Scientific Computing. Error Analysis
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1 ECE257 Numerical Methods and Scientific Computing Error Analysis
2 Today s s class: Introduction to error analysis Approximations Round-Off Errors
3 Introduction Error is the difference between the exact solution and a numerical method solution. In most cases, you don t t know what the exact solution is, so how do you calculate the error. Error analysis is the process of predicting what the error will be even if you don t t know what the exact solution Errors can also be introduced by the fact that the numerical algorithm has been implemented on a computer
4 Falling Object Velocity From Numerical Methods for Engineers, Chapra and Canale, Copyright The McGraw-Hill Companies, Inc.
5 Significant digits Can a number be used with confidence? How accurate is the number? How many digits of the number do we trust?
6 Significant digits From Numerical Methods for Engineers, Chapra and Canale, Copyright The McGraw-Hill Companies, Inc.
7 Significant digits The significant digits of a number are those that can be used with confidence The digits that are known with certainty plus one estimated digit Exact numbers vs. measured numbers Exact numbers have an infinite number of significant digits π is an exact number but usually subject to round-off
8 Significant digits 49.0 mph - 3 significant digits From Numerical Methods for Engineers, Chapra and Canale, Copyright The McGraw-Hill Companies, Inc miles 7 significant digits
9 Accuracy and Precision Accuracy is how close a computed or measured value is to the true value Precision is how close individual computed or measured values agree with each other Reproducability Inaccuracy/Bias vs. Imprecision/Uncertainty
10 Accuracy and Precision From Numerical Methods for Engineers, Chapra and Canale, Copyright The McGraw-Hill Companies, Inc.
11 Accuracy and Precision The level of accuracy and precision required depend on the problem
12 Error Definitions Two general types of errors Truncation errors - due to approximations of exact mathematical functions E t = True value - approximation Round-off errors - due to limited significant digit representation of exact numbers E t = True value - representation
13 Error Definitions E t does not capture the order of magnitude of error 1 V error probably doesn t t matter if you re measuring line voltage, but it does matter if you re measuring the voltage supply to a VLSI chip. Therefore, its better to normalize the error relative to the value ε t = true error true value 100%
14 Error definitions Example: Line voltage E t =120V 119V =1V ε t = 1V 100% = 8.33% 120V Chip supply voltage E t = 3.3V 2.3V =1V ε t = 1V 100% = 30.3% 3.3V
15 Error definitions What if we don t t know the true value? Use an approximation of the true value ε t = approximate error approximate value 100% How do we calculate the approximate error? Use an iterative approach Approximate error = current approximation - previous approximation Assumes that the iteration will converge
16 Error Definitions For most problems, we are interested in keeping the error less than specified error tolerance ε a < ε s How many iterations do you do, before you re satisfied that the result is correct to at least n significant digits? ε s = ( 0.5 x 10 2 n )%
17 Example Infinite series expansion of e x e x =1+ x + x 2 2! + x 3 3! +L+ x n n! As we add terms to the expansion, the expression becomes more exact Using this series expansion, can we calculate e 0.5 to three significant digits?
18 Example Calculate the error tolerance ε s = 0.5 x ( )% = 0.05% First approximation e 0.5 =1 Second approximation e 0.5 = =1.5 Error approximation ε a = = 33.3% > 0.05% 1.5
19 Example Third approximation e 0.5 = =1.625 Error approximation ε a = = 7.69% > 0.05% 1.625
20 Example Terms Result ε t ε a % % 33.3% % 7.69% % 1.27% % 0.158% % % True value is
21 Round-Off Errors Round-off errors are caused because exact numbers can not be expressed in a fixed number of digits as with computer representations For example, a 32-bit number only has a range from to For larger numbers or fractional quantities, we can use floating point numbers, but we will encounter round-off errors
22 Floating-Point What can be represented in N bits? Unsigned 0 to 2 N 2s Complement - 2 N-1 1s Complement -2-2 N-1 N-1 to N-1 +1 to 2 N to 2 N-1-1 N BCD 0 to 10 N/4-1 But, what about? very large numbers? 9,349,398,989,787,762,244,859,087,678 very small number? rationals 2/3 irrationals transcendentals 2 e
23 Recall Scientific Notation decimal point exponent Sign, magnitude x x 10 Issues: Mantissa Sign, magnitude Arithmetic (+, -, *, / ) Representation, Normal form Range and Precision Rounding radix (base) Exceptions (e.g., divide by zero, overflow, underflow) Errors IEEE F.P. ± 1.M x 2 e Properties ( negation, inversion, if A B then A - B 0 )
24 Floating-Point Numbers Representation of floating point numbers in IEEE 754 standard: single precision sign S E M actual exponent is e = E exponent: excess 127 binary integer mantissa: sign + magnitude, normalized binary significand w/ hidden integer bit: 1.M 1 < E < 254 (E=0,255 are used for special values) S E-127 N = (-1) 2 (1.M) 0 = =
25 Floating point numbers Problems: Limited range of representable numbers Overflow and underflow Finite number of representable numbers within range Interval between numbers increases as numbers grow in magnitude
26 Floating point numbers From Numerical Methods for Engineers, Chapra and Canale, The McGraw-Hill Companies, Inc.
27 Overflow and underflow errors Range of numbers for IEEE single-precision is from to ( ) or 1.18x10-38 to 3.40x10 38 Overflow becomes infinity and underflow becomes zero. Double precision extends range to ( )
28 Infinity and NaNs result of operation overflows, i.e., is larger than the largest number that can be represented overflow is not the same as divide by zero (raises a different exception) +/- infinity S It may make sense to do further computations with infinity e.g., X/0 > Y may be a valid comparison Not a number, but not infinity (e.q. sqrt(-4)) invalid operation exception (unless operation is = or =) NaN S non-zero NaNs propagate: f(nan) = NaN HW decides what goes here
29 Quantization Finite number of representable numbers due to round-off or chopping π becomes or instead of Due to base-conversion, rational numbers may become unrepresentable as well becomes Round-off is better than chopping. Upper bound error of x/2 instead of x.
30 Increasing Interval x Allows floating point numbers to preserve significant digits Quantization errors will be proportional to magnitude of number Δx x Δx ε is the machine epsilon - measure of the precision of the mantissa ε for chopping x ε 2 for rounding ε = b 1 t with no implicit 1 ε = b t with implicit 1
31 Round-off errors Large numbers of interdependent computations Adding a large and small number Subtractive calculation Subtracting two nearly equal numbers
32 Real world examples Why does round-off error matter? Real world examples (from Vancouver Stock Exchange Ariane rocket Patriot missile
33 Vancouver Stock Exchange In 1982, the index was initiated with a starting value of with three digits of precision and truncation After 22 months, the index was at The index should have been at Successive additions and subtractions introduced truncation error that caused the index to be off so much.
34 Ariane rocket Ariane 5 rocket was launched in June The rocket was on course for 36 seconds and then veered off and crashed The internal reference system was trying to convert a 64-bit floating point number to a 16-bit integer. This caused an overflow which was sent to the on- board computer The on-board computer interpreted the overflow as real flight data and bad things happened.
35 Patriot missile The Patriot missile had an on-board timer that incremented every tenth of a second Software accumulated a floating point time value by adding 0.1 seconds Problem is that 0.1 in floating point is not exactly 0.1. With a 23 bit representation it is really only Thus, after 100 hours (3,600,000 ticks), the software timer was off by.3433 seconds. Scud missile travels at 1676 m/s. In.3433 seconds, the Scud was 573 meters away from where the Patriot thought it was.
36 Next Lecture Taylor Series Truncation Error Read Chapter 4
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