Numerical Analysis By Mollah Pdf Free 15 ((TOP))
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The computability of numerical analysis and error control are central concerns for the use of numerical methods. In the early decades of numerical analysis, a great emphasis was placed on computability. As the applicability of numerical methods was widened to the non-linear case, the emphasis shifted towards the control of error, so that the two major concerns of numerical analysis at the beginning of the 21st century are the development of new, efficient and robust algorithms and error control. Careful error control and in some cases error estimation can allow the use of fewer samples than would otherwise be required for a given level of precision.
A numerical method is called numerically exact if it can compute the exact value of the given function up to a given absolute precision. For example, Newton's method is numerically exact.
Numerical methods are often used to solve differential equations, by discretizing these equations, where the discretization results in a difference equation. Solving difference equations is much faster than solving differential equations, and difference equations are easier to solve than their differential counter parts. This renders numerical methods useful for simulating the time evolution of difference equations. Numerical methods are also used for differential equations that are not easily discretizable. Thus we have developed numerical methods for non-linear partial differential equations as well as for ordinary differential equations. Furthermore, numerical methods are used to approximate the solutions of differential equations by polynomials, trigonometric functions, rational functions or complex numbers.
However, some numerical methods may give the wrong answer (the so called root of the problem), even if they are numerically exact. Thus the importance of the existence of such methods increases with the level of precision required.
A numerical method is called exact if it can compute the exact value of a given function at any given point. For example, the Tauber algorithm is exact for finding the zeros of a given function.
Even before computers, some of the major advances in numerical analysis were made by hand calculation. Among others, the Gauss-Seidel method for solving linear systems, the Fast Fourier Transform algorithm, and the 3D interpolation scheme to fill a cube with discrete values were invented by hand calculation, and were probably used first in practice.
A numerical analysis algorithm is a sequence of steps used to solve a numerical problem. To perform numerical analysis, the following steps are usually followed:
Specify the problem: the set of operations to be performed on the data to solve the problem. These operations may be arithmetic (addition, multiplication, division) or algebraic (factoring polynomials, solving linear equations, etc.)
Specify the number of significant figures: an approximate estimate of the results of the operations to be performed, obtained by rounding the arithmetic results to the desired degree of accuracy.
Specify a precision: a measure of the quality of the approximation of the results of the operations to be performed.
Specify the desired approximation: the desired precision of the approximation.
Specify an initial guess: an approximation of the value of the solution that the algorithms is supposed to converge to. 827ec27edc