15 Nov 2018
1 sur 15

• 1. Topic – General Quadrature Formula Subject Incharge: Dr. Dharm Raj Singh Name: Mrinal Dev
• 2. Basic Topics  Calculus  Function  Differentiation  Limit  Range Main Topic  Integration  Quadrature  Quadratic Equation  Quadrature Formula  Derivation
• 3. 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.
• 6. 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: ∫ xn dx = 1/n+1 (xn+1) + c Examples: ∫ x5 dx = 1/6 (x6) + c
• 8.  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. An equation where the highest exponent of the variable (usually "x") is a square (2). So it will have something like x2, but not x3 etc. A Quadratic Equation is usually written ax2 + bx + c = 0. Example: 2x2 + 5x − 3 = 0. Quadrature Equation is also known as Newton’s Forward Interpolation Formula.
• 10. 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 = ∫ 𝒇 𝒙 = 𝒚 𝟎 + 𝒖 𝒚 𝟎 𝒖(𝒖−𝟏) 𝟐! 𝟐 𝒚 𝟎 + 𝒖(𝒖−𝟏)(𝒖−𝟐) 𝟑! 𝟑 𝒚 𝟎 + ⋯
• 11. Let I = ydx where y = f (x) Also assume that f (x) be given for certain equidistant values of x, say x0, x1, x2, x3,….xn. Let the range (b-a) be divided into n equal parts, each of width h, so that h = 𝑏 − 𝑎 𝑛 Thus, we have x0 = a, x1 = a + h, x2 = a + 2h, … xn = a + nh = b Now , let yk = f (xk), k = 0,1,2…n Consider, I = y dx = ydx b a  b a  0 0 x nh x  
• 12. We have y = ∫ 𝒇 𝒙 = 𝒚 𝟎 + 𝒖 𝒚 𝟎 + 𝒖(𝒖−𝟏) 𝟐! 𝟐 𝒚 𝟎 + 𝒖(𝒖−𝟏)(𝒖−𝟐) 𝟑! 𝟑 𝒚 𝟎 + ⋯ where u = du = dx = hdu 𝑥 − 𝑥0 ℎ 1 ℎ  
• 13. Approximating y by Newton’s forward formula taking limit of integration becomes 0 to n. I = h y0 + u y0+ 𝑢(𝑢−1) 2! 2 𝑦0 + 𝑢(𝑢−1)(𝑢−2) 3! 3 𝑦0 + ⋯ du n = h y0u + 𝒖 𝟐 2 y0 + 𝒖 𝟑 𝟑 − 𝒖 𝟐 𝟐 2! 2 y0 + 𝑢4 4 −𝑢3 +𝑢2 3! 3y0 + … 0 =h ny0 + 𝒏 𝟐 2 y0 + 𝒏 𝟑 𝟑 − 𝒏 𝟐 𝟐 2! 2y0 + 𝑛4 4 −𝑛3 +𝑛2 3! 3y0 + … 0 n 
• 14. I = nh y0 + 𝒏 2 y0 + 𝒏(𝟐𝒏−𝟑) 12 2y0 + 𝒏 𝒏−𝟐 𝟐 24 3y0+… This is called General Quadrature formula.