5.2 Errrors

Why Study Errors First? Nearly all our modeling is done on digital computers (aside: what would a non-digital analog computer look like?)

Analog Computer for Fitting a Line to a Set of Points

Why Study Errors First? Nearly all our modeling is done on digital computers (aside: what would a non-digital analog computer look like?) A better set of terms is discrete vs. continuous (recall the discrete/continuous distinction from 1.2). Discrete devices have finite precision - we cannot represent all the different values we’d like. Finite precision leads to errors

Finite Precision with Bits Bit = value of 0 or 1 (aside: why base 2 ?) With n bits, we can distinguish 2 n different values E.g., with three bits we get eight possible values. These values can correspond to anything we want to represent… 0000AGreg 0011BWade 0102CSam 0113DJordan 1004EMark 1015FJason  1106GGaurav  1117HAlex

Floating-Point Numbers History: origins of computers in finance (banking) and military (physics). Financial calculations are (usually) fixed- point: the decimal place is fixed in one position, two digits from right: $129.95, $0.59, etc. In physical calculations, we need the decimal place to “float”: , 0.333, × 10 23, etc.

Exponential Notation Format: Mantissa X 10 Exponent, or a X 10 n Abbreviated a e n or a E n; for example, e23 Mantissa (a.k.a. significand, a.k.a. fractional part) is a floating-point number; exponent is an integer. So someone has to decide the precision: –IEEE 754 standard: with 64-bit numbers, use 52 bits for mantissa, 11 for exponent, 1 for sign (+ or −) Rough conversion: 3 bits  1 decimal digit (why?) E.g., 53 bits  16 decimal digits of precision

Normalized Notation Normalization is a common operation: e.g., percentage normalizes everything to 100: 2/4 = 75/50 = 50% Scientists usually prefer to normalize to 1: 2/4 = 75/150 = 0.5 So a normalized number in exponential notation has the decimal place immediately before the first nonzero digit: e1, e24, etc. Significant digits of a normalized floating-point number are all of the digits except leading zeros.

Precision and Magnitude Precision is the number of significant digits. Magnitude is the power of e 1 6 digits of precision Magnitude = 1

Quick Review Question 1 For the number , give a.the significand in normalized exponent notation b.the magnitude in normalized exponent notation c.the precision

Absolute and Relative Error If correct is the answer and result is the result obtained, then absolute error = | correct - result | relative error = (absolute error) | correct | = (absolute error) | correct | X 100%

Truncation To truncate a normalized number to k significant digits, eliminate all digits of the significand beyond the kth digit. E.g., truncated to 5 significant digits =.?????

Truncation To truncate a normalized number to k significant digits, eliminate all digits of the significand beyond the kth digit. E.g., truncated to 5 significant digits = Relative error = | correct - result | | correct | X 100%

Truncation To truncate a normalized number to k significant digits, eliminate all digits of the significand beyond the kth digit. E.g., truncated to 5 significant digits = Relative error = | | | | X 100%

Truncation To truncate a normalized number to k significant digits, eliminate all digits of the significand beyond the kth digit. E.g., truncated to 5 significant digits = Relative error = X 100%

Truncation To truncate a normalized number to k significant digits, eliminate all digits of the significand beyond the kth digit. E.g., truncated to 5 significant digits = Relative error = %

Rounding To round a normalized number to precision k, consider the (k+1)th significant digit d. If it is less than 5, round down the normalized number by truncating the significand to k significant digits. If d is greater than or equal to 5, round up the normalized number by truncating the significand to k significant digits and then adding 1 to the kth significant digit of the significand, carrying as necessary to digits on the left.

Quick Review Question 3 Round each of the following so that the significand has a precision of 2: a e-5 b e-5 c e-5

Round-off Error Round-off error is the problem of not having enough bits (or digits) to store an entire floating- point number and approximating the result to the nearest number that can be represented. E.g., 1/3 = ≠ 0.3 ≠ 0.33 ≠ ≠ etc.

Avoiding Round-off Error Some values (1/3) will always result in roundoff; however, we can help reduce error: –When adding numbers whose magnitudes are drastically different, accumulate smaller number before combining them with larger ones. –When multiplying and dividing, perform all multiplications in the numerator before dividing by the denominator.

Avoiding Round-off Error E.g. consider computing (x/y)z on a machine with three significant digits of precision, where x = 2.41y = 9.75z = 1.54 (x / y)z = (2.41 / 9.75)(1.54) = (0.247)(1.54) = But (x / y)z = (xz) / y: (xz) / y = ((2.41) )(1.54)) / 9.75 = 3.71 / 9.75 = 0.381

Overflow and Underflow Overflow is an error condition that occurs when there are not enough bits to express a value in a computer. Underflow is an error condition that occurs when the result of a computation is too small for the computer to represent.

Overflow and Underflow in Excel

Looping and Error Propagation A loop is a segment of code [computer instructions] that is executed repeatedly. Accumulating values in a loop (repeatedly) can produce error E.g., Patriot Missile failure in 1991 Gulf War

Patriot Missile Failure Patriot’s internal clock measured time in 1/10 of a second. Each tick of the clock added 1/10 s to current time. But 1/10 cannot be represented in a finite number of bits (just as 1/3 cannot be represented in a finite number of decimal digits)

Representing Fractions in Binary Consider ordinary Base-10 notation: what does e.g mean? Binary works the same way; e.g., what is ? = /4 = 5.025

1676 meters 0.34 sec 1 sec Patriot Missile Failure Each one-tenth increment produced an error of about seconds. After ten hours of running Patriot’s computer: = 0.34 sec 10 hr 60 min 60 sec sec 1 hr 1 min 1 sec Scud travels 1,676 meters / sec. So error = = 570 meters Result Patriot missed Scud, 28 Americans died

+ Looping and Error Propagation in Excel* Highlight; click corner; drag downward: Enter values: See apparent uniform sequence: Double click for actual values: *Courtesy of Bob Panoff

Avoiding Error Propagation in Excel* Enter first value; highlight rest: *Courtesy of Bob Panoff Do Fill/Series:

Other Ways Up?

LA Times, 1 October 1999