CE 402 Written Homework Number 4
Due On or Before Wednesday, February 17, 2010 Part One
Derive the Gaussian Quadrature formula for 4 unequally spaced sample points. Please show details in work. The results should be given in terms of the weights, wi , fro
CE 402 Written Homework Number 5
Due On or Before Monday, March 1, 2010
(1) Derive the interpolation formula for 4 equally spaced sample points, f1 , f0 , f1 , f2 , as: f (x0 + h) p(p 1)(p 2) (p2 1)(p 2) p(p + 1)(p 2) p(p2 1) f1 + f0 f1 + f2 6 2 2 6 ,
in
CE 402 Written Homework Number 6
Due On or Before Wednesday, March 10, 2010
(1) Expand the function ex in the interval [-1,1] in a series of Legendre Polynomials ex = a0 P0 (x) + a1 P1 (x) + a2 P2 (x) + a3 P3 (x) + . . . (a) Find the coefcients a0 through
CE 402 Written Homework Number 7
Due On or Before Monday, March 29, 2010
(1) Put in integer order the 32 sample points, from 0 to 31, so they would be ready for an in-place Fast Fourier Transform algorithmic calculation.
(2) Perform the Fast Fourier Trans
CE 402 Written Homework Number 8
Due On or Before Monday, April 12, 2010
(1) Given a 3 3 symmetric matrix [A] as: 4 2 1 [A] = 1 -1 0 1 1 1 .
Decompose [A] to a product [Q][R] using the Gram-Schmidt Orthogonalization process. Perform the decomposition usin
CE 402 Written Homework Number 9
Due On or Before Wednesday, April 21, 2010
(1) Given the 4 4 symmetric matrix: 4 1 -2 2 1 1 2 0 [A] = -2 0 3 -2 2 1 -2 -1
.
(a) Find the Householder reflection matrices, Q1 and Q2 , required to reduce A into a Hessenberg
CE 402 Written Homework Number 10
Due On or Before Wednesday, April 28, 2010 Part One
Using the Second Order Runge Kutta method, find the numerical solution of the first order differential equation, dy + cos y = sin x , dx for the initial condition, y(0)
Application of Binary Arithmetics
Consider an example in which binary arithmetics can help develop an efcient algorithm, the famous FFT (Fast Fourier Transform) algorithm was developed using a binary scheme. For example, the term x365 needs 364 mulitplica
Gaussian Quadrature
For an high order polynomial, f (x), of degree 2m + 1, it can be written in two part as f (x) = q (x)Pm+1 (x) + r(x) ,
in which pm+1 (x) is the Legendre Polynomial of order m +1, q (x) is the quotient of f (x)/Pm+1 (x) and r(x) is the
CE 402 Written Homework Number 3
Due On or Before Wednesday, February 3, 2010
Numerical Differentiation
(1) Derive, algebraically, the 4-th order central difference derivatives using 5 sample points, f2 , f1 , f0 , f1 , f2 , with an equally spaced increme
CE 402 Written Homework Number 2
Due On or Before Wednesday, January 27, 2010
Numerical Integration
(1) Derive Newton-Cotes integration formulas of order 4 using Lagrange Interpolation Polynomials with 5 equally spaced sample points. The increment in x is
Magnitude of 1st column: = 42 + 12 + 12 = 4.2426 Divide old column 1 by , save alpha as R11 for restoration.
Dot Product: = [0.9428, 0.2357, 0.2357] [2, -1, 1] = 1.8856 Perform (new column 2)=(old column 2)-(new column 1) Put into R12 so the original col
CE 402 Computer Project Number 1
Due On or Before Wednesday, January 20, 2010
(1) Download the executable le, terminal.exe, from the ce402 website,
http:/www-classes.usc.edu/engr/ce/402
and execute it to answer the following questions (use escape or contr
CE 402 Written Homework Number 1
Due On or Before Wednesday, January 20, 2010
(1) Convert the 24-bit color code: R=197, G=132, B=76 into a 6-hexadecimal-digit, i.e., "#rrggbb", color code for html (Hyper Text Markup Language). Hint, convert R, G and B sep