AMSC/CMSC 660
Quiz 9
,
Fall 2006
1. (10) Recall that a Hamiltonian system is a system of ODEs for which there exists a scalar Hamiltonian function H(y) so that y = D y H(y) where D is a block-diagonal matrix with blocks equal to 0 1 -1 0 Derive t

AMSC/CMSC 660
Quiz 8
,
Fall 2006
1. (10) Let i = -1, and suppose we have a system of differential equations y = y(t, y) with 3 components. Suppose the system has a Jacobian matrix J(t, y) with eigenvalues 4 - t2 , -t - it, -t + it. For what valu

AMSC/CMSC 660
Quiz 7
,
Fall 2006
1. (10) Write Matlab code to apply 5 steps of Newton's method to the problem x2 y 3 + xy = 2 , 2xy 2 + x2 y + xy = 0 , starting at the point x = 5, y = 4. Answer: F(x) = and J(x) = x2 y 3 + xy - 2 2xy 2 + x2 y + x

AMSC/CMSC 660
Quiz 6
,
Fall 2006
1. (10) Let f (x) = 1 xT Hx - xT b, where H and b are constant, independent 2 of x, and H is symmetric positive definite. Given vectors x(0) and p(0) , find the value of the scalar that minimizes f (x(0) + p(0) )

AMSC/CMSC 660
Quiz 4
,
Fall 2006
1. (10) Write Matlab code using rand to generate a random number from the following distribution: The probability that the number is 0 is 0.6. The probability that the number is 1 is 0.4. (In other words, if p(x)

AMSC/CMSC 660
Quiz 3
,
Fall 2006
1. (10) Suppose we have factored A = LU and now we need to solve a linear system (A - ZVT )x = b, where Z and V have dimension n k and k is much less than n. Write Matlab code to do this accurately and efficientl

AMSC/CMSC 660
Quiz 2
,
Fall 2006
1. (10) Suppose we have factored the m n matrix A = QR (m n), and ^ let x be the solution to the least squares problem min Ax - b .
x
Show that A^ - b 2 = c2 2 , where c2 is the vector consisting of the last x

AMSC/CMSC 660
Quiz 1
,
Fall 2006
Show all work. You may leave arithmetic expressions in any form that a calculator could evaluate. By putting your name on this paper, you agree to abide by the university's code of academic integrity in completing

AMSC/CMSC 660 Scientific Computing I Fall 2006 Unit 3: Optimization Dianne P. O'Leary c 2002,2004,2006 Optimization: Fundamentals
Our goal is to develop algorithms to solve the problem Problem P: Given a function f : S R, find min f (x)
xS
with so

AMSC/CMSC 660 Scientific Computing I Fall 2006 UNIT 5: Numerical Solution of Ordinary Differential Equations Dianne P. O'Leary c 2002,2004,2006 The Plan
Initial value problems (IVPs) for ordinary differential equations (ODEs) Review 460 ODE notes

AMSC/CMSC 660 Scientific Computing I Fall 2006 UNIT 4: Nonlinear Systems and the Homotopy Method Part 2 Dianne P. O'Leary c 2002,2004,2006 Case study: Octahedral variable geometry truss
Note: This case study concerns solving a system of polynomial e

AMSC/CMSC 660 Scientific Computing I Fall 2006 UNIT 4: Nonlinear Systems and the Homotopy Method Dianne P. O'Leary c 2002, 2004, 2006 The problem
Given a function F : Rn Rn , find a point x Rn such that F(x) = 0 .
Note: The one-dimensional case (

AMSC/CMSC 660 Scientific Computing I Fall 2006 UNIT 4: Monte Carlo Methods Dianne P. O'Leary c 2002,2004,2006 What is a Monte-Carlo method? In a Monte-Carlo method, the desired answer is formulated as a quantity in a stochastic model and estimated by

AMSC/CMSC 660 Scientific Computing I Fall 2006 Dense Matrix Computations Dianne P. O'Leary c 2002,2004, 2006 Goals
We'll consider computations involving dense matrices, those that don't have a large number of zero elements.
We need to 1. be able to