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Dear students, this is to inform that GDB will be started on February 6, 2013 at 00:00 and will be closed on February 7, 2013 at 23:59.

The topic for Graded Moderated Discussion Board is

“Compare the efficiency and characteristics of methods available to you for numerical integration.”

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NUMERICAL INTEGRATION (“Compare the efficiency and characteristics of methods available to you for numerical integration.”)

1. NEWTON-COTES INTEGRATION FORMULAE
2. THE TRAPEZOIDAL RULE ( COMPOSITE FORM)
3. SIMPSON’S RULES (COMPOSITE FORM)
4. ROMBERG’S INTEGRATION
5. DOUBLE INTEGRATION

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In numerical analysis, the Newton–Cotes formulae, also called the Newton–Cotes quadrature rules or simply Newton–Cotes rules, are a group of formulae for numerical integration (also called quadrature) based on evaluating the integrand at equally-spaced points. They are named after Isaac Newton and Roger Cotes.

Newton–Cotes formulae can be useful if the value of the integrand at equally-spaced points is given. If it is possible to change the points at which the integrand is evaluated, then other methods such as Gaussian quadrature and Clenshaw–Curtis quadrature are probably more suitable.

In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. (See numerical integration for more on quadrature rules.) An n-point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree 2n − 1 or less by a suitable choice of the points xi and weights wi for i = 1,...,n. The domain of integration for such a rule is conventionally taken as [−1, 1], so the rule is stated as

In mathematics, the Euler–Maclaurin formula provides a powerful connection between integrals (see calculus) and sums. It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and infinite series using integrals and the machinery of calculus. For example, many asymptotic expansions are derived from the formula, andFaulhaber's formula for the sum of powers is an immediate consequence

some imp methods

Forward Euler

Hamming Midpoint

Backward Euler

Trapezoidal

Parabolic

Sirnpson's Rule

Corrected Trapezoidal

Romberg Integration

Gear's Integration Methods

Bode's Integration Methods

Milne's PredictorICorrector

Newton-Cotes

runge-KtUta

mth603 gdb# 1

Numerical ordinary differential equations is the part of numerical analysis which studies the numerical solution of ordinary differential equations (ODEs). This field is also known under the name numerical integration, but some people reserve this term for the computation of integrals.
Many differential equations cannot be solved analytically; however, in science and engineering, a numeric approximation to the solution is often good enough to solve a problem. The algorithms studied here can be used to compute such an approximation. An alternative method is to use techniques from calculus to obtain a series expansion of the solution.
Ordinary differential equations occur in many scientific disciplines, for instance in physics,chemistry, biology, and economics. In addition, some methods in numerical partial differential equations convert the partial differential equation into an ordinary differential equation, which must then be solved.
Numerical integration aims at approximating definite integrals using numerical techniques. There are many situations where numerical integration is needed. For example, several well defined functions do not have an anti-derivative, i.e. their anti-derivative cannot be expressed in terms of primitive function. A popular example is the function e��x2 whose anti-derivative does not exist. This function arises in a variety of applications such as those related to probability and statistics analyses. Furthermore, many applications in science and engineering are represented by integral differential equations that require a special treatment for the integral terms (e.g. expansion, liberalization, closure ...).

Therefore, numerical integration does not only provide a means for evaluating integrals numerically, but also grants us the ability to approximate special functions that are defined in terms of integrals. Without loss of generality, there are two classes of problems where numerical integration is needed. In the first class, one wishes to evaluate the integral of a well defined function. In this case, the integrand can be evaluated a various points because and numerical integration techniques help define the optimum number of these points as well as their locations. The second class of problems for applying numerical integration is found in differential equations the most common of which are those that express conservation principles. For example, the population balance equation, a well known partial differential equation encountered in process modeling and biological systems, exhibits source terms that are represented as integrals of the solution variable (e.g. the number density function). The most common technique for numerical integration is called quadrature. The recipe for quadrature consists of three steps

1. Approximate the integrand by an interpolating polynomial using a specified number of points or nodes

2. Substitute the interpolating polynomial into the integral

3. Integrate