MTH603: Numerical Analysis
Numerical analysis is a field of mathematical and computer science that creates, analyzes, and uses numerical algorithms for solving numerical problems that go on. MTH603 Handouts pdf
MTH603 Handouts pdf
Course Category: Mathematics MTH603 Handouts pdf
Number systems, Errors in computation, Methods of solving non-linear equations, Solution of a linear system of equations and matrix inversion, Eigenvalue problems, power method, Jacobi’s method, Different techniques of interpolation, Numerical differentiation and integration, Numerical integration formulas, different methods of solving ordinary differential equations. MTH603 Handouts pdf
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MTH603: Numerical Analysis
Errors in Computations
Numerically, computer-generated solutions are subject to certain errors. There may be fruit in identifying the source of the error and its magnitude while distinguishing the errors by numbers to count. These are
2-Local cycle errors
3-Location reduction errors
That is the number of errors in the statement of the problem itself before it can be resolved. It arises as a result of a simplified guess in the mathematical model of the problem. It may also arise when data are obtained from specific physical measurements of the problem parameters.
Local round-off errors
Every computer has a limited font length so it is possible to keep only the fixed digit of the input number provided. Since computers store information in a binary form, storing an accurate decimal number in its binary form in computer memory provides an error. This error depends on the computer. At the end of the calculation of a particular problem, the final results on the computer, apparently in binary form, must be converted into a decimal form — a form that is understandable to the user – before it can be printed. Therefore, an additional error has been performed in this category as well. This error is called a local error.
Local truncation error
It is usually easy to extend a task into a power series using the Taylor series extension and test it by keeping the first few terms. For example, we might limit the function f (x) = cos x by series to simplify the task into a power series using the Taylor series extension and test it by keeping the first few terms. For example, we might estimate function f (x) = cos x in series.
Descartes’s rule of signs
This rule shows the relationship between the characteristics of the coefficients of the equation and its roots. “The number of positive algebraic root roots f (x) = 0 with real coefficients cannot exceed the number of coefficient character variables in polynomial f (x) =. x) = 0 ”
Intermediate value property
If f (x) is a continuous function with a real value in the axb of the closed interval ≤ ≤ if f (a) and f (b) have the same opposite sign; i.e. f (x) = 0 has at least one root β so that b ≤ ≤ β Simply If f (x) = 0 is a polynomial number and if f (a) and f (b) have different signals, then f (x) = 0 must have at least one true root between a and b. Numerical methods for solving algebraic or transcendental equations are divided into two groups.
Those methods that do not require any information about the initial root measurement to start the solution are known as straightforward methods. Examples of direct methods are the Graefee root squaring method, the Gauss elimination method, and the Gauss Jordan method. All of these methods do not require any kind of initial measurement.