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**Unformatted text preview: **all (5.9.8), p. 386), and
(7.10.43) insures that R (P) = R (B) = R (M) and N (P) = N (C) = N (M).
Thus (7.10.39) is proved. If (7.10.38) is used to produce a full-rank factorization
∗
M = U1 (DV1 ), then, because D is nonsingular,
∗
∗
∗
∗
P = (U1 D)(V1 (U1 D))−1 V1 = U1 (V1 U1 )−1 V1 . Equations (7.10.41) and (7.10.42) follow from (5.9.11), p. 386.
Formulas (7.10.40) and (7.10.42) are useful because all good matrix computation packages contain numerically stable SVD implementations from which
∗
U1 and V1 can be obtained. But, of course, the singular values are not needed
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7.10 Diﬀerence Equations, Limits, and Summability
http://www.amazon.com/exec/obidos/ASIN/0898714540 635 Example 7.10.8
Shell Game. As depicted in Figure 7.10.2, a pea is placed under one of four
shells, and an agile manipulator quickly rearranges them by a sequence of discrete
moves. At the end of each move the shell containing the pea has been shifted
either to the left or right by only one position according to the following rules. It is illegal to print, duplicate, or distribute this material
Please report violations to meyer@ncsu.edu 1 #1 1/2 1/2 #2 D
E #3 1/2 #4 1/2 1 T
H Figure 7.10.2 When the pea is under shell #1, it is moved to position #2, and if the pea is
under shell #4, it is moved to position #3. When the pea is under shell #2 or
#3, it is equally likely to be moved one position to the left or to the right. IG
R Problem 1: Given that we know something about where the pea starts, what
is the probability of ﬁnding the pea in any given position after k moves?
Problem 2: In the long run, what proportion of time does the pea occupy each
of the four positions? Y
P Solution to Problem 1: Let pj (k ) denote the probability that the pea is in
position j after the k th move, and translate the given information into four
diﬀerence equations by writing
p2 (k −1)
2 O
C p1 (k ) = ⎛ p1 (k ) ⎞⎛ 0 ⎟⎜
⎜
p3 (k −1)
⎜ p2 (k...

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