Improve Taylor1{TaylorN{T}} mixtures (#333)

* Add tests for  Taylor1{TaylorN{T}}

* Add fixorder to Taylor1{TaylorN{T}}

* Add methods for arithm opers for Taylor1{TaylorN{T}}

* Add methods of ^, sqrt, pow!, sqr!, sqrt! for Taylor1{TaylorN{T}

* Add specialized methods for most mutating functions

for Taylor1{TaylorN{T}}

* Implement suggestions by @PerezHz

* Fix error in docs

* Use TS

* Improvements for evaluate

* Clean up commented (old) code

* Remove some superfluous methods in evaluate

* More fixes

* Fixes and clean-up of `evaluate` methods

* Returning a zero Homogeneouspolynomial is of order 0

* Scattered fixes and clean up

* Fix documentation

* Remove evaluate methods for TaylorN with Varargs,
and fix tests

* Fix _evaluate for mixtures, and new tests

* Methods of evaluate for Taylor1 involving TaylorN
(fixing certain type instability)

* Add tests for new evaluate methods

* Bump minor version

* Remove a `@show` left there

Project.toml

@@ -1,7 +1,7 @@
 name = "TaylorSeries"
 uuid = "6aa5eb33-94cf-58f4-a9d0-e4b2c4fc25ea"
 repo = "https://github.com/JuliaDiff/TaylorSeries.jl.git"
-version = "0.15.4"
+version = "0.16.0"
 
 [deps]
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"

docs/src/api.md

@@ -77,7 +77,7 @@ div!
 pow!
 square
 sqr!
-sqr!(::HomogeneousPolynomial{T}, ::HomogeneousPolynomial{T}) where {T<:NumberNotSeriesN}
+accsqr!
 sqrt!
 exp!
 log!
@@ -105,6 +105,5 @@ _populate_dicts!
 
 ```@index
 Pages = ["api.md"]
-Module = ["TaylorSeries"]
 Order = [:type, :function]
 ```

ext/TaylorSeriesIAExt.jl

@@ -41,7 +41,7 @@ function ^(a::Taylor1{Interval{T}}, r::S) where {T<:Real, S<:Real}
     c_order = l0 == 0 ? a.order : min(a.order, trunc(Int,r*a.order))
     c = Taylor1(zero(aux), c_order)
     for k = 0:c_order
-        TaylorSeries.pow!(c, aa, r, k)
+        TS.pow!(c, aa, r, k)
     end
 
     return c
@@ -66,7 +66,7 @@ function ^(a::TaylorN{Interval{T}}, r::S) where {T<:Real, S<:Real}
 
     c = TaylorN( a0r, a.order)
     for ord in 1:a.order
-        TaylorSeries.pow!(c, aa, r, ord)
+        TS.pow!(c, aa, r, ord)
     end
 
     return c
@@ -84,7 +84,7 @@ for T in (:Taylor1, :TaylorN)
         aa[0] = one(aux) * a0
         c = $T( aux, order )
         for k in eachindex(a)
-            TaylorSeries.log!(c, aa, k)
+            TS.log!(c, aa, k)
         end
         return c
     end
@@ -101,7 +101,7 @@ for T in (:Taylor1, :TaylorN)
         c = $T( aux, order )
         r = $T( sqrt(1 - a0^2), order )
         for k in eachindex(a)
-            TaylorSeries.asin!(c, aa, r, k)
+            TS.asin!(c, aa, r, k)
         end
         return c
     end
@@ -118,7 +118,7 @@ for T in (:Taylor1, :TaylorN)
         c = $T( aux, order )
         r = $T( sqrt(1 - a0^2), order )
         for k in eachindex(a)
-            TaylorSeries.acos!(c, aa, r, k)
+            TS.acos!(c, aa, r, k)
         end
         return c
     end
@@ -135,7 +135,7 @@ for T in (:Taylor1, :TaylorN)
         c = $T( aux, order )
         r = $T( sqrt(a0^2 - 1), order )
         for k in eachindex(a)
-            TaylorSeries.acosh!(c, aa, r, k)
+            TS.acosh!(c, aa, r, k)
         end
         return c
     end
@@ -152,14 +152,14 @@ for T in (:Taylor1, :TaylorN)
             """Series expansion of atanh(x) diverges at x = ±1."""))
 
         for k in eachindex(a)
-            TaylorSeries.atanh!(c, aa, r, k)
+            TS.atanh!(c, aa, r, k)
         end
         return c
     end
 end
 
 
-function evaluate(a::Taylor1, dx::Interval)
+function evaluate(a::Taylor1{T}, dx::Interval{S}) where {T<:Real, S<:Real}
     order = a.order
     uno = one(dx)
     dx2 = dx^2
@@ -201,7 +201,7 @@ end
 function _evaluate(a::HomogeneousPolynomial, dx::IntervalBox{N,T}, ::Val{true} ) where {T<:Real,N}
     a.order == 0 && return a[1] + Interval{T}(0, 0)
 
-    ct = TaylorSeries.coeff_table[a.order+1]
+    ct = TS.coeff_table[a.order+1]
     @inbounds suma = a[1]*Interval{T}(0,0)
 
     Ieven = Interval{T}(0,1)