Absolute Error And Relative Error In Numerical Analysis Pdf

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The difference between the measured or inferred value of a quantity and its actual value , given by. The absolute error of the sum or difference of a number of quantities is less than or equal to the sum of their absolute errors. Abramowitz, M.

Numerical Analysis - Errors

However, result from numerical analysis is an approximation, in general, which can be made as accurate as desired. The reliability of the numerical result will depend on an error estimate or bound, therefore the analysis of error and the sources of error in numerical methods is also a critically important part of the study of numerical technique. Errors and Approximations in Computation A computer has a finite word length and so only a fixed number of digits are stored and used during computation.

This would mean that even in storing an exact decimal number in its converted form in the computer memory, an error is introduced. This error is machine dependent and is called machine epsilon.

Example: 5, ,. They contain infinitely many non- recurring digits. Therefore, the numbers obtained by retaining a few digits are called approximation numbers. Example: , 0. Types of Errors 1. Inherent Error: The inherent error is that quantity which is already present in the statement of the problem before its solution.

The inherent error arises either due to the simplified assumptions in the mathematical formulation of the problem or due to the errors in the physical measurements of the parameters of the problem. Inherent error can be minimized by obtaining better data, by using high precision computing aids and by correcting obvious errors in the data. Round-off error: is the quantity, which arises from the process of rounding off numbers.

It sometimes also called numerical error. The round-off the number is the process of dropping unwanted digits is called round-off. Example: The number can be written as 0.

Example: 8. Example: 5. Example: Truncation Error: These types of errors caused by using approximate formulae in computation or on replace an infinite process by a finite one. Absolute error: is the numerical difference between the true value of a quantity and its approximate value.

The unit of exact or unit of approximate values expresses the absolute error. Where is the approximate value of quantity. The relative error is independent of units. The percentage error is also independent of units. Example: Find the absolute, relative and percentage errors if is rounded-off to three decimal digits.

Exercise: a. Find the relative error of the number 8. Perform the following computations I. Exactly II. Using 3-digit chopping III. Using 3-digit round-off IV. General Formula for errors In this section, we derive a general formula for the error committed in using a certain formula for a functional relation.

Find the product of the numbers Find also in the result. These are called closed form solution. Similar formula also available for cubic and bi-quadratic polynomial equations but we rarely remember them. For higher order polynomial equations and non-polynomials it is difficult and in many cases impossible to get closed form solutions. Iterative method: those methods also known as trial and error methods are based on the idea of successive approximation starting with one or more initial approximation to the value of the root to obtain the accurate solution.

The above quadratic equation can be found by iterative method in the following manner. Bisection Bolzano method This is one of the simplest iterative methods and is strongly based on the property of intervals bracketing. Finding a root using this method, let be continuous between and. The procedure for the Bisection method Step1: Choose two initial guess values approx. Find the minimum number of iteration that the bisection method requires to get the approximate root of the given function with a maximum absolute error of Solution: a.

Therefore, a root lies between 0 and 1. False Position Regula-falsi method The false position method retains the main features of the Bisection method, that the root is trapped in a sequence of intervals of decreasing size. This method uses the point where the secant lines intersect the -axis. Hence the approximate root is 1. Fixed-point iteration Iteration method This method is also known as substitution method or method of fixed iterations.

Hence the required root is given by 0. Let be an approximation root of. If this tangent meets the -axis at , then is the better approximation to the root. Exercise 1. Given these three equations i. Regula-Falsi method b. Secant method c. Newton Raphson method 5. Find the root of the equations i. Direct Methods Consider a system of linear equations in unknowns.

Therefore, matrix is solvable if it can be transformed in to any one of the forms ,. Gaussian Elimination Method This is a method reduction of equation 3. Therefore, the elimination procedure describe above is called Gauss-elimination method. Additionally, to reduce round-off error; it is often necessarily to perform row interchanges even when the pivot elements are not zero. Example: Using the four decimal places computer, solve 0.

In theory, the solution can always be obtained by Gauss elimination. However, there are two major pitfalls in the application of Gauss elimination or its variations a. Round-Off Errors Round-off errors occur when exact infinite precision numbers are approximated by finite precision numbers.

Example: Consider the following system of linear algebraic equations 0. Round-off errors can never be completely eliminated.

However, they can be minimized by using high precision arithmetic and pivoting. On the other hand if small changes in and give small changes in the solution, the system is said to be stable, or, well conditioned.

Thus in an ill-conditioned system, even the small round off errors effect the solutions very badly. Unfortunately it is quite difficult to recognize an ill-conditioned system. Therefore, the system is ill conditioned. Matrix Decomposition Method i.

Indirect Iteration Methods All the previous methods seen in solving the system of simultaneous algebraic linear equations are direct methods. Now we will see some indirect methods or iterative methods. This iterative method is not always successful to all systems of equations. If this method is to succeed, each equation of the system must possess one large coefficient and the large coefficient must be attached to a different unknown in that equation.

This condition will be satisfied if the large coefficients are along the leading diagonal of the coefficient matrix. When this condition is satisfied, the system will be solvable by the iterative method. This system in equation 3. The condition is sufficient but not necessary. Under the category of iterative method, we shall describe the following two methods: i. Jacobi Iteration Method Solving of the system of equations, we assume that the quantities in 3.

The equations 3. Note: Gauss—Seidel method is a modification of Jacobi method.

Approximation error

The approximation error in some data is the discrepancy between an exact value and some approximation to it. An approximation error can occur because:. In the mathematical field of numerical analysis , the numerical stability of an algorithm indicates how the error is propagated by the algorithm. Given some value v and its approximation v approx , the absolute error is. In words, the absolute error is the magnitude of the difference between the exact value and the approximation.

This work develops a computational approach for boundary and initial-value problems by using operational matrices, in order to run an evolutive process in a Hilbert space. Besides, upper bounds for errors in the solutions and in their derivatives can be estimated providing accuracy measures. Differential equations are ubiquitous in engineering daily life but their solutions are sometimes very difficult, mainly if they are nonlinear. Several of them do not have an analytical solution describable by a finite combination of elementary functions, or even by an unlimited series with a determinable recurrence relation. In previous works, analytical and numerical results were obtained for nonlinear differential equations [ 1 — 3 ], and an algorithm called SIV Solving Initial Value was developed.

For example, is an approximation to with an absolute error of 1 and a relative error of 10−6, while 2 is an approximation to 1.

Numerical Methods For Engineers CHAPTER ONE BASIC CONCEPTS IN ERROR ESTIMATION

Data: Errors in measurement or estimation of raw data Numerical Method: Errors based on some approximation. Representation of Numbers: for example, p cannot be represented exactly by a finite number of digits. Arithmetic: Mistakes in carrying out operations such as addition or multiplication. Number Representation: Numerical calculation can involve numbers that cannot be represented exactly by a finite number of digits. Absolute Error: Absolute difference between the exact number x and the approximate number X.

This paper is concerned with the numerical solution of the general initial value problem for linear recurrence relations. An error analysis of direct recursion is given, based on relative rather than absolute error, and a theory of relative stability developed. Miller's algorithm for second order homogeneous relations is extended to more general cases, and the propagation of errors analysed in a similar manner.

Introduction

There are two techniques for measuring error: the absolute error of an approximation and the relative error of the approximation. The first gives how large the error is, while the second gives how large the error is relative to the correct value. Given an approximation a of a correct value x , we define the absolute value of the difference between the two values to be the absolute error. We will represent the absolute error by E abs , therefore.

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1.2 Absolute and Relative Error

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