Essays on algorithms and data structures by Ali Khalilli. Binary search, numerical methods, dynamic programming, graphs, strings. Zola 2021-12-21T00:00:00+00:00 https://invariant.khalilli.ai/series/numerical-methods/atom.xml 2021-12-21T00:00:00+00:00 2021-12-21T00:00:00+00:00 alikhalilli https://invariant.khalilli.ai/blog/newton-raphson/

<p>Using the square-root finding technique as an example, this chapter will examine the Newton-Raphson approach in further detail. The in-depth examination and distinctions between <strong>Gradient Descent</strong> and <strong>Newton-Raphson</strong> will be the subject of a future season. The Newton-Raphson technique is another numerical method for determining the square root. The root of a nonlinear equation must be bracketed by two estimations using methods like the bisection technique and the false position method. Bracketing approaches are used to accomplish this. Because they reduce the interval between the two estimations in order to zero in on the equation’s root, these approaches are always convergent.</p> <link rel="stylesheet" href="https://invariant.khalilli.ai/admonition.css?h=98c1477488c4b2c9d71a" type="text/css"> <div class="my-4 flex flex-col rounded-lg bg-(--admonition-bg)" style="--admonition-bg: var(--admonition-tip);" > <div class="flex items-center rounded-t-lg bg-(--admonition-bg) p-1"> <div class="mx-2 h-4 w-4 text-[0px] [background:var(--url)_center_center_no-repeat] dark:invert" style="--url: url(icons/tip.svg);" > tip </div> <span><strong>Key idea</strong></span> </div> <div class="pl-4"><p>The root is not bracketed in the Newton-Raphson approach. When it comes to solving an equation, just one initial guess of the root is required to get the iterative process started. As a result, it might be considered an open approach. Open approaches may or may not converge, but if they do, it will be substantially faster than with bracketing.</p> </div> </div> <table><thead><tr><th></th><th>Bracketing (bisection, false position)</th><th>Newton-Raphson (open)</th></tr></thead><tbody> <tr><td>Initial guesses</td><td>two estimations</td><td>one initial guess</td></tr> <tr><td>Convergence</td><td>always convergent</td><td>may or may not converge</td></tr> <tr><td>Speed</td><td>slower</td><td>substantially faster</td></tr> </tbody></table>

2021-12-20T00:00:00+00:00 2021-12-20T00:00:00+00:00 alikhalilli https://invariant.khalilli.ai/blog/error-analysis/

<p>While solving a mathematical model using numerical approaches, we can use errors to minimize errors. When doing a numerical analysis, mistakes will inevitably occur. To address the problem of errors, we must first</p>

2021-12-08T00:00:00+00:00 2021-12-08T00:00:00+00:00 alikhalilli https://invariant.khalilli.ai/blog/bisection-method/

<p>Consider the following scenario: we have a nonlinear equation and are attempting to solve it using a computer program. For the nonlinear equation, what strategy can the CPU employ in order to compute a solution? Alternatively, from what point of view would we address that challenge in terms of efficacy?</p>

2021-12-01T00:00:00+00:00 2021-12-01T00:00:00+00:00 alikhalilli https://invariant.khalilli.ai/blog/numerical-methods-why/

<p>Have you ever wondered how computer software determines the square root of a given value?</p>