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mid1-review

# mid1-review - First Midterm Review Introduction to...

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First Midterm: Review

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Introduction to Numerical Computing Concerns when solving a numerical calculation on a computer: numbers must be represented using a finite number of bits; this can lead to errors in number representation numerical error can be accumulated and amplified when quantities are used and modified in subsequent arithmetic calculations
Introduction to Numerical Computing Significant digits of precision dependence on measuring instrument Absolute and relative error, signed and unsigned absolute_error = | exact_x approx_x | relative_error = absolute_error / | exact_x | signed_total_error = approx_x exact_x signed_relative_error = signed_total_error / | exact_x | Achieving finite precision by chopping or rounding; different methods of tie-breaking in rounding, including round-half-up and round-half-to-even

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Introduction to Numerical Computing Nested multiplication f ( x ) = a 0 + x ( a 1 + x ( a 2 + L x ( a n ) L )) f ( x ) = a 0 + a 1 x + a 2 x 2 + L + a n x n
Review of Taylor Series can approximate the value of a function at a point using an infinite sum of terms involving the values of higher order derivatives of the function evaluated at a (nearby) single point of expansion f ( x ) " f ( c ) + # f ( c )( x \$ c ) + # # f ( c ) 2 ( x \$ c ) 2 + L + f ( k ) k ! ( x \$ c ) k + L

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Review of Taylor Series Some classical Taylor series: e x = 1 + x + x 2 2 + x 3 3! + L = x k k ! k = 0 " # , x < " sin x = x " x 3 3! + x 5 5! " + L = ( " 1) k x 2 k + 1 (2 k + 1)! k = 0 # \$ , x < # cos x = 1 " x 2 2! + x 4 4! " + L = ( " 1) k x 2 k (2 k )! k = 0 # \$ , x < # 1 1 " x = 1 + x + x 2 + x 3 L = x k k = 0 # \$ , x < 1 ln(1 + x ) = x " x 2 2 + x 3 3 " + L = ( " 1) k " 1 x k k # \$ % % & ( ( k = 0 ) * , " 1 < x + 1
Review of Taylor Series Role and importance of the point of expansion a Taylor series converges quickly for x near the point of expansion but more slowly, or not at all, for more distant x Taylor’s Theorem using Taylor’s theorem to determine the range of x for which a Taylor series will converge, and to bound the error in a Taylor Series approximation of a function f ( x ) = f ( k ) ( c ) k ! k = 0 n " ( x # c ) k + f ( n + 1) ( \$ ) ( n + 1)! ( x # c ) n + 1

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Review of Taylor Series Alternating Series Theorem If a 1 a 2 0 and then the alternating series a 1 a 2 + a 3 a 4 converges, and the difference between the n th partial sum and the exact sum does not exceed the size of a n +1 lim n "# a n = 0
Taylor Series Demo

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Number Representation on a Computer Normalized floating point number representation Single-precision floating point number representation on a computer x = ± (0. b 1 b 2 K ) 2 " 2 n x = ( " 1) s # (1. f ) 2 # 2 c " 127 s c f 8 bits 23 bits
Number Representation on a Computer Double-precision floating point number representation on a computer Subnormal numbers x = ( " 1) s # (1. f ) 2 # 2 c " 1023 s c f 11 bits 52 bits x = ( " 1) s # (0. f ) 2 # 2 m

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Number Representation on a Computer Representing signed and unsigned integers on a
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