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However, the extra computations required for each step can slow down the overall performance relative to Newton's method, particularly if or its derivatives are computationally expensive to evaluate.
Using this approximation would result in something like the secant method whose convergence is slower than that of Newton's method.
The essence of Vieta's method can be found in the work of the Persian mathematician Sharaf al-Din al-Tusi, while his successor Jamshīd al-Kāshī used a form of Newton's method to solve (Ypma 1995).
A special case of Newton's method for calculating square roots was known since ancient times and is often called the Babylonian method.
To overcome this problem one can often linearise the function that is being optimized using calculus, logs, differentials, or even using evolutionary algorithms, such as the stochastic funnel algorithm.
Good initial estimates lie close to the final globally optimal parameter estimate.