An exploration of a fundamental numerical optimization technique, with a focus on its geometrical interpretation
Gradient descent is widely regarded as one of the fundamental numerical optimization techniques, and the Newton-Raphson method stands out as a significant component within this domain. This method possesses notable qualities in terms of its simplicity, elegance, and computational power, warranting an in-depth exploration.
Within this article, our objective is to elucidate the geometric principles underlying the functioning of the Newton-Raphson method. This elucidation aims to provide readers with an intuitive understanding of its mechanics and dispel any potential complexities associated with its mathematical foundations.
Subsequently, in order to establish a robust mathematical framework for our discussion, we will delve into the mathematical intricacies of the method, accompanied by a practical implementation in the Python programming language.
Following this presentation, we will distinguish between the two primary applications of the Newton-Raphson method: root finding and optimization. This differentiation will clarify the distinct contexts in which the method finds utility.
To conclude, we will conduct a comparative analysis between the Newton-Raphson method and the gradient descent method, offering insights into their respective strengths and weaknesses.
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Fundamentally, the Newton-Raphson method is an iterative procedure designed for the numerical determination…
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