General Quadrature Equation

Content
Basic Topics
Calculus
Function
Differentiation
Limit
Range
Main Topic
Integration
Quadrature
Quadratic Equation
Quadrature Formula
Derivation

1. Numerical methods bca – s304 Topic – General Quadrature Formula Subject Incharge : Dr. Dharm Raj Singh Name: Mrinal Dev

2. content Basic Topics Calculus Function Differentiation Limit Range Main Topic Integration Quadrature Quadratic Equation Quadrature Formula Derivation

3. Basic Topics Calculus: Calculus , is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations . Function: A function was originally the idealization of how a varying quantity depends on another quantity. Differentiation: Differentiation is a process of finding a function that outputs the  rate of change  of one variable with respect to another variable.

4. Limit: A  limit  is the value that a function (or sequence) "approaches" as the input (or index) "approaches" some  value. Limits are essential to   calculus  (and mathematical analysis in general) and are used to define continuity, derivatives, and integrals . It is of Two Types and they are: 1- Lower Limit:- The  lower  class  limit  of a class is the smallest data value that can go into the class.   2- Upper Limit:- The upper class  limit  of a class is the largest data value that can go into the class.  Range: Values can occur between the smallest and largest values in a set of observed values or data points. Given a set of values, or data points, the range is determined by subtracting the smallest value from the largest value.

5. Main topic

6. INTEGRATION Integration is the reverse of differentiation. However: If y = 2x + 3, dy /dx = 2 If y = 2x + 5, dy /dx = 2 If y = 2x, dy /dx = 2 So the integral of 2 can be 2x + 3, 2x + 5, 2x, etc. For this reason, when we integrate, we have to add a constant. So the integral of 2 is 2x + c, where c is a constant. A "S" shaped symbol is used to mean the integral of, and dx is written at the end of the terms to be integrated, meaning "with respect to x". This is the same "dx" that appears in dy /dx .

7. To integrate a term, increase its power by 1 and divide by this figure. In other words: ∫ x n dx = 1/n+1 (x n+1 ) + c Examples: ∫ x 5 dx = 1/6 (x 6 ) + c

8. WHAT IS QUADRATURE In   mathematics ,  quadrature  is a historical term which means determining area. Quadrature problems served as one of the main sources of problems in the development of calculus, and introduce important topics in  mathematical   analysis.

9. Quadratic Equations An equation where the highest exponent of the variable (usually "x") is a square (2). So it will have something like x 2 , but not x 3  etc. A  Quadratic  Equation is usually written ax 2  + bx + c = 0. Example : 2x 2  + 5x − 3 = 0 . Quadrature Equation is also known as Newton’s Forward Interpolation Formula.

10. QUADRATURE FORMULAE The Gauss– Kronrod   quadrature formula  is an adaptive method for numerical integration. ... It is an example of what is called a nested  quadrature rule : for the same set of function evaluation points, it has two  quadrature  rules, one higher order and one lower order (the latter called an embedded   rule ). The Formula is given by: y =  

11. Derivation Let I = y dx where y = f (x) Also assume that f (x) be given for certain equidistant values of x, say x , x 1 , x 2 , x 3 ,…. x n . Let the range (b-a) be divided into n equal parts, each of width h, so that Thus, we have Now , let y k = f ( x k ), k = 0,1,2…n Consider, I = y dx = y dx  

12. We have y = where u = du = dx = hdu      

13. Approximating y by Newton’s forward formula taking limit of integration becomes 0 to n. I = h y + u y + du n = h y u + y + 2 y + 3 y + … =h ny + y + 2 y + 3 y + …  

14. I = nh y + 2 y + 3 y +… This is called General Quadrature formula.  

15. Thank You