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hw4.pdf
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hw4.pdf
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Homework Math 2071 5 problems: Pick 2 (your choice)
Due Wednesday April 1, 2020. Please scan it in and email the PDF to me.
It will be VERY IMPORTANT to practice writing math well. That means:
Write legibly and BIG. Display all equations and number al lequations.
Structure your HW like a math book or math paper.
1. For non symmetric A two functionals to minimize are
A
T
Ax = A
T
b ⇒ J
ne
(x) = (1/2)x
T
A
T
Ax − x
T
A
T
b
J
mr
(x) = (b − Ax)
T
(b − Ax)
Find and prove a relationship between these 2 functionals.
2. Consult the Templates book, http://www.netlib.org/templates/templates.pdf
, or some other source. Pick a generalized CG method (for A not symmetric)
not c9onsidered in my lectures. a. Write it out in pseudocode Explain how it
differs from CG pseudocode and how much needs to be changed to convert a
CG program to your method. b. Explain its mathematical basis and what
properties of CG fail for the method.
3.Nesterov iteration is a current ’hot topic’ in big data. It solves an opti-
mization problem and thus gives an iterative method for A spd when applied to
J(x), Look it up. Present it in pseudocode and explain its convergence properties
when applied to Ax=b for A spd.
4. Consider the 2d MPP with ADI splitting of the coefficient matrix A:
A = A
x
+ A
y
You can assume that A
x
, A
y
commute [they do under some
conditions]. The ADI preconditioner is M = (ρI + A
x
)(ρI + A
y
). Prove an
estimate of cond(M
−1
A). You can pick ρ to be the optimal value for Peaceman-
Rachford ADI [given in the text book].
5. Reconsider the 2d discrete convection diffusion equation from an earlier
homework. Pick a generalized CG method and text its convergence for the
equation by repeating the test you did earlier for CG. Describe what you see.
1
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