# CS514: Numerical Analysis

Fall '00.
Tu, Th, 10:30 - 11:50 AM

## Course Announcements:

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## Course Contents:

• MACHINE ARITHMETIC, ERROR PROPAGATION AND THE CONDITIONING OF PROBLEMS

Real numbers, machine numbers, rounding.
Machine arithmetic.
Propagation of rounding errors, cancellation errors.
Conditioning of problems, examples.

• APPROXIMATION AND INTERPOLATION

Least squares approximation and data fitting.
Orthogonal polynomials.
Polynomial interpolation, Lagrange's formula.
Interpolation error and convergence.
Interpolation at Chebyshev points, Chebyshev polynomials.
Newton's form of the interpolation polynomial.
Hermite interpolation.
Inverse interpolation.
Interpolation by means of spline functions; minimal properties of spline interpolants.

• NUMERICAL DIFFERENTIATION AND INTEGRATION

Finite difference approximation of derivatives.
Numerical integration by composite trapezoidal and Simpson rules.
Newton-Cotes formulae.
Approximation of linear functionals, methods of interpolation and undetermined coefficients.
Extrapolation methods, Romberg integration.

• NONLINEAR EQUATIONS

Examples.
Iterative methods, order of convergence.
Bisection method.
Secant method and its convergence properties.
Newton's method, local and global convergence.
Algebraic equations.
Systems of nonlinear equations

• ORDINARY DIFFERENTIAL EQUATIONS

One-step methods, local and global error.
Runge-Kutta methods.
Stiff equations.
Multistep methods.

Notes on solving ODES from the CSEP book

• PARTIAL DIFFERENTIAL EQUATIONS

Notes on solving PDES from the CSEP book