Graphics Programs Reference
In-Depth Information
Roots of Equations
4
Find the solutionsof
f
(
x
)
=
0, where the function
f
is given
4.1
Introduction
A commonproblemencounteredinengineering analysis isthis: givenafunction
f
(
x
),
determine the values of
x
forwhich
f
(
x
)
=
0
.
The solutions(values of
x
) areknown as
the
roots
of the equation
f
(
x
)
.
Before proceeding further, it might be helpfultoreview the conceptofa
function
.
The equation
=
0, or the
zeroes
of the function
f
(
x
)
=
y
f
(
x
)
containsthree elements: an input value
x
, an output value
y
and the rule
f
for comput-
ing
y
. The functionissaid to be givenif the rule
f
isspecified. Innumericalcomputing
the rule is invariablyacomputeralgorithm. It may be a function statement, such as
f
(
x
)
=
cosh(
x
) cos(
x
)
−
1
oracomplexprocedurecontaining hundredsor thousandsoflines of code.Aslong
as the algorithmproduces an output
y
for each input
x
, itqualifies as a function.
The roots of equations may be realor complex. The complexroots are seldom
computed,since theyrarely have physicalsignificance.Anexceptionis the polynomial
equation
a
1
x
n
a
2
x
n
−
1
+
+···+
+
a
n
+
1
=
a
n
x
0
where the complexroots may be meaningful(as in the analysisofdamped vibrations,
for example).For the time being, we will concentrate on finding the real roots of
equations.Complex zeroes of polynomials aretreatednear the end of thischapter.
143
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