Overview

• Overview: Reals and computations.
  • Computing over the Reals [2006, Braverman+].
  • Computing over the Reals [2004, Blum].

Numbers (Reals)

• Computable Reals: Definitions, examples.
  • Computability and Analysis [2014, Avigad+] (§1-2).
  • A Tutorial on Computable Analysis [2008, Brattka+] (§1-3).
• Computable Operations: Comparison, functions.
  • Recursive Real Numbers [1954, Rice].
  • Computing over the Reals [2006, Braverman+].
  • A Tutorial on Computable Analysis [2008, Brattka+] (§4).

Numbers (Floats)

• Floating-Point Arithmetic (Basics): Floating-point arithmetic, floating-point errors.
  • Handbook of Floating-Point Arithmetic [2018, Muller+] (§2, 3.1, 5.1-5.3).
• Floating-Point Arithmetic (Advanced): Automatic error analysis, error-free computations.
  • Handbook of Floating-Point Arithmetic [2018, Muller+] (§4.2-4.4, 12.4).

Functions

• Fixed-Precision Evaluation (Basics): Polynomial approximation, range reduction.
  • Handbook of Floating-Point Arithmetic [2018, Muller+] (§10.1-10.3).
  • Elementary Functions (3ed) [2016, Muller] (§3, 4, 6.1-6.2, 11.1, 14).
• Fixed-Precision Evaluation (Advanced): Accurate computation, correct rounding.
  • Handbook of Floating-Point Arithmetic [2018, Muller+] (§10.2, 10.4-10.6).
  • Elementary Functions (3ed) [2016, Muller] (§5, 11.2, 12).
• ~~Arbitrary-Precision Evaluation: Integer multiplication, power series.~~
  • ~~Elementary Functions (3ed) [2016, Muller] (§7.1-7.2, 7.4).~~
  • ~~Modern Computer Arithmetic (link) [2012, Brent+] (§1.3, 3.3, 4.1, 4.3-4.4, 4.9).~~

Derivatives

• Numerical & Symbolic Differentiation: Derivatives, accuracy/efficiency.
  • The Elements of Differentiable Programming [2024, Blondel+] (§2, 7).
  • A Hitchhikers Guide to Automatic Differentiation [2016, Hoffmann] (§5).
• Automatic Differentiation (Basics): Forward/reverse mode, dual numbers.
  • The Elements of Differentiable Programming [2024, Blondel+] (§8).
  • Backpropagation in the Simply Typed Lambda Calculus [2020, Brunel+] (§1-3).
• Automatic Differentiation (Advanced): Higher-order derivatives, non-differentiability.
  • The Elements of Differentiable Programming [2024, Blondel+] (§9).
  • On Correctness of Automatic Differentiation for Non-Differentiable Functions [2020].

Random Numbers

• Discrete Distributions: Inversion/alias method, entropy optimality.
  • Darts, Dice, and Coins: Sampling from a Discrete Distribution [2011, Schwarz].
  • Non-Uniform Random Variate Generation [1986, DeVroye] (§15).
• Continuous Distributions: Transformation/inversion method, rejection/mixture method.
  • Monte Carlo Theory, Methods and Examples [2013, Owen] (§??).
  • Lecture Notes of “Monte Carlo Methods and Applications” [2024, Iyer] (§??).

Integrals