it holds there -- by
  induction, which is how Sullivan and Mackey do it (Ex 5.3.2: {0, 1} \/ ZZ /\ [5; +oo)).
  Over ZZ the other tail is proved the same way, downwards. A tail the quantifier
  leaves undecided leaves the set as written: a window is evidence about the window.
  https://github.com/asc-community/AngouriMath/issues/1409
  

• TrigonometricSolver