NUMERICAL INTEGRATION : ERROR FORMULA, GAUSSIAN QUADRATURE FORMULA

ACTIVE LEARNING ASSIGNMENT PRESENTATIONCOMPLEX VARIABLES & NUMERICAL METHODS

(2141906)

TOPICNUMERICAL INTEGRATION : ERROR

FORMULAE, GAUSSIAN QUADRATURE FORMULAE

PREPARED BYMECHANICAL – 4B2

DEVANSU KHORASIYA (150120119066)

GUIDED BYPROF. RAVI PANCHAL

PREPARED BY : DEVANSU KHORASIYA(GIT-150120119066)

INTRODUCTIONWhat is Numerical Integration? In numerical analysis, numerical

integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations.

The basic problem considered by numerical integration is to compute an approximate solution to a definite integral. It is different from analytical integration in two ways: first it is an approximation and will not yield an exact answer; Error analysis is a very important aspect in numerical integration. Second it does not produce an elementary function with which to determine the area given any arbitrary bounds; it only produces a numerical value representing an approximation of area.

Let’s Start The Journey…PREPARED BY : DEVANSU KHORASIYA(GIT-150120119066)

First Of All…

History of Numerical Integration The Beginnings Of Numerical Integration Have Its Roots In Antiquity. A Prime Example Of How Ancient These Methods Are Is The Greek Quadrature Of The Circle By Means Of Inscribed And Circumscribed Regular Polygons. This Process Led Archimedes To An Upper Bound And Lower Bound For The Value Pi. These Methods Were Used Widely Due To The Lack Of Formal Calculus.

The Method Of The Sum Of An Infinitesimal Area Over A Finite Range Was Unknown Until The Sixteenth Century When Newton Formalized The Concepts Of What We Know Now Know As Calculus. The Earliest Forms Of Numerical Integration Are Similar To That Of The Greek Method Of Inscribing Regular Polygons Into Curved Functions.

One could improve accuracy by choosing a better fitting shape. Later methods decided to improve upon estimating area under a curve decided to use more polygons but smaller in area. Such an example is the use of rectangles evenly spaced under a curve to estimate the area. Even further improvements saw the use oftrapezoids instead of rectangles to better fit the curvature of the function being analyzed. Today the best methods for numerical integration are known as quadrature methods that have a very small error.PREPARED BY : DEVANSU KHORASIYA(GIT-

150120119066)


Elements of Numerical IntegrationIf f(x) is a smooth well behaved Function, integrated over a small number ofdimensions and the limits of integration are bounded, there are many methods ofapproximating the integral with arbitrary precision. We consider an indefinite integral: Numerical integration methods can generally be described as combining evaluations of the integrand to get an approximation to the integral.

In this example the definite integral is thus approximated using areas of rectangles. The integration points and weights depend on the specific method used and the accuracy required from the approximation. An important part of the analysis of any numerical integration method is to study the behavior of the approximation error as a function of the number of integrand evaluations.


An Easy Method of Numerical Integration: Trapezoid RuleThe Trapezoid Rule calls for the approximation of area under a curve by fitting trapezoids under the curve and regularly spaced intervals. This method is very common in beginning calculus courses used as a transition into analytical integration. The method uses the outputs of the function as the two legs of the trapezoid and the specified interval is the height. The area of a trapezoid is one half the height multiplied by the sum of the two bases:

It would be more advantageous to use more trapezoids of smaller height to better fit the curvature of the graph. As we increase the number of trapezoids by increasing the number of divisions inthe interval, accuracy increases.


SIMPSON’S RULE

dx = + ) + 4+ ) + 2(+ ) ]

SIMPSON’S RULE

dx = + ) + 2+ ) + 3( + ) ]


Gaussian Quadrature FormulaeAn n- Point Gaussian Quadrature Formula is a Quadrature Formula Constructed to give an Exact Result For Polynomials degree 2n-1 or Less by a Suitable choice of the Points and Weight for I = 1, 2, 3,….n.The Gaussian Quadrature Formula is..

dx =f()

One-Point Gaussian Quadrature FormulaeConsider a Function f(x) over the Interval [-1, 1] with Sampling Point . and Weight Respectively.The One Point Gaussian Quadrature Formula is..

dx = f()

For n = 1 We Find Polynomials 2n-1 = 2(1)-1 = 1 and For f(x) = 1 or x…

dx = 2f(0)

𝑥1


Two-Point Gaussian Quadrature Formulae

Consider a Function f(x) over the Interval [-1, 1] with Sampling Points , . and Weights , Respectively.

The Two Point Gaussian Quadrature Formula is..

dx = f() + f()

This Formula Will be Exact for Polynomials of Degree up to 2n-1 = 2(2)-1 = 3, i.e. It is Exact for f(x)= 1, x,

dx = f(- ) + f()


Three-Point Gaussian Quadrature FormulaeConsider a Function f(x) over the Interval [-1, 1] with Sampling

Points , , . and Weights , Respectively.

The Three Point Gaussian Quadrature Formula is..

dx = f() + f() +f()

This Formula Will be Exact for Polynomials of Degree up to 2n-1 = 2(3)-1 = 5, i.e. It is Exact for f(x)= 1, x,

dx =f(-) + f(0) + f()


EXAMPLE : by One Point, Two Point and Three point Gaussian Formulae.



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NUMERICAL INTEGRATION : ERROR FORMULA, GAUSSIAN QUADRATURE FORMULA