#
MTH 471 Numerical Analysis

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Semester: *Fall 99*

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Text:
*Numerical Analysis, by D. Kincaid and W. Cheney -*Second Edition

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Evaluation: *Homework and other assignments (60%) and
2 Exams (20% each)*

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About the course

This is an introductory course to the algorithms and methods used in scientific
computing, for students of Mathematics, Sciences, Engineering, Computer
Science. You should be familiar with calculus, linear algebra,
and one programming language (e.g. matlab, maple, mathematica, fortran, etc.)
Algorithms will be discussed in pseudocode, so they
are easy to write in any computer language.
The exposition is going to be mathematical,
and it will include the statement and proofs
of theorems.
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Topics

**Mathematical Preliminaries**

Basic Concepts and Taylor's Theorem. Order of Convergence.

**Computer Arithmetic.**

Floating Point Numbers and Roundoff Errors. Absolute and Relative Errors.
Stable and Unstable Computations. Conditioning.

**Solution of Nonlinear Equations.**

Bisection Method. Newton's Method. Secant Method.

**Solving Systems of Linear Equations.**

Matrix Algebra. The LU and Cholesky Factorizations. Pivoting
and Constructing an Algorithm. Norms and the Analysis of Errors. Neumann
Series and Iterative Refinement.

**Selected Topics.**

Eigenvalues and the Power Method. Schur's and Gershgorin's Theorems.

**Approximating Functions**

Polynomial Interpolation. Divided Differences. Hermite
Interpolation. Spline Interpolation.

**Numerical Differentiation and Integration.**

Numerical Differentiation and Richardson Extrapolation. Numerical Integration
Based on Interpolation. Gaussian Quadrature. Romberg Integration. Adaptive
Quadrature.