Course Outline
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1)
Approximation by polynomials
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- The approximation problem: good approximations and norms of
function
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Class notes
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- Taylor’s theorem in several dimensions and the Weierstrass
Approximation theorem
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Atkinson, 4.1
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- Polynomial interpolation: existence and formula
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Atkinson 3.1
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- Pointwise error representation, maximum norm bounds. Is the
error small?
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Atkinson 3.1, 3.4, 3.5
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- Piecewise polynomial interpolation: basis functions in one
and two dimensions, error representations, error bound
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Atkinson 3.7, Class notes
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- Least square approximations: orthonormal polynomials and orthogonal
projections, error result
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Atkinson 4.3-4.5, Class notes
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- Approximation of functions in Sobolev space
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Class notes
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2)
Finding roots of functions
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Atkinson 2.1, Estep and Johnson 10, 12
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- Fixed points and the fixed point iteration
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Atkinson 2.5, Estep and Johnson 8, 13
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- Local and Global convergenc
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Estep and Johnson 13, 26
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Estep and Johnson 13, 26
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- A higher order convergent method: Newton’s metho
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Atkinson 2.2, 2.10, 2.11, Estep and Johnson 26
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- Practical considerations: variations of Newton’s metho
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Notes
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3)
Numerical integration
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- Simple rules: trapezoidal and Simpson’s rule
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Atkinson 5.1
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- Interpolatory quadrature: Newton-Cotes formula
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Atkinson 5.2
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- Precision, error representations, error bound
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Atkinson 5.2, 5.4
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Atkinson 5.3
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- Numerical integration in two dimension
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Notes
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4)
Solving initial value problems for ordinary differential
equations
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Atkinson 6.1
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- Existence and uniqueness of a solution of an initial value
problem by an a posteriori convergence result
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Notes
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- Methods for numerical solution of differential equations:
finite difference and finite element method
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Atkinson 6.2, 6.10, 6.3, Notes
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- A priori convergence result
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Notes
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- Error estimation and a posteriori error result
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Notes
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Notes
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Atkinson 6.9, Notes
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