Chapter 22 — Further Reading

Numerical differentiation and integration are two of the oldest, best-understood corners of numerical analysis, so the literature is deep and settled. These are the resources worth your time, grouped by why you would open them.

The canonical numerical-methods references

• Press, Teukolsky, Vetterling & Flannery, Numerical Recipes: The Art of Scientific Computing (3rd ed., Cambridge University Press, 2007). The chapters on numerical integration and on the evaluation of derivatives are the standard practitioner's account — trapezoid, Simpson, Romberg, Gauss quadrature, and the round-off-vs-truncation trade-off for differentiation are all here, with unusually candid advice. Read it for judgement, not just formulas.
• Burden, Faires & Burden, Numerical Analysis (10th ed., Cengage, 2015). A rigorous undergraduate textbook with clean derivations of every rule in this chapter, full error terms, and the convergence theory behind the halving test. The place to go when you want the proof that Simpson is $O(h^4)$.
• Dahlquist & Björck, Numerical Methods in Scientific Computing, Volume I (SIAM, 2008). A graduate-level treatment; its discussion of the optimal step size for numerical differentiation and of Richardson extrapolation is especially clear.

Gauss quadrature and adaptive integration, deeper

• Trefethen, "Is Gauss Quadrature Better than Clenshaw–Curtis?" (SIAM Review 50(1), 2008). A short, readable paper that will change how you think about "optimal" nodes; it argues the near-equivalence of the two rules in practice. Excellent once §22.3 feels comfortable.
• Piessens, de Doncker-Kapenga, Überhuber & Kahaner, QUADPACK: A Subroutine Package for Automatic Integration (Springer, 1983). The book documenting the Fortran library that still underlies scipy.integrate.quad. Read its introduction to see production adaptive Gauss–Kronrod quadrature — the §22.3 idea, industrial strength.

Fortran-specific: procedure arguments and pure

• Metcalf, Reid & Cohen, Modern Fortran Explained: Incorporating Fortran 2018 (Oxford University Press, 2018). The authoritative reference for passing procedures as arguments, abstract interfaces, and the pure/elemental attributes that let the integrators in §22.2 be optimized — the exact language features this chapter leans on.
• Chapman, Fortran for Scientists and Engineers (4th ed., McGraw-Hill, 2018). A gentler companion with worked examples of procedure arguments and numerical routines.

Free and online

• The GNU Scientific Library (GSL) reference manual — "Numerical Integration" chapter (gnu.org/software/gsl). Documents gsl_integration_qag and friends; a clear map of which adaptive routine fits which integrand, and a good model for a well-designed quadrature API.
• The SciPy documentation, scipy.integrate (docs.scipy.org). Read it knowing that quad calls QUADPACK (Fortran) and simpson/trapezoid implement the rules of §22.2 — the Python side of the Fortran-under-the-notebook story.
• fortran-lang.org and the Fortran stdlib documentation. The community hub; check the stdlib for interpolation and (evolving) numerical utilities, and the "Learn" pages for procedure-argument idioms.

Primary source

• ISO/IEC 1539-1:2018 (the Fortran 2018 standard). The definitive word on procedure arguments, abstract interfaces, and the constraints on pure procedures used throughout this chapter's code. Consult it when a compiler and a tutorial disagree.

Suggested order

1. Skim the Numerical Recipes integration and differentiation sections for the practitioner's overview.
2. Work the derivations and error terms in Burden & Faires until the halving test feels obvious.
3. Read the Metcalf, Reid & Cohen treatment of procedure arguments and pure to make the Fortran idioms second nature.
4. When you need production integration, read the GSL or QUADPACK interface and call the library rather than shipping your own — the same lesson as Chapter 21's LAPACK.
5. Finish with Trefethen (2008) for a sophisticated view of what "best" quadrature even means.