Explain The Secant Method With An Example?

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The secant method of root-finding is an iterative method that is a variation on Newton's iteration formula. Newton's iterator is   xn+1 = xn - f(xn)/f'(xn) The secant method replaces f'(xn) with a finite difference, so two starting values are required.   f'(xn) ≈ (f(xn) - f(xn-1))/(xn - xn-1) So the iterator becomes   xn+1 = xn - f(xn)*((xn - xn-1)/(f(xn) - f(xn-1))) Example   Suppose we have f(x) = x^2 - 2, for which we would like to find a root. (We know that one root is √2 ≈ 1.4142.) Further suppose that we want to start with x0=1 and x1=2.   x2 = x1 - (x1^2 - 2)*(x1 - x0)/((x1^2 - 2) - (x0^2 - x))    (the secant method iterator)   x2 = x1 - (x1^2 - 2)/(x1 + x0)    (factor the denominator & cancel the common factor from the numerator)   x2 = (2+x0*x1)/(x0+x1)    (we simplify the iterator first, so the steps below are not so difficult)   Substituting our starting values, we get   x2 = (2+1*2)/(1+2) = 4/3    (≈1.3333)   x3 = (2 + (2)*(4/3))/(2 + 4/3) = 7/5    (= 1.4000)   x3 = (2 + (4/3)*(7/5))/(4/3 + 7/5) = 58/41    (≈1.4146)   x4 = (2 + (7/5)*(58/41))/(7/5 + 58/41) = 816/577    (≈1.4142)
Under some conditions, the method will not converge, so various refinements have been suggested by different authors.

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