Seating Plan

ID	Seat Number
09531703	103
20089844	3
20090219	22
20124644	66
20126549	93
20143951	102
20153774	39
20175289	88
20176403	71
20176611	87
20182816	5
20186795	18
20187309	42
20187799	6
20191958	37
20191984	72
20192574	69
20193396	21
20193425	68
20194285	20
20198920	61
20199132	92
20200797	1
20200876	94
20200943	19
20203191	85
20217051	100
20219190	95
20253756	63
20253897	24
20255651	64
20265747	48
20266090	7
20268218	44
20268311	17
20268335	41
20268414	40
20271186	23
20271863	70
20273586	8
20273603	65
20273706	67
20274358	43
20274592	86
20274645	45
20274657	62
20275613	46
20276007	4
20278237	38
20278471	47
20279554	104
20414489	90
20415342	99
20415524	101
20415548	96
20415550	98
20415562	91
20417704	2
20418162	89
20419465	97

Lecture 25 (Nov 30)

Wrapping up.

PPT: [click here]

Topic Covered in the Textbook

Chapter 1
1.2 Round-off errors and computer arithmetic

YES: Concept of machine epsilon; limitation of finite precision calculations; ways
to avoid the problem of loss of significant digits.
NO: Number of bits/bytes used in the IEEE single/double precision. 

Chapter 2
2.1 Bisection method
2.2 Fixed-point iteration
2.3 Newton's method and its extensions
2.4 Error analysis for iterative methods

YES: Mean Value Theorem; Taylor series expansion; Intermediate Value Theorem; how to use Bisection method, fixed point iteration and Newton’s method; application of the fixed-point theorem (Theorem 2.3); order of convergence of a sequence. 
NO: Bisection/Newton’s method for more than 3 iterations. 

Chapter 3
3.1 Interpolation and the Lagrange polynomial
3.3 Divided differences

YES: Interpolation using the Lagrange interpolating polynomials and the Newton’s divided difference; error bound for using the polynomial interpolation; piecewise interpolation. 
NO: Interpolation with more than 3 data points. 

Chapter 4
4.1 Numerical Differentiation
4.3 Elements of numerical integration
4.4 Composite numerical integration
4.7 Gaussian quadrature

YES: Numerical differentiation; derivation of approximation rules using polynomial interpolation and Taylor expansion. Trapezoidal rule, Simpson's rule and Midpoint rule; Derivation of their error term; Degree of accuracy; Composite Trapezoidal/Simpson's/Midpoint rules. 
NO: Invert a linear system of size larger than 3-by-3. 

Chapter 6
6.1 Linear systems of equations
6.2 Pivoting strategies
6.3 Linear algebra and matrix inversion
6.5 Matrix factorization
6.6 Special types of matrices

YES: Gaussian elimination (GE) with backward substitution; with partial pivoting; LU decomposition; PA=LU decomposition.
NO: Number of operations for GE; Invert a linear system of size larger than 3-by-3. 

Chapter 7
7.1 Norms of vectors and matrices
7.3 The Jacobi and Gauss-Siedel iterative techniques
7.4 Relaxation techniques for solving linear systems
7.5 Error bounds and iterative refinement

YES: Vector and matrix norms; Jacobi, Gauss-Seidel and SOR iteration; Their matrix representations; Convergence of the classical iterative methods.  
NO: Invert a linear system of size larger than 3-by-3. 

Chapter 8
8.1 Discrete least squares approximation

YES: Normal equation; least squares approach with general set of basis.
NO: Invert a linear system of size larger than 3-by-3.

Chapter 5
5.1 The elementary theory of initial-value problems
5.2 Euler's method
5.3 High order Taylor methods
5.4 Runge-Kutta methods
5.9 Higher-order equations and systems of differential equations
5.11 Stiff ODE's

YES: Forward Euler method, backward Euler method, Trapezoidal method; their derivation, derivation of the local truncation error and the global error; Lipschitz constant; converting a high order ODE to a system of first order ODE; high order Taylor method; Runge Kutta; Interval of absolute stability
NO: The exact expressions for the local truncation error and the global error; table form of RK; RK4

Lecture 24 (Nov 25)

Stiff ODE. Interval of absolute stability. A-stable methods.

Numerical methods to check the accuracy in a numerical scheme.

Lecture 23 (Nov 23)

High order ODE's.

High order Taylor methods.

Runge-Kutta methods.

Lecture 22 (Nov 18)

Local truncation error for the forward Euler and the backward Euler methods.

Global error for the forward Euler method.

Lecture 21 (Nov 16)

Forward Euler Method, Backward Euler Method and Trapezoidal Method.

Lecture 20 (Nov 11)

Least squares fitting.

Normal equation. Other basis functions for least squares.

Lecture 19 (Nov 9)

Convergence of the iterative methods. Convergence for some special matrices.

Error bounds of the iterative method.

HW5 and HW6

HW5
(Postponed from HW4) Chapter 1: 32,33,34(a-d),35
Chapter 1: 34(e),36,37,38,39,41,43
Chapter 2: 10, 11

BackwardSubstitution.m: [click here]
ForwardSubstitution.m [click here]

Due: Nov 22 (Tue) or 23 (Wed) in your tutorial.


HW6
Chapter 1: 44,45,46,48,49(a)
Chapter 2: 12,13

Due: Dec 2 (Fri) to the TA.

Lecture 18 (Nov 4)

Induced norm: matrix 1-norm and inf-norm

Spectral radius: rho(A)=max |e-value(A)|

Lecture 17 (Nov 2)

Classical iterative methods: Jacobi, Gauss-Seidel and SOR.

Matrix Analysis. Vector norm: 1-, 2-, inf-norms

Lecture 16 (Oct 28)

Midterm

Lecture 15 (Oct 26)

Gaussian elimination with partial pivoting.

LU decomposition, i.e. A=LU.

(P^T)LU decomposition, i.e. PA=LU.

Lecture 14 (Oct 21)

Lecture cancelled due to the weather condition.

HW4 due Oct 25 (Tue) or 26 (Wed):
Chapter 1: #29,30,31,32,33,34(a-d),35

Midterm will cover everything up to, including, materials from Lecture 12. Will NOT cover Gaussian Elimination.

Lecture 13 (Oct 19)

Gaussian elimination with backward substitution.

Operations count.

Lecture 12 (Oct 14)

Composite rules. Gaussian Quadrature.

Midterm

Date: Oct 28 (Friday)
Time: 130pm-230pm
Venue: Room 2407

Closed book and closed note.
No calculator.
Test up to and including materials from the lecture on Oct 21 (Friday) Oct 14 (Friday).

Lecture 11 (Oct 12)

Numerical integration: Trapezoidal rule, Midpoint rule, Simpson's rule. Their derivation based on polynomial interpolation and Taylor's expansion.

Degree of accuracy.

Composite rules.

HW4

Chapter 1: #29-33,34(a-d),35

Due: Oct 25 (Tue) or 26 (Wed) in your tutorial.

Midterm will cover every up to, and including, this HW set.

Lecture 10 (Oct 7)

Numerical differentiation.

Approach 2: Taylor's expansion.

Higher order derivatives.

Lecture 9 (Oct 5)

Numerical differentiation:

Approach 1: Given data points -> interpolating polynomial -> differentiate it and evaluate it at a given location.

Forward difference, backward difference, central difference.

Error bound based on polynomial interpolation.

HW3

Chapter 1: #18-19,21-28
Chapter 2: #8

Due: Oct 18 (Tue) or 19 (Wed) in your tutorial.

Lecture 8 (Sep 30)

Piecewise linear interpolation.

Interpolation in two dimensions.

Scattered data: triangulation and then piecewise linear interpolation.

A MATLAB implementation can be found here: [click here]

Structured data: bilinear interpolation

Lecture 7 (Sep 28)

Error estimate of using polynomial interpolation.

Newton's divided difference.

Lecture 6 (Sep 23)

Modified Newton's method for finding root with multiplicity m.

Interpolation: Polynomial interpolation.

Lagrange Interpolating Polynomial. We will prove the error estimate in the next lecture.

Lecture 5 (Sep 21)

Order of convergence.

General fixed point iteration: For a converging iteration and if g'(p)!=0, the method converges only linearly.

Otherwise, if g'(p)=0 and g'' continuous and strictly bounded, we have at least quadratic convergence.

Modified Newton's method for finding root with multiplicity $m$.

HW2

Chapter 1: #10,12-17
Chapter 2: #5-7
Data file for #2.6: Schrodinger.mat

Due: Oct 4 (Tue) or 5 (Wed) in your tutorial.

Note: For Q1.16, the theorem is the one we discuss in Lecture 7 on Sep 28.

Note: For Q2.7, the statement in the bracket is not right. Please ignore it.

Lecture 4 (Sep 14)

Error estimates for fixed point iterations.

Newton's method. Geometrical construction. Convergence.

Secant method.

HW1

HW set: [click here]

Chapter 1: #1-8
Chapter 2: #1,3,4

Due: Sep 20 (Tue) or 21 (Wed) in your tutorial.

Lecture 3 (Sep 9)

Existence and Uniqueness of a fixed point. Example.

Fixed Point Iteration: Method.

Fixed Point Theorem (Thm 2.3 from the textbook).

Example.

Error estimates.

Lecture 2 (Sep 7)

Root-finding
1) Bisection method. Algorithm. Convergence and an error estimate.
2) Fixed point iteration. Definition of fixed point. Existence and Uniqueness of a fixed point.

Lecture 1 (Sep 1)

Course overview

Numerical/Computer Arithmetic
IEEE 754 standard
Integers (4 bytes), single precision (4 bytes), double precision (8 bytes)
Machine epsilon

HW, Past Papers

[Click here]

Welcome

Syllabus: [click here]