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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
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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
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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
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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
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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
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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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