 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2019/discrete-mathematics-continuous-mathematics-and-probability-resit/5/a/i/solution

x_{0}, x_{1}

. Replace the function near its zero of a function by the secant line through

(x_{k - 1}, f (x_{k - 1}))

and

(x_{k}, f (x_{k}))

; the zero of that line is the next approximation:

x_{k + 1} = x_{k} - f (x_{k}) \frac{x _{k} - x _{k - 1}}{f ( x _{k} ) - f ( x _{k - 1} )} .

Stop immediately if

f (x_{k}) = 0

; otherwise the denominator must be nonzero. Each further step needs one new function evaluation and no derivative. There is no automatic bracketing or global convergence guarantee.^[ref]

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