73 c = Q\Qc
c = 11.5094
11.5094 19.0566
16.9983 11.5094
7.7 The problem can be written in matrix form as
⎪ ⎪
⎭ ⎪
⎪ ⎬
⎫
⎪ ⎪
⎩ ⎪
⎪ ⎨
⎧ −
= ⎪
⎪ ⎪
⎭ ⎪⎪
⎪ ⎬
⎫
⎪ ⎪
⎪ ⎩
⎪⎪ ⎪
⎨ ⎧
⎥ ⎥
⎥ ⎥
⎥
⎦ ⎤
⎢ ⎢
⎢ ⎢
⎢
⎣ ⎡
− −
− −
− −
− −
1000
1 866
. 5
. 1
1 5
. 1
1 866
. 866
. 5
. 5
. 866
.
3 2
2 3
2 1
V V
H F
F F
MATLAB can then be used to solve for the forces and reactions, A = [0.866 0 -0.5 0 0 0;
0.5 0 0.866 0 0 0; -0.866 -1 0 -1 0 0;
-0.5 0 0 0 -1 0; 0 1 0.5 0 0 0;
0 0 -0.866 0 0 -1] b = [0 -1000 0 0 0 0];
F = A\b F =
-500.0220 433.0191
-866.0381 -0.0000
250.0110 749.9890
Therefore, F
1
= –500 F
2
= 433 F
3
= –866 H
2
= 0 V
2
= 250 V
3
= 750
7.8 The problem can be written in matrix form as
⎪ ⎪
⎭ ⎪
⎪ ⎬
⎫
⎪ ⎪
⎩ ⎪
⎪ ⎨
⎧ =
⎪ ⎪
⎪ ⎭
⎪⎪ ⎪
⎬ ⎫
⎪ ⎪
⎪ ⎩
⎪⎪ ⎪
⎨ ⎧
⎥ ⎥
⎥ ⎥
⎥
⎦ ⎤
⎢ ⎢
⎢ ⎢
⎢
⎣ ⎡
− −
− −
− −
− −
−
200 20
10 5
5 15
10 10
1 1
1 1
1 1
1 1
1 1
43 54
65 32
52 12
i i
i i
i i
MATLAB can then be used to solve for the currents,
74 A = [1 1 1 0 0 0 ;
0 -1 0 1 -1 0; 0 0 -1 0 0 1;
0 0 0 0 1 -1; 0 10 -10 0 -15 -5;
5 -10 0 -20 0 0]; b = [0 0 0 0 0 200];
i = A\b i =
6.1538 -4.6154
-1.5385 -6.1538
-1.5385 -1.5385
7.9
k1 = 10;k2 = 40;k3 = 40;k4 = 10; m1 = 1;m2 = 1;m3 = 1;
km = [1m1k2+k1, -k2m1,0; -k2m2, 1m2k2+k3, -k3m2;
0, -k3m3,1m3k3+k4]; x = [0.05;0.04;0.03];
kmx = kmx kmx =
0.9000 0.0000
-0.1000 Therefore,
1
x
= −0.9,
2
x
= 0 , and
3
x
= 0.1 ms
2
.
75
CHAPTER 8
8.1
The flop counts for the tridiagonal algorithm in Fig. 8.6 can be summarized as
MultDiv AddSubtr
Total Forward elimination
3n – 1 2n – 1
5n – 1 Back substitution
2n – 1 n – 1
3n – 2
Total 5n – 4
3n – 3 8n – 7
Thus, as n increases, the effort is much, much less than for a full matrix solved with Gauss elimination which is proportional to n
3
.
8.2
The equations can be expressed in a format that is compatible with graphing x
2
versus x
1
:
6 34
6 1
3 5
.
1 2
1 2
+ −
= +
= x
x x
x
which can be plotted as
1 2
3 4
5 6
7 8
9 10
2 4
6 8
10 12
Thus, the solution is x
1
= 4, x
2
= 5. The solution can be checked by substituting it back into the equations to give
34 30
4 5
6 4
24 40
16 5
8 4
4 =
+ =
+ −
= −
= −
8.3 a
The equations can be expressed in a format that is compatible with graphing
x
2
versus
x
1
:
10 114943
. 12
11 .
1 2
1 2
+ =
+ =
x x
x x
76 which can be plotted as
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