This is an undergraduate course introducing most of the basic and
classical numerical algorithms. The course is more focused on the
study and implementation of these basic algorithms. The students will
have to complete several computing projects in addition to other
homeworks.
- Computer arithmetic
- Representation of numbers in different bases
- Floating point representation
- Loss of significiance
- Numerical solution of nonlinear equations.
- Bisection method
- Newton's method
- Secant method
- Polynomial interpolation
- Lagrange interpolation
- Errors in Polynomial Interpolation
- Estimating derivatives and Richardson Extrapolation
- Numerical integration
- Trapezoid rule
- Romberg Algorithm
- Adaptive Simpson's quadrature scheme
- Gaussian quadrature formulae
- Direct methods for linear systems.
- Gaussian elimination.
- Gaussian Elimination with scaled partial pivoting
- Tri-diagonal and banded systems
- LU-Factorization
- Piece-wise polynomial interpolation. Splines
- First degree and second degree splines
- Natural cubic splines
- Numerical solution of ordinary differential equations (ODE)
- Taylor series methods for ODE
- Runge Kutta methods
- Methods for first order systems of ODE
Textbook
- Numerical Mathematics and Computing, by Ward Cheney and David
Kincaid, published by Brooks/Cole publishing Company, 2000. ISBN
0-534-35184-0
The course will provide an introduction to the basics of the modern
numerical analysis and its techniques when applied to various problems
of analysis and algebra. Various numerical techniques and algorithms
for some classical problems will be considered. The focus will be on
their efficient computer implementation, robustness and
reliability. Some essential theoretical properties of these numerical
techniques will also be studied in more detail.
MATH/CSE 455--Introduction to Numerical Analysis I
- Computer arithmetic
- Floating point numbers and roundoff errors.
- Absolute and relative errors: Loss of significance.
- Conditioning: Stable and unstable computations.
- Numerical methods for nonlinear equations:
- Bisection, Newton's and Secant methods.
- Fixed point iterations and convergence rate.
- Direct methods for systems of linear equations
- Matrices and vectors. Norms, condition numbers and convergence
matrices.
- The Cholesky Factorization.
- Gaussian elimination with scaled partial pivoting.
- Direct methods for banded and sparse matrices.
- Approximation of functions
- Polynomial interpolation.
- Errors in polynomial interpolation.
- Splines.
- Numerical differentiation and integration
- Numerical differentiation
- Trapezoid rule. Romberg Algorithm.
- Simpson's rule.
- Gaussian quadrature formulae.
Textbook
- Numerical Analysis: Mathematics of Scientific
Computing, Second Edition, by David Kincaid and Ward Cheney,
Brooks/Cole Publishing Co. 1996, ISBN 0-534-33892-5.
This course is a follow-up to Introduction to Numerical Analysis I. It
will provide further introduction to the basics of the modern
numerical techniques and the supporting mathematical theory.
- Polynomial approximation
- Polynomial approximation. Weierstrass' approximation theorem.
- Best approximation and least squares method
- Trigonometric interpolation and Fast Fourier Transform.
- Iterative methods for systems of linear equations.
- Basic iterative methods for linear systems.
- Convergence of the basic iterative methods.
- Numerical methods for eigenvalue problems
- Power method.
- Schur theorem.
- The symmetric and non-symmetric QR algorithm.
- Numerical methods for ordinary differential equations
- Taylor series methods for ODEs.
- Runge--Kutta methods for ODEs.
- Local and global errors: stability.
- Boundary value problems. Finite differences.
- Variational principle and introduction to finite element method.
Textbook
- Numerical Analysis: Mathematics of Scientific
Computing, Second Edition, by David Kincaid and Ward Cheney,
Brooks/Cole Publishing Co. 1996, ISBN 0-534-33892-5.