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
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 identit
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 gi
M572 - Numerical Methods for Scientic Computing II - 2014
Assignment # 7.
Due: Tuesday, April 1, 2014.
1. Given the discrete sequence uj , j = 0, 1, 2, . dene the Fourier transform
u(h) by
h
u(h) =
u
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 )
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 t
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 ne
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)
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 va
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 so
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 Sch
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 fo