the geometry is far less clear, but linear approximation still makes sense 2 3 The Convergence of the Newton Method The argument that led to Equation 1 used
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[PDF] Newtons Method - Philadelphia University
Context Newton's (or the Newton-Raphson) method is one of the most powerful The Taylor series derivation of Newton's method points out the importance of
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Let p be a root of the function f ∈ C2[a, b] (i e f(p)=0), and p0 be an approximation to p If p0 is su ciently close to p, the expansion of f(p) as a Taylor series in
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dx2 ) and f(k)(x) is the kth derivative of f evaluated at x As we have function f : R R by a simpler function is to use the Taylor series representation for This method is also known as the Newton-Raphson method and is based on the approx-
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Performance of Numerical Optimization Routines Derivation of the Newton- Raphson Method • The Taylor polynomial for f(x) is • As the function approaches a
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Newton-Raphson formula consists geometrically of extending the tangent line at familiar Taylor series expansion of a function in the neighborhood of a point,
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3 New iterative methods Let f : X → R, X ⊂ R is a scalar function then by using Taylor series expansion one can obtain generalized Newton Raphson's method:
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The Newton-Raphson Method
1 Introduction
The Newton-Raphson method, or Newton Method, is a powerful technique for solving equations numerically. Like so much of the dierential calculus, it is based on the simple idea of linear approximation. The Newton Method, properly used, usually homes in on a root with devastating eciency. The 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.2 Using Linear Approximations to Solve Equa-
tions Letf(x) be a well-behaved function, and letrbe a root of the equation f(x) = 0. We start with an estimatex 0 ofr.Fromx 0 , we produce an improved|we hope|estimatex 1 .Fromx 1 , we produce a new estimate x 2 .Fromx 2 , we produce a new estimatex 3 . We go on until we are `close enough' tor|or until it becomes clear that we are getting nowhere. The above general style of proceeding is callediterative.Of the many it- erative root-nding procedures, the Newton-Raphson method, with its com- bination of simplicity and power, is the most widely used. Section 2.4 de- scribes another iterative root-nding procedure, theSecant Method.Comment.The initial estimate is sometimes calledx
1 , but most mathe- maticians prefer to start counting at 0. Sometimes the initial estimate is called a \guess." The Newton Method is usually very very good ifx 0 is close tor,andcanbehorridifitisnot.The \guess"x
0 should be chosen with care. 12.1 The Newton-Raphson Iteration
Letx 0 be a good estimate ofrand letr=x 0 +h.Sincethetruerootisr, andh=r-x 0 ,thenumberhmeasures how far the estimatex 0 is from the truth. Sincehis `small,' we can use the linear (tangent line) approximation to conclude that0=f(r)=f(x
0 +h)f(x 0 )+hf 0 (x 0 and therefore, unlessf 0 (x 0 )iscloseto0, h-f(x 0 f 0 (x 0It follows that
r=x 0 +hx 0 -f(x 0 f 0 (x 0Our new improved (?) estimatex
1 ofris therefore given by x 1 =x 0 -f(x 0 f 0 (x 0The next estimatex
2 is obtained fromx 1 in exactly the same way asx 1 was obtained fromx 0 x 2 =x 1 -f(x 1 f 0 (x 1Continue in this way. Ifx
n is the current estimate, then the next estimate x n+1 is given by x n+1 =x n -f(x n f 0 (x n (1)2.2 A Geometric Interpretation of the Newton-Raphson It-
eration In the picture below, the curvey=f(x) meets thex-axis atr.Letabe the current estimate ofr. The tangent line toy=f(x)atthepoint(a;f(a)) has equation y=f(a)+(x-a)f 0 (a):Letbbe thex-intercept of the tangent line. Then
b=a-f(a) f 0 (a): 2 abrCompare with Equation 1:bis just the `next' Newton-Raphson estimate of r. The new estimatebis obtained by drawing the tangent line atx=a,and then sliding to thex-axis along this tangent line. Now draw the tangent line at (b;f(b)) and ride the new tangent line to thex-axis to get a new estimate c. Repeat. We 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 atx=awe can makebfly far away from r. When a Newton Method calculation is going badly, a picture can help us diagnose the problem and x it. It 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, wherethe geometry is far less clear, but linear approximation still makes sense.2.3 The Convergence of the Newton Method
The argument that led to Equation 1 used the informal and imprecise symbol . We probe this argument for weaknesses. No numerical procedure works forallequations. For example, letf(x)= x 2 +17ifx6=1,andletf(1) = 0. The behaviour off(x)near1givesno clue to the fact thatf(1) = 0. Thus no method of successive approximation can arrive at the solution off(x) = 0. To make progress in the analysis, we need to assume thatf(x) is in some sense smooth. We will suppose that f 00 (x) (exists and) is continuous nearr. The tangent line approximation is|an approximation.Let's try to get a handle on the error. Imagine a particle travelling in a straight line, and letf(x) be its position at timex.Thenf