GTPSA.jl

A full-featured Julia interface to the Generalised Truncated Power Series Algebra library.

GTPSA.jl is a full-featured Julia interface to the Generalised Truncated Power Series Algebra (GTPSA) library, which computes Taylor expansions, or Truncated Power Series (TPSs), of real and complex multivariable functions to high orders.

Truncated Power Series Algebra (TPSA) performs forward-mode automatic differentation (AD) similar to the typical dual-number implementation, as in ForwardDiff.jl. However, instead of nesting derivatives for higher orders, TPSA naturally extends to arbitary orders by directly using the power series expansions. Furthermore, because TPSA is designed for high order AD, the storage, indexing, and propagation of all nonzero partial derivatives has been highly optimized.

GTPSA.jl provides several advantages over current Julia AD packages:

1. Speed: GTPSA.jl is significantly faster than ForwardDiff.jl for 2nd-order calculations and above, and has very similar performance at 1st-order. See our example where GTPSA was x3.5 faster than ForwardDiff to 2nd order, and x19.8 faster to 3rd order.

2. Easy Monomial Indexing: Beyond 2nd order, accessing/manipulating the partial derivatives in an organized way can be a significant challenge. GTPSA provides three simple indexing schemes for getting/setting monomial coefficients in a truncated power series, as well as a cycle! function for cycling through all nonzero monomials.

3. Easy TPS slicing: Suppose you would like to get all terms in a multivariable truncated power series proportional to a specific variable; GTPSA.jl provides a simple syntax similar to array slicing to extract parts of a TPS.

4. Complex Numbers: GTPSA.jl natively supports complex numbers and allows for mixing of complex and real truncated power series

5. Custom Orders in Individual Variables: Other AD packages use a single maximum order for all variables. With GTPSA, the maximum order can be set differently for individual variables, as well as for a separate part of the monomial. For example, computing the Taylor expansion of $f(x_1,x_2)$ to 2nd order in $x_1$ and 6th order in $x_2$ is possible.

6. Distinction Between State Variables and Parameters: Distinguishing between dependent variables and parameters in the solution of a differential equation expressed as a power series in the dependent variables/parameters can be advantageous in analysis

Setup

To use GTPSA.jl, in the Julia REPL run

import Pkg; Pkg.add("GTPSA")

Basic Usage

julia> using GTPSA

julia> const D = Descriptor(2, 6); # 2 variables to 6th order

julia> Δx = @vars(D)  # Get truncated power series (TPSs) corresponding to the variables

julia> f = cos(Δx[1]) + im*sin(Δx[2]) # Manipulate TPSs as you would any other number
ComplexTPS64{GTPSA.Dynamic}:
Descriptor(NV=2, MO=6)
 Real                     Imag                       Order   Exponent
  1.0000000000000000e+00   0.0000000000000000e+00      0      0   0
  0.0000000000000000e+00   1.0000000000000000e+00      1      0   1
 -5.0000000000000000e-01   0.0000000000000000e+00      2      2   0
  0.0000000000000000e+00  -1.6666666666666666e-01      3      0   3
  4.1666666666666664e-02   0.0000000000000000e+00      4      4   0
  0.0000000000000000e+00   8.3333333333333332e-03      5      0   5
 -1.3888888888888887e-03   0.0000000000000000e+00      6      6   0

The GTPSA library currently only supports truncated power series representing Float64 and ComplexF64 number types.

Acknowledgements

Much thanks must be given to Laurent Deniau, the creator of the C GTPSA library, for his time and great patience in explaining his code.

Advanced users are referred to this paper discussing the inner workings of the C GTPSA library.