Graphics Programs Reference
In-Depth Information
formulainEq. (4.3) with
m
f
(
x
i
)
f
(
x
i
)
x
i
+
1
=
x
i
−
where
m
is the multiplicity of the root (
m
2inthis problem).Aftermaking the
change in the above program, weobtained the result in 5 iterations.
=
4.6
Systems of Equations
Introduction
Up to this point, weconfined our attention to solving the single equation
f
(
x
)
=
0
.
Let us now consider the
n
-dimensional version of the same problem, namely
f
(
x
)
=
0
or, using scalar notation
f
1
(
x
1
,
x
2
,...,
x
n
)
=
0
f
2
(
x
1
,
x
2
,...,
x
n
)
=
0
(4.4)
.
f
n
(
x
1
,
x
2
,...,
x
n
)
=
0
The solution of
n
simultaneous, nonlinear equations is amuch moreformidable task
thanfinding the root of a single equation. The trouble is the lack of areliablemethod for
bracketing the solutionvector
x
. Therefore, wecannot provide the solutionalgorithm
with a guaranteedgood starting valueof
x
, unless such avalue issuggestedbythe
physics of the problem.
The simplest and the most effective meansofcomputing
x
is the Newton-
Raphsonmethod. It works well with simultaneousequations, provided that it issup-
pliedwith agood starting point. There are othermethodsthathave betterglobalcon-
vergence characteristics, but all of themare variants of the Newton-Raphsonmethod.
Newton-Raphson Method
In order to derive the Newton-Raphsonmethod forasystem of equations, westart
with the Taylor series expansion of
f
i
(
x
) about the point
x
:
n
∂
f
i
x
2
)
f
i
(
x
+
x
)
=
f
i
(
x
)
+
x
j
x
j
+
O
(
(4.5a)
∂
j
=
1
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