Welcome to OTIlib and pyoti documentation!

The OTIlib and pyoti library is a C and Python library to support the Order Truncated Imaginary (OTI) algebra. OTI numbers were developed in the Ph.D. thesis “Order Truncated Imaginary Algebra for Computation of Multivariable High-Order Derivatives in Finite Element Analysis” (link). OTI numbers are an extension of Dual numbers. Dual numbers are a hypercomplex algebra that allows computing a single first order derivative per evaluation. OITs, is also a hypercomplex algebra, but in contrast to dual numbers, it can be used to compute efficient high-order derivatives with respect to multiple variables. In contrast to the Dual numbers ( \(a + b \epsilon, \epsilon^2 = 0\) ), OTI numbers set a truncation condition other than \(\epsilon^2 = 0\), and considers multiple imaginary basis. For instance \(\epsilon_1^5, \epsilon_1^2\epsilon_2, \ldots, \epsilon_m\) can be non-truncated imaginary directions.

More information on HYPAD

• Theory, references, lecture notes and other information on Hypercomplex-based Automatic Differentiation HYPAD.

Warning

This website is under construction. Documentation placed here may be incomplete.

Development is constantly being done to improve documentation.

Quick example:

import pyoti.sparse as oti
import numpy as np

# Define variables and apply perturbations
x = np.pi/4 + oti.e(1,order=2)

# Evaluate function.
f = oti.sin(x)

print(f)
# >>>  0.707107 + 0.707107 * e([1]) - 0.353553 * e([[1,2]])

# Extract the function and its derivatives:

print(f'f(x):      {f.real}')
print(f'df/dx:     {f.get_deriv([1  ])}')
print(f'd^2f/dx^2: {f.get_deriv([1,1])}')

# >>> f(x):      0.7071067811865475
# >>> df/dx:     0.7071067811865476
# >>> d^2f/dx^2: -0.7071067811865475

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Contents

Contents:

• Installation
  • Prerequisites
  • Conda environment installation instructions:
  • Adding the local environment to the global jupyter notebook server
  • MacOS additional instructions:
  • Compiling the library
• Tutorials
  • core Tutorial 01: Core module utilities
  • sparse Tutorial 01: Sparse module basic usage.
  • sparse Tutorial 02: Computing arbitrary-order derivatives.
  • sparse Tutorial 03: Linear algebra support.
  • static Tutorial 01: Basic usage.
  • static Tutorial 02: Computing derivatives.
• Examples
  • Basic examples
  • Jupyter Notebooks
• About
• Contributors
• Acknowledgements

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Module documentation

Module documentation:

• pyoti package

Indices and tables

• Index

• Module Index

• Search Page