L21 - Review The three approximation methods we will...

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Unformatted text preview: Review The three approximation methods we will discuss, for the FOOIVP y ( t ) = f ( t,y ( t )) , y ( t ) = y , all consist in constructing a sequence ( t ,y ) ( t 1 ,y 1 ) ( t 2 ,y 2 ) ( t n ,y n ) ( t n +1 ,y n +1 ) ( t fin ,y fin ) of points in the t,y plane. The polygon obtained by joining these points by straight lines between them is an approximation to the real solution ( t ) . The (systemic) error that comes from go- ing along a straight line from ( t n ,y n ) to ( t n +1 ,y n +1 ) instead of along a solution of the differential equation is called The Lo- cal Error e n . The local errors accrue to the Global Er- ror E def = y fin- ( t fin ) in a compounding fashion. The methods we will discuss all have a fixed step size h , which means that t- n +1 = t n + h , n = 0 , 1 , 2 ,... . These methods have variants with vari- able step sizes, which have their advan- tages but which we wont discuss. 2 ( t,y ) t y ( t n ,y n ) ( t 1 ,y 1 ) t n ( t n +1 ,y n +1 ) The slope is f ( t,y ) The solution has y ( t ) = f ( t,y ( t )) t 1 = t + h t t n +1 = t n + h t t 1 = t + h Euler Approximation: t n +1 =...
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This note was uploaded on 09/07/2010 for the course PHY 303L taught by Professor Turner during the Spring '08 term at University of Texas at Austin.

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L21 - Review The three approximation methods we will...

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