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When can forward stable algorithms be composed stably?

Carlos Beltrán, Vanni Noferini, Nick Vannieuwenhoven

TL;DR

This work addresses when forward-stable numerical algorithms can be stably composed to solve a composed problem $f=g\circ h$ under floating-point arithmetic. It introduces amenability and compatibility as mild, problem-structural conditions that guarantee stable composition, and proves a composition theorem showing forward stability is preserved when $g$ and $h$ are compatible amenable and their algorithms are forward stable. The paper also establishes a chain of implications from backward to forward stability under amenability, and provides a broad catalog of amenable problems with forward-stable algorithms, including elementary operations, linear maps, and tensor arithmetic, while presenting a non-amenable sine example to delineate limits. The results offer a practical framework for designing stable composite algorithms and understanding numerical instability sources in complex computations.

Abstract

We state some widely satisfied hypotheses, depending only on two functions $g$ and $h$, under which the composition of a stable algorithm for $g$ and a stable algorithm for $h$ is a stable algorithm for the composition $g \circ h$.

When can forward stable algorithms be composed stably?

TL;DR

This work addresses when forward-stable numerical algorithms can be stably composed to solve a composed problem under floating-point arithmetic. It introduces amenability and compatibility as mild, problem-structural conditions that guarantee stable composition, and proves a composition theorem showing forward stability is preserved when and are compatible amenable and their algorithms are forward stable. The paper also establishes a chain of implications from backward to forward stability under amenability, and provides a broad catalog of amenable problems with forward-stable algorithms, including elementary operations, linear maps, and tensor arithmetic, while presenting a non-amenable sine example to delineate limits. The results offer a practical framework for designing stable composite algorithms and understanding numerical instability sources in complex computations.

Abstract

We state some widely satisfied hypotheses, depending only on two functions and , under which the composition of a stable algorithm for and a stable algorithm for is a stable algorithm for the composition .

Paper Structure

This paper contains 32 sections, 11 theorems, 72 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Why forward errors?
  3. Outline and contributions
  4. Bornemann's approach
  5. The condition number in coordinatewise relative error
  6. Numerical stability of algorithms
  7. Numerical algorithm
  8. Scalable functions
  9. Backward, mixed, and forward stability
  10. Stability of numerical algorithms for amenable problems
  11. The amenability condition
  12. The compatibility condition
  13. The composition theorem
  14. A chain of implications
  15. The stacking theorem
  16. ...and 17 more sections

Key Result

Corollary 1.1

Let $f = g \circ h$ be a composition of functions. The composition of forward stable algorithms for $g$ and $h$ results in a forward stable algorithm for $f$ if all of the following conditions hold: The first condition is called compatibility and the last two conditions are collectively called amenability of $g$ and $h$.

Figures (1)

  • Figure 1: The loss of precision when computing the sine of $x = 1 + \pi 2^k$ for $k=1,\ldots,100$ in Matlab. The reference value was computed using variable precision arithmetic with $10000$ digits.

Theorems & Definitions (34)

  • Corollary 1.1: Informal version of \ref{['th:composition']}
  • Definition 3.1: Condition number
  • Lemma 3.2
  • proof
  • Definition 3.3: Coordinatewise relative error metric Pryce1984
  • Lemma 3.4
  • proof
  • Proposition 3.5: Condition number in coordinatewise relative error
  • proof
  • Remark 3.6
  • ...and 24 more