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CE33500 – Computational Methods in Civil Engineering Differentiation Provided by : Shahab Afshari safshar00@citymail.cuny.edu

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Introduction Numeric differentiation is the computation of values of the derivative of a function f from given values of f In mathematics, finite-difference methods (FDM) are numerical methods for approximating the solutions to differential equations using finite difference equations to approximate derivatives

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Assuming the function whose derivatives are to be approximated is properly-behaved, by Taylor's theorem, we can create a Taylor Series expansion: where n! denotes the factorial of n, and R n (x) is a remainder term, denoting the difference between the Taylor polynomial of degree n and the original function.

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Finite Difference Schemes Consider the diagram below of a typical function f(x). It is desired to evaluate the derivative at point A where x=x 0 i.e. to find the gradient of the tangent at this point.

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Finite Difference Schemes For a function f(x) and at a point x=x 0, We will derive an approximation for the first derivative of the function by first truncating the Taylor polynomial as following which is forward difference scheme, first-order-accurate: While, central difference scheme, second-order-accurate

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Example 1 Using a step-size of h = 0.01, estimate the derivative of ln(x) at x = 3. Find the value of x where the derivative is really equal to this value. Work to eight decimal places. Set f(x)=ln(x). Now, use the formula with x 0 =3 and h=0.01. So The actual derivative of ln(x) is 1/x which equals 0.3333333 rather than 0.3333345 at x=3. It is at x=2.9999895 that the actual derivative is 0.3333345.

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Finite Difference Schemes Second-order accurate finite-difference approximations to higher derivatives (which can also be derived from Taylor's theorem) are

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Finite Difference Schemes Non-centered (forward or backward) finite-difference approximations can be derived which are useful for estimating a derivative at the edge of a function's range. For example, a second-order accurate forward finite- difference approximation for the first derivative is

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Richardson Extrapolation Richardson’s extrapolation can be viewed as a general procedure for improving the accuracy of approximations when the structure of the error is known. In numerical analysis, Richardson extrapolation is a sequence acceleration method, used to improve the rate of convergence of a sequence.

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Richardson Extrapolation

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Example 3

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Interpolation Interpolation means finding (approximate) values of a function f(x) for an x between different x-values x 0, x 1, …, x n at which the values of f(x) are given. Lagrange Interpolation Given (x 0, f 0 ), (x 1, f 1 ), …, (x n, f n ) with arbitrarily spaced x j, Lagrange had the idea of multiplying each f j by a polynomial that is 1 at x j and 0 at the other n nodes and then taking the sum of these n + 1 polynomials. Clearly, this gives the unique interpolation polynomial of degree n or less.

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Linear Interpolation is interpolation by the straight line through (x 0, f 0 ), (x 1, f 1 ); Thus the linear Lagrange polynomial p 1 is a sum p 1 = L 0 f 0 + L 1 f 1 with L 0 the linear polynomial that is 1 at x 0 and 0 at x 1 ; similarly, L 1 is 0 at x 0 and 1 at x 1. Linear Interpolation

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Example 4

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General Lagrange Interpolation For general n we obtain

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Given data (x 0, f 0 ), (x 1, f 1 ), …, (x n, f n ) can be interpolated by a polynomial P n (x) passes through these n+1 points (x j, f j ); Now if n is large, there may be trouble: P n (x) may tend to oscillate for x between the nodes x 0, x 1, …, x n. Hence we must be prepared for numeric instability. Cubic Spline Interpolation

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Determination of Cubic Spline

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Example 4

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Example 4 - continued

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