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I'm looking for some guidance on utilizing the secant method to calculate a particular value within a function. Here's the formula I'm using:

$$x_n = x_{n-1} - f(x_{n-1}) \cdot \frac{x_{n-1} - x_{n-2}}{f(x_{n-1}) - f(x_{n-2})}$$

I was suggested using the equation (f'(x) = f(x) - 400) to simplify the problem. By applying the secant method to (f'), we aim to find its root where (f'(x_n) = 0). I was led to the understanding that (f'(x_n)) and (f'(x_{n-1})) can be substituted into the secant formula.

However, I'm unsure how to determine the specific values of (f'(x_{n-1})) and (f'(x_{n-2})) without the ability to differentiate the function. Is it so that (f') is not a derivative, but rather (f(x) - 400)?

I've attempted adding -400 directly to the formula previously but encountered errors, indicating I must have made a mistake. Could anyone provide insights into the correct implementation of this modified secant method?

Bishop_1
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1 Answers1

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The secant iteration formula to solve $f(x)=y$ has the simple modification $$ x_n = x_{n-1} - [f(x_{n-1})-y] \cdot \frac{x_{n-1} - x_{n-2}}{f(x_{n-1}) - f(x_{n-2})} $$ In the function value difference in the denominator the value shifts cancel, so that that remains unmodified.

Lutz Lehmann
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