MA 2213 Numerical Analysis I
Chapter 1.1: Review of Calculus
Dr. Andy M. Yip
Department of Mathematics
National University of Singapore
20122013 Semester 2
A M Yip (NUS Math)
MA 2213 Chapter 1.1
1 / 23
Limit of a function
Denition (1.1)
A function f dened
Motivation
Numeric vs. Symbolic
What is Numerical Analysis?
Other Information
MA 2213 Numerical Analysis I
Chapter 0: Introduction
Dr. Andy M. Yip
Department of Mathematics
National University of Singapore
20122013 Semester 2
A M Yip (NUS Math)
MA 2213 Ch
MA 2213 Numerical Analysis I
Chapter 6.6: Some Special Types of Matrices
Dr. Andy M. Yip
Department of Mathematics
National University of Singapore
20122013 Semester 2
A M Yip (NUS Math)
MA 2213 Chapter 6.6
1 / 36
Objectives of This Section
Objectives
1
P
MA 2213 Numerical Analysis I
Chapter 6.5: Matrix Factorization (Supplementary)
Dr. Andy M. Yip
Department of Mathematics
National University of Singapore
20122013 Semester 2
Source: Lecture notes by Prof. Tom Lyche, University of Oslo
A M Yip (NUS Math)
M
Assembling of L (CH05 5A.pdf, p.41p.45)
A(1)
0
0 1
1
1 1
=
1 1 2
1
2
0
1
2
0
2
L(0)
1
0
=
0
0
0
1
0
0
0
0
1
0
0
0
0
1
The rst diagonal entry of A(1) is zero, which cant be used as a pivot. We swap the 1st and
2nd row of A(1) to obtain a nonzero pivot. (M
Section 4.1
Problem: Approximate f (x0 ), f (x0 ), etc.
Ideas of numerical methods:
1
2
3
Replace f (n) (x0 ) with P (n) (x0 ) where P is the (Lagrange) interpolant.
Combine several Taylors polynomials of the form f (x0 + i h) to
express f (n) (x0 ) in te
Recall: Setting up equations for natural splines
Example (n = 3):
Data: (x0 , f (x0 ), (x1 , f (x1 ), (x2 , f (x2 ), (x3 , f (x3 )
(x0 < x1 < x2 < x3 )
(nodes for one-piece interpolation need not be sorted, but nodes for
piece-wise interpolation have to b
Section 3.1
Problem: Given n + 1 distinct (possibly unordered) points and data
x1
xn1
xn
x0
f (x0 ) f (x1 ) f (xn1 ) f (xn )
nd a polynomial P of degree n or less that interpolates f at the
given points.
Existence of P : Lagrange interpolation formula
Uni
Pseudocode
Rate of convergence
MA 2213 Numerical Analysis I
Chapter 1.3: Algorithms and Convergence
Dr. Andy M. Yip
Department of Mathematics
National University of Singapore
20122013 Semester 2
The section Characterizing Algorithms of the textbook is ski
Section 1.1
Existence theorems: Existence of
a tangent line parallel to the secant (Rolles and Mean Value Thms)
a maximum and a minimum in closed and bounded interval (Extreme
Value Thm)
all intermediate values (Intermediate Value Thm)
a value equals to t
MA 2213 Numerical Analysis 1
Year 20122013 Semester II
Laboratory 5
Objectives
1. Task 1: investigate the time eciency of Gaussian elimination
2. Task 2: compare Gaussian elimination and inverse multiplication
Background
1. It has been shown in Tutorial 5
MA 2213 Numerical Analysis 1
Year 20122013 Semester II
Laboratory 4
Objectives
1. Task 1: determine the rate of convergence of Romberg method
2. Task 2: compare the Gaussian quadrature and composite trapezoidal rule
Background
1. The theoretical rate of c
MA 2213 Numerical Analysis I
2012-2013 Semester 2
Introduction to Matlab
1. Introduction
Matlab stands for Matrix Laboratory. It is a high-level language for numerical computation and
visualization licensed by MathWorks Inc. in USA. It is based on the LAP
MA 2213 Numerical Analysis 1
Year 20122013 Semester II
Laboratory 2
Objectives
1. Task 1: to reduce the interpolation error by adjusting the locations of the nodes
2. Task 2: to build a function that computes the coecients of natural cubic splines
Backgro
MA 2213 Numerical Analysis 1
Year 2012–2013 Semester II
Laboratory 1
Objectives
1. Task 1: to estimate the rate of convergence of |f (x) − PN (x)| as x → x0 experimentally
2. Task 2: to compare two diﬀerent ways to compute x2 − y 2
Background
1. Let PN (x
MA 2213 Numerical Analysis 1
Year 20122013 Semester II
Laboratory 3
Objectives
1. Task 1: test the performance of various dierentiation formulas
2. Task 2: test the performance of the composite Simpsons rule
Background
1. Let h > 0. We have the following
Trigonometric Polynomial Interpolation
Discrete Fourier Transform
Fast Fourier Transform
MA 2213 Numerical Analysis I
Chapter 8.6: Fast Fourier Transform
Dr. Andy M. Yip
Department of Mathematics
National University of Singapore
20122013 Semester 2
A M Yi
The following essay appeared in the November, 1992 issue of SIAM News and the March,
1993 issue of the Bulletin of the Institute for Mathematics and Applications.]
THE DEFINITION OF NUMERICAL ANALYSIS
Lloyd N. Trefethen
Dept. of Computer Science
Cornell U
IEEE 754
Decimal Machine Numbers
Finite-digit Arithmetic
Catastrophic Cancelation
Nested Arithmetic
MA 2213 Numerical Analysis I
Chapter 1.2: Round-o Errors and Computer Arithmetic
Dr. Andy M. Yip
Department of Mathematics
National University of Singapore