IPS PLANET EARTH (BPS)
IPS PLANET EARTH (BPS) USU 1360
Utah State University
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This 33 page Class Notes was uploaded by Lorna Mayer on Wednesday October 28, 2015. The Class Notes belongs to USU 1360 at Utah State University taught by Vicki Allan in Fall. Since its upload, it has received 28 views. For similar materials see /class/230485/usu-1360-utah-state-university in University Studies at Utah State University.
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Date Created: 10/28/15
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using English units but NASA used metric units Precision Floating Point numbers with a decimal and exponent are stored in three pieces sign 0 or 1 mantissa or significand value exponent or magnitude Example 44355 would be written as 44355 X 105 we say it is normalized sign Oasis positive mantissa or significand value is 44355 exponent 5 The precision is the number of significant digits Example 00004500 In normalized form what is the precision magnitude and exponent 00004500 45 x 10394 what is the precision 2 digits of accuracy magnitude 4 mantissa 45 Recall the magnitude is the number of places you moved the decimal If to convert to normal form you moved the exponent left the exponent is positive If to convert to normal form you moved the exponent right the exponent is negative RecaH 105 100000 104 10000 103 1000 102 100 101 10 100 1 anything to the zero power is 1 10391 1 1101 one tenth 10392 01 1102 one hundredth 10393 001 1103 one thousandth 10394 0001 1104 one ten thousandth Example 874200 In normalized form what is the precision magnitude and exponent 874200 8742 x 106 what is the precision 4 digits of accuracy magnitude 6 mantissa 8742 A representation of a continuous real variable or process can never be exact because the finite number of represented points cannot cover the infinity of real points Representation error is the difference between a real number and its nearest represented number With careful planning algorithms can be organized so that representation errors of outputs are limited Mathematical software libraries are designed this way In a poorly organized long computation representation errors can accumulate culminating in a floating point result with a large error For example a sum of n floating point numbers can have an error as high as n23 Errors in small differences can also cause problems For example an algorithm that computes 1AB may give a dividebyzero error if A and B are within 23932 of each other Propagation of Error When you add two numbers that each have error what is the uncertainty of the sum When you multiply two numbers that each have error what is the uncertainty of the product Absolute and relative error Absolute error the absolute value throw away sign of the difference between the exact and the computed answer Example Area of a polygon is actually 4525 square pixels We estimated 4700 What is the absolute error 4525 4700 175 Absolute and relative error Absolute error the absolute value throw away sign of the difference between the exact and the computed answer Example Area of a polygon is actually 272 square pixels We estimated 150 What is the absolute error 272150 122 Which of the two errors is more significant If I am off by 175 out of 4525 or if I am off by 122 out of272 Relative error divides the absolute error by the actual answer 1754525 038 or 38 error 122272 448 or 448 error Truncation With computers we only have so much room to store a number If we are short of memory what should we not store Sign txponentr mantissa The first digits or the last To truncate to k significant digits remove all digits after the kth one Truncate 35443552 to five digits 35443552 35444000 Truncate to 3 significant digits 874200 In normalized form what is the precision magnitude and exponent 874200 8742 x 106 what is the desired precision 3 digits of accuracy magnitude 6 mantissa 874 after truncation Round off error Instead of truncating a number to fit in storage the computer might round 874600 In normalized form what is the recision magnitude and exponent rounded to three digits 874600 8746 x 106 what is the precision 3 digits of accuracy magnitude 6 mantissa 875 after rounding Rounding As you expect rounding doesn t always make the number larger but goes larger or smaller whichever is closer minimizes the error To round to k digits consider the k1st digit If it is below five round the kth digit down If it is five or above round the kth digit up Round off error The error between what you wanted to store and what you actually could store Actually just quotregular error when you are rounding In Binary In binary these same kinds of problems occur Suppose we have a simple integer with no fractional part and that we have 4 bits for each integer Now lets add two numbers 1101 13 in decimal 1101 13in decimal 11010 but we have overflowed our memory as we have a four bit limit 1010 is all that will be stored 1313 10 You ask why doesn t the machine check for the error and warn you Well it could but it would slow things down Every operation would be followed by a check to see if overflowed We call this quotoverkillquot In Binary the errors sometimes look different Suppose we have a simple integer with no fractional part and that we have 8 bits for each integer Let s assume that the first bit is the sign 00001111is 15 10001111is15 Now lets add two numbers 01010101 85 in decimal 01010101 85 in decimal 10101010 oops this is 42 Similarly we can add two negative numbers and have the overflow make it look positive Suppose we add a small number to a large one 1 Suppose we have four digits of accuracy 157823 is stored as 157800 1578 x 106 2 If we add a small number to it we get 157800 1 157801 157800 with four digits of accuracy 3 We may be fine with that but if we add the small number a million times it doesn t seem right that the original number never changes We call this error propagation or cumulative error It started out small but after repeated additions it had become significant From Lawcumulative error is premised on the existence of errors no one of which merits reversal but in combination they necessitate the reversal of a finding or sentence In navigation a 1 error in direction becomes significant if you are flying 10000 miles Cumulative Error Cumulative error can also occur with truncation Suppose I need to add 433333 multiple times Suppose we have three digits of accuracy In theory 433333 433333 866666 which rounds to 867 With just three digits of accuracy however 4330 4330 8660 which is off by one digit in a single operation We have to be careful with floating point numbers When we store numbers in the computer they are truncated so that expressions that are numerically equivalent do not hold with computer arithmetic Xv 2 XZv right if x 241 y 975 z 154 and we have three significant digits of accuracy use Excel or a caIUIator put round after each operation xy z 380 xzy 381 The rule Never compare floating point numbers for equality 1 Write the following numbers in normalized form a 63850 b 88 c oooosz using excel or some other tool evaluate the following in multiple steps 1000000000000000000 1 1000000000000000000 That is 1 x1018 1 1 x1018 What should the result be What is the result Why Note that excel writes it as 1E18 Consider 6279 Round it to two decimal places and compute the relative and absolute error Consider 6279 Truncate it to two decimal places and compute the relative and absolute error Super For a machine that rounds to three significant digits give an example where a floating point number x has no multiplicative inverse ie y where xy 1