LECTURE 05 - Lecture 5 Lecture 5 MEEN 357 Engineering...

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Lecture 5 MEEN 357 Engineering Analysis for Mechanical Engineers 1 Lecture-05-MEEN357-2009.doc RMB Lecture 5 Roundoff and Truncation Errors Chapter 4 of the textbook. The Taylor’s Theorem Theorem : Given a continuous function [ ] :, fa b R that is differentiable of order 1 K + on () , ab and a point , ca b , then at any point ( ) , x ( ) 1 0 ! k K kK k fc fx xc O xc k + = =− + (5.1) Examples: 2 0 24 2 0 35 21 1 2 0 1 11 1 !2 1 cos 1 2! 2 4 ! 1 sin ! 3 ! 5 ! 1 cosh 1 ! 1 sinh ! 3 ! 5 ! xn n n n n n n n n n n n ex x x n xx n x n n x n = = + = = + = == + + + = = − + +⋅⋅⋅ + + + = = + + = + + (5.2) Numerical Differentiation Section 4.3.4 of the textbook. If you are given a real valued function ( ) N N Real Open interval numbers b R (5.3) which is differentiable at a point ( ) , x , then the derivative is defined by ( ) ( ) lim h f xh fx h →∞ +− = (5.4)
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Lecture 5 MEEN 357 Engineering Analysis for Mechanical Engineers 2 Lecture-05-MEEN357-2009.doc RMB The formulas we derived during Lecture 4 made use of a partition of the interval [ ] , ab into equal step sizes . We built this partition by first selecting a positive integer N which represents the number of partitions. Given N , we defined the step size h by the formula ba h N = (5.5) and divided the interval into equal segments by the formulas 0 10 21 1 NN xa xh x x x hx b = =+ = += (5.6) The following figure illustrated this construction Difference Approximation to the First Derivative Forward Difference : () ( ) ( ) 1 ' ii i fx f xO h h + (5.7) 0 ax = N bx = ( ) f x h i f x 1 i f x +
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Lecture 5 MEEN 357 Engineering Analysis for Mechanical Engineers 3 Lecture-05-MEEN357-2009.doc RMB When one adopts (5.7) as the derivative the resulting error is ( ) Oh . The following figure illustrates this error. Backward Difference : () ( ) ( ) 1 ' ii i fx f xO h h =+ (5.8) We again reached the conclusion is that when one adopts (5.8) as the derivative the resulting error is ( ) . The following figure illustrates this error. Centered Difference ( ) ( ) 11 2 ' 2 i f h h +− (5.9) i x 1 i x f x Approximation True Slope i x 1 i x + f x Approximation True Slope
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Lecture 5 MEEN 357 Engineering Analysis for Mechanical Engineers 4 Lecture-05-MEEN357-2009.doc RMB Equation (5.9) shows that if one adopts the first centered difference as the first derivative, the resulting error is () 2 Oh . This error is an improvement of that associated with the forward and backward differences introduced above. The following figure illustrates the error associated with use of the centered difference. Finite Difference Approximations to Higher Derivatives Second and higher derivatives can also be approximated with the use of Taylor’s theorem. An application of Taylor’s theorem (5.1) yields ( ) () () ( ) ( ) 23 22 2 2 '' ' 2! i ii i i i i i fx f x f xf x xx xxO ++ + + =+ + + (5.10) Also, (see (5.1)) ( ) 11 1 1 '' ' 2! i i i i i i i i f x f x O + + + + (5.11) In terms of the step size h , these two equations can be written ( ) () () 2 3 2 '' '2 2 2!
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This note was uploaded on 02/23/2009 for the course MEEN 357 taught by Professor Anamalai during the Fall '07 term at Texas A&M.

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LECTURE 05 - Lecture 5 Lecture 5 MEEN 357 Engineering...

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