Numerical Methods Course Notes Version 0.1 (UCSD Math 174, Fall ...

54 CHAPTER 4. FINDING ROOTS
Algorithm 4: Algorithm for finding a square root using simple operations.
Input: a number
Output: its square root
sqrt(z)
(1) if z < 0 throw an error.
(2) Let x ← 1, n ← 0.
(3) while n ≤ 50
(4) Let x ← (x + z/x) /2.
(5) Let n ← n + 1.
f(xk+1)
PSfrag replacements
f(x k−1 )
f(x k )
x k
x k−1
x k+1
Figure 4.2: One iteration of the Secant method is shown for some quadratic function f(x). The
secant line through (x k−1 , f(x k−1 )) and (x k , f(x k )) is shown. It happens to be the case that
|f(x k+1 )| is smaller than |f(x k )| , i.e., x k+1 is a better guess than x k .
line at (x k , f(x k )) , which is f ′ (x k ) is approximated by the slope of the secant line passing through
(x k−1 , f(x k−1 )) and (x k , f(x k )) , which is
f(x k ) − f(x k−1 )
x k − x k−1
Thus the iterate x k+1 is the root of this secant line. That is, it is a root to the equation
f(x k ) − f(x k−1 )
x k − x k−1
(x − x k ) = y − f(x k ).