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AngouriMath.Functions.Algebra.AnalyticalSolving

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Classes within the AngouriMath.Functions.Algebra.AnalyticalSolving namespace

• AnalyticalEquationSolver

• AnalyticalInequalitySolver

• CommonDenominatorSolver

• ExponentialSolver

• FractionedPolynoms

• LambertWSolver

  Summary

  K (alpha x + beta) b^(gamma x + delta) = c, a linear function times an exponential of
  another, solved by the Lambert W function. https://github.com/asc-community/AngouriMath/issues/2182
  

  Remarks

  Written e^((gamma x + delta) ln b), the exponent is u = g x + d, and with
  s = beta g / alpha - d the equation is (u + s) e^(u + s) = A,
  A = c g e^s / (K alpha), so u + s = W(A) and x = W(A) / g - beta / alpha.
  Over the complex numbers every branch is a root, W_n for a whole n, as a periodic
  equation's roots are written. Over the reals two branches can be: W_0 where
  A >= -1/e and W_(-1) where -1/e <= A < 0, and none where
  A < -1/e, since w e^w >= -1/e for every real w.
  

• 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.
  

• LogarithmSolver

  Summary

  An equation linear in logarithms of polynomials to one base,
  k_1 log_b(P_1) + ... + k_n log_b(P_n) = c with rational k_i and at least two
  logarithms, solved by exponentiating.
  

  Remarks

  Every root makes b to either side the same, so it is a root of
  P_1^(k_1 L) ... P_n^(k_n L) = b^(c L), for L the common denominator of the
  k_i: a polynomial equation, with finitely many roots, which holds at every root of the
  original. The converse does not hold. Over the complex numbers log(u) + log(v) is
  log(u v) only up to a multiple of 2 pi i, and a factor can be zero, so each
  candidate is put back into the equation as written, under the principal logarithm, and kept
  only where it holds. ln(x) + ln(x + 1) = 0 is x (x + 1) = 1, whose roots are
  (-1 +- sqrt(5)) / 2; at the negative one both logarithms carry i pi and their sum
  is 2 pi i, so the answer is { (sqrt(5) - 1) / 2 }. Newton's method answered it
  in decimals, and ln(1 + 3x^2) - ln(4x) = 1 with one of its two roots.
  

• ModulusSolver

  Summary

  Equations in moduli: where one is real for every complex x, how one in moduli of
  polynomials is solved over the reals, and how one in and out of moduli, conjugates, real
  and imaginary parts is solved over the complex numbers.
  

  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
  

• NonSquareSystemSolver

  Summary

  A system with more or fewer equations than unknowns that neither the triangularisation nor
  LinearSystemSolver takes, such as x^2 + y^2 = 1 for x and
  y, or sin(x) = 0 and x = 0 for x. Each unknown is eliminated in
  turn, an unknown that nothing is left to constrain becomes a parameter, and every family
  found is substituted back into every equation.
  #212

  Remarks

  An answer is given only where it is shown to hold. The elimination trusts each solve
  it makes, and a solve can answer for a case its equation does not cover. So each family is
  kept only where every equation, with the family substituted, simplifies to 0. A family
  that says where it holds is checked there: sqrt(y) = x is solved for y as
  x^2 provided x is where sqrt takes its values, and sqrt(t^2) - t is zero wherever that holds.
  A decline declines the whole system. An equation left with no unknown in it is zero,
  a number away from zero, or neither as far as can be told, like (a + b)/2 - c. A
  branch where one is away from zero has no solutions and is dropped. A branch where one
  cannot be told gives no answer for the system, rather than leaving the other branches to
  read as all of its solutions.
  A square system does not come here. It is the eliminator's in
  InSolveSystem(System.Collections.Generic.List{AngouriMath.Entity},System.ReadOnlySpan{AngouriMath.Entity.Variable},AngouriMath.Entity,AngouriMath.Core.Budgets.BudgetLedger), whose answers this would change.
  

• 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