Expanding exp(x) with taylor_expand()

The taylor_expand function takes the function to expand as it's first argument, and the point to expand about as the second argument. A keyword argument order determines which order to expand to:

julia> using TaylorSeries
julia> taylor_expand(exp, 0, order=2) 1.0 + 1.0 t + 0.5 t² + 𝒪(t³)

And voìla! It really is that simple to calculate a simple taylor polynomial. The next example is slightly more complicated.

Expanding exp(x) with a symbolic object

An alternative way to compute the single-variable taylor expansion for a function is by defining a variable of type Taylor1, and using it in the function you wish to expand. The argument given to the Taylor1 constructor is the order to expand to:

julia> using TaylorSeries
julia> x = Taylor1(2) 1.0 t + 𝒪(t³)
julia> exp(x) 1.0 + 1.0 t + 0.5 t² + 𝒪(t³)

Let's also get rid of the printed error for the next few examples, and set the printed independent variable to x:

julia> displayBigO(false)false
julia> set_taylor1_varname("x")"x"
julia> exp(x) 1.0 + 1.0 x + 0.5 x²

Changing point to expand about

A variable constructed with Taylor1() automatically expands about the point x=0. But what if you want to use the symbolic object to expand about a point different from zero? Because expanding exp(x) about x=1 is exactly the same as expanding exp(x+1) about x=0, simply replace the x in your expression with x+1 to expand about x=1:

julia> p = exp(x+1) 2.718281828459045 + 2.718281828459045 x + 1.3591409142295225 x²
julia> p(0.01)2.7456005608350584
julia> exp(1.01)2.7456010150169163

More examples

You can even use custum functions

julia> f(a) = 1/(a+1)f (generic function with 1 method)
julia> f(x) 1.0 - 1.0 x + 1.0 x²

Functions can be nested

julia> sin(f(x)) 0.8414709848078965 - 0.5403023058681398 x + 0.11956681346419151 x²

and complicated further in a modular way

julia> sin(exp(x+2))/(x+2)+cos(x+2)+f(x+2) 0.364113974242596 + 0.41259243488717107 x - 11.843864409375039 x²

Four-square identity

The first example shows that the four-square identity holds:

\[(a_1^2 + a_2^2 + a_3^2 + a_4^2)\cdot(b_1^2 + b_2^2 + b_3^2 + b_4^2) = \\ \qquad (a_1 b_1 - a_2 b_2 - a_3 b_3 -a_4 b_4)^2 + (a_1 b_2 - a_2 b_1 - a_3 b_4 -a_4 b_3)^2 + \\ \qquad (a_1 b_3 - a_2 b_4 - a_3 b_1 -a_4 b_2)^2 + (a_1 b_4 - a_2 b_3 - a_3 b_2 -a_4 b_1)^2,\]

which was originally proved by Euler. The code can also be found in this test of the package.

First, we reset the maximum degree of the polynomial to 4, since the RHS of the equation has a priori terms of fourth order, and define the 8 independent variables.

julia> using TaylorSeries
julia> # Define the variables α₁, ..., α₄, β₁, ..., β₄ make_variable(name, index::Int) = string(name, TaylorSeries.subscriptify(index))make_variable (generic function with 1 method)
julia> variable_names = [make_variable("α", i) for i in 1:4]4-element Vector{String}: "α₁" "α₂" "α₃" "α₄"
julia> append!(variable_names, [make_variable("β", i) for i in 1:4])8-element Vector{String}: "α₁" "α₂" "α₃" "α₄" "β₁" "β₂" "β₃" "β₄"
julia> # Create the TaylorN variables (order=4, numvars=8) a1, a2, a3, a4, b1, b2, b3, b4 = set_variables(variable_names, order=4)8-element Vector{TaylorN{Float64}}: 1.0 α₁ 1.0 α₂ 1.0 α₃ 1.0 α₄ 1.0 β₁ 1.0 β₂ 1.0 β₃ 1.0 β₄
julia> a1 # variable a1 1.0 α₁

Now we compute each term appearing in Eq. (\ref{eq:Euler})

julia> # left-hand side
       lhs1 = a1^2 + a2^2 + a3^2 + a4^2 ;
julia> lhs2 = b1^2 + b2^2 + b3^2 + b4^2 ;
julia> lhs = lhs1 * lhs2;
julia> # right-hand side rhs1 = (a1*b1 - a2*b2 - a3*b3 - a4*b4)^2 ;
julia> rhs2 = (a1*b2 + a2*b1 + a3*b4 - a4*b3)^2 ;
julia> rhs3 = (a1*b3 - a2*b4 + a3*b1 + a4*b2)^2 ;
julia> rhs4 = (a1*b4 + a2*b3 - a3*b2 + a4*b1)^2 ;
julia> rhs = rhs1 + rhs2 + rhs3 + rhs4;

We now compare the two sides of the identity,

julia> lhs == rhstrue

The identity is satisfied. $\square$

Fateman test

Richard J. Fateman, from Berkeley, proposed as a stringent test of polynomial multiplication the evaluation of $s\cdot(s+1)$, where $s = (1+x+y+z+w)^{20}$. This is implemented in the function fateman1 below. We shall also consider the form $s^2+s$ in fateman2, which involves fewer operations (and makes a fairer comparison to what Mathematica does).

julia> using TaylorSeries
julia> const order = 2020
julia> const x, y, z, w = set_variables(Int128, "x", numvars=4, order=2order)4-element Vector{TaylorN{Int128}}: 1 x₁ 1 x₂ 1 x₃ 1 x₄
julia> function fateman1(degree::Int) T = Int128 s = one(T) + x + y + z + w s = s^degree s * ( s + one(T) ) endfateman1 (generic function with 1 method)

(In the following lines, which are run when the documentation is built, by some reason the timing appears before the command executed.)

julia> @time fateman1(0);  0.044044 seconds (55.04 k allocations: 41.074 MiB, 14.80% gc time, 50.90% compilation time)
julia> @time f1 = fateman1(20); 1.687105 seconds (1.86 k allocations: 49.837 MiB, 0.30% gc time)

Another implementation of the same, but exploiting optimizations related to ^2 yields:

julia> function fateman2(degree::Int)
           T = Int128
           s = one(T) + x + y + z + w
           s = s^degree
           s^2 + s
       endfateman2 (generic function with 1 method)
julia> fateman2(0);
julia> @time f2 = fateman2(20); # the timing appears above 0.853971 seconds (1.99 k allocations: 53.986 MiB, 0.29% gc time)

We note that the above functions use expansions in Int128. This is actually required, since some coefficients are larger than typemax(Int):

julia> getcoeff(f2, (1,6,7,20)) # coefficient of x y^6 z^7 w^{20}128358585324486316800
julia> ans > typemax(Int)true
julia> length(f2)41
julia> sum(TaylorSeries.size_table)135751

These examples show that fateman2 is nearly twice as fast as fateman1, and that the series has 135751 monomials in 4 variables.


The functions described above have been compared against Mathematica v11.1. The relevant files used for benchmarking can be found here. Running on a MacPro with Intel-Xeon processors 2.7GHz, we obtain that Mathematica requires on average (5 runs) 3.075957 seconds for the computation, while for fateman1 and fateman2 above we obtain 2.15408 and 1.08337, respectively.

Then, with the current version of TaylorSeries.jl and using Julia v0.7.0, our implementation of fateman1 is about 30%-40% faster. (The original test by Fateman corresponds to fateman1 above, which avoids some optimizations related to squaring; the implementation in Mathematica is done such that this optimization does not occur.)