082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

12.4 The Newton--Raphson method for solving equations 421
Animprovedestimateofthevalueoftherootis2.73(2d.p.).Themethodisusedagain,
takingx 1
= 2.73 asthe initial approximation:
x 1
= 2.73 f(x 1
) = 0.441 f ′ (x 1
) = 11.439
x 2
= 2.73 − 0.441
11.439 = 2.691
An improved estimateisx = 2.69 (2d.p.). The method isused again:
So
x 1
= 2.69 f (x 1
) = −0.007 f ′ (x 1
) = 10.948
x 2
= 2.69 − (−0.007)
10.948 = 2.691
Thereisnochangeinthevalueoftheapproximaterootandsototwodecimalplacesthe
rootof f(x) =0isx =2.69.
The calculation can be performed inMATLAB ® usingtherootsfunction:
roots([1 -2 0 -5])
which will produce the real root of 2.69, agreeing with our numerical calculation using
the Newton–Raphson method. The two complex roots of the equation will also be returned.Technicalcomputinglanguagesmakeuseofavarietyofnumericalmethodsand
to some extent the user has to take it on trust that they are correctly implemented and
testedso thatthey always produce the correctresult.
Theprevious examples illustrate the generalNewton--Raphson formula.
Ifx = x n
is an approximate root of f (x) = 0, then an improved estimate,x n+1
, is
given by
x n+1
=x n
− f(x n )
f ′ (x n
)
TheNewton--Raphsonformulaiseasytoprograminaloopstructure.Exitfromtheloop
is usually conditional upon |x n+1
−x n
| being smaller than some prescribed very small
value. This condition shows that successive approximate roots are very close to each
other.
Engineeringapplication12.3
Seriesdiode--resistorcircuit
Consider the circuit of Figure 12.15(a). A diode is in series with a resistor with resistanceR.ThevoltageacrossthediodeisdenotedbyV
andthecurrentthroughthe
diode is denoted byI. TheI--V relationship for the diode is non-linear and is given
by
I =I s
(e 40V −1)
where I s
is the reverse saturation current of the diode. Given that the supply voltage,V
s
,is2V,I s
is10 −14 AandRis22k,calculatethesteady-statevaluesofIandV.
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