07 Aug 2017

Newton’s method can be computationally heavy if we cannot easily find what \(f’(x)\) is. A remedy to this is the secant method. Where we use the definition of the derivative to make an approximation for \(f’(x)\).

In other words, this is an alternative way to find \(f(x) = 0\) if you are given initial approximations of \(p_{o}\) and \(p_{1}\).

\[f’(p_{n-1}) = \lim_{x \to p_{n-1}} \dfrac{f(x) - f(p_{n-1})}{x - p_{n-1}}\]

If \(p_{n-2}\) is close to \(p_{n-1}\) then:

\[ f’(p_{n-1}) \approx \dfrac{f(p_{n-2})-f(p_{n-1})}{p_{n-2} - p_{n-1}} \]

\[= \dfrac{f(p_{n-1}) - f(p_{n-2})}{p_{n-1} - p_{n-2}} \]

Using the approximation for \(f’(p_{n-1})\) in Newton’s Formula gives:

\[ p_{n} = p_{n-1} - \dfrac{f(p_{n-1})(p_{n-1} - p_{n-2})}{f(p_{n-1} - f(p_{n-2})} \]

```
INPUT initial approximation p_o;tolerance TOL;maximum number of iterations N_o
OUTPUT approximate solution p or message of failure.
Step 1 Set i = 2;
q_0 = f(p_0);
q_1 = f(p_q)
Step 2 While i =< N_o do steps 3-6
Step 3 Set p = p_1 - q_1(p_q - p_o)/(q_1-q_o). (compute p_i)
Step 4 Set if |p - p_1 | < TOL
OUTPUT (p); (The procecure was successful.)
STOP.
Step 5 Set i = i + 1;
q = f(p)
Step 6 If q times q_1 < 0 then set p_o = p_1:
q_o = q_1
Step 7 Set p_1 = p;
q_1 = q
Step 8 OUTPUT ('Method failed after N_o iterations')
STOP
```

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