MATH3076/3976 Mathematical Computing

Lecturer: Sheehan Olver
Time: Tuesday–Thursday, 3pm
Location: Carlsaw 375
Consultation: Wednesday, 4pm, Carslaw 624



Assignment 1: 3076, due 14 April (Solutions)
3976, due 14 April (Solutions)
Assignment 2: 3076/3976, due 16 May (Solutions)
Assignment 3: 3076/3976, due 6 June (Solutions)
3976 Project: Draft due 6 June
Final due 1 July

Lecture notes

Lecture 1: Integers and Bits
Lecture 2: Strings and Vectors
Lecture 3: Types and Functions
Lecture 4: IEEE Floating Point Arithmetic
Lecture 5: Rounding and Arithmetic in IEEE Floating Point

Lecture 6: Linear Algebra
Lecture 7: Interpolation
Lecture 8: Abstract Types
Lecture 9: Norms
Lecture 10: Permutation, Orthogonal and Triangular Matrices
Lecture 11: Matrix Factorizations
Lecture 12: PLU Decomposition
Lecture 13: Givens' Rotations
Lecture 14: QR Factorization and Least Squares

Lecture 15: Computational Cost and Complexity
Lecture 16: Matrix Norms
Lecture 17: Problems and Absolute Condition Numbers
Lecture 18: Relative and Matrix Condition Numbers
Lecture 19: Error Analysis
Lecture 20: Backward Error Analysis For Dot Products

Lecture 21: Images
Lecture 22: The Singular Value Decomposition (SVD)
Lecture 23: The SVD and Condition Numbers, Matrix Exponentials
Lecture 24: Schur decomposition, The QR Algorithm and Companion Matrices

Lecture 25: Continuous Problems
Lecture 26: Numerical Integration/Quadrature
Lecture 27: Error of Quadrature Rules
Lecture 28: Bernoulli Polynomials
Lecture 29: The Euler–Maclaurin Formula
Lecture 30: The Trapezium Rule and Fourier Modes
Lecture 31: The Discrete Fourier Expansion
Lecture 32: The Discrete Fourier Transform

Lecture 33: Linear ODEs
Lecture 34: Discretizing Derivatives
Lecture 35: Forward Euler, Backward Euler and Midpoint Methods
Lecture 36: Boundary Value Problems
Lecture 37: Time-evolution PDEs with Periodic Boundary Conditions
Lecture 38: Nonlinear Problems
Lecture 39: CFL Condition, Nonlinear Boundary Value Problems


Lab 1: REPL, Jupyter and Plotting (Solutions)
Lab 2: Creating a Rational Type (Solutions)
Lab 3: Structures (Solutions)
Lab 4: Linear Systems for Structures (Solutions)
Lab 5: Stable Structures (Solutions)
Lab 6: QR Decomposition using Householder Reflections (Solutions)
Lab 7: Conditioning and Error Analysis (Solutions)
Lab 8: Function Compression and the SVD (Solutions)
Lab 9: Simpson's Rule (Solutions)
Lab 10: Romberg Quadrature (Solutions)
Lab 11: The Discrete Cosine Transform (Solutions)
Lab 12: Time-evolution PDEs with Boundary Conditions (Solutions)

Reading list

Lecture 1: Overton, Chapters 1–3 Introduction, The Real Numbers, Computer Representation of Numbers
Julia Documentation Variables, Mathematical Operations and Elementary Functions
Lecture 2: Julia Documentation Integers and Floating Point Numbers, Strings
Lecture 3: Overton, Chapter 4 IEEE Floating Point Representation
Julia Documentation Functions, Types
Lecture 4: Overton, Chapter 5–6 Rounding, Correctly Rounded Floating Point Operations
Lecture 5: Overton, Chapter 11 Cancellation
Lecture 6: Julia Documentation Multi-dimensional Arrays, Linear Algebra
Quantitative Economics Vectors, Arrays and Matrices, Linear Algebra
Lecture 7:Trefethen & Bau, Lecture 1–2 Matrix-Vector Multiplication, Orthogonal Vectors and Matrices
Wikipedia Interpolation, Least squares
Lecture 9:Trefethen & Bau, Lecture 3 Norms (Section Vector Norms)
Lecture 10:Corless & Fillon, Sections 4.2–4.3 Solving Unitary or Orthogonal Systems, Solving Triangular Systems
Lecture 11:Wikipedia LU Decomposition, QR Decomposition
Corless & Fillon, Sections 4.4–4.5 Factoring as a Step Toward Solution, The QR Factoring
Lecture 12:Corless & Fillon, Sections 4.7 Solving Ax=b with the LU Factoring
Lecture 13:Corless & Fillon, Sections 4.5.3 Householder Reflections
Lecture 14:Corless & Fillon, Sections 4.5.5 Solving Overspecified Systems with the QR Factoring
Lecture 15:Corless & Fillon, Section 1.5 Complexity and Cost of Algorithms
Lecture 16:Trefethen & Bau, Lecture 3 Norms (Section Matrix Norms Induced by Vector Norms)
Lecture 17: Overton, Chapter 12 Conditioning of Problems
Corless & Fillon, Sections 1.1 Mathematical Problems and Computability of Solutions
Lecture 18:Corless & Fillon, Sections 1.4 Perspectives on Error Analysis
Lecture 19: Overton, Chapter 13 Stability of Algorithms
Lecture 20:Corless & Fillon, Sections 1.3 Error Accumulation and Catastrophic Cancellation
Lecture 22:Trefethen & Bau, Lecture 4 The Singular Value Decomposition
Lecture 23:Trefethen & Bau, Lecture 5 More on the SVD
Wikipedia Matrix Exponential
Lecture 24:Wikipedia Schur Decomposition , Companion Matrix , QR Algorithm
Lecture 25: 18.330 Notes Numerical Integration, Part 1
Lecture 30: 18.330 Notes The FFT and its Applications
Lecture 33: 18.330 Notes Integration of Ordinary Differential Equations
Lecture 36: 18.330 Notes Boundary Value Problems
Lecture 38: 18.330 Notes Nonlinear Root Finding and a Glimpse at Optimization

Further reading