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Class Note for COSC 3361 at UH

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This 18 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 12 views.

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Date Created: 02/06/15
Basic concepts of integration I Antidifferentiation Given a function fX find a function FX with the property that F x f x 1 Example fxax gt Fx axquot1c n1 fxex gt Fxex Calculating the Integral of a function from a to b l fxdx Fltbgt Flta 0080 3361 Numerical Analysis Edgar Gabriel C J Basic concepts of integration ll Graphical Interpretation 0080 3361 Numerical Analysis Edgar Gabriel C Basic concepts of integration III Some rules i6 fxdx ijxdx jgx f xdx f f 206126 j gxdx jfxdx jfxdx f fxdx jfxdx j fxdx 0080 3361 Numerical Analysis Edgar Gabriel C J Basic concepts of integration III Some rules 2 g 1ltbgt I fxdx I fgtg tdt g 1ltagt Ig xfxdx gxfxZ I gxf xdx 0080 3361 Numerical Analysis Edgar Gabriel C S Problems Some antiderivatives can not be determined easily eg Do 2k1 x2 x Fm I6 dx 2k1k some can not be determined at all FxI dx In these situations numerical approximations for the integrals are required 0080 3361 Numerical Analysis Edgar Gabriel C J Numerical integration 15t solution use a polynomial to approximate the function in the integration domain ifxdx z ipxdx Using the Lagrange Form of the interpolation function b b n rt jpxdx IZfxilixdx with lix H x x a i0 jOj i xi xj xnllmdx 71 0080 3361 Numerical Analysis Edgar Gabriel C J Numerical integration ll Equation 71 is often expressed as b n b j fxdx z 2 Aifxi with A139 j remix 81 a i0 a If the nodes in equation 81 are equally spaced it is called the NewtonCotes formula 0080 3361 Numerical Analysis Edgar Gabriel C J Trapezoid Rule I Applying the NewtonCotes formula for n1 assuming that x0 a and x1 19 the cardinal functions are 9 10xb x and 11xx a Thus a 5 61 A0 10xdx b a A1 llxdxb a and b a 2 I fxdxe fafb Please note the trapezoid rule produces exact results for all polynomials of degree at most 1 0080 3361 Numerical Analysis Edgar Gabriel C J Trapezoid Rule ll 1 The error term can be expressed as e Eba3f 5 If the interval ab is partitioned into several pieces eg ax0 ltx1ltltxn 17 the trapezoid rule can be applied to each subinterval l fxdx i fxdx z gim x1fX1fxi 101 1 1 xi 101 is also called the composite trapezoid rule Can also be applied for spline functions 0080 3361 Numerical Analysis Edgar Gabriel C J Trapezoid Rule Ill Graphical interpretation YA 0080 3361 Numerical Analysis Edgar Gabriel C S J NewtonCotes example For n2 and ab0 1 apply the NewtonCotes formula For equally spaced points x0 ox1lx2 1 The according cardinal functions are 10x 2x x 1 11x 4xx 1 12 x 2xx A0j10xdxl A1j11xdxE A2j12xdxl Thus 0 6 o 3 o 6 from z Aron lflt03fltlgtlflt1gt 0 i0 1 l 6 3 2 6 121 Note the NewtonCotes formula is exact for polynomials of degree at most n 0080 3361 Numerical Analysis Edgar Gabriel C J Method of undetermined coefficients Equation 121 can also be derived differently from z Aoflt0gt A1f Agra If the polynomial is of degree 2 use trial functions to determine 140241242 The trial functions for n2 would be foo 1xx2 0080 3361 Numerical Analysis Edgar Gabriel C J Method of undetermined coefficients ll Applying the trial function one obtains 1 Idxxg1 1AOA1A2 141 0 1 1 1 1 1 xdxax2g5 gt E A1A2 142 1 1 1 1 1 2d31gt A x x 13 10 3 3 41A2 143 Solving 141142 and 143 leads to 1 2 1 A 2 6 1 3A2 6 0080 3361 Numerical Analysis Edgar Gabriel C J Example 1 For fx3x21 determine Ifxdx Using antiderivative O Fx x3 x C andthus 1 jfxdx2 O Using NewtonCotes for n2 1 1 2 1 1 1 27 1 1 42 remix 6f03f26f 6346 0080 3361 Numerical Analysis Edgar Gabriel C J Simpson s Rule NewtonCotes of order 2 for an arbitrary interval ab b a ab 6 fa4f 2 i foodxz fb Error term i 5 lt4 5 6 90W a2f 98 011 0080 3361 Numerical Analysis Edgar Gabriel C J Composite Simpson s rule For an even number of subintervals and even spacing hb an xiaih OSiSn Then b nZ xzi Ifxdx Tfxdxxffxdx Tfxdx Z Ifxdx i1 x214 Applying the Simpson Rule to each subinterval leads to nZ foodx fay 2 4fx2i 1 fltx2i Error term e iltb agth4f lt gt 0014 1 80 0080 3361 Numerical Analysis Edgar Gabriel C J Exercises I Derive the NewtonCotes formula for n3 based on the nodes O13231 using the method of undetermined coefficients 3 3 Solution AO 1A1 A2 A3 l 8 8 8 Use the formula obtained from the prevrous exercrse to calculate ln2 using 1 1 dx Ox1 Solution ln2z069765625 errorz00045 0080 3361 Numerical Analysis Edgar Gabriel C J 0080 3361 Numerical Analysis Numerical Integration and Differentiation ll Numerical Integration of Polynomials Edgar Gabriel Fall 2005 0080 3361 Numerical Analysis Edgar Gabriel C

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