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Notes and course material for MATH50003 Numerical Analysis
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MATH50003NumericalAnalysis
Notes and course material for MATH50003 Numerical Analysis
Lecturer: Dr Sheehan Olver
Problem Classes: 2–4pm Thursdays, Huxley 340–342
Overview lecture: 10–11am Fridays, Clore
Q&A: 9:50–10:30 Mondays, 10:40–11:20 Tuesdays on Teams
Course notes
Background material
1. Introduction to Julia: we introduce the basic features of the Julia language.
2. Asymptotics and Computational Cost: we review Big-O, little-o and asymptotic to
notation, and their usage in describing computational cost.
Part I: Computing with numbers
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README.md
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1. Numbers: we discuss how computers represent integers and real numbers, as well as
their arithmetic operations.
2. Differentiation: we discuss ways of approximating derivatives, including automatic
differentiation, which is essential to machine learning.
Part II: Computing with matrices
1. Structured Matrices: we discuss types of structured matrices (permutations, orthogonal
matrices, triangular, banded).
2. Decompositions: we discuss algorithms for computing matrix decompositions (QR and
PLU decompositions) and their use in solving linear systems.
3. Singular values and condition numbers: we discuss vector and matrix norms, and
condition numbers for matrices, and the singular value decomposition.
4. Differential Equations: we discuss the numerical solution of linear differential equations,
including both time-dependent ordinary differential equations and boundary value
problems, by reduction to linear systems.
Part III: Computing with functions
1. Fourier series: we discuss Fourier series and their usage in numerical computations via
the fast Fourier transform.
2. Orthogonal Polynomials: we discuss orthogonal polynomials—polynomials orthogonal
with respect to a prescribed weight.
3. Interpolation and Gaussian quadrature: we discuss polynomial interpolation, Gaussian
quadrature, and expansions in orthogonal polynomials.
Assessment
1. Practice computer-based exam (Solutions)
2. Computer-based exam (released on Blackboard): 18 March 2022, 3–5pm (1 hour exam,
1 hour upload/download)
3. Practice final exam (pen-and-paper, not for credit): Summer Term (TBC)
4. Final exam (pen-and-paper): Summer Term (TBC)
Problem sheets
1. Week 1 (Solutions): Binary representation, integers, floating point numbers, and interval
arithmetic
2. Week 2 (Solutions): Finite-differences, dual numbers, and Newton iteration
3. Week 3 (Solutions): dense, triangular, banded, permutation, rotation and reflection
matrices
4. Week 4 (Solutions): least squares, QR and PLU decompositions