Error: Finding Creative ways to Screw Up CS 170: Computing for the Sciences and Mathematics
Error: Finding Creative ways to Screw Up
CS 170:Computing for the Sciences
and Mathematics
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 698.043990
103? 9 significant digits
Precision Number of significant digits
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
Truncation error Error that occurs when truncated, or finite, sum is
used as approximation for sum of infinite series
ex 1 xx2
12x3
123
x4
1234
xn
n!
e 111
12
1
123
1
1234
1
20!
1
21!
1
22!
1
23! 2.0510 20
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
average velocitychange in position
change in times(b) s(a)
b a
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?
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
Derivative
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):
instantaneous velocity at 1 sec limt 0
s(1t) s(1)
t
s'(a) dy
dt ta
limt 0
s(at) s(a)
t
Derivative at a point is the slope of the curve at that point
Differential Equation
Equation that contains a derivativeVelocity 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