Math 5485 October 13, 2006
Homework #5
Problems: 3.3 3.6 3.7 2a, 3a, b(a), c, 5b, d (also, what is the spectral radius?), 6b, c, 7a, 10. 2,10. 14b, 19.
Due: Friday, October 20 Text: B. Bradie, A Friendly Introduction to Numerical Analysis.
First Midterm:
Chapter 16 Complex Analysis
The term "complex analysis" refers to the calculus of complex-valued functions f (z) depending on a single complex variable z. On the surface, it may seem that this subject should merely be a simple reworking of standard real v
AIMS Lecture Notes 2006
Peter J. Olver
6. Eigenvalues and Singular Values
In this section, we collect together the basic facts about eigenvalues and eigenvectors. From a geometrical viewpoint, the eigenvectors indicate the directions of pure stretch and t
AIMS Exercise Set # 7
Peter J. Olver
1. In this exercise, you are asked to find "one-sided" finite difference formulas for derivatives. These are useful for approximating derivatives of functions at or near the boundary of their domain. (a) Construct a se
AIMS Exercise Set # 6
Peter J. Olver
1. Prove that the Midpoint Method (10.58) is a second order method. 2. Consider the initial value problem du = u(1 - u), dt for the logistic differential equation. solution for t > 0. (b) Use the Euler Method with step
AIMS Exercise Set # 5
Peter J. Olver
1. Use the power method to find the dominant eigenvalue and associated 4 1 0 1 -2 0 1 1 4 1 0 eigenvector of the following matrices: (a) -3 -2 0 , (b) . 0 1 4 1 -2 5 4 1 0 1 4 2. Use Newton's Method to find all points
AIMS Exercise Set # 4
Peter J. Olver
1. Find the explicit formula for the solution to the following linear iterative system: u(k+1) = u(k) - 2 v (k) , v (k+1) = - 2 u(k) + v (k) , u(0) = 1, v (0) = 0.
2. Determine whether or not the following matrices are
AIMS Exercise Set # 3
Peter J. Olver
1. Which of the following matrices are regular? If reguolar, write down its L U 1 -2 3 2 1 0 -1 factorization. (a) , (b) , (c) -2 4 -1 . 1 4 3 -2 3 -1 2 2. In each of the following problems, find the A = L U factorizat
AIMS Exercise Set # 2
Peter J. Olver
1. Explain why the equation e- x = x has a solution on the interval [ 0, 1 ]. Use bisection to find the root to 4 decimal places. Can you prove that there are no other roots? 2. Find 6 3 to 5 decimal places by setting
AIMS Exercise Set # 1
Peter J. Olver
1. Determine the form of the single precision floating point arithmetic used in the computers at AIMS. What is the largest number that can be accurately represented? What is the smallest positive number n1 ? The second
AIMS Lecture Notes 2006
Peter J. Olver
1. Computer Arithmetic
The purpose of computing is insight, not numbers. - R.W. Hamming, [23]
The main goal of numerical analysis is to develop efficient algorithms for computing precise numerical values of mathemati
Chapter 21 The Calculus of Variations
We have already had ample opportunity to exploit Nature's propensity to minimize. Minimization principles form one of the most wide-ranging means of formulating mathematical models governing the equilibrium configurat
Math 5485 October 4, 2006
Homework #4
Problems: 3.1 3.2 3.5 1, 8, 10, 12b. 7a, 14, 18b. 10a, 11.
Due: Friday, October 13 Text: B. Bradie, A Friendly Introduction to Numerical Analysis.
First Midterm: Wednesday, November 1 Will cover chapters 1, 2, 3. You
Math 5485 September 27, 2006
Homework #3
Problems: 2.5 2.6 1d, 6, 11a. 1, 5, 8.
Due: Wednesday, October 4 Text: B. Bradie, A Friendly Introduction to Numerical Analysis.
Chapter 14 Vibration and Diffusion in OneDimensional Media
In this chapter, we study the solutions, both analytical and numerical, to the two most important equations of one-dimensional continuum dynamics. The heat equation models the diffusion of thermal
AIMS Lecture Notes 2006
Peter J. Olver
4. Gaussian Elimination
In this part, our focus will be on the most basic method for solving linear algebraic systems, known as Gaussian Elimination in honor of one of the all-time mathematical greats - the early nin
Chapter 12 Fourier Series
Just before 1800, the French mathematician/physicist/engineer Jean Baptiste Joseph Fourier made an astonishing discovery. As a result of his investigations into the partial differential equations modeling vibration and heat propa
Chapter 13 Fourier Analysis
In addition to their inestimable importance in mathematics and its applications, Fourier series also serve as the entry point into the wonderful world of Fourier analysis and its wide-ranging extensions and generalizations. An
AIMS Lecture Notes 2006
Peter J. Olver
3. Review of Matrix Algebra
Vectors and matrices are essential for modern analysis of systems of equations - algebrai, differential, functional, etc. In this part, we will review the most basic facts of matrix arithm
AIMS Lecture Notes 2006
Peter J. Olver
12. Minimization
In this part, we will introduce and solve the most basic mathematical optimization problem: minimize a quadratic function depending on several variables. This will require a short introduction to pos
AIMS Lecture Notes 2006
Peter J. Olver
14. Finite Elements
In this part, we introduce the powerful finite element method for finding numerical approximations to the solutions to boundary value problems involving both ordinary and partial differential equa
Orthogonal Bases and the QR Algorithm
by Peter J. Olver University of Minnesota
1. Orthogonal Bases.
Throughout, we work in the Euclidean vector space V = R n , the space of column vectors with n real entries. As inner product, we will only use the dot pr
Chapter 15 The Planar Laplace Equation
The fundamental partial differential equations that govern the equilibrium mechanics of multi-dimensional media are the Laplace equation and its inhomogeneous counterpart, the Poisson equation. The Laplace equation i
AIMS Lecture Notes 2006
Peter J. Olver
5. Inner Products and Norms
The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number of other important norms that are used in numerical analysis
AIMS Lecture Notes 2006
Peter J. Olver
7. Iterative Methods for Linear Systems
Linear iteration coincides with multiplication by successive powers of a matrix; convergence of the iterates depends on the magnitude of its eigenvalues. We discuss in some det
Chapter 17 Dynamics of Planar Media
In this chapter, we continue our ascent of the dimensional ladder for linear systems. In Chapter 6, we embarked on our journey with equilibrium configurations of discrete systems - massspring chains, circuits, and struc
Math 5485 November 27, 2006
Homework #8
Problems: 4.4 4.5 1ac, 4a, 5b, 8. 6 (ignore the Wilkinson shift), 12 (compare the convergence rate of the direct QR algorithm with that based on tridiagonalization).
Due: Monday, December 4 Text: B. Bradie, A Friend
Stat 310 Homework 6 Key
Chapter 5, problems 16, 18. Chapter 7, problems 3, 4, 12 (explain your answers!), 26, 33.
Chapter 8, problems 2, 6, 11. Due 11/4/99.
5.16 Suppose that X ; : : :; X are independent random variables with density functions
f (x) = 2x;