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MTH718: Topics in Numerical Methods

In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The introduction of a numerical method and checking the correct combination of programming language is called a numerical algorithm. MTH718 Handouts pdf

MTH718 Handouts pdf

Course Category: Mathematics MTH718 Handouts pdf

Course Overview

In this study, emphasis will be placed on learning basic and advanced Numerical techniques for solving Equal and Non-Line Mathematics. It also includes Interpolation and Different Numerical Methods for Solving Problems, Differences, and Different Mathematical Problems that can be solved accurately by Integration and Differences techniques. Therefore, the use of Numerical Techniques in these types of problems will be very helpful. In addition, MAPLE software will be taught to find non-linear mathematical solutions as well as to evaluate the efficiency and order of integration. MTH718 Handouts pdf

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MTH718 Handouts pdf
MTH718 Handouts pdf

MTH718 HANDOUTS

MTH718: Topics in Numerical Methods

Bisection Method

The dual division method is the easiest to use numerically and is almost always effective. The downside is that the reunion is slow. If the dual-split method results in a slower computer system, other faster methods may be selected; otherwise, it is a good decision of the way. We want to build a sequence x0, x1, x2,. . . that shifts to x = r which solves f (x) = 0. We select x0 and x1 that x0 <r <x1.

We say x0 and x1 in parentheses. With f (r) = 0, we want f (x0) and f (x1) to be the opposite symbol, so that f (x0) f (x1) <0. We then give x2 to be the center point of x0 and and x1, i.e. x2 = (x0 + x1) / 2. The f (x2) symbol can then be determined.

Secant Method

The Secant method is the second most advanced in Newton’s Method and is used when desired for faster integration than Bisection, but it is very difficult or impossible to take out of other analysis of the function f (x).
f 0 (xn) ≈ f (xn) – f (xn − 1) / xn-xn-1
Starting the Secant Method requires your guessing of both x0 and x1.

Gaussian Elimination

The Gauss termination, with line and multi-line algebra, is a process of finding solutions for a line mathematical system simultaneously by first solving one of the equations of one variable (according to all the others) and then replacing the phrase with the rest of the arithmetic.
EXAMPLE

x + y = 2 and 2x + 2y-4 = 0. If you multiply any non-zero constant by both numbers, you will find that always the x-values ​​and the variable y-values ​​are canceled or deleted.

Partial Pivoting

When you make a Gaussian finish, the diagonal thing one uses during the removal process is called a pivot. To find the correct multiplication, one uses pivot as a subdivision of sub-pivot objects. Completing the Gaussian in this form will not succeed if the pivot is zero. In this case, a change of lines should be made. Even if the pivot is not equal to zero, a small amount can lead to large circulatory errors.

For very large matrices, one can easily lose all accuracy in the solution. To avoid these rotation errors that occur in small pivots, line rotation is performed, and this process is called partial pivoting (partial pivoting is opposite completion, where both lines and columns alternate). We will illustrate for example the decay of LU using partial pivoting.

Operation Counts

To estimate how much calculation time is required in the algorithm, one can calculate the number of tasks required (multiplication, division, addition, and subtraction). Usually, it is interesting how the algorithm measures the problem size. For example, suppose one wants to duplicate two n x n matriculants. The calculation of each element requires the multiplication of n and n – 1 added, or 2n – 1 functions. There are n 2 items that must be calculated so that the total working calculation is n 2 (2n – 1).

If n is greater, we would like to know what will happen during the calculation if n is doubled. Most important is the fast-growing term, which leads to job counts. In this matrix multiplication example, the operating value is n 2 (2n – 1) = 2n 3 – n 2, and the lead order time is 2n 3. Feature 2 is not important for measurement, and we say the algorithm scale is the same and O (n 3), which reads “Oh big of n cubed.” If we use the big-Oh notation, we will discontinue both low-order terms and frequent repetitions

Least-Squares approximation

The least-square method is often used to add a parameter break to test data. In general, the appropriate curve is not expected to exceed the data points, making the problem very different from that of the interpreter. Here we look only at the simple case of the same test error in all data points. Allow the input data to be given by (xi, yi), by i = 1 to n.

MTH718 HANDOUTS pdf

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MTH718: Topics in Numerical Methods Notes (pdf)