numerical differential & integration. introduction if a function f(x) is defined as an...
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Numerical Differential & Integration
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Introduction• If a function f(x) is defined as an expression, its
derivative or integral is determined using analytical techniques.
• If f(x) is complicated or when it is given in a tabular form, numerical methods are used to determine the derivative or integral.
• The accuracy of these methods would depend on the given function & the order of the polynomial used. If the fitted polynomial is exact the error is likely to be zero.
• The numerical methods would be avoided if an alternative exists.
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Numerical Differentiation
• Find a suitable interpolating polynomial to represent the function.
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This provides the value of dy/dx at any which is not in the table
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Putting x = a and u = 0 in 3
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Maxima & Minima of Tabulated Functions
Maxima/Minima of a y=fx) can be found by equating dy / dx = 0
Differentiating it w. r. t u we get
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For simplicity truncate terms after the third term and by solving this quadratic equation we get two values of u.
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Reading Assignment
• Read on errors in numerical differntiation and write a comprehensive report.
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Numerical Integration
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Newton-Cote’s Quadrature Formula
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This is a Newton-Cote’s Formula. Trapezoidal Rule, Simpson’s one-third and three-eigth rules, and Weddle’s rule all can be deduce from this by putting n=1,2,3 & 6.
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Assignment No. 4