UC Berkeley Math 221, Fall 2016: Problem Set 4
Due October 21
1. (Trefethen/Bau 22.2) Experiment with solving 60 60 systems of equations Ax = b by Gaussian
elimination with partial pivoting, with A having the form
1
1
1 1
1
A = 1 1 1
1 .
1 1 1 1 1
1 1 1 1
Prof. Ming Gu, 861 Evans, tel: 2-3145
Oce Hours: TuWTh 12:00-1:30PM
Email: [email protected]
http:/www.math.berkeley.edu/mgu/MA221
Math221: Matrix Computations
Homework #2 Solutions
2.3: We need relevent vectors in equation (2.1) to have the same dir
Prof. Ming Gu, 861 Evans, tel: 2-3145
Oce Hours: TuWTh 12:00-1:30PM
Email: [email protected]
http:/www.math.berkeley.edu/mgu/MA221
Math221: Matrix Computations
Solutions to Homework #6
Problem 3.8: P and Q can never be equal. The determinent of a Hou
Prof. Ming Gu, 861 Evans, tel: 2-3145
Oce Hours: TuWTh 12:00-1:30PM
Email: [email protected]
http:/www.math.berkeley.edu/mgu/MA221
Math221: Matrix Computations
Homework #7 Solutions
Let A Rnn be non-singular. The QR factorization with column pivoting
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1
Introduction
1.1. Basic Notation
In this course we will refer frequently to matrices, vectors, and scalars. A
matrix wil
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7
Iterative Methods for Eigenvalue
Problems
7.1. Introduction
In this chapter we discuss iterative methods for finding eig
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Linear Least Squares Problems
3.1. Introduction
Given an m-by-n matrix A and an m-by-1 vector b, the linear least squares
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0
Nonsymmetric Eigenvalue Problems
4.1. Introduction
We discuss canonical forms (in section 4.2), perturbation theory (in
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2
Linear Equation Solving
2.1. Introduction
This chapter discusses perturbation theory, algorithms, and error analysis for
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The Symmetric Eigenproblem and
Singular Value Decomposition
5.1. Introduction
We discuss perturbation theory (in section 5
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6
Iterative Methods for Linear Systems
6.1. Introduction
Iterative algorithms for solving Ax = b are used when methods suc
Lecture Slides Numerical Linear Algebra
Per-Olof Persson
[email protected]
Department of Mathematics
University of California, Berkeley
Math 221 Advanced Matrix Computations
Matrix-Vector Multiplication
Matrix-vector product b = Ax
bi =
n
X
aij xj ,
i
UC Berkeley Math 221, Fall 2016: Problem Set 7
Due December 2
This last problem set is different from the previous ones, in the sense that you can pick one of several
proposed problems. They are slightly more extensive problems, and also more open-ended,
UC Berkeley Math 221, Fall 2016: Problem Set 6
Due November 18
1. Let L, U be an ILU(0) factorization of A, and set B = LU . Show that if aij 6= 0 then aij = bij , that
is, A and B only differ in elements outside the non-zero pattern of A.
2. (Trefethen/B
UC Berkeley Math 221, Fall 2016: Problem Set 2
Due September 23
1. (Trefethen/Bau 6.3) Given A Cmn with m n, show that A A is nonsingular if and only if A has
full rank.
2. (Trefethen/Bau 6.4) Consider the matrices
1 0
A = 0 1 ,
1 0
1 2
B = 0 1 .
1 0
(1)
UC Berkeley Math 221, Fall 2016: Problem Set 5
Due November 4
In problems 1 and 2 below you will study inverse iteration and classical iterative methods for a linear
elasticity problem. Some utility functions in MATLAB are available on the course web page
UC Berkeley Math 221, Fall 2016: Problem Set 1
Due September 9
1. (Trefethen/Bau 2.3) Let A Cmm be hermitian. An eigenvector of A is a nonzero vector x Cm
such that Ax = x for some C, the corresponding eigenvalue.
(a) Prove that all eigenvalues of A are r
UC Berkeley Math 221, Fall 2016: Problem Set 3
Due October 7
1. (Trefethen/Bau 12.1) Suppose A is a 202 202 matrix with kAk2 = 100 and kAkF = 101. Give the
sharpest possible lower bound on the 2-norm condition number (A).
2. (Trefethen/Bau 13.1) Between a
Prof. Ming Gu, 861 Evans, tel: 2-3145
Oce Hours: TuWTh 12:00-1:30PM
Email: [email protected]
http:/www.math.berkeley.edu/mgu/MA221
Math221: Matrix Computations
Homework #4 Selected Solutions
2.7: Since A is nonsingular, all diagonal entries of D must
Prof. Ming Gu, 861 Evans, tel: 2-3145
Oce Hours: TuWTh 12:00-1:30PM
Email: [email protected]
http:/www.math.berkeley.edu/mgu/MA221
Math221: Matrix Computations
Selected Solutions to Homework #5
Problem 3.3:
The system
I
AT
A
0
r
x
=
b
0
(1)
is equiv
1
Math 221, Fall 2012: Problem Set 02
Please hand in detailed solutions to 5 of the following 6 problems.
Exercise 1 Prove the following inequalities and for each give an example of
a nonzero n-vector or m n matrix for which equality is achieved:
(a)
(b)
1
Math 221, Fall 2012: Problem Set 03
Please hand in detailed solutions to 5 of the following 6 problems.
Exercise 1 Let P be a nonzero projection matrix. Show that P 1 for
any induced matrix norm, and that the 2-norm P 2 = 1 i P is orthogonal.
Exercise 2
1
Math 221, Fall 2012: Problem Set 04
Please hand in codes (where needed), output and discussion for the following
problems.
Exercise 1 Implement classical Gram-Schmidt and use both classical and
modied Gram-Schmidt to repeat experiments 1-3 in Trefethens
1
Math 221, Fall 2012: Problem Set 05
Please hand in solutions for the following problems.
Exercise 1 Given an n-vector x, show that oating-point computation of the
Householder vector v such that P x = (I 2vv T )x = x 2 e1 gives a forward
stable result v
1
Math 221, Fall 2012: Problem Set 06
Please hand in solutions for the following problems.
Exercise 1 Let A = D(I + B )D where D is diagonal with dii = 2i and
B is a random matrix with entries uniformly distributed between 1 and 1.
Let b be a random vecto
1
Math 221, Fall 2012: Problem Set 07
Please hand in solutions for the following problems.
Exercise 1 Newtons method for a nonlinear equation f (x) = 0 consists
of Taylor-expanding f (x) about an approximate solution xn to get a linear
equation f (xn ) +
1
Math 221, Fall 2012: Problem Set 08
Please hand in solutions for the following problems.
Exercise 1 Write a program which nds the eigenvalues of the rank-one
update T = D + uuT of a diagonal matrix D with distinct diagonal entries
di by a vector u with
c 2011 Society for Industrial and Applied Mathematics
SIAM REVIEW
Vol. 53, No. 2, pp. 217288
Finding Structure with Randomness:
Probabilistic Algorithms for
Constructing Approximate
Matrix Decompositions
N. Halko
P. G. Martinsson
J. A. Tropp
Abstract. Low
1
Math 221, Fall 2012: Problem Set 09
Please hand in solutions for the following problems.
Exercise 1 Let A be the n2 n2 symmetric matrix with aii = i and
ai,i+1 = ai,i+n = 1. Find the largest and smallest eigenvalues of A by the
Lanczos iteration with a