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