27.1 ALGEBRAIC AND TRANSCENDENTAL EQUATIONS
nxn f(xn)
1 1.7 5.42
2 1.545 01 1.03
3 1.498 87 7. 20 × 10 −^2
4 1.495 13 4. 49 × 10 −^4
5 1.495 106 40 2. 6 × 10 −^8
6 1.495 106 40 —Table 27.4 Successive approximations to the root of (27.1) using the Newton–
Raphson method.of (notionally) constructing the chord between two points on the curve off(x)
againstx, the tangent to the curve is notionally constructed at each successive
value ofxn, and the next value,xn+1, is taken as the point at which the tangent
cuts the axisf(x) = 0. This is illustrated in graph (d) of figure 27.2.
If thenth value isxn, the tangent to the curve off(x) at that point has slopef′(xn) and passes through the pointx=xn,y=f(xn). Its equation is thus
y(x)=(x−xn)f′(xn)+f(xn). (27.10)The value ofxat whichy= 0 is then taken asxn+1; thus the conditiony(xn+1)=0
yields, from (27.10), the iteration scheme
xn+1=xn−f(xn)
f′(xn). (27.11)
This is theNewton–Raphson iteration formula. Clearly, ifxnis close toξthenxn+1
is close toxn, as it should be. It is also apparent that if any of thexncomes close
to a stationary point off,sothatf′(xn) is close to zero, the scheme is not going
to work well.
For our standard example, (27.11) becomesxn+1=xn−x^5 n− 2 x^2 n− 3
5 x^4 n− 4 xn=4 x^5 n− 2 x^2 n+3
5 x^4 n− 4 xn. (27.12)
Again taking a starting value ofx 1 =1.7, we obtain in succession the entries
in table 27.4. The different values are given to an increasing number of decimal
places as the calculation proceeds;f(xn) is also recorded.
It is apparent that this method is unlike the previous ones in that the increasein accuracy of the answer is not constant throughout the iterations but improves
dramatically as the required root is approached. Away from the root the behaviour
of the series is less satisfactory, and from its geometrical interpretation it can be
seen that if, for example, there were a maximum or minimum near the root then
the series could oscillate between values on either side of it (instead of ‘homing
in’ on the root). The reason for the good convergence near the root is discussed
in the next section.