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23- Numerical Differentiation

# 23- Numerical Differentiation - Numerical differentiation...

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Numerical differentiation Errors & accuracy Rounding and truncation errors Numerical Differentiation Dhavide Aruliah UOIT MATH 2070U c ± D. Aruliah (UOIT) Numerical Differentiation MATH 2070U 1 / 27 Numerical differentiation Errors & accuracy Rounding and truncation errors Numerical Differentiation 1 Numerical differentiation formulas 2 Errors and accuracy of numerical differentiation formulas 3 Rounding and truncation errors in numerical differentiation c ± D. Aruliah (UOIT) Numerical Differentiation MATH 2070U 2 / 27 Numerical differentiation Errors & accuracy Rounding and truncation errors Numerical differentiation From calculus, the derivative of f at x = x is f 0 ( x ) = lim h 0 f ( x + h ) - f ( x ) h Whole array of techniques (product rule, chain rule, etc.) compute exact derivatives Given numerical function f , numerical differentiation is ﬁnding formulas for computing approximate derivatives Two major motivations for numerical differentiation formulas I Nonlinear equations: Newton’s method (approximation to f 0 through numerical differentiation) I Finite-difference approximations to solutions of differential equations (ordinary and partial) c ± D. Aruliah (UOIT) Numerical Differentiation MATH 2070U 4 / 27 Numerical differentiation Errors & accuracy Rounding and truncation errors Taylor’s theorem Theorem (Taylor) Let f have m + 1 continuous derivatives on [ a , b ] R for some m 0 . Let x , x [ a , b ] and deﬁne h : = x - x. Then, f ( x ) = f ( x + h ) = T m ( x ) + R m + 1 ( x ) where T m ( x ) : = m k = 0 f ( k ) ( x ) k ! h k = Taylor polynomial = f ( x ) + f 0 ( x ) h + f 00 ( x ) 2 h 2 + ··· + f ( m ) ( x ) m ! h m , R m + 1 ( x ) : = f ( m + 1 ) ( ξ ) ( m + 1 ) ! h m + 1 = remainder or truncation error , and ξ lies between x and x = x + h. c ± D. Aruliah (UOIT) Numerical Differentiation MATH 2070U 5 / 27

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Numerical differentiation Errors & accuracy Rounding and truncation errors Forward-difference approximation for f 0 ( x ) Use a Taylor polynomial of f centred at x = x to derive a forward-difference approximation to f 0 ( x ) . f ( x + h ) = f ( x ) + f 0 ( x ) h + f 00 ( ξ ) 2 h 2 f ( x ) = f ( x ) f ( x + h ) - f ( x ) = f 0 ( x ) h + f 00 ( ξ ) 2 h 2 Thus, we arrive at the forward-difference formula f 0 ( x ) = f ( x + h ) - f ( x ) h - f 00 ( ξ ) 2 h 1 ξ ( x , x + h ) c ± D. Aruliah (UOIT) Numerical Differentiation MATH 2070U 6 / 27 Numerical differentiation Errors & accuracy Rounding and truncation errors Backward-difference approximation for f 0 ( x ) Use a Taylor polynomial of f centred at x = x to derive a backward-difference approximation to f 0 ( x ) . f ( x ) = f ( x ) f ( x - h ) = f ( x ) - f 0 ( x ) h + f 00 ( ξ ) 2 h 2 f ( x ) - f ( x - h ) = f 0 ( x ) h - f 00 ( ξ ) 2 h 2 Thus, we arrive at the backward-difference formula f 0 ( x ) = f ( x ) - f ( x - h ) h + f 00 ( ξ ) 2 h 1 ξ ( x - h , x ) c ± D. Aruliah (UOIT) Numerical Differentiation MATH 2070U 7 / 27 Numerical differentiation Errors & accuracy Rounding and truncation errors Centred-difference approximation for f 0 ( x )
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