Refactor functions (#99)

* More homegeneous structure for * and / for Taylor1 and TaylorN

Among others, mulHomogCoeff -> mul! and divHomogCoeff -> div!, and
internal changes.

* Fix getindex for Taylor1 and TaylorN and get rid of fixorder in * and /

Also introduce div! for TaylorN polynomials

* Fix an evaluate method and remove fixorder

fixorder is not needed anymore since getindex was fixed and returns
zero of appropriate type whenever the requested coefficient is well
inside the maximum order, but not defined as such.

* Use Base.power_by_squaring instead of defining more methods

* Metaprogramming for * and mul!, and cleanup code

* Refactor ^, square and sqrt and related functions for homogeneous coefficients

The following functions for the homogeneous coefficients
have been renamed: powerHomogCoef -> pow!, squareHomogCoef -> sqr!,
sqrtHomogCoef -> sqrt!

* Metaprogramming for the elementary functions and recurrence relations

This allows a more smooth approach for Taylor1 and TaylorN functions.

* Some metaprogramming in other_functions.jl

* Adapt docs/src/api.md to changes in the naming of functions

* Fix two tests

* Add `add!` and `subst!` for completeness, and tests

* Add methods for zero and one and metaprogramming for auxiliary functions

* Add constant_term and max_order, and reduce repeated code

* Add zero_korder and reduce code duplication

Also, eliminate an unused function (fdot) in src/other_functions

* Address comments pointed out in code review, and rename reverse -> inverse

Some tests have been modified to reflect current functionality.

* Add tests related to new functions, some docstrings, and update api.md

* Improve memory allocation mainly for TaylorN using mul!

* Two more small changes

* Add more tests for TaylorN

docs/src/api.md

@@ -31,22 +31,6 @@ integrate
 gradient
 jacobian
 hessian
-*
-/
-^
-sqrt
-exp
-log
-sin
-cos
-tan
-asin
-acos
-atan
-sinh
-cosh
-tanh
-abs
 ```
 
 ### Internals
@@ -57,20 +41,24 @@ generate_index_vectors
 in_base
 make_inverse_dict
 order_posTb
+max_order
+resize_coeffs1!
+resize_coeffsHP!
+zero_korder
+constant_term
 mul!
-mulHomogCoef
-divHomogCoef
-powHomogCoef
-squareHomogCoef
-sqrtHomogCoef
-expHomogCoef
-logHomogCoef
-sincosHomogCoef
-tanHomogCoef
-asinHomogCoef
-acosHomogCoef
-atanHomogCoef
-sinhcoshHomogCoef
-tanhHomogCoef
+div!
+pow!
+sqr!
+sqrt!
+exp!
+log!
+sincos!
+tan!
+asin!
+acos!
+atan!
+sinhcosh!
+tanh!
 A_mul_B!
 ```

src/TaylorSeries.jl

@@ -32,13 +32,13 @@ import Base: zero, one, zeros, ones, isinf, isnan,
     rem, mod, mod2pi, abs, gradient,
     sqrt, exp, log, sin, cos, tan,
     asin, acos, atan, sinh, cosh, tanh,
-    reverse, A_mul_B!,
+    A_mul_B!,
     getindex, setindex!, endof
 
 export Taylor1, TaylorN, HomogeneousPolynomial, AbstractSeries
 
 export get_coeff, derivative, integrate,
-    evaluate, evaluate!,
+    evaluate, evaluate!, inverse,
     show_params_TaylorN,
     get_order, get_numvars,
     set_variables, get_variables,