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AngouriMath.Numerics

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Classes within the AngouriMath.Numerics namespace

• ComplexInterval

  Summary

  A rectangle of two Intervals holding a complex value.

• Interval

  Summary

  A closed interval of doubles, [Low, High], that holds a real value: every operation
  rounds its bounds outward, so the exact value of what was evaluated is inside.
  

• IntervalEvaluation

  Summary

  A number an expression stands for, as a ComplexInterval that holds it: in
  double arithmetic rounded outward, with no settings read -- neither the precision of the
  decimal evaluation nor the downcasting -- so that a question needing a dozen digits is
  answered in a dozen and gets the same answer however the caller's settings are set. The
  pilot of #1019's item 10, evaluation to a requested accuracy, for the checks inside the
  library; EvalNumerical stays as it is beside it.
  

  Remarks

  An interval is an answer with its error attached, so a comparison has three outcomes, not
  two: the values agree to the accuracy asked, they differ by more, or the intervals are too
  wide to tell -- where cancellation has eaten the digits, or a node is not one this reads,
  and the caller asks the decimal evaluation instead.
  https://github.com/asc-community/AngouriMath/issues/1019
  

• PreciseComplexInterval

  Summary

  A rectangle of two PreciseIntervals.

• PreciseEvaluation

  Summary

  IntervalEvaluation at a requested number of significant digits, for the
  points where double intervals are too wide to tell -- the digits lost to cancellation --
  or land on a cut by a rounding. The precision is an argument, never a setting.
  

• PreciseInterval

  Summary

  An interval of decimals at a working precision the evaluation carries: every operation
  rounds its lower bound down and its upper bound up in that many digits, so the exact value
  is inside. Interval's second tier, for where doubles cannot tell.
  

• SpecialFunctions

  Summary

  The special functions in double precision: erf, erfc, erfi, Ei,
  li, Si, Ci, Shi and Chi. Both compilers call them, and they
  sit beside the numerical evaluation so that its fast tier can take them up too.
  The interpreter's kernels, Erf(AngouriMath.Entity.Number.Complex) and the
  others, work in arbitrary precision. At 20 digits they take 0.7 to 5.5 ms a call. Newton's
  method makes thousands of calls. These take a microsecond or two.
  https://github.com/asc-community/AngouriMath/issues/1607
  

  Remarks

  Each is the kernel's function, with the kernel's branch cuts and the kernel's values on the
  axes. They were measured against the kernels at 30 digits on some ten thousand points: a
  grid of [-14, 14]^2, both axes out to 40 and down to 1e-4, either side of the
  negative real axis, and circles out to 300. The integrals agree to within 6e-14 of the
  value. The error functions agree to within 3e-13, which is the rounding of
  z^2 in their e^(-z^2) where |z| is large. An imaginary part of
  either sign of zero is read as the axis, as the kernels read it.
  The error functions come from the Faddeeva function
  w(z) = e^(-z^2) erfc(-i z), by Weideman's rational approximation with 40 terms
  (J. A. C. Weideman, Computation of the complex error function, SIAM J. Numer. Anal.
  31, 1994), and near the origin from the Taylor series of erf.
  Ei is summed as its series where the terms cancel little, which is where
  |z| - Re z is at most 3 or |z| at most 2. Out to |z| = 40 it is
  -E1(-z) otherwise, by E1's continued fraction, plus the branch term. From
  40 on it is the asymptotic series. li is Ei of the logarithm. Shi and
  Chi are summed as their own series by the same rule with |Re z|, and
  otherwise come from Ei(z) and Ei(-z). Si and Ci are those at
  i z. Every relation is one the kernels state.