cs357-slides6 - 6 Taylor Series Expansions Approximations...

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# 6 Taylor Series: Expansions, Approximations and Error L. Olson September 15, 2015 Department of Computer Science University of Illinois at Urbana-Champaign 1
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motivation All we can ever do is add and multiply with our Floating Point Unit (FPU) We can’t directly evaluate e x , cos ( x ) , x What can we do? Use Taylor Series approximation 2
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taylor series definition The Taylor series expansion of f ( x ) at the point x = c is given by f ( x ) = f ( c ) + f 0 ( c )( x - c ) + f 00 ( c ) 2 ! ( x - c ) 2 + · · · + f ( n ) ( c ) n ! ( x - c ) n + . . . = X k = 0 f ( k ) ( c ) k ! ( x - c ) k 3 If a function f(x) can be represented by an infinite power series, then it has a Taylor series representation.
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an example The Taylor series expansion of f ( x ) about the point x = c is given by f ( x ) = f ( c ) + f 0 ( c )( x - c ) + f 00 ( c ) 2 ! ( x - c ) 2 + · · · + f ( n ) ( c ) n ! ( x - c ) n + . . . = X k = 0 f ( k ) ( c ) k ! ( x - c ) k Example ( e x ) We know e 0 = 1, so expand about c = 0 to get f ( x ) = e x = 1 + 1 · ( x - 0 ) + 1 2 · ( x - 0 ) 2 + . . . = 1 + x + x 2 2 ! + x 3 3 ! + . . . 4 1. How close the expansion point is to the evaluation point. 2. The convergence of the Taylor Series based on that expansion. 3. Whether evaluation of f ^ (n)(x _0 ) is straightforward.
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taylor approximation So e 2 = 1 + 2 + 2 2 2 !
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