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, extra focus is given to the data structure storing all partial derivatives, as well as indexing/propagating each of the nonzero partial derivatives in an efficient way. With these features, GTPSA.jl is significantly faster than ForwardDiff.jl for 2nd-order calculations and above, and has similar performance to 1st-order. See the benchmark/track.jl example for a speed comparison in calculating the partial derivatives for a system with 58 inputs and 6 outputs. In this example, GTPSA was x3.5 faster than ForwardDiff to 2nd order, and x19.8 faster to 3rd order.

GTPSA 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
2. Easy Monomial Indexing: Beyond 2nd order, accessing/manipulating the partial derivatives in an organized way can be a significant challenge when using other AD packages. In GTPSA, three simple indexing schemes for getting/setting monomial coefficients in a truncated power series is provided, as well as a cycle! function for cycling through all nonzero monomials
3. Custom Orders in Individual Variables: Other 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
4. Complex Numbers: GTPSA natively supports complex numbers and allows for mixing of complex and real truncated power series
5. 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

First, a Descriptor must be created specifying the number of variables, number of parameters, and truncation order(s) for the variables/parameters in the TPSA. A TPS can then be created based on the Descriptor. TPSs can be manipulated using all of the arithmetic operators (+,-,*,/,^) and math functions (e.g. abs, sqrt, sin, exp, log, tanh, etc.).

TPSs can be viewed as structures containing the coefficients for all of the monomials of a multivariable Taylor expansion up to the orders specified in the Descriptor. As an example, to compute the truncated power series of a function $f(x_1, x_2) = \cos{(x_1)}+i\sin{(x_2)}$ to 6th order in $x_1$ and $x_2$:

using GTPSA

# Descriptor for TPSA with 2 variables to 6th order
d = Descriptor(2, 6)

# Get the TPSs corresponding to each variable based on the Descriptor
x = vars(d)
# x[1] corresponds to the first variable and x[2] corresponds to the second variable

# Manipulate the TPSs as you would any other mathematical variable in Julia
f = cos(x[1]) + im*sin(x[2])
# f is a new ComplexTPS64

Note that scalars do not need to be defined as TPSs when writing expressions. Running print(f) gives the output

ComplexTPS64:
 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.

For more details, including using TPSs with differing truncation orders for each variable, see the GTPSA documentation.

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.