AMS 526 Homework 3, Fall 2014
Due: Monday 10/13 in class
1. (15 points) Show that for Gaussian elimination with partial pivoting applied to any matrix A Rnn ,
the growth factor = maxi,j |uij |/ maxi,j |aij | satises 2n1 .
(0)
Solution: Let A = A(0) = aij
AMS 526 Homework 2
Due: Wednesday 09/28 in class
1. (15 points) Suppose that A Rmn has rank n.
(a) (10 points) Show that A(AT A)1 AT is an orthogonal projection matrix onto range of A.
(b) (5 points) Show that kA(AT A)1 AT k2 = 1.
Solution:
(a) Let P = A(
AMS526: Numerical Analysis I
(Numerical Linear Algebra)
Lecture 2: Algorithms and Efficiency;
Block Matrices and Algorithms;
Range and Null Space
Xiangmin Jiao
Stony Brook University
Xiangmin Jiao
Numerical Analysis I
1 / 34
Outline
1
Algorithms and Effic
AMS526: Numerical Analysis I
(Numerical Linear Algebra)
Lecture 5: More on Projectors;
Conditioning and Condition Number
Xiangmin Jiao
SUNY Stony Brook
Xiangmin Jiao
Numerical Analysis I
1 / 19
Outline
1
More on Projectors (NLA6)
2
Conditioning and Condit
AMS526: Numerical Analysis I
(Numerical Linear Algebra)
Lecture 8: Accuracy and Stability of
Gaussian Elimination
Xiangmin Jiao
Stony Brook University
Xiangmin Jiao
Numerical Analysis I
1 / 22
Outline
1
Condition Number of Gaussian Elimination (NLA22)
Per
AMS526: Numerical Analysis I
(Numerical Linear Algebra)
Lecture 7: LU Factorization;
Gaussian Elimination with Pivoting
Xiangmin Jiao
Stony Brook University
Xiangmin Jiao
Numerical Analysis I
1 / 27
Outline
1
Solution of Linear Systems (MC3.2, NLA20)
2
Ga
Sample Solutions for AMS 526 Sample Test 1
September 29, 2016
Note: The exam is closed-book. However, you can have a single-sided, one-page, letter-size cheat sheet.
1. Answer true or false and give a brief justification. (No credit without justification.
AMS 526 Sample Questions for Test 2
October 31, 2016
Note: The exam is closed-book. However, you can have a single-sided, one-page, letter-size cheat sheet.
1. Answer true or false and give a brief justification. (No credit without justification.)
(a) Whe
AMS526 Homework 2 Solutions
October 8, 2014
1. (10 points) Let P Rnn be an orthogonal projection matrix (projector).
(a) Show that I P is also an orthogonal projection matrix.
(b) Show that I 2P is an orthogonal matrix.
Solution
(a) A projector is a squar
AMS 526 Homework 4, Fall 2014
Due: Wednesday 10/29 in class
1. (10 points)
(a) Let A Rnm and B = A+ Rmn . Show that the following four relationships hold:
BAB = B
ABA = A
(BA)T = BA
(AB)T = AB
(b) Conversely, show that if A and B satisfy the above four co
AMS 526 Homework 6, Fall 2014
Due: Wednesday 12/3 in class
1. (10 points) In our lectures, it was pointed out that the eigenvalues of a symmetric matrix A Rmm
are the stationary values of the Rayleigh quotient r(x) = xT Ax / xT x for x Rm . Show that the
AMS 526 Homework 5, Fall 2014
Due on Monday 11/17 in class
1. (15 points) The preliminary reduction to tridiagonal form would be of little use if the steps of the QR
algorithm did not preserve this structure. Fortunately, they do.
(a) In the QR factorizat
AMS526 Homework 2 Solutions
Due : Wednesday 09/23
1. (10 points) Let P Rnn be an orthogonal projection matrix (projector).
(a) Show that I P is also an orthogonal projection matrix.
(b) Show that I 2P is an orthogonal matrix.
Solution
(a) A projector is a
AMS 526 Sample Questions for Final Exam
December 3, 2014
1. (20 points) Answer true or false with a brief justication. (No credit without justication.)
(a) If A Cmm is Hermitian positive denite, then the eigenvalues of BAB T are all real and positive for
Sample Solutions for AMS 526 Sample Test 1
September 24, 2014
Note: The exam is closed-book. However, you can have a single-sided, one-page, letter-size cheat sheet.
1. Answer true or false and give a brief justication. (No credit without justication.)
(a
AMS526: Numerical Analysis I
(Numerical Linear Algebra)
Lecture 6: Floating Point Arithmetic;
Accuracy and Stability; Triangular Systems
Xiangmin Jiao
Stony Brook University
Xiangmin Jiao
Numerical Analysis I
1 / 19
Outline
1
Floating Point Arithmetic (NL
AMS526: Numerical Analysis I
(Numerical Linear Algebra)
Lecture 11: QR Factorization; Gram-Schmidt Process
Xiangmin Jiao
Stony Brook University
Xiangmin Jiao
Numerical Analysis I
1 / 15
Outline
1
QR Factorization (NLA7)
2
Modified Gram-Schmidt Orthogonali
AMS 526 Homework 1 Sample Solutions
1. (10 points) Show that if A Rmn has rank p, then there exists an X Rmp and Y Rnp
such that A = XY T , where rank(X) = rank(Y ) = p.
Solution:
Let A Rmn be a rank p matrix. This means that there are p linearly independ
AMS 526 Homework 3 Solutions, Fall 2016
Due: Monday 10/19 in class
1. (15 points) Let A = LU be the LU factorization of n-by-n A with |`ij | 1. Let aTi and uTi denote the
ith row of A and U , respectively. Verify the equation
uTi = aTi
i1
X
`ij uTj ,
j=1
AMS 526 Homework 4 Solutions, Fall 2016
Due: Wednesday 11/02 in class
1. (10 points)
(a) Let A Rnm and B = A+ Rmn . Show that the following four relationships hold:
BAB = B
ABA = A
(BA)T = BA
(AB)T = AB
(b) Conversely, show that if A and B satisfy the abo
AMS 526 Homework 5, Fall 2015
Due on Wednesday 11/18 in class
1. (15 points) The preliminary reduction to tridiagonal form would be of little use if the steps of the QR
algorithm did not preserve this structure. Fortunately, they do.
(a) In the QR factori
AMS 526 Homework 6 Solutions, Fall 2015
Due: written part due on Wednesday 12/02 in class;
programming part due on Friday 12/04 by 11:59pm
1. (10 points) In our lectures, it was pointed out that the eigenvalues
matrix A Rmm
of Ta symmetric
T
are the stat
AMS 526 Homework 3 Solutions, Fall 2015
Due: Monday 10/13 in class
1. (15 points) Let A = LU be the LU factorization of n-by-n A with |`ij | 1. Let aTi and uTi denote the
ith row of A and U , respectively. Verify the equation
uTi = aTi
i1
X
`ij uTj ,
j=1
AMS 526 Homework 4 Solutions, Fall 2015
Due: Wednesday 10/28 in class
1. (10 points)
(a) Let A 2 Rnm and B = A+ 2 Rmn . Show that the following four relationships hold:
BAB = B
ABA = A
(BA)T = BA
(AB)T = AB
(b) Conversely, show that if A and B satisfy the