Newton Raphson Method

7 Key excerpts on "Newton Raphson Method"

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  eBook - ePub

  Maths in Chemistry

  Numerical Methods for Physical and Analytical Chemistry

  • Prerna Bansal(Author)
  • 2020(Publication Date)
  • De Gruyter

    (Publisher)

  Chapter 9  Numerical root-finding methods

  9.1  Introduction

  In chemistry, we often come across lengthy and complicated polynomial equations, which are difficult to solve analytically. According to algebra, a root is the zero of the function, that is, where the function f(x) is zero. There are three ways to solve the equations, namely analytically, graphically and numerically. Numerical methods of finding roots of the equations is the most robust way of solving even very complicated equation with a great degree of ease. The most important technique in any numerical method is the iteration. Generally, an approximation of an expected value is taken and an algorithm is applied which further improves the approximation. This step is repeated until the approximation yields almost the same value. Numerical methods are particularly useful while solving the intensive polynomial for their roots.

  These are the following numerical methods to find roots of an equation:

  1. Newton–Raphson method
  2. Iteration method
  3. Binary bisection method
  4. Secant method
  5. Regula-Falsi method

  Each of the above-mentioned methods is discussed at length in the subsequent sections with their drawbacks and advantages.

  9.2  Newton–Raphson method

  Newton–Raphson (also called Newton’s iteration or Newton’s technique) is the most widely used root-finding algorithm of nonlinear equations or real-valued single variable functions (f(x) = 0). It uses an iterative method to approach the root of equation by arbitrarily choosing any root which is close to the real root.

  N-R method converges quadratically as we approach the root. It needs only one initial guess value for the root. This method involves expansion of Taylor series of a function f(x

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  Applied Numerical Methods Using MATLAB

  • R. V. Dukkipati, R. V. Dukkipati(Authors)
  • 2023(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  = 1.8919

  ea =

  1.8494e− 005

  iter =

  9

  3.4 NEWTON-RAPHSON METHOD

  The Newton-Raphson method is the best-known method of finding roots of a function f(x). The method is simple and fast. One drawback of this method is that it uses the derivative f′ (x) of the function as well as the function f(x). Hence, the Newton-Raphson method is usable only in problems where f′ (x) can be readily computed. The Newton-Raphson method is also called Newton’s method . Here, again we assume that f(x) is continuous and differentiable and the equation is known to have a solution near a given point. Figure 3.4 illustrates the procedure used in Newton-Raphson method. The solution process starts by selecting point x1 as the first estimate of the solution. The second estimate x2 is found by drawing the tangent line to f(x) at the point (x1 , f(x1 )) and determining the intersection point of the tangent line with the x-axis. The next estimate x3 is the intersection of the tangent line to f(x) at the point (x2 , f(x2 )) with the x-axis, and so on. The slope, f′ (x1 ), of the tangent at point (x1 , f(x1 )) is written as

  (3.9)

  FIGURE 3.4  The Newton-Raphson method.

  Rewriting Equation (3.9) for x2 gives

  (3.10)

  Equation (3.10) can be generalized for determining the next solution xi + 1 from the current solution xi as

  (3.11)

  The solution is obtained by repeated application of the iteration formula given by Equation (3.11) for each successive value of i. Algorithm for the Newton-Raphson Method:

  1. Select a point x1 as an initial guess of the solution.

  2. For i = 1, 2, …, until the error is smaller than a specified value, compute xi + 1 by using Equation (3.11).

  Two error estimates that are generally used in Newton-Raphson method are given below:

  The iterations are stopped when the estimated relative error is smaller than a specified value ∈ .

  (3.12)

  The iterations are stopped when the absolute value of f(xi ) is smaller than some number δ :

  |f(xi )| ≤

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  Computer Methods for Engineering with MATLAB Applications

  • Yogesh Jaluria(Author)
  • 2011(Publication Date)
  • CRC Press

    (Publisher)

  The resulting convergence is termed quadratic , or second order , and is more rapid than that for the other methods discussed earlier in this chapter. Most of these methods have a linear convergence, that is, the error after a given iteration is proportional to that obtained after the preceding iteration. Thus, they have first-order convergence. The order of convergence for the secant method can similarly be shown to be 1.62, implying that its convergence is faster than methods like search and bisection, but not as fast as Newton’s method. Because of its high rate of convergence, applicability to a variety of equations, and simplicity in programming, the Newton–Raphson method is used extensively in engineering applications. It is also used as a correction scheme in the solution of ODEs, for satisfying the boundary conditions, and in the iterative solution of a system of nonlinear equations. These applications are considered in Chapters 6 and 9. The modified Newton’s method is generally employed if the derivative f ′ ( α ) Roots of Equations 143 goes to zero or becomes very small in the vicinity of the root. The following exam-ples illustrate the use of these methods. Example 5.4 The water mass flow rate w , in kg/s, in a heating equipment that transfers energy from condensing steam to water, is to be obtained from energy balance consider-ations. If 250 kW of thermal energy are to be exchanged between the two fluids, the equation for the conservation of energy is given as 250 294 1 1000 21 5 20 = --+ uni239B uni239D uni239C uni239E uni23A0 uni239F uni23A1 uni23A3 uni23A2 uni23A2 uni23A4 uni23A6 uni23A5 uni23A5 w w exp ( ) Find the root of this equation by the Newton–Raphson method. The flow rate is known to be less than 5 kg/s. SOLUTION It is evident from the above outline of the physical problem under consideration that the root to be obtained is real and positive, being in the range 0 to 5 kg/s.

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  Structural Dynamic Systems Computational Techniques and Optimization

  Finite Element Analysis Techniques

  • Cornelius T. Leondes(Author)
  • 2021(Publication Date)
  • Routledge

    (Publisher)

  3 NEWTON-RAPHSON TECHNIQUES IN FINITE ELEMENT METHODS FOR NON-LINEAR STRUCTURAL PROBLEMS DOI: 10.1201/9780203738108-3 CHANG-NEW CHEN Department of Naval Architecture and Marine Engineering, National Cheng Kung University, Tainan, Taiwan, Republic of China

  SUMMARY

  Mathematical procedures of Newton-Raphson techniques in finite element methods for non-linear structural problems are summarized These techniques are the Newton-Raphson method, quasi-Newton methods, modified Newton-Raphson methods and accelerated modified Newton-Raphson methods. Numerical results are used to study the convergency performances of these techniques for material non-linearity problems and deformation non-linearity problems, separately. Converged finite element results are used to study various non-linear structural problems having plastic failure, non-linear fracture or non-linear buckling mechanics behaviours.

  3.1. INTRODUCTION

  The incremental/iterative method is admitted to be an effective technique for solving non-linear finite element problems. Therefore, it has become very popular to implement an iterative algorithm into a finite element computer code to solve various engineering problems with any type of non-linearity.

  As the iterative procedure is a unified approach for accurately capturing the response of an engineering system with non-linear response behaviour, it is essential that an efficient and reliable algorithm is adopted for the analysis. Unfortunately, it is difficult for us to select an iterative scheme which will provide optimal computational efficiency with high reliability to solve any type of non-linear problem. And the performance of an iterative algorithm to the non-linear analysis is problem-dependent. Among the existing schemes available, the Newton-Raphson method possesses quite good popularity because of its promising quadratic convergence. In applying this method to non-linear finite element analysis, the tangential stiffness matrix has to be formed and factorized at each iteration step. In certain cases, this will either require too many arithmetical operations to get a converged solution or cause numerical instability.

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  Numerical Analysis for Engineers and Scientists

  • G. Miller(Author)
  • 2014(Publication Date)
  • Cambridge University Press

    (Publisher)

  Note also that effective order near convergence is also not 1.618, because numerical errors dominate. The theory presented above did not account for numerical error. 162 Iterative methods and the roots of polynomials 6.5 Newton–Raphson The Newton–Raphson iteration is designed to exploit the Taylor series analysis of local convergence described in Section 6.1. This method says that to determine a root of f (x ), we use the iteration (x ) = x − f (x ) f  (x ) . The method is attributed to Sir Isaac Newton and Joseph Raphson, who published ver- sions of this algorithm in the seventeenth century, although an early version of the method specialized to computing square roots can be traced to first-century Greek scholar Heron of Alexandria. Raphson published the method in 1690 [187]. Newton de- veloped it first around 1669 and published several versions of it; a history is presented by Ypma [247]. The Newton–Raphson method may be derived using the Taylor series in the following way: f (x ) = f (ξ)    0 +(x − ξ) f  (ξ) + · · · ξ ≈ x − f (x ) f  (ξ) ≈ x − f (x ) f  (x ) . If ξ is a simple root of f (x ), then f  (ξ)  = 0, and clearly (ξ) = ξ − f (ξ) f  (ξ) = ξ, so the root of f () is a fixed point of the iteration. If x and f are vectors in R n , then we can write x (k +1) = x (k ) −  ∇f(x (k ) )  −1 f(x (k ) ), where ∇f is an n × n matrix with elements (∇f) i j = ∂ f i /∂ x j . Note that (6.7) implies that the secant method is equivalent to Newton–Raphson with the derivative estimated by finite differences using the previous two iteration values: x (n+1) = x (n) − f (x (n) ) f [x (n) , x (n−1) ] = x (n) − f (x (n) ) f  (ξ) , for some ξ ∈ [x (n) , x (n−1) ]. The local order of convergence comes from the Taylor series analysis (ξ) = ξ   (ξ) = 1 − f  (ξ) f  (ξ) + f (ξ) ( f  (ξ )) 2 f  (ξ) = 0 if f  (ξ)  = 0. (6.9) So, from the discussion of (6.3) we conclude that Newton–Raphson is a method of local second order if ξ is a simple root.

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  Algorithms, Graphs, and Computers

  • Bellman(Author)
  • 1970(Publication Date)
  • Academic Press

    (Publisher)

  The Newton-Raphson Method: Square Roots 63 This method is a very famous one, called the Newton-Raphson method. Sometimes, and quite unjustly as far as Raphson is concerned, it is abbre- viated and called merely “Newton’s method.” We shall assume an elementary knowledge of differential calculus in what follows, namely that required to determine the equation of a tangent to the curve y = f ( x ) at a specified point. We confess that it is a bit embarrassing that the simplest examples we can conjure up to illustrate the method of successive approximations re- quire more mathematical training than we require for the subsequent treat- ment of Eq. (1.1). This illustrates, however, a fundamental point that “elementary” is not synonymous with “simple. ” In compensation, however, we feel that it is important for the student both to appreciate the versatility of the method of successive approxima- tions and to become aware of the problems usually encountered in its ap- plication. 7. The Newton-Raphson Method : Square Roots To illustrate the method in one of its simplest and most ancient forms, let us consider the problem of finding the square root of a positive number a. In analytic terms, we wish to find the positive solution of the equation x2 = a ( 1 ) y = x 2 -a ( 2 ) In Fig. 3, we have a graph of the parabola, Our objective is to determine numerically the coordinates of P, the point (d7, 0). Since (1) does not have the form x = f ( x ) , we cannot im- mediately apply the algorithm discussed in 5 6 . I Fig. 3 64 The Method of Successive Approximations Suppose we make an initial guess to the x-coordinate of P, say xI. For example, if a = 5 , we might guess x, = 3 (see Fig. 4). How do we find a better estimate? We suppose that the parabola is approximately, for our purposes, the straight line which is tangent to it at the point Q,. If the first guess is good enough, this is a reasonable procedure as we shall see.

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  Problems and Solutions in Scientific Computing with C++ and Java Simulations

  • Willi-Hans Steeb, Yorick Hardy;Alexandre Hardy;Ruedi Stoop;(Authors)
  • 2004(Publication Date)
  • WSPC

    (Publisher)

  Consider the equation f(x) = 0 (1) where it is assumed that / is at least twice differentiable. Let / be some in-terval containing a root of /. The Newton-Raphson method can be derived by taking the tangent line to the curve y = f(x) at the point (x n , f(x n )) corresponding to the current estimate, x n , of the root. The intersection of this line with the rr-axis gives the next estimate to the root, x n +i-The gra-dient of the curve y = f(x) at the point (x n , f(x n )) is f'{x n ). The tangent line at this point has the form y — f'(x n )x + b. Since it passes through (x n ,f{x n )) we see that b = f(x n ) — x n f'(x n ). Therefore the tangent line is y = f'(x n )x + f(x n ) — x n f'(x n ). To determine where this line cuts the rc-axis, we set y = 0. Taking this point of intersection as the next estimate, x n +i, to the root we have 0 = f'(x n )x n+1 + f(x n ) = —x n f'(x n ). It follows that -» -* -$ & • < 2 ) (2) 206 Problems and Solutions This is the Newton-Raphson method. This method has the form next estimate = current estimate + correction term. The correction term is —f(x n )/f'(x n ) and this must be small when x n is close to the root if convergence is to be achieved. This will depend on the behavior of f'(x) near the root and, in particular, difficulty will be encountered when f'(x) and f(x) have roots close together. Since the method is of the form x n+ i = g(x n ) with g(x) = x — f(x)/f'(x) the order of the method can be examined. Differentiating g leads to g'(x) — (f(x)f(x))/(f'(x)) 2 . For convergence we require that (fix)) 2 < l (3) for all x in some interval I containing the root. Since f(a) = 0, the above condition is satisfied at the root x = a provided that f'(a) =£ 0. Then, pro-vided that g{x) is continuous, an interval / must exist in the neighbourhood of the root and over which (3) is satisfied. Difficulty is sometimes encoun-tered when the interval / is small because the initial guess must be taken from the interval.

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