WIP: Explicit basis

[jverzani]

Currently the polynomial type implies both a basis and a backend storage type. E.g., ImmutablePolynomial uses the standard basis and a tuple for storage. This PR breaks this up so the basis is explicit -- the equivalent being ImmutableDensePolynomial{StandardBasis}. If all goes to plan, this will be aliased to ImmutablePolynomial and the PR should be non breaking to the end user.

The main types are

• MutableDenseUnivariatePolynomial backed by a vector.
• MutableDensePolynomial backed by a vector. This type has an offset, so it will allow LaurentPolynomial to be MutableDenseLaurentPolynomial{StandardBasis}.
• MutableDenseViewPolynomial non copying when possible
• ImmutableDensePolynomial backed by an ntuple. For better type stability, trailing zeros may be present, unlike for the mutable dense polynomial type
• MutableSparsePolynomial backed by a dictionary

Some simple benchmarking shows some issues have been improved.

 Row │ poly_eval  scalar_add  scalar_mul  scalar_div  poly_add  poly_mul  mixed_var_add  mixed_var_mul  poly_diff  poly_int  types         ⋯
     │ String     String      String      String      String    String    String         String         String     String    String        ⋯
─────┼──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
*  1 │ 10.5ns     61.2ns      54.8ns      55.2ns      70.1ns    221.4ns   87.0ns         81.8ns         131.2ns    149.8ns   MutableDenseP ⋯
   2 │ 13.7ns     46.0ns      44.7ns      91.3ns      65.9ns    242.9ns   73.4ns         69.3ns         353.0ns    397.4ns   Polynomial
   3 │ 11.3ns     77.5ns      115.0ns     113.5ns     79.5ns    272.9ns   100.0ns        139.9ns        303.2ns    1.3μs     LaurentPolyno
*  4 │ 32.8ns     27.4ns      33.7ns      34.6ns      25.0ns    97.9ns    26.9ns         35.3ns         36.8ns     35.8ns    ImmutableDens
   5 │ 33.3ns     31.1ns      33.6ns      1.3μs       29.5ns    101.3ns   26.1ns         29.7ns         1.6μs      1.7μs     ImmutablePoly ⋯
*  6 │ 192.1ns    135.0ns     429.1ns     423.0ns     782.4ns   9.8μs     173.8ns        476.6ns        771.6ns    796.7ns   MutableSparse
   7 │ 192.9ns    139.7ns     420.1ns     911.5ns     787.2ns   10.3μs    173.4ns        453.9ns        865.8ns    792.1ns   SparsePolynom

TODO:

[] currently common.jl and abstract-polynomial.jl do the same thing and should be consolidated. (For later, if at all)
[x] the types ImmutablePolynomial, SparsePolynomial, LaurentPolynomial can be removed (Polynomial too)
[x] The Chebyshev polynomial example needs to be rewritten to use MutableDensePolynomial as a backend
[x] use this to close #510 and #512. (#511 to be addressed later
[x] check that this doesn't break downstream packages!!

The plan is for this to be followed by another PR which should
[] replaces Polynomial type using new framework
[] makes FactoredPolynomial <: AbstractPolynomial
[] drop polynomials/standardbasis.jl (merging into standard-basis/standardbasis.jl)

[codecov]

Codecov Report

Patch coverage: 85.48% and project coverage change: -1.89% ⚠️

Comparison is base (f4d90d9) 79.78% compared to head (7b3fe93) 77.90%.

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #513      +/-   ##
==========================================
- Coverage   79.78%   77.90%   -1.89%     
==========================================
  Files          26       35       +9     
  Lines        3216     3838     +622     
==========================================
+ Hits         2566     2990     +424     
- Misses        650      848     +198     

Files Changed	Coverage Δ
src/Polynomials.jl	100.00% <ø> (ø)
src/rational-functions/common.jl	71.63% <ø> (ø)
src/standard-basis/polynomial.jl	0.00% <0.00%> (ø)
src/abstract.jl	49.12% <20.00%> (-4.73%)	⬇️
src/promotions.jl	33.33% <33.33%> (ø)
src/show.jl	76.74% <36.36%> (-4.09%)	⬇️
...ynomial-basetypes/mutable-dense-view-polynomial.jl	40.00% <40.00%> (ø)
src/polynomials/multroot.jl	57.95% <44.44%> (+0.32%)	⬆️
src/polynomials/standard-basis.jl	74.41% <44.44%> (-5.95%)	⬇️
.../polynomial-basetypes/mutable-sparse-polynomial.jl	81.48% <81.48%> (ø)
... and 14 more

... and 1 file with indirect coverage changes

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[jverzani]

Closing, cf #517