SYSTEMATIC MATHEMATICS

Numerical Analysis

A computer returns a number; why should anyone trust it? Across 37 developed chapters, begin with arithmetic that can be checked by hand and separate four possible problems: inaccurate input, a sensitive mathematical problem, rounding by the machine, and error introduced by the chosen method. Each approximation is accompanied by a residual, an error estimate or bound, and a second calculation using more precision, a smaller step, or a different method. Only after that foundation does the course solve equations, full-rank and rank-deficient systems, general sparse systems, eigenvalue problems, fitting, interpolation, fixed and adaptive integration, initial- and boundary-value dynamics, and simple PDE discretizations. It always distinguishes finding an approximate answer, verifying the numerical calculation, regularizing unresolved information, and validating the original model.

Before this course: Completed Single-Variable Calculus I & II, Multivariable Calculus, Linear Algebra, Differential Equations & Dynamical Systems, and Partial Differential Equations. The final weak-form chapter also uses ideas from Vector Calculus and benefits from Measure & Lebesgue Integration. Learners should be able to use Taylor’s theorem, norms, eigenvalues, orthogonality, initial and boundary data, and basic proof arguments.

COURSE FACTSLevel, chapters, units, prerequisite, and outcome
Chapter 1

Absolute, relative, truncation, and data error

Objective: Can relative error be used unchanged when the exact answer is zero?

Every computation begins with an exact target and an approximation; different error measures answer different questions and must not be interchanged without scale information.

Read the target and error measure before running the algorithm: For exact x and approximation x̂, absolute error is |x−x̂| and relative error is |x−x̂|/|x| when x≠0; truncation error comes from replacing an infinite process, while data error is already present in the input. The guarantee to establish is: If |x−x̂|≤ε and |x|≥m>0, then relative error is at most ε/m; without a lower bound on |x| no such conversion is possible. Keep the data scale, norm, precision, stopping rule, and discretization attached to every reported digit.

The worked question is: Approximate √2 by 1.414 and compute absolute and relative error. Start from “Use √2≈1.414213562 and subtract 1.414.” and finish at “Divide by √2 to obtain relative error about 1.51×10⁻⁴.” Then use the stated residual, bound, invariant, or refinement comparison to decide what the digits actually certify.

If |x−x̂|≤ε and |x|≥m>0, then relative error is at most ε/m; without a lower bound on |x| no such conversion is possible.

Start from |x−x̂|≤ε and divide by the known positive quantity |x|.

Use |x|≥m to obtain 1/|x|≤1/m.

Combine the inequalities to get |x−x̂|/|x|≤ε/m.

Approximate √2 by 1.414 and compute absolute and relative error.

  1. Use √2≈1.414213562 and subtract 1.414.
  2. The absolute error is approximately 0.000213562.
  3. Divide by √2 to obtain relative error about 1.51×10⁻⁴.

Result: The absolute error is about 2.14×10⁻⁴ and the relative error about 0.0151%; the reference value itself is rounded, so the displayed last digits are approximate.