[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-326413-105":59,"doc-detail-326413-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","the-newton-raphson-method-section-2-introduction","The Newton-Raphson Method - Section 2 Introduction","","Newton-Raphson method notes explain how to solve nonlinear equations numerically using linear (tangent line) approximation. Starting from an initial estimate x0, the method iteratively produces improved approximations xn+1 = xn − f(xn)/f'(xn) by approximating the function near the current estimate. The document adds a geometric interpretation via tangent line x-intercepts and analyzes convergence, discussing when smoothness is needed, and how large second derivatives or derivatives near zero can cause failure.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/the-newton-raphson-method-section-2-introduction/326413/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/the-newton-raphson-method-section-2-introduction/326413.png","ImageObject",300,407,{"name":92,"@type":93},"Ophelia","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-02","2026-09-21",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",12,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What is the Newton-Raphson method used for?","Question",{"text":113,"@type":114},"It is used to solve equations numerically by finding roots of f(x)=0 through iterative refinement of an estimate.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How does the Newton-Raphson iteration update the current estimate?",{"text":118,"@type":114},"Given a current estimate xn, the next estimate is xn+1 = xn − f(xn)/f'(xn), derived from linear (tangent line) approximation near xn.",{"name":120,"@type":111,"acceptedAnswer":121},"Why can Newton’s method fail to converge?",{"text":122,"@type":114},"Convergence can break down when the function is not sufficiently smooth near the root, when second derivatives are large, or when f'(xn) is close to 0, leading to poor predictions of the next step.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},326413,1790530482,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":145,"read_time":146,"preview_page_count":4},7971461741311,"https://ap-avatar.wpscdn.com/avatar/74000253aff267980c6?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779345379180704826","The Newton-Raphson Method  \n1 Introduction  \nThe Newton-Raphson method, or Newton Method, is a powerful technique for solving equations numerically. Like so much of the di􀀋erential calculus, it is based on the simple idea of linear approximation. The Newton Method, properly used, usually homes in on a root with devastating e􀀎ciency.  \nThe essential part of these notes is Section 2.1, where the basic formula is derived, Section 2.2, where the procedure is interpreted geometrically, and|of course|Section 6, where the problems are. Peripheral but perhaps interesting is Section 3, where the birth of the Newton Method is described.  \n2 Using Linear Approximations to Solve Equations  \nLet f (x) be a well-behaved function, and let r be a root of the equation f (x) = 0 . We start with an estimate x0 of r. From x0 , we produce an improved|we hope|estimate x 1 . From x 1 , we produce a new estimate x2 . From x2 , we produce a new estimate x3 . We go on until we are `close enough' to r|or until it becomes clear that we are getting nowhere.  \nThe above general style of proceeding is called iterative. Of the many iterative root-􀀌nding procedures, the Newton-Raphson method, with its combination of simplicity and power, is the most widely used. Section 2.4 describes another iterative root-􀀌nding procedure, the Secant Method.  \nComment. The initial estimate is sometimes called x 1 , but most mathematicians prefer to start counting at 0 .  \nSometimes the initial estimate is called a \\guess.\" The Newton Method is usually very very good if x0 is close to r, and can be horrid if it is not. The \\guess\" x0 should be chosen with care.  \n2.1 The Newton-Raphson Iteration  \nLet x0 be a good estimate of r and let r = x0 + h. Since the true root is r , and h = r 􀀀 x0 , the number h measures how far the estimate x0 is from the truth.  \nSince h is `small,' we can use the linear (tangent line) approximation to conclude that  \n0 = f(r) = f(x0 + h) 􀀙 f (x0 )+ hf0 (x0 ) ; and therefore, unless f 0 (x0 ) is close to 0,  \nf (x0 ) h 􀀙 􀀀 f 0 (x0 ) :  \nIt follows that  \nr = x0 + h 􀀙 x0 􀀀 ff0(~~(~~xx00)~~)~~ : Our new improved (?) estimate x 1 of r is therefore given by  \nx 1 = x0 􀀀 ff0(~~(~~xx00)~~)~~ :  \nThe next estimate x2 is obtained from x 1 in exactly the same way as x 1 was obtained from x0 :  \nf (x1 ) x2 = x 1 􀀀 f 0 (x1 ) :  \nContinue in this way. If x n is the current estimate, then the next estimate xn+1 is given by  \nxn+1 = xn 􀀀 ff0((xxnn)) (1)  \n2.2 A Geometric Interpretation of the Newton-Raphson Iteration  \nIn the picture below, the curve y = f (x) meets the x-axis at r. Let a be the current estimate of r. The tangent line to y = f (x) at the point (a; f(a)) has equation  \ny = f (a)+(x 􀀀 a)f 0 (a):  \nLet b be the x-intercept of the tangent line. Then  \nf (a) b = a 􀀀 f 0 (a) :  \nr b a  \nCompare with Equation 1: b is just the `next' Newton-Raphson estimate of r. The new estimate b is obtained by drawing the tangent line at x = a, and then sliding to the x-axis along this tangent line. Now draw the tangent line at (b; f(b)) and ride the new tangent line to the x-axis to get a new estimate c. Repeat.  \nWe can use the geometric interpretation to design functions and starting points for which the Newton Method runs into trouble. For example, by putting a little bump on the curve at x = a we can make b 􀀍y far away from r. When a Newton Method calculation is going badly, a picture can help us diagnose the problem and 􀀌x it.  \nIt would be wrong to think of the Newton Method simply in terms of tangent lines. The Newton Method is used to 􀀌nd complex roots of polynomials, and roots of systems of equations in several variables, where the geometry is far less clear, but linear approximation still makes sense.  \n2.3 The Convergence of the Newton Method  \nThe argument that led to Equation 1 used the informal and imprecise symbol 􀀙 . We probe this argument for weaknesses.  \nNo numerical procedure works for all equations. For example, let f(x) = x2 + 17 if x  1, and le","cbCaiqTA8Iy53NDG","https://ap.wps.com/l/cbCaiqTA8Iy53NDG","pdf",181522,11,"English","# 1 Introduction\n# 2 Using Linear Approximations to Solve Equations\n## 2.1 The Newton-Raphson Iteration\n## 2.2 A Geometric Interpretation of the Newton-Raphson Iteration\n## 2.3 The Convergence of the Newton Method","[{\"question\":\"What is the Newton-Raphson method used for?\",\"answer\":\"It is used to solve equations numerically by finding roots of f(x)=0 through iterative refinement of an estimate.\"},{\"question\":\"How does the Newton-Raphson iteration update the current estimate?\",\"answer\":\"Given a current estimate xn, the next estimate is xn+1 = xn − f(xn)/f'(xn), derived from linear (tangent line) approximation near xn.\"},{\"question\":\"Why can Newton’s method fail to converge?\",\"answer\":\"Convergence can break down when the function is not sufficiently smooth near the root, when second derivatives are large, or when f'(xn) is close to 0, leading to poor predictions of the next step.\"}]","The Newton-Raphson Method - Section 2 Introduction | PDF",1789974032,28]