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Unformatted text preview: d b2 e bc
The examples 4 3 9 and b d e lead to
and
.
7 32
e bc f c2
890
cef
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#
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1
1
1
0
1
1
1
1
2
0
1
1;
1 AD 1 1
1
1
22 1
AD 0 1
1
231
1
1
201
1
2
3
0001
This cyclic P exchanges rows 12 then
61 0 0 07
23 A D 4
D P and L D U D I .
0 1 0 05
rows 23 then rows 34.
0010
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1
012
1
2
1
1
1
038D0 1
3
8 . If we wait
24 PA D LU is
1
211
0 1=3 1
2 =3
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1
1
211
1
0 1 2.
to exchange and a12 is the pivot, A D L1 P1 U1 D 3 1
11
002
25 The splu code will not end when abs.A.k; k// < tol line 4 of the slu code on page 100. Instead splu looks for a nonzero entry below the diagonal in the current column k , and
executes a row exchange. The 4 lines to exchange row k with row r are at the end of
Section 2.7 (page 113). To ﬁnd that nonzero entry A.r; k/, follow abs.A.k; k// < tol
by locating the ﬁrst nonzero (or the largest A.r; k/ out of r D k C 1; : : : ; n). 26 One way to decide even vs. odd is to count all pairs that P has in the wrong order. Then P is even or odd when that count is even or odd. Hard step: Show that an exchange
always switches that count! Then 3 or 5 exchanges will leave that count odd.
#
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"
1
100
T
31
puts 0 in the 2; 1 entry of E21 A. Then E21 AE21 D 0 2 4
27 (a) E21 D
049
1
#
"
1
1
is still symmetric, with zero also in its 1, 2 entry. (b) Now use E32 D
41
T
T
to make the 3, 2 entry zero and E32 E21 AE21 E32 D D also has zero in its 2, 3 entry.
Key point: Elimination from both sides gives the symmetric LDLT directly.
2
3
0123
61 2 3 07
28 A D 4
D AT has 0; 1; 2; 3 in every row. (I don’t know any rules for a
2 3 0 15
3012
symmetric construction like this) Solutions to Exercises 26
29 Reordering the rows and/or the columns of a b c d will move the entry a. So the result
cannot be the transpose (which doesn’t move a).
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1
0
1
yBC
yBC C yBS
1
1
0
yCS
yBC C yCS .
30 (a) Total currents are AT y D
D
0
1
1
yBS
yCS yBS
(b) Either way .Ax /T y D x T .AT y / D xB yBC C xB yBS xC yBC C xC yCS
xS yCS xS yBS .
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" 700 #
1
50
x1
1
40
2
6820
1 truck
T
3
31 40 1000
D Ax ; A y D
D
x2
50 1000 50
188000 1 plane
2
50
3000 32 Ax y is the cost of inputs while x AT y is the value of outputs.
33 P 3 D I so three rotations for 360ı ; P rotates around .1; 1; 1/ by 120ı .
34 1
4
2
10 1
D
9
21 2 35 L.U T /
2
D EH D (elementory matrix) times (symmetric matrix).
5 1 is lower triangular times lower triangular, so lower triangular. The transpose
of U DU is U T D T U T T D U T DU again, so U T DU is symmetric. The factorization
multiplies lower triangular by symmetric to get LDU which is A.
T 36 These are groups: Lower triangular with diagonal 1’s, diagonal invertible D , permuta1 tions P , orthogonal matrices with QT D Q . 1
1
is southeast: 1 1
D 0 1.
10
1
The rows of B are in reverse order from a lower triangular L, so B D PL. Then
B 1 D L 1 P 1 has the columns in reverse ord...
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 Spring '12
 Minki
 Mass

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