Numerical Methods by K. Das pdf Numerical analysis is the study of the approximate solution to mathematical problems, or methods for obtaining them. The term numerical method refers to a variety of techniques used to solve numerical problems involving time series data analysis, regression analysis, stochastic processes, differential equations and other types of equations. A numerical method uses some idea about how to represent the unknowns in an equation as numbers for calculation purposes. Calculations are then performed either by specific formulas or iteratively until a result is obtained that satisfies certain requirements. Some approximate methods are valid only for small problems, as the error is greater as the problem size decreases.
Technique used to solve a given problem that does not depend upon a theoretical understanding of the topic but upon numerical techniques.
This type of method is applied when exact analytic solutions are not available for a problem. "For example:"
As a general rule, NUMERICAL METHODS give the approximate solution to problems under consideration with negligible error.
ALGORITHM: The sequence of instructions for solving specific types of problems in computer programs. "An algorithm is represented by an unambiguous definition written down in advance; it can then be executed on any computer without error.
The following is a list of some numerical analysis algorithms:
Computers and numbers: A number in mathematics is called an integer if it can be positive, negative or zero. For example: -3, 0 ,5, 12 ……
Usually when we divide two numbers like x and y each with many digits in the process of calculation the answer obtained will also contain many digits. So we need to truncate (round off ) this answer to some reasonable number of decimals.This process is called rounding off. Let us take an example: Consider x = 10 and y = 9 then x/y = 1 (rounded off to two decimals.
If we use floating point numbers to store data, then some inaccuracies will arise due to the limited number of bits used to represent the number. This can be overcome by using (constructed) approximating functions that are very close to the actual function.
The terms approximation and approximation error are used more conservatively in numerical analysis than in more established sciences like physics. Numerical analysis does not set its sights on extreme accuracy; it aims at developing algorithms for solving problems that are good enough for the purpose at hand.
The accuracy of a numerical solution can be quantified by its "error". The most common types of errors are given below:
ROUND-OFF ERROR: If the difference between the exact and the approximate (numerical) solution is large it is said to contain "round-off error." The ratio between the magnitude of this error and the magnitude of the exact solution at that point, is called round-off error.
The following steps describe the basic idea behind how numerical algorithms work.
Example 1 (digital root): Consider 2x = 1.5 x = 7r + 0.
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