Mathematics
Iterative Methods
Iterative methods in mathematics are algorithms used to approximate solutions to equations or systems of equations. Instead of finding an exact solution in a single step, iterative methods repeatedly update an initial guess to approach the true solution. These methods are commonly used in numerical analysis and are particularly useful for solving large, complex problems.
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3 Key excerpts on "Iterative Methods"
- Rizwan Butt(Author)
- 2011(Publication Date)
- Mercury Learning and Information(Publisher)
Chapter 2Iterative Methods for Linear Systems2.1 IntroductionThe methods discussed in Chapter 1 for the solution of the system of linear equations have been direct, which required a finite number of arithmetic operations. The elimination methods for solving such systems usually yield sufficiently accurate solutions for approximately 20 to 25 simultaneous equations, where most of the unknowns are present in all of the equations. When the coefficients matrix is sparse (has many zeros), a considerably large number of equations can be handled by the elimination methods. But these methods are generally impractical when many hundreds or thousands of equations must be solved simultaneously.There are, however, several methods that can be used to solve large numbers of simultaneous equations. These methods, called Iterative Methods, are methods by which an approximation to the solution of a systemof linear equations may be obtained. The Iterative Methods are used most often for large, sparse systems of linear equations and they are efficient in terms of computer storage and time requirements. Systems of this type arise frequently in the numerical solutions of boundary value problems and partial differential equations. Unlike the direct methods, the Iterative Methods may not always yield a solution, even if the determinant of the coefficients matrix is not zero.The Iterative Methods to solve the system of linear equationsstart with an initial approximation x(0) to the solution x of the linear system (2.1) and generate a sequence of vectors {x( k)}1k=0that converges to x. Most of these Iterative Methods involve a process that converts the system (2.1) into an equivalent system of the formfor some square matrix T and vector c. After the initial vector x(0)- eBook - ePub
- Jin Keun Seo, Eung Je Woo(Authors)
- 2012(Publication Date)
- Wiley(Publisher)
Chapter 5 Numerical MethodsQuantitative analyses are essential elements in solving forward and inverse problems. To utilize computers, we should devise numerically implementable algorithms to solve given problems, which include step-by-step procedures to obtain final answers. Formulation of a forward as well as an inverse problem should be done bearing in mind that we will adopt a certain numerical algorithm to solve them. After reviewing the basics of numerical computations, we will introduce various methods to solve a linear system of equations, which are most commonly used to obtain numerical solutions of both forward and inverse problems. Considering that most forward problems are formulated by using partial differential equations, we will study numerical techniques such as the finite difference method and finite element method. The accuracy or consistency of a numerical solution as well as the convergence and stability of an algorithm to obtain the solution need to be investigated.5.1 Iterative Method for Nonlinear Problem
Recall the abstract form of the inverse problem in section 4.1, where we tried to reconstruct a material property P from knowledge of the input data X and output data Y using the forward problem (4.1 ). It is equivalent to the following root-finding problem:where X and Y are given data.We assume that both the material property P and F(P, X) are expressed as vectors in N-dimensional space . Imagine that P* is a root of G(P) = 0 and P0 is a reasonably good guess in the sense that P0 + ΔP = P* for a small perturbation ΔP. From the tangent line approximation, we can approximatewhere J(P0 ) denotes the Jacobian of G(P) at P0 . Then, the solution P* = P0 + ΔP can be approximated byThe Newton–Raphson algorithm is based on the above idea, and we can start from a good initial guess P0 to perform the following recursive process:5.1The convergence of this approach is related to the condition number of the Jacobian matrix J(P*). If J(P*) is nearly singular, the root-finding problem G(P) = 0 is ill-posed. According to the fixed point theorem, the sequence Pnconverges to a fixed point P* provided that Φ(P): = P − [J(P)]−1 G(P - eBook - ePub
Numerical Analysis
An Introduction
- Timo Heister, Leo G. Rebholz, Fei Xue(Authors)
- 2019(Publication Date)
- De Gruyter(Publisher)
5Solving nonlinear equationsThere should be no doubt that there is a great need to solve nonlinear equations with numerical methods. For most nonlinear equations, finding an analytical solution is impossible or very difficult. Just in one variable, we have equations such asex= x 2 which cannot be analytically solved. In multiple variables, the situation only gets worse. The purpose of this chapter is to study some common numerical methods for solving nonlinear equations. We will quantify how they work, when they work, how well they work, and when they fail.5.1Convergence criteria of Iterative Methods for nonlinear systems
This section will develop and discuss algorithms that will (hopefully) converge to solutions of a nonlinear equation. Each method we discuss will be in the form of: Step 0: Guess at the solution with x 0 (or two initial guesses x 0 and x 1 ) Step k : Use x 0 , x 1 , x 2 , . . . ,xk−1 to generate a better approximationxkThese algorithms all build sequencesWe now define the notions of linear, superlinear, and quadratic convergence.{x kk = 1∞{x 0,x 1,x 2, ...ek= |xk− x ∗ |), that is, the error is the distance betweenxkand the solution. Once this is sufficiently small, the algorithm can be terminated, and the last iterate becomes the solution.Definition 31. Suppose an algorithm generates a sequence of iterates {x 0 , x 1 , x 2 , . . .} which converges to x ∗
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