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Linear Programming
Published in Michael W. Carter, Camille C. Price, Ghaith Rabadi, Operations Research, 2018
Michael W. Carter, Camille C. Price, Ghaith Rabadi
Round-off error is a natural consequence of using finite precision computing devices. As was pointed out in Chapter 1, this inability to store computed results exactly is particularly pronounced when we perform arithmetic operations on numeric values of very different magnitudes, where we are often unable to record that portion of a result contributed by the smaller value. In an attempt to remove the source of some of these numerical inaccuracies, most commercial linear programming systems apply some kind of scaling before beginning the Simplex method. Rows and columns of the matrix A may be multiplied by constants in order to make the largest element of each row and column the same (Murtagh 1981). To improve the condition of a matrix (and, therefore, obtain greater accuracy of its inverse), all the elements of A should be kept within a reasonable range, say within a factor of 106 or 108 of each other (Orchard-Hays 1968). More elaborate and specific mechanisms for scaling have been devised. In general, a healthy awareness of the limitations of computer arithmetic and numerical computation is essential in understanding and interpreting computed results.
Steps in Performing Finite Element Analyses
Published in Victor N. Kaliakin, Introduction to Approximate Solution Techniques, Numerical Modeling, and Finite Element Methods, 2018
The finite element method is a numerical (approximate) method that is used to analyze a mathematical model which itself is an approximation to some physical problem. As such, in using any finite element program the following questions must be answered: Is the mathematical model of the physical problem correct?Is the mesh used fine enough?Are there any errors in the program itself? The specific problem may not have been considered previously using the particular program — “bugs” may thus have been discovered.Is roundoff error a problem? Even if the coding and the theory are correct, as noted in Chapter 2 roundoff errors can lead to incorrect results. A potential way to check on the effect of roundoff error is to re-analyze a particular problem using double the precision.
Simulation
Published in Devendra K. Chaturvedi, ®, 2017
The round off error is an error which is present in every type of computer due to the fact that the computer can represent the quantities with a finite number of digits. For example, we know that the value of π is 3.141592653897285… and if we are using a computer that can retain only seven significant figures so this computer might store or use π as π = 3.141592 which omitted the term resulting in an error called round off error. This error equals to 0.00000065. Every floating-point operation incurs a round off error of O(p) which arises from the finite accuracy to which floating-point numbers are stored by the computer. Suppose that we use Euler’s method to integrate our ODE over a given interval of time. This entails O(h−1) integration steps, and, therefore, O(h−1) floating-point operations. If each floating-point operation incurs an error of O(p) and the errors are simply cumulative, then the net round off error is O(p(h)).
The role of falsification in the validation of numerical models
Published in Civil Engineering and Environmental Systems, 2023
Round-off errors occur because computers have limited capacity to store exact numbers. These errors can have cumulative effect in a numerical process. When an exact number is stored a round-off error arises once. When repeated arithmetic operations are performed, round-off error may occur in each operation and these errors add up.