Quiz I, Math 572, 4 February, 2010
Name:
Solutions
1. (15 points:) Suppose that y (t) = f (t, y(t) on the interval [t0 , t1 ] with y(t0 ) = y0 .
Assume that a unique solution y exists such that it and all of its derivatives up to
and including the third o
Homework 8, Math 572
18 March, 2010
Name:
1. Let m be a positive integer. Dene Am Rmm via
4 1
0 0
.
.
.
1 4
.
0 . . 1
Am =
0
.
.
.
1 4 1
0 0
1 4
,
and let Om , Im Rmm denote the zero and identity matrices, respectively. Now
2
2
we construct the (sti
M572 - Numerical Methods for Scientic Computing II - 2014
Assignment # 2.
Due: Thursday, February 6, 2014.
1. Show that the eigenvectors and eigenvalues of the matrix
A=
1
h2
2
1
1 2
1
1 2 1
k
are given by rj = sin kjh
1 2
1
1 2
and k =
2
(cos kh 1) , k,
M572 - Numerical Methods for Scientic Computing II - 2014
Assignment # 4.
Due: Thursday, February 20, 2014.
1. Gaussian Elimination Operation Count
Show that
n
n(n + 1)(2n + 1)
k2 =
6
k=1
2. Consider the linear system
2 1
1
2
x1
x2
=
1
1
.
I. Set up the i
M572 - Numerical Methods for Scientic Computing II - 2014
Assignment # 6.
Due: Thursday, March 13, 2014.
1. The trapezoid method is un+1 = un + k (f (un ) + f (un+1).
2
I. Show that the LTE is O(k 2 ).
II. Show that the method is A-stable.
III. Show direc
Homework 1, Math 572
21 January, 2010
Name:
1. Suppose A, B Cmm are similar matrices, and let C be a common eigenvalue.
Prove that the geometric multiplicities of , with respect to both A and B, are the
same.
A
Hint: Suppose A = XBX 1 . Verify that if E i
M572 - Numerical Methods for Scientic Computing II - 2014
Assignment # 8.
Due: April 17, 2014.
1. Consider the linear advection equation ut + aux = 0, where the constant a
may be either positive or negative. The Lax-Friedrichs scheme is
1
un+1 2 (un + un
Quiz V, Math 572, 22 April, 2010
Name:
Solutions
1. (30 points:) Consider the scheme
un+1 = un + un1 2un + un+1
bx n
u +1 un1
2
for the convection-diusion problem
u
2u
u
=
b
t
x2
x
u(0, t) = u(1, t) = 0
u(x, 0) = g(x)
for
0x1,
for
for
0tt ,
0tt ,
0x1,
1
M572 - Numerical Methods for Scientic Computing II - 2014
Assignment # 3.
Due: Thursday, February 13, 2014.
1. Variable Coecients
In class we considered a symmetric dierence scheme for the BVP with variable coecients (k(x)u ) = f , u(0) = , u(1) = .
1
kj1
Quiz III, Math 572, 4 March, 2010
Name:
Solutions
1. (15 points:) Prove that the BDF3 scheme
yn+3
18
9
2
6
yn+2 + yn+1 yn = hf (tn+3 , yn+3 )
11
11
11
11
satises the root condition and, therefore, must be convergent.
Hint: One of the roots is w = 1.
Solu
Quiz II, Math 572, 18 February, 2010
Name:
Solutions
1. (15 points:) Suppose that y (t) = f (t, y(t) on the interval [t0 , t ] with y(t0 ) = y0 .
Dene tn = t0 + hn, with h > 0. Assume that a unique solution y exists such
that it and all of its derivatives
Midterm Exam, Math 572
25 March, 2010
Name:
Solutions
1. (10 points): Suppose A Cmm and 1 , . . . , m C are the eigenvalues of A.
Prove that
m
tr(A) =
j .
j=1
Solution: Every matrix A Cmm admits a Schur Factorization (Theorem 24.9,
T&B). In other words, t
M572 - Numerical Methods for Scientic Computing II - 2014
Assignment # 5.
Due: Thursday, February 27, 2014.
1. Consider Eulers method applied to the linear ode u = au + b, u(0) = u0 .
Find a closed form for the numerical solution un and show directly that