In newton raphson method if x not is the initial approximation the first approximation is
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A method for finding the roots of an “arbitrary” function that uses the derivative was first circulated
by Isaac Newton in 1669. John Wallis published Newton’s method in 1685, and in 1690 Joseph
Raphson (1648–1715) published an improved version, essentially the form in which we use it today.
The idea is as follows. We are given a function:
f(x) = x
3 − 4x
2 − x + 2.
Even a rough graph shows that this function has three roots: near −1, 1, and 4 (see figure 1).
Those are not the exact values of the roots. The value of f(−1) is −2. We shall have to move a
little to the right of −1. To see how far we should move, we note that the slope of our function
at x = −1 is f
0
(−1) = 10. Our function does not have a constant slope between (−1, −2) and the
root we are seeking, but it looks as if the slope does not change by too much. If we increase x
by 1/5, the y value should increase by about 2. Our next guess for the location of the root is at
x = −1 + 1/5 = −0.8 (see figure 2).
Again, we are not quite there: f(−0.8) = −0.272. Our slope has changed slightly. At x = −0.8
it is only f
0
(−0.8) = 7.32. To increase the y value by 0.272 (∆ y = 0.272), we need to increase
x by approximately 0.272/7.32 = 0.037158 [∆ x ≈ ∆ y/f0
(x)]. The next guess is pretty good:
f(−0.8 + 0.037158) = −0.0088, and so the root is close to x = −0.762842.
What we have found is an iterative procedure that will get us progressively closer to our root. If
xk was our last guess and neither f(xk) nor f
0
(xk) is zero, then the next approximation will be
xk+1 = xk −
f(xk)
f
0(xk)
. (1)
This kind of iteration is easily programmed. Starting with x1 = −1, the successive iterations (with
ten-digit accuracy) are
x2 = −0.8,
x3 = −0.7628415301,
x4 = −0.7615586962,
x5 = −0.7615571818,
x6 = −0.7615571818.
1
Newton-Raphson Method 2
Figure 1: f(x) = x
3 − 4x
2 − x + 2.
Figure 2: A close-up of f(x) = x
3 − 4x
2 − x + 2.
Appendix to A Radical Approach to Real Analysis 2
nd edition. c 2006 David M. Bressoud
Newton-Raphson Method 3
Figure 3: f(x) = x/p
|x|.
This is as close as we are going to get to the root using a ten-digit decimal approximation. This is the Newton–Raphson method.
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