Location & Time

TR 8.00 AM-9.15 AM
169 Willard Building

Instructor

Kamesh Madduri
Office hours:
TR 9.30 AM-11.00 AM (343E IST)
Appointment (cal)

Important Dates

Jan 08. First class
Jan 17. HW1 out
Jan 29. HW1 due
Feb 05. HW2 out
Feb 12. HW2 due
Feb 14. Midterm exam 1
Mar 12. HW3 out
Mar 21. HW3 due
Mar 26. HW4 out
Apr 02. HW4 due
Apr 02. Make-up midterm 1
Apr 04. Midterm exam 2
Apr 16. HW5 out
Apr 25. HW5 due
May 02. Final exam

Overview

This class introduces students to key topics in numerical analysis, with emphasis on the computational mathematics behind numerical methods and on proving theorems. CMPSC/MATH 451 is a related course that has a significant overlap with topics studied in CMPSC/MATH 455 and CMPSC/MATH 456. Students may obtain credit for either this class or CMPSC/MATH 451, but not both.

Topics

Floating Point Arithmetic, Approximations in Scientific Computing, Non-linear equations in one variable, Linear Systems: Direct Methods, Polynomial Interpolation, Piecewise Polynomial Interpolation, Numerical Differentiation, Numerical Integration.

Syllabus

Last updated April 18. pdf icon

Schedule

Lecture 1, Jan 8
Course Logistics
Introduction to Scientific Computing
Lecture 2, Jan 10
Error types
Taylor series
Big-O notation for error analysis
Lecture 3, Jan 15
Conditioning of a problem
Assessing Algorithms
Floating-point systems
Lecture 4, Jan 17
Floating-point systems, properties
Roundoff error
Octave introduction
Lecture 5, Jan 22
Rounding unit
Sources of roundoff error
Decimal to binary conversion
Lecture 6, Jan 24
IEEE floating-point system
Avoiding cancellation and roundoff error
Introduction to nonlinear methods
Lecture 7, Jan 29
Interval bisection
Fixed point theorem
Fixed point iteration
Lecture 8, Jan 31
Speed of convergence
Newton iteration
Secant iteration
Lecture 9, Feb 5
Nonlinear equations Octave tutorial
Multiple roots
Function minimization
Lecture 10, Feb 7
Matrices review
Vector and matrix norms
Lecture 11, Feb 12
Midterm 1 review
Lecture 12, Feb 19
Special classes of matrices
Operation counts
Backward substitution
Lecture 13, Feb 21
Forward substitution
Gaussian Elimination
Lecture 14, Feb 26
Gaussian Elimination algorithm, operation count
Partial pivoting
Lecture 15, Feb 28
Complete pivoting
Gauss-Jordan elimination
Uniqueness of LU decomposition
Cholesky factorization
Lecture 16, Mar 12
Midterm 1 solutions, discussion
Lecture 17, Mar 14
Cholesky factorization example
Banded matrices
Error, residual, conditioning
Linear systems: Octave tutorial
Lecture 18, Mar 19
Linear systems review
Polynomial Interpolation: Introduction
Monomial basis
Lecture 19, Mar 21
Lagrange basis
Newton basis
Lecture 20, Mar 26
Divided differences
Polynomial interpolation error
Chebyshev interpolation
Lecture 21, Mar 28
Octave interlation tutorial
Piecewise polynomial interpolation introduction
Lecture 22, Apr 9
Midterm 2 solutions discussion
Hermite cubic interpolation
Cubic splines
Lecture 23, Apr 11
Finite difference formulas
Richardson Extrapolation
Lecture 24, Apr 16
Numerical Quadrature introduction
Newton-Cotes quadrature rules
Lecture 25, Apr 18
Precision of a quadrature rule
Deriving error with a quadrature formula
Clenshaw-Curtis quadrature
Composite quadrature
Lecture 26, Apr 23
Octave tutorial on numerical integration
Gaussian quadrature
Adaptive quadrature
Lecture 27, Apr 25
Final exam review

Textbook

Uri M. Ascher and Chen Greif, A First Course in Numerical Methods, SIAM, 2011.

Prerequisites

MATH 220; MATH 230 or MATH 231; Three credits of programming (CMPSC 201 or CMPSC 202 or CMPSC 121).

Class material

Presentations, lecture notes, and homework assignments will be posted on Angel.



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