Technology & Engineering
Numerical Root Finding
Numerical root finding is a computational method used to locate the roots or solutions of equations that cannot be solved algebraically. It involves iterative algorithms that approximate the roots of a function within a specified tolerance. This technique is widely used in engineering and technology for solving complex equations and optimizing designs.
Written by Perlego with AI-assistance
Related key terms
1 of 5
5 Key excerpts on "Numerical Root Finding"
eBook - ePubMaths in Chemistry
Numerical Methods for Physical and Analytical Chemistry
- Prerna Bansal(Author)
- 2020(Publication Date)
- De Gruyter(Publisher)
Chapter 9 Numerical root-finding methods9.1 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:- Newton–Raphson method
- Iteration method
- Binary bisection method
- Secant method
- Regula-Falsi method
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
- Owen Jones, Robert Maillardet, Andrew Robinson(Authors)
- 2014(Publication Date)
- Chapman and Hall/CRC(Publisher)
CHAPTER 10 Root-finding 10.1 Introduction The next few chapters introduce numerical algorithms for solving some com-mon applied mathematical problems. In each case we present motivating ex-amples, some underpinning theory, and applications in R. This chapter fo-cuses on root-finding, and covers fixed-point iteration, the Newton–Raphson method, the secant method, and the bisection method. Suppose that f : R → R is a continuous function. A root of f is a solution to the equation f ( x ) = 0 (see Figure 10.1 for example). That is, a root is a number a ∈ R such that f ( a ) = 0. If we draw the graph of our function, say y = f ( x ), which is a curve in the plane, a solution of f ( x ) = 0 is the x -coordinate of a point at which the curve crosses the x -axis. The roots of a function are important algebraically; for example, we use the 0 2 4 6 8 10 -2 0 2 4 6 x f(x) root at x=1 root at x=7 Figure 10.1 The roots of the function f. 181 182 ROOT-FINDING roots of a polynomial to factorise it. Moreover the solution to a physical prob-lem can often be expressed as the root of a suitable function. Root-finding is also a classical numerical or computational problem, and provides a good introduction to important issues in numerical mathematics. 10.1.1 Example: loan repayments Suppose that a loan has an initial amount P , a monthly interest rate r , a duration of N months, and a monthly repayment of A . The remaining debt after n months is given by P n , where P 0 = P ; P n +1 = P n (1 + r ) − A. That is, each month you pay interest on the previous balance, then reduce the balance of the loan by amount A . This is a first-order recurrence equation, and has the following solution (check that it works): P n = P (1 + r ) n − A ((1 + r ) n − 1) /r. Putting P N = 0, we get A P = r (1 + r ) N (1 + r ) N − 1 . Suppose that we know P , N , and A ; then we can find r by finding the root(s) of the following function: f ( x ) = A P − x (1 + x ) N (1 + x ) N − 1 .
eBook - PDF- Andrew S. Glassner(Author)
- 2013(Publication Date)
- Morgan Kaufmann(Publisher)
NUMERICAL AND PROGRAMMING TECHNIQUES ROOT FINDING SUMMARY Finding the roots of an equation is a common task in computer graphics. In 3D, the operation is very common in ray tracing, when object/ray intersections are computed using the implicit form for an object; the intersections of the object and ray are then represented by the zeros of the equation formed by plug-ging the explicit form of the ray into the explicit form of the object. In 2D, some applications of root-finding include the determination of bounding boxes and accurate tracing of curves. The following Gems discuss Numerical Root Finding. Linear and quadratic equations are trivial, and the solutions well-known. Cubic and quartic equations may also be solved analyti-cally, but it takes some care to make sure the solution is stable; the first Gem addresses that question. The second and third Gems are more general numerical solutions, which are designed to find the zeros of a function efficiently and robustly. See also Ray Tracing (383); Distance Measures Summary (423) 403 CUBIC AND QUARTIC ROOTS Jochen Schwarze ISA GmbH Stuttgart, Federal Republic of Germany / The ray-object intersection used in ray tracing requires the solution of cubic and quartic equations as soon as more complex objects like splines and tori are to be supported. Iterative algorithms are often slow and numerically unstable. Start values and the number of noncomplex roots are hard to determine. An approach to finding cubic and quartic roots analytically is presented in the following. Sample code in C shows a possible implementation with intermediate variables and case decisions for time efficiency. It uses double precision arithmetic; no complex numbers and operations are required. Solution of the Cubic A cubic equation of the form CotX/ I CnX I C-i X I C r ^ J is first divided by c 3 , giving the normal form x 3 + Ax 2 + Bx + C = 0. Substitution of / A 404
eBook - PDF- Xin-She Yang(Author)
- 2008(Publication Date)
- WSPC(Publisher)
Part II Numerical Algorithms This page intentionally left blank This page intentionally left blank Chapter 5 Roots of Nonlinear Equations Many problems such as finding an optimal solution to a particular problem are often related to find the critical points and extreme points. In order to find these critical points, we have to solve the stationary conditions when the first derivatives are zero. On the other hand, sometimes, we do have to solve a nonlinear equation to find its roots. Therefore, root-finding algorithms are important. Close-form solutions are rare, and in most cases, only approximate solutions are possible. In this chapter, we will introduce the fundamentals of root-finding algorithms. 5.1 Bisection Method The bisection method is a classic method of finding roots of a nonlinear function f ( x ) in the interval [ a,b ]. It works in the following way as shown in Fig. 5.1. The iteration procedure starts with two initial bounds x a (lower bound), and x b (upper bound) so that the true root x = x ∗ lies between these two bounds. This requires that f ( x a ) and f ( x b ) have different signs. In our case shown in Fig. 5.1, f ( x a ) > 0 and f ( x b ) < 0, but f ( x a ) f ( x b ) < 0. The obvious choice is x a = a and x b = b . The next estimate is just the midpoint of A and B , and we have x n = 1 2 ( x a + x b ) . (5.1) We then have to test the sign of f ( x n ). If f ( x n ) < 0 (having the same sign as f ( x b )), we then update the new upper bound as x b = x n . If f ( x n ) > 0 (having the same sign as f ( x a )), we update the new lower bound as x a = x n . In a special case when f ( x n ) = 0, we have found the true root. The 59 60 Introduction to Computational Mathematics a45 0 a54 x f ( x ) A x a B x b x ∗ x n Fig. 5.1 Bisection method for finding the root x ∗ of f ( x ∗ ) = 0 between two bounds x a and x b in the domain x ∈ [ a, b ].
- Jaan Kiusalaas(Author)
- 2015(Publication Date)
- Cambridge University Press(Publisher)
140 Roots of Equations All methods of finding roots are iterative procedures that require a starting point, that is, an estimate of the root. This estimate can be crucial; a bad starting value may fail to converge, or it may converge to the “wrong” root (a root different from the one sought). There is no universal recipe for estimating the value of a root. If the equation is associated with a physical problem, then the context of the problem (physical insight) might suggest the approximate location of the root. Otherwise, the function must be plotted (a rough plot is often sufficient), or a systematic numerical search for the roots can be carried out. One such search method is described in the next article. It is highly advisable to go a step further and bracket the root (determine its lower and upper bounds) before passing the problem to a root finding algorithm. Prior bracketing is, in fact, mandatory in the methods described in this chapter. The number of iterations required to reach the root depends largely on the intrinsic order of convergence of the method. Letting e k be the error in the computed root after the k th iteration, an approximation of the error after the next iteration has the form E k + 1 = cE m k | c | < 1 The order of convergence is determined by m . If m = 1, the the method is said to con-verge linearly. Methods for which m > 1 are called superlinearly convergent. The best methods display quadratic convergence ( m = 2). 4.2 Incremental Search Method The approximate locations of the roots are best determined by plotting the function. Often a very rough plot, based on a few points, is sufficient to give us reasonable start-ing values. Another useful tool for detecting and bracketing roots is the incremental search method. It can also be adapted for computing roots, but the effort would not be worthwhile, since other methods described in this chapter are more efficient for that.
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
