First, see review_topics_for_midterm.txt
Material AFTER exam 1
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Chapter 4, sections 4.1-4.3,4.5
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Taylor series, simplification tricks, error bounds
Asymptotic approximations, Newton hulls
Interpolation
Constant-time nearest-neighbor table lookup
Linear interpolation
Lagrange polynomial interpolation
barycentric implementation
creation efficiency, evaluation efficiency
Runge phenomena
Chebyshev interpolation
Minimize oo-norm of error
node calculation formula
Pade approximations
Chapter 7, Quadrature, sections 7.1-7.3, 7.5
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Riemann Rule
Newton-Cotes rules
trapezoidal, simpson, 3/8's simpson, boole
local and composite forms
Truncation error, Local Precision, composite precision
use of taylor series in calculation
Other rules
Romberg integration
Richardson Extrapolation
precision
Midpoint rule
Gauss-Legengre vs Clenshaw--Curtis
problems created by singularities (jumps, infinite derivatives)
Practical: quad(), quad8()
Chapter 5, sections 5.1-5.2
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Least squares for overdetermined linear systems
measures of "best fit"
Least squares solutions of over-determined systems
Normal equations, interpretation, residual
rank, column and row spaces
conditions for existence and uniqueness of solutions
Coefficient of Determination
b2 in ColNull(A) implies A^T b2 = 0
b1 in ColSpace(A) implies b1 = A z for some z
Performance
Applications - Kepler, Taylor
how to fit a line, plane, power function, exponential
How to solve with matlab, also polyfit
Cholesky Factorization, efficency
QR version, efficency
Chapter 11, Eigenvalues and eigenvectors, 11.1-11.2
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hand methods for 2x2 systems
Power method for symmetric matrices
how to approximate eigenvalue
Convergence rate
Improvements
inverse theorem
shifting theorem
cubic convergence of shifted inverse method
Francis Algorithm
Chapter 9, Differential equations, 9.1-9.2, 9.7
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Examples
Eigenvalues and Matrix exponential expm for x' = A x
General concept of a vector field
nth-order ordinary equations can be written as
system of n first-order equations
Approximations to a Derivative
Taylor series method for f' and f''
Truncation error (same as for quadrature)
Forward Euler Method
Backward Euler Method
ode45, Runga--Kutta
Chapter 8, Optimization, 8.1, first part of 8.3
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Discrete problem
Brute force
argmin_{x \in U} f(x)
setup, existence, uniqueness
Calculus methods
Gradient descent
Convex, strictly convex
definitions, existence, uniqueness
Quasi-concave
Golden Ratio method
performance
line search, Gradient search, steepest decent, greedy methods
Mona Lisa examples
Costs and benefits
When is simplex algorithm useful?
Monte carlo
Simulated annealing
algorithm, good and bad
Montecarlo integration
area of a circle
volume of a 4-sphere
performance
Black-Scholes-Merton
Fast matrix multiplication and the FFT