ppapp 1.0.3

Piecewise polynomial approximation: code generator for Chebyshev approximation

ppapp - Piecewise Polynomial Approximation

Code generator for piecewise Chebyshev approximation of mathematical functions.

Overview

This package generates C source code containing polynomial coefficients that approximate a given function f(x) over a specified domain. The approximation uses piecewise Chebyshev polynomials with octave-based domain subdivision for optimal accuracy.

Reference: Joachim Wuttke and Alexander Kleinsorge, "Algorithm 1XXX: Code generation for piecewise Chebyshev approximation"

Installation

pip install ppapp

Requirements

• Python 3.9+
• python-flint (for arbitrary-precision arithmetic)

Quick Start

Command Line

# Compute a function value
ppapp v ppapp.demo_functions.imwofx 1.0

# Compute error bound for given M and N
ppapp e ppapp.demo_functions.imwofx 5 10

# Generate C source code
ppapp s ppapp.demo_functions.imwofx 5 10 > coefficients.c

Using Demo Functions

The package includes several demo functions:

# Im w(x) = exp(-x^2) * erfi(x)
ppapp e ppapp.demo_functions.imwofx 5 10

# erfcx(x) = exp(x^2) * erfc(x)
ppapp e ppapp.demo_functions.erfcx 5 10

# Polynomial x^3 - x^2 + x - 1
ppapp e ppapp.demo_functions.polynomial 8 15

Python API

from ppapp import (
    set_reference_function,
    compute_subdomains,
    cheb_pm,
    analyse_row,
    error_bound,
)
from ppapp.demo_functions.imwofx import my_arb_f, my_domain

# Set the reference function
set_reference_function(my_arb_f)

# Compute subdomains
a, b = my_domain
j0, l0, D = compute_subdomains(a, b, M=5)

# Compute coefficients for each subdomain
for d in D:
    d.coeffs = cheb_pm(d.a, d.b, N=10)
    d.s, d.r = analyse_row(d.coeffs)
    err = error_bound(d, N=10)
    print(f"Subdomain [{d.a}, {d.b}): error = {err:.2f} epsilon")

Creating Custom Functions

Create a Python module with:

from flint import arb

# Domain [a, b)
my_domain: tuple[float, float] = (0.5, 12.0)

def my_arb_f(X: arb, prec: int) -> arb:
    """Evaluate f(x) with given precision."""
    # Your implementation here
    return result

# Test cases: (x, expected_value, tolerance)
my_testcases: list[tuple[float, float, float]] = [
    (1.0, 0.123456, 1e-5),
]

Then use it:

ppapp s path/to/my_function.py 5 10 > output.c

Modes

Mode	Command	Description
v	ppapp v <f> <x>	Compute f(x)
n	ppapp n <f> <M> <Nmax> <E>	Find minimal N for error ≤ E
e	ppapp e <f> <M> <N>	Compute error bound
c	ppapp c <f> <M> <N> [E]	Print Chebyshev coefficients
p	ppapp p <f> <M> <N> [E]	Print monomial coefficients
s	ppapp s <f> <M> <N> [E]	Generate C source code
t	ppapp t <f> <M> <Nxo> <E>	Generate test cases

Documentation

The user manual is included in the package:

from ppapp import get_user_manual_path
print(get_user_manual_path())  # Path to userManual.pdf

License

GNU General Public License v3 or later (GPLv3+)

Authors

• Joachim Wuttke (Forschungszentrum Jülich GmbH)
• Alexander Kleinsorge (Technische Hochschule Wildau)

Links

• Repository
• Documentation (PDF)