Week
Topic
1
Floating point arithmetic. Approximating sequences. Order of convergence. Summation order and floating point non-associativity. Finite difference approximation of derivative and catastrophic loss of precision. Condition number.
2
Linear combinations. Vector spaces and subspaces. Bases. Dimension. Orthogonal matrices. Matrix subspaces. Fundamental theorem of linear algebra. Rank-nullity.
3
Vector and matrix norms. Singular value decomposition theorem & proof. Link to Karhunen-Loève. Rank-1 expansions. Operator approximation.
4
Linear statement of principal applied mathematics problems: coordinate changes (linear systems), reduced-order models (least squares), operator invariants (eigenproblems). Solution by the SVD. Pseudo-inverse. Midterm 1.
5
Additional operator representations: 
,
, 
, 
.
Computational complexity.
6
Projection: Householder, Givens. Hessenberg form.
least-squares solution. Stability
7
Interpolation in the monomial basis. Newton form (matrix triangularization). Lagrange form, including barycentric. Continuum limits. Taylor series. Residuals. Finite difference calculus.
8
Interpolation in 
-spline
basis. Bezier curves and surfaces. Computational geometry.
Simplicia. Midterm 2.
9
Approximation in monomial basis. Continuum norms. Stationarity
conditions & extrema. 
approximants
(momentum, energy, constraints of physical systems).
10
Interpolation in orthogonal bases: trigonometric, Legendre, Laguerre, Hermite, Bessel, spherical harmonics. Fast decompositions.
11
Linear operator approximation 1: quadrature (
). Newton-Cotes. Moments. Gauss. Convergence.
Stability.
12
Linear operator approximation 2: differentiation (
), linear ODE (
). Convergence. Stability.
13
Non-linear operator approximation 1: 
.
0-degree approximant (bisection), 1-degree approximants (secant,
Newton). Convergence, fixed points.
14
Non-linear operator approximation 2: Non-linear operator composition
, Quasi-linear
approximants 
. Artificial
neural networks.
15
Non-linear operator approximation 3: 
.
0,1,2-degree approximants. Convexity, steepest descent, sampled
descent directions (stochastic gradient). Recurrent, generative,
adversarial neural networks.
,
).