Numerical Analysis Spring 2011 1 of 5 Department of Mathematics

Numerical Analysis Spring 2011 1 of 5 Department of Mathematics
Numerical Analysis Spring 2011
1 of 5
Department of Mathematics
California State University Los Angeles
Master’s Degree Comprehensive Examination in
NUMERICAL ANALYSIS
SPRING 2011
Instructions:
• Do exactly 2 problems from Part A AND 2 problems from Part
B. If you attempt more than two problems in either Part A or Part B,
and do not clearly indicate which two are to count, only the first two
problems will be counted toward your grade.
• No calculators.
• Closed books and closed notes.
PART A: Do only 2 problems
1. (a) [8 points] Find the LU decomposition of the matrix


1 1 2
B =  2 4 7 ,
3 11 19
that is, find a unit lower-triangular matrix L and an upper-triangular
matrix U such that B = LU .
(b) Given the linear system Ax = b, where A has been factored as
A = LU , and




 
1 0 0
2 4 4
2





1 1 0 , U=
0 1 2 , b=
0 .
L=
1 0 1
0 0 1
2
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Numerical Analysis Spring 2011
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i. [8 points] Solve Ax = b without multiplying the matrices L
and U .
ii. [3 points] Give matrices L, D, and U such that A = LDU ,
where L is a unit lower-triangular matrix, D is a diagonal
matrix, and U is a unit upper-triangular matrix.
(c) [3 points each] Give one advantage of Gaussian elimination with
partial pivoting over each of the following techniques for solving
a nonsingular system of n linear equations in n unknowns:
• Gaussian elimination without partial pivoting
• Jacobi iteration
2. (a) [6 points] Let

 

 
1
1
4 1 0
(0)





1 .
2 , u =
1 4 2 , b=
A=
1
3
0 2 4
Perform one Jacobi iteration for solving Au = b, with starting
vector u(0) , to find u(1) .
(b) [7 points] Let B be an arbitrary 4 × 4 upper-triangular matrix
with nonzero diagonal entries. Show that Gauss-Seidel iteration
converges for Bx = b, for arbitrary b.
(c) Let
(
C=
3 1
5 7
)
.
C has eigenvalues 2 and 8, and corresponding eigenvectors [−s, s]T
and [s, 5s]T , respectively, where s ̸= 0.
i. [6 points] Apply two iterations of the Power Method to the
matrix C with initial vector x(0) = [1, 0]T to obtain x(2) , an
approximation to the eigenvector of C corresponding to eigenvalue 8.
ii. [6 points] Explain why the Power Method converges when
applied to the matrix C using the initial value [1, 0]T .
3. (a) [7 points] Let A be an n × n matrix. Show that if ρ(A) < 1, then
I − A is non-singular (invertible).
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Numerical Analysis Spring 2011
(b) [7 points] Let
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
3
 1
B=
 0
0
1
3
1
0
0
1
3
1

0
0 
.
1 
3
Use Gershgorin’s circle theorem, together with part (a), to show
that B is non-singular.
(c) [3 points] Give an example to show that a diagonally-dominant
matrix need not be non-singular. Be sure to show that your matrix
is non-singular.
(d) [8 points] Let
(
C=
1 1
2 2
)
.
Find the matrix resulting from performing one iteration of the QR
method (for approximating eigenvalues) on C.
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PART B: Do only 2 problems
1. For the initial boundary value problem
∂u
∂ 2u
, 0 < x < 1, t > 0
=
∂t
∂x2
∂u
= u + 1, x = 0, t > 0
∂x
∂u
= −u, x = 1, t > 0
∂x
u(x, 0) = x, 0 ≤ x ≤ 1
(a) [10 points] Show that the scheme
ui,j+1 − ui,j
ui−1,j − 2ui,j + ui+1,j
=
k
h2
is consistent with the differential equation.
(b) [15 points] Let h = 1/10 and k = 1/200. Using central differencing
for the boundary data and the scheme above, compute u0,2 and
u10,2 .
2. (a) [7 points] By finding the truncation error in approximating the
ordinary derivative f ′′ (x) by
Dh (f ) =
f (x + 2h) − 2f (x) + f (x − 2h)
h2
determine whether or not this is a consistent approximation to
f ′′ (x).
(b) [2 points each] Consider the parabolic PDE ut = uxx .
i. Explain what it means for a difference scheme for this PDE
to be stable.
ii. Can a consistent explicit scheme approximating this PDE be
stable but not convergent? Explain why or why not?
iii. Give an example of an explicit scheme approximating this
PDE which is not stable. (You need not prove that your
scheme is not stable.)
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iv. Explain why the concept of stability does not apply to a difference scheme that is approximating an elliptic PDE.
∂2u
∂ 2u
(c) Consider the PDE x
− 2 = 0.
∂y∂x ∂ y
i. [3 points] Determine the values of (x, y) for which this PDE
is hyperbolic.
ii. [4 points] Determine the characteristic curves for this PDE.
iii. [3 points] Can an elliptic PDE generate characteristic curves?
Why or why not?
3. (a) Consider the finite difference scheme
ui,j+1 − ui,j ui+1,j − ui,j
+
=0
k
h
for the PDE ut + ux = 0.
i. [6 points] Show that the scheme is unstable.
ii. [3 points] Modify the scheme to make it stable.
iii. [4 points] Write down an explicit consistent scheme for ut +
ux = 0 that is unconditionally stable if there is any. If there
is none, explain why.
(b) Consider the PDE
∂u
∂u
+ e−y
= −u2
∂x
∂y
u(x, 0) = 1, 0 < x < ∞
x2 u
i. [6 points] Find the equation for the characteristic curve through
the point (s, 0).
ii. [6 points] Find the equation for the characteristic curve through
the point (1, 1) and the value of the exact solution of the given
IVP at (1, 1).
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