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Fundamentals of Numerical Computation Book Club
Welcome
Book club meetings
Pace
1
Introduction
Preface
Why (teach) Julia?
Julia cons
What to expect from this book
Condition number and unstable algorithms
Floating-point numbers
Machine epsilon
Precision and accuracy
Demo
Double Precision
Floating-point arithmetic
Problems and conditioning
Condition numbers, cont.
Polynomial roots:
Algorithms
Horner’s algo
Writing your own
julia
functions
Stability
Meeting Videos
Cohort 1
2
Linear systems of equations
Row pivoting
Permutations and Stability
Vector norms
Matrix norms
Meeting Videos
Cohort 1
3
Overdetermined linear systems
3.1
Fitting Functions to Data
Least Squares
3.1.1
Defintion {-} 3.1.3:
Change of Variables
Exercise 3.1.7
3.2
The Normal Equations
Pseudoinverse and definiteness
Implementation
Conditioning and Stability
Exercise 3.2.4
3.3
QR factorization
Orthogonal and ONC matrices
Orthogonal Factorization
Least squares and QR
Exercise 3.3.7
3.4
Computing QR factorizations
Householder reflections
Factorization Algorithm
Q-less QR and least squares
Exercise 3.4.1
Meeting Videos
Cohort 1
4
Roots of nonlinear equations
The rootfinding problem
The rootfinding problem, Exercise 1
Fixed-point iteration
Fixed-point iteration, Exercise 2
Newton’s method
Newton’s method, Exercise 4
Interpolation-based methods
Newton for nonlinear systems
Quasi-Newton methods
Nonlinear least squares
Meeting Videos
Cohort 1
5
Piecewise interpolation
SLIDE 1
Meeting Videos
Cohort 1
6
Initial-value problems for ODEs
SLIDE 1
Meeting Videos
Cohort 1
7
Matrix analysis
What’s new in Julia
7.1
From matrix to insight
7.2
Eigenvalue Decomposition
7.3
Singular Value Decomposition
7.4
Symmetry and definiteness
7.5
Dimension Reduction
Meeting Videos
Cohort 1
8
Krylov methods in linear algebra
SLIDE 1
Meeting Videos
Cohort 1
9
Global function approximation
SLIDE 1
Meeting Videos
Cohort 1
10
Boundary-value problems
New Julia
Two-point BVP
Demo 10.1.5
Shooting Demo 10.2.1
Instability of Shooting Method
Differentiaion Matrices
Demo 10.3.2
Spectral differentiation
Demo 10.3.4
Exercise 10.1.5 and 10.2.3
Exercise 10.2.3
Exercise 10.3.3
Exercise 10.1.5 with BVP
Collocation for linear problems
Demo 10.4.2
Accuracy and Stability for Collocation
Exercise 10.4.5
Nonlinearity and boundary conditions
Exercise 10.5.7
The Galerkin method
Galerkin conditions
Finite Elements
Exercise 10.6.1
Meeting Videos
Cohort 1
11
Diffusion equations
SLIDE 1
Meeting Videos
Cohort 1
12
Advection equations
SLIDE 1
Meeting Videos
Cohort 1
13
Two-dimensional problems
Tensor-product discretizations
Exercises
Two-dimensional diffusion and advection
Exercise 13.2.2
Exercise 13.2.5
Laplace and Poisson equations
Sylvester equation
Exercise
Nonlinear elliptic PDEs
Exercise 13.4.2
Meeting Videos
Cohort 1
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Fundamentals of Numerical Computation Book Club
Matrix norms
matrix norms are defined via vector norms
Frobenius norm is the vector 2-norm extension to matrices
induced norm is geometrically interpreted as the radius of the smallest sphere that encloses the projection of unit vectors