# 1.2 Round-off Errors and Computer Arithmetic

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1 1.2 Round-off Errors and Computer Arithmetic 1

2 In a computer model, a memory storage unit word is used to store a number. A word has only a finite number of bits. These facts imply: 1. Only a small set of real numbers (rational numbers) can be accurately represented on computers. 2. (Rounding) errors are inevitable when computer memory is used to represent real, infinite precision numbers. 3. Small rounding errors can be amplified with careless treatment. So, do not be surprised that (9.4)10= ( )2 can not be represented exactly on computers. Round-off error: error that is produced when a computer is used to perform real number calculations. 2

3 Binary numbers and decimal numbers Binary number system: A method of representing numbers that has 2 as its base and uses only the digits 0 and 1. Each successive digit represents a power of 2. ( bb 3 bb 2 bb 1 bb 0. bb 1 bb 2 bb 3 ) 2 where 0 bb ii 1, for each ii = 2,1,0, 1, 2 Binary to decimal: ( bb 3 bb 2 bb 1 bb 0. bb 1 bb 2 bb 3 ) 2 = ( bb bb bb bb bb bb bb ) 10 3

4 Binary machine numbers IEEE (Institute for Electrical and Electronic Engineers) Standards for binary and decimal floating point numbers For example, double type in the C programming language uses a 64-bit (binary digit) representation 1 sign bit (s), 11 exponent bits characteristic (c), 52 binary fraction bits mantissa (f) 1. 0 cc =

5 This 64-bit binary number gives a decimal floating-point number (Normalized IEEE floating point number): 1 ss 2 cc 1023 (1 + ff) where 1023 is called exponent bias. Smallest normalized positive number on machine has ss = 0, cc = 1, ff = 0: (1 + 0) Largest normalized positive number on machine has ss = 0, cc = 2046, ff = : ( ) Underflow: nnnnnnnnnnnnnn < (1 + 0) Overflow: nnnnnnnnnnnnnn > ( ) Machine epsilon εε mmmmmmm = 2 52 : this is the difference between 1 and the smallest machine floating point number greater than 1. 5

6 Positive zero: ss = 0, cc = 0, ff = 0. Negative zero: ss = 1, cc = 0, ff = 0. Inf: ss = 0, cc = 2047, ff = 0 NaN: ss = 0, cc = 2047, ff 0 6

7 Example a. Convert the following binary machine number (P)2 to decimal number. PP 2 = Example b. What s the next largest machine number of (P)2? 7

8 Decimal machine numbers Normalized decimal floating-point form: ±0. dd 1 dd 2 dd 3 dd kk 10 nn where 1 dd 1 9 and 0 dd ii 9, for each ii = 2 kk. A. Chopping arithmetic: 1. Represent a positive number yy as 0. dd 1 dd 2 dd 3 dd kk dd kk+1 dd kk+2 10 nn 2. chop off digits dd kk+1 dd kk+2. This gives: ffff yy = 0. dd 1 dd 2 dd 3 dd kk 10 nn B. Rounding arithmetic: 1. Add 5 10 nn (kk+1) to yy 2. Chop off digits dd kk+1 dd kk+2. Remark. ffff yy represents normalized decimal machine number. 8

9 Example Compute 5-digit (a) chopping and (b) rounding values of ππ = Definition Suppose pp is an approximation to pp. The actual error is pp pp. The absolute error is pp pp. The relative error is pp pp pp 0., provided that pp Remark. Relative error takes into consideration the size of value. Definition The number pp is said to approximate pp to tt significant digits if tt is the largest nonnegative integer for which pp pp pp 5 10 tt. 9

10 Example Find absolute and relative errors, and number of significant digits for: (a) pp = and pp = (b) pp = and pp = Example c. Find a bound of relative error for k-digit chopping arithmetic. 10

11 Finite-Digit arithmetic Arithmetic in a computer is not exact. Let machine addition, subtraction, multiplication and division be,,,. xx yy = ffff ffff xx + ffff yy xx yy = ffff(ffff xx ffff(yy)) xx yy = ffff(ffff xx ffff(yy)) xx yy = ffff(ffff xx ffff(yy)) Example xx = 5, yy = 1, uu = , vv = Use 5-digit chopping arithmetic to compute xx yy, xx uu, (xx uu) vv. Compute relative error for xx uu. 11

12 Calculations resulting in loss of accuracy 1. Subtracting nearly equal numbers gives fewer significant digits. 2. Dividing by a number with small magnitude or multiplying by a number with large magnitude will enlarge the error. Example d. Suppose zz is approximated by zz + δδ. where error δδ is introduced by previous calculation. Let εε = 10 nn, nn > 0. Estimate the absolute error of zz εε. 12

13 Technique to reduce round-off error Reformulate the calculation. Example e. Compute the most accurate approximation to roots of xx xx + 1 = 0 with 4-digit rounding arithmetic. Nested arithmetic Purpose is to reduce number of calculations. Example evaluate ff xx = xx 3 6.1xx xx at xx = 4.71 using 3-digit chopping arithmetic. 13

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