Graphics Programs Reference
In-Depth Information
18.
+120 V
-120 V
1
Ω
Ω
50
30
3
2
Ω
25
Ω
20
4
Ω
30
Determine the loop currents
i
1
to
i
4
in the electrical network shown.
19.
Consider the
n
simultaneousequations
Ax
=
b
, where
−
n
1
j
)
2
A
i j
=
(
i
+
b
i
=
A
i j
,
i
=
0
,
1
,...,
n
−
1,
j
=
0
,
1
,...,
n
−
1
j
=
0
1
T
.Write aprogram that solves these equations
for any given
n
(pivoting is recommended).Runthe programwith
n
11
The solutionis
x
=
···
=
2, 3 and 4,
and commenton the results.
∗
2.6
Matrix Inversion
Computing the inverse of amatrix and solving simultaneousequations are related
tasks. The most economical way to invert an
n
×
n
matrix
A
istosolve the equations
AX
=
I
(2.33)
×
×
where
I
is the
n
n
, will be the
inverse of
A
. The proof issimple:afterwe premultiplyboth sides of Eq. (2.33) by
A
−
1
wehave
A
−
1
AX
n
identitymatrix. The solution
X
, also of size
n
A
−
1
I
, which reduces to
X
A
−
1
.
Inversion of largematrices shouldbe avoidedwheneverpossible due its high cost.
As seen fromEq. (2.33), inversion of
A
isequivalent to solving
Ax
i
=
=
=
n
,
where
b
i
is the
i
th column of
I
. If LU decompositionisemployedinthe solution, the
solutionphase (forward and back substitution) must be repeated
n
times, once for
each
b
i
.Since the cost of computationis proportionalto
n
3
for the decomposition
phase and
n
2
for each vector of the solutionphase, the cost of inversionisconsiderably
moreexpensivethan the solution of
Ax
b
i
,
i
=
1
,
2
,...,
=
b
(single constant vector
b
).
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