Error: Finding Creative ways to Screw Up

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CS 170Computing for the Sciences and Mathematics

• Error Finding Creative ways to Screw Up

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Administrivia

• Last time
• Basics of modeling
• Assigned HW 1
• Today
• HW1 due!
• Assign HW 2
• Monday 9/13 NO CLASS

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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

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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

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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

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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 isnt 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

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Implementation Errors

• Incorrect programming (a bug)
• Didnt 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!

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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

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Exponential notation

• Example 698.043990 ? 103
• Fractional part or significand?
• 698043990
• Exponent?
• 3
• Normalized?
• 6.98043990 ? 105

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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

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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
  arent!
• i.e. 0.2 is an infinite repeating series when
  expressed in binary

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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

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Relative error

• (correct - result) / correct
• Example (correct - result) 0.990 and correct
  0.698043990 ? 105
• 0.990/(0.698043990 ? 105) 1.4182487 ? 10-5

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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?

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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)

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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

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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

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Error Propagation (Accumulated Error)

• Example repeatedly executing
• t t dt
• Better to repeatedly increment i and calculate
• t i dt

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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

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Truncation error

• Truncation error
• Error that occurs when truncated, or finite, sum
  is used as approximation for sum of infinite
  series

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Error is Not Inherently Bad

• Almost all of these issues can be managed and
  controlled!
• A certain amount of error is normal
• Whats important is that we
• Know how much error there might be
• Keep it within a bound that allows the results to
  still be valid

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Rates The Basics of Calculus
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Calculus

• Mathematics of change
• Two parts
• Differential calculus
• Integral calculus

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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

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Height (y) in m vs time (t) in sec of ball thrown
up from bridge
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Average velocity
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

• 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?

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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)

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Derivative at a point is the slope of the curve
at that point
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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

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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

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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

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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