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CS 170: Computing for the Sciences and Mathematics Error: Finding Creative ways to Screw Up Administrivia Last time Basics of modeling Assigned HW 1 Today HW1 due! Assign HW 2 Monday 9/13– NO CLASS Error What is the value of a model… That is completely wrong That is a perfect match to a physical system That may be off by as much as 5% Types of Error Input Data Errors Faulty/inaccurate sensors, poorly calibrated, mis-read results.. Modeling Errors Poor assumptions, bad math, mis-understanding of system Implementation Errors Bug in computer program, poor programming Precision The limits of finite number representation Input Data Errors NSIDC Sensor drift led to their real-time sea ice estimates to be off by over 500,000 km2 Still only off by 4% http://nsidc.org/arcticseaicenews/2009/022609.html Modeling Errors Obvious type: math formulation errors, mis-writing a formula, etc. By virtue of making assumptions and simplifications, models will have “error” versus reality. This isn’t necessarily a bad thing, as long as we manage it well Lord Kelvin used his knowledge of temperature dissipation to estimate that the earth was between 20-40 million years old. Actual: ~12 billion Assumed there was no heat source but the sun Implementation Errors Incorrect programming (a “bug”) Didn’t implement the model correctly Wrong equations Mis-defined inputs/outputs Implemented the solution to a different problem Precision Errors Computers only have so much space to store numbers This limits the range and precision of values! Precision Errors In a computer, a number is stored in a number of bits (binary digits) IEEE 754 standard floating point Variation on standard normalized scientific notation single-precision is 32 bits double-precision is 64 bits Stored in 3 parts sign (1 bit) – is it positive or negative? magnitude – what is the exponent? mantissa/significand – what is the number? i.e. 6.0221415 × 1023 Exponential notation Example: 698.043990 103 Fractional part or significand? 698043990 Exponent? 3 Normalized? 6.98043990 105 Significant digits Significant digits of floating point number All digits except leading zeros Number of significant digits in 9 significant digits Precision Number of significant digits 698.043990 103? Round-off error Problem of not having enough bits to store entire floating point number Example: 0.698043990 105 if only can store 6 significant digits, rounded? 0.698044 105 Do not test directly for equality of floating point variables Note that many numbers that seem “safe” really aren’t! i.e. 0.2 is an infinite repeating series when expressed in binary Absolute error |correct – result| Example: correct = 0.698043990 105 and result = 0.698043 105 Absolute error = ? |0.698043990 105 - 0.698043 105| = 0.00000990 105 = 0.990 Relative error |(correct - result) / correct| Example: (correct - result) = 0.990 and correct = 0.698043990 105 0.990/(0.698043990 105) = 1.4182487 10-5 Relative error Why consider relative errors? 1,000,000 vs. 1,000,001 1 vs. 2 Absolute error for these is the same! If exact answer is 0 or close to 0, use absolute error Why? Addition and subtraction errors Beware if there is a big difference in the magnitude of numbers! Example: (0.65 105) + (0.98 10-5) = ? 65000 + 0.0000098 = 65000.0000098 Suppose we can only store 6 significant digits? 65000.0 = (0.650000 105) Associative property Does not necessarily hold! Sum of many small numbers + large number may not equal adding each small number to large number Similarly, distributive property does not necessarily hold To reduce numerical errors Round-off errors Use maximum number of significant digits If big difference in magnitude of numbers Add from smallest to largest numbers Error Propagation (Accumulated Error) Example: repeatedly executing: t = t + dt Better to repeatedly increment i and calculate: t = i * dt Overflow/underflow Overflow - error condition that occurs when not enough bits to express value in computer Underflow - error condition that occurs when result of computation is too small for computer to represent Truncation error 2 3 4 n x x x x e x 1 x 1 2 1 2 3 1 2 3 4 n! 1 1 1 1 e 11 1 2 1 2 3 1 2 3 4 20! Truncation error Error that occurs when truncated, or finite, sum is used as approximation for sum of infinite series 1 1 1 21! 22! 23! 2.051020 Error is Not Inherently Bad Almost all of these issues can be managed and controlled! A certain amount of error is normal What’s important is that we: Know how much error there might be Keep it within a bound that allows the results to still be valid Rates: The Basics of Calculus Calculus Mathematics of change Two parts Differential calculus Integral calculus Rates “Rate of Change” Often depends on current amounts Rates are important in a lot of simulations growth/decay of… populations materials/concentrations radioactivity money motion force pressure Height (y) in m vs time (t) in sec of ball thrown up from bridge change in position s(b) s(a) average velocity change in time b a Time (t) in seconds Height (y) in meters 0.00 11.0000 0.25 14.4438 0.50 17.2750 0.75 19.4938 1.00 21.1000 1.25 22.0938 1.50 22.4750 1.75 22.2438 2.00 21.4000 2.25 19.9437 2.50 17.8750 2.75 15.1937 3.00 11.9000 3.25 7.9938 3.50 3.4750 3.75 -1.6563 Average velocity between 0 sec & 1 sec? between 1 sec & 2 sec? between 0.75 sec & 1.25 sec? Estimate of instantaneous velocity at t = 1 sec? Derivative s(1 t) s(1) instantaneous velocity at 1 sec lim t 0 t Derivative of y = s(t) with respect to t at t = a is the instantaneous rate of change of s with respect to t at a (provided limit exists): dy s'(a) dt s(a t) s(a) lim t 0 t t a Derivative at a point is the slope of the curve at that point Differential Equation Equation that contains a derivative Velocity function v(t) = dy/dt = s'(t) = -9.8t + 15 Initial condition y0 = s(0) = 11 Solution: function y = s(t) that satisfies equation and initial condition(s) in this case: s(t) = -4.9t2 + 15t + 11 Second Derivative Acceleration - rate of change of velocity with respect to time Second derivative of function y = s(t) is the derivative of the derivative of y with respect to independent variable t Notation s''(t) d2y/dt2 Systems Dynamics Software package that makes working with rates much easier. Components include: “Stocks” – collections of things (noun) “Flow” – activity that changes a stock (verb) “Variables” – constants or equations – converter “Connector” – denotes input/information being trasmitted HOMEWORK! READ pages 17-48 in the textbook On your own Work through the Vensim PLE tutorial Turn-in the final result files on W: Vensim is being deployed tonight. If there is a problem, I will notify everyone ASAP. NO CLASS on Monday
