Name: newton’s method for finding roots
Uploaded: 2026-01-12T12:24:54.000Z
Duration: 1 min 30 s
Channel: JoVE

Newton Yöntemi, gerçek değerli, türevlenebilir fonksiyonların yaklaşık köklerini bulmak için yinelemeli bir tekniktir.Standart cebirsel yöntemler için çok karmaşık olan doğrusal olmayan denklemleri çözmeye yardımcı olur.Örneğin, Newton Yöntemi, araba kredisi geri ödemesini modelleyen doğrusal olmayan bir denklemden faiz oranını tahmin edebilir. Bu denklemler y eşit f x olarak yazılır ve formülü geliştirmek için genellikle grafiksel olarak gösterilir.Süreç, kökün kabaca tahminine dayalı bir ilk tahminle başlar.Tahmin edilen noktada, fonksiyonun eğimi kullanılarak bir teğet çizgisi çizilir. Bu çizginin x-kesişmesi yeni bir tahmin haline gelir ve görsel olarak gerçek köke daha yakındır.Bu yeni tahmin doğrusal yaklaşımdan gelir. Başlangıç tahmininden fonksiyonun değerinin o tahmindeki türevine bölünmesine eşittir.Bu süreç yeni tahmin kullanılarak tekrarlanıyor. Her tekrarla, değerler genellikle köke daha da yaklaşır.Bu genel formüle yol açar: yeni tahmin, önceki tahminin fonksiyon değerinin türevine bölünmesine eşittir.Her adım yaklaşımı daha da geliştirir ve Newton Yöntemi'ni doğrusal olmayan denklemleri çözmek için etkili bir yineleme aracı haline getirir.

newton yöntemi

Newton’s Method

Newton Y&#xF6;ntemi, &#xF6;zellikle analitik &#xE7;&#xF6;z&#xFC;mlerin uygulanabilir olmad&#x131;&#x11F;&#x131; durumlarda, ger&#xE7;el say&#x131;l&#x131; ve t&#xFC;revlenebilir fonksiyonlar&#x131;n k&#xF6;klerini yakla&#x15F;&#x131;k olarak bulmak i&#xE7;in kullan&#x131;lan g&#xFC;&#xE7;l&#xFC; bir yinelemeli tekniktir. Bu yakla&#x15F;&#x131;m, denklemlerin geleneksel cebirsel y&#xF6;ntemlerle ele al&#x131;namayacak kadar karma&#x15F;&#x131;k olabildi&#x11F;i bilimsel hesaplama, m&#xFC;hendislik ve finans alanlar&#x131;nda yayg&#x131;n bi&#xE7;imde kullan&#x131;lmaktad&#x131;r. Y&#xF6;ntem, fonksiyonun t&#xFC;revinden yararlanarak ba&#x15F;lang&#x131;&#xE7;taki bir tahmini yinelemeli bir s&#xFC;re&#xE7; i&#xE7;inde iyile&#x15F;tirmeye, b&#xF6;ylece ger&#xE7;ek &#xE7;&#xF6;z&#xFC;me kademeli olarak yakla&#x15F;maya dayan&#x131;r. Matematiksel olarak a&#x15F;a&#x11F;&#x131;daki &#xF6;zyinelemeli ba&#x11F;&#x131;nt&#x131; ile verilir:<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:50%;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19849/19849_PT_Equation_1_editor_19849.svg" alt="19849_PT_Equation_1.svg" /></figure>Burada:x_n = k&#xF6;k&#xFC;n mevcut yakla&#x15F;&#x131;k de&#x11F;erif(x_n) = x_n noktas&#x131;ndaki fonksiyon de&#x11F;erif&#x2032;(x_n) = x_n noktas&#x131;ndaki fonksiyonun t&#xFC;revix_(n+1) = mevcut tahmin kullan&#x131;larak hesaplanan bir sonraki yakla&#x15F;&#x131;k de&#x11F;er.Ba&#x15F;lang&#x131;&#xE7; tahmini yeterince uygun se&#xE7;ildi&#x11F;i ve fonksiyon d&#xFC;zg&#xFC;n davrand&#x131;&#x11F;&#x131; s&#xFC;rece, her yineleme ger&#xE7;ek k&#xF6;ke giderek daha fazla yakla&#x15F;&#x131;r.Newton Y&#xF6;ntemi&#x2019;nin pratik uygulamalar&#x131;ndan biri, &#xF6;rne&#x11F;in do&#x11F;rusal olmayan geri &#xF6;deme denklemlerinden faiz oranlar&#x131;n&#x131;n tahmin edilmesi gibi bir uygulamada kullan&#x131;labilir. Bu t&#xFC;r ba&#x11F;lamlarda denklemler a&#xE7;&#x131;k &#xE7;&#xF6;z&#xFC;mlere elveri&#x15F;li olmayabilir; ancak uygun bir ba&#x15F;lang&#x131;&#xE7; tahmini se&#xE7;ildi&#x11F;inde Newton Y&#xF6;ntemi, az say&#x131;da hesaplama ad&#x131;m&#x131;yla verimli bi&#xE7;imde bir k&#xF6;ke yak&#x131;nsayabilmektedir.Verimlili&#x11F;i ve h&#x131;zl&#x131; yak&#x131;nsama &#xF6;zellikleri nedeniyle Newton&#x2013;Raphson y&#xF6;ntemi, uygulamal&#x131; matematik ve hesaplama bilimleri alan&#x131;nda k&#xF6;k bulma ve denklem &#xE7;&#xF6;zme amac&#x131;yla kullan&#x131;lan en g&#xFC;&#xE7;l&#xFC; tekniklerden biri olmaya devam etmektedir.Bununla birlikte, Newton Y&#xF6;ntemi her durumda yak&#x131;nsamay&#x131; garanti etmez. E&#x11F;er f&#x2032;(x_n) s&#x131;f&#x131;r ya da s&#x131;f&#x131;ra &#xE7;ok yak&#x131;nsa, g&#xFC;ncelleme form&#xFC;l&#xFC; k&#xFC;&#xE7;&#xFC;k bir say&#x131;ya b&#xF6;lmeye yol a&#xE7;arak say&#x131;sal karars&#x131;zl&#x131;&#x11F;a neden olabilir. Ayr&#x131;ca, uygunsuz ba&#x15F;lang&#x131;&#xE7; tahminleri y&#xF6;ntemin bir k&#xF6;ke yakla&#x15F;mak yerine &#x131;raksamas&#x131;na ya da bir d&#xF6;ng&#xFC;ye girmesine yol a&#xE7;abilir. Ayr&#x131;ca, b&#xFC;k&#xFC;m noktalar&#x131;, yerel u&#xE7; noktalar veya t&#xFC;revinde s&#xFC;reksizlik bulunan fonksiyonlarda y&#xF6;ntem k&#xF6;ke yakla&#x15F;amayabilir ya da istenmeyen bir &#xE7;&#xF6;z&#xFC;me yak&#x131;nsayabilir. Bu nedenle, Newton Y&#xF6;ntemi&#x2019;nin ba&#x15F;ar&#x131;l&#x131; bir bi&#xE7;imde uygulanabilmesi i&#xE7;in fonksiyonun dikkatle analiz edilmesi ve uygun bir ba&#x15F;lang&#x131;&#xE7; tahmininin se&#xE7;ilmesi kritik &#xF6;neme sahiptir.

Newton Yöntemi

Newton Y&#xF6;ntemi, &#xF6;zellikle analitik &#xE7;&#xF6;z&#xFC;mlerin uygulanabilir olmad&#x131;&#x11F;&#x131; durumlarda, ger&#xE7;el say&#x131;l&#x131; ...

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JoVE Core

Calculus

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Unit

Birim

Applications of Differentiation

Türevin Uygulamaları

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

newton’s method for finding roots

Newton’s Method is an iterative way to find the roots of real-valued, differentiable functions. A root is the value of x that makes a function equal to zero. This method is useful when an exact algebraic answer is hard to find.

The process starts with an initial guess. Newton’s Method then uses the function value and its derivative, which is the slope of the function at that point, to build a better estimate. The next approximation is found from the recursive formula x n+1 = x n - f(x n) / f'(x n). Each step can move the estimate closer to the true root when the first guess is reasonable and the function behaves well.

This method is widely used in scientific computing, engineering, and finance. One practical example is financial modeling, where it can help estimate interest rates from nonlinear repayment equations. These problems often do not have a simple explicit solution, so Newton’s Method can reach a root in only a few steps.

Newton’s Method does not always converge. If the derivative f'(x n) is zero or very close to zero, the formula can become unstable because it divides by a small number. A poor starting guess can also make the method diverge or cycle instead of approaching a root.

The method can also fail near inflection points, local extrema, or places where the derivative is discontinuous. In some cases, it may converge to the wrong solution. That is why a careful look at the function and a well-chosen starting point are important for successful root finding.

Newton’s Method for Finding Roots

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Absolute and local extreme values show the highest and lowest points on a function graph. Absolute maximum and absolute minimum values describe the highest and lowest values across the entire domain. Local maximum and local minimum values describe the highest and lowest values within a smaller part of the graph. Periodic functions such as sine and cosine can have extreme values at many points. Their repeating pattern creates multiple local extrema. In these cases, the derivative becomes zero...

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rolle’s theorem

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Rolle’s Theorem explains when a differentiable function must have a flat tangent inside an interval. It applies to a real-valued function on a closed interval, which means the endpoints are included. The function must also be continuous on the whole interval and differentiable on the open interval, where the interior points lie. The theorem uses one more key condition. The function must have the same value at both endpoints. When that happens, Rolle’s Theorem guarantees at least one point...

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first-derivatives-and-the-shape-of-a-graph

Using First Derivatives to Find Graph Changes

The first derivative shows how a function changes across its domain. In calculus, the derivative is written as f'(x). It tells us whether the graph is rising, falling, or turning at a given point. A function is increasing on an interval when its derivative is positive. That means the slope of the tangent line is positive. A function is decreasing on an interval when its derivative is negative. In that case, the tangent line slopes downward. These sign changes help describe the overall shape...

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second derivatives and the shape of a graph

second-derivatives-and-the-shape-of-a-graph

Using Second Derivatives to Read Curvature

The second derivative shows how a graph bends. It gives key information about curvature and how that curvature changes over an interval. These ideas help with graph analysis in math and in real-world settings such as road elevation, population growth, and economic trends. A function f(x) is concave upward on an interval when its graph lies above all of its tangent lines. In this case, the second derivative is positive. That means the slope of the tangent line is increasing, and the graph bends...

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Classifying Critical Points with the Second Derivative

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Using Derivatives to Read a Graph

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guidelines for sketching a curve

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Rational Functions with Slant Asymptotes

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optimization problems

Pipe Length Around a Corner

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application of differentiation to business

application-of-differentiation-to-business

Maximizing Bakery Revenue with Calculus

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newton’s method

Newton’s Method is an iterative way to find the roots of real-valued, differentiable functions. A root is the value of x that makes a function equal to zero. This method is useful when an exact algebraic answer is hard to find. The process starts with an initial guess. Newton’s Method then uses the function value and its derivative, which is the slope of the function at that point, to build a better estimate. The next approximation is found from the recursive formula x n+1 = x n - f(x n) /...

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Antiderivative Graph Shape and Slope

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application of antiderivatives: linear motion

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Finding Car Stopping Distance with Integration

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key terms and concepts

Learn how Newton’s Method uses derivatives and repeated guesses to find roots of equations, plus why a good starting point matters for convergence.

Newton’s Method for Finding Roots | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Newton’s Method for Finding Roots

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