CSE 541 Numerical Methods

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CSE 541Numerical Methods

• Machine Representation of Numbers

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Speed

• The computer vs the human
• Who can count to 1 trillion the fastest?
• Human count from one to 1,000,000,000,000 by one
• - or -
• Computer Execute on a Pentium-4 2GHz machine
• int i 1
• while (1)
• printf( d/n, i )
• Computers are extremely fast at performing simple
  arithmetic

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Blazing Fast

• Assume a human can count on average one number
  per second (bad assumption).
• Implies 1 trillion seconds, or about 32,000
  years!!!!
• Assume that a 2GHz machine can increment by one
  each clock cycle.
• Implies 500 seconds!!!!

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Machine Precision

• What is our current National Debt?
• (see http//www.brillig.com/debt_clock/)
• What is the annual revenue of a Fortune 100
  company?
• How many molecules are in one mole of gas (under
  ideal assumptions)
• How many pixels are displayed in a two hour HDTV
  movie?
• What is the current balance of Bill Gates
  checking account?
• How many ounces of Pepsi are consumed per year?
• What is the smallest measurable mass in the
  universe that is not zero?

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Machine Precision

• Problem
• We live in a world with a wide range of scales
• Computers have limited capabilities

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Machine Precision

• Computers have limited capabilities!!!

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Accuracy and Range

• More bits you have to represent numbers means
• A wider range and more accuracy
• Integer Representation
• 16-bit integers (common in the late 1990s)
• 216 65,536
• 32-bit integers
• 232 4,294,967,296
• 64-bit integers
• 264 ? 1.84 1019

ltlt 1,000,000,000,000
Still no Avagadros number 6.022 1023
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Back to Our Challenge

• A 2GHZ 32-bit machine may be blazing fast, but it
  will never reach one trillion either, if care
  isnt taken!!!
• Overflow
• Computer Forever
• Human 32,000 years
• Forever gtgt 32,000 years, humans win!!!!

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Accuracy and Standards

• What is int i and what is its range?
• Machine dependent
• ANSI C standard
• int machine dependent
• short machine dependent

Not much of a standard, huh?
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Advice

• Use the preprocessor to define your own types
• define CSEint int \\ I want this to be 32-bit
  signed integer
• define CSEshort short \\ 16-bit signed integer
• Hence, if you port to another architecture and
  need to change it, only change it in one place.
• Likewise, use each packages types glFloat,

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Numbers

• Whole numbers 0,1,2,
• Signed integers ,-1,0,1,2,
• Rational numbers a/b, 1/3, etc.
• Irrational numbers v2, e, pi, etc.
• Is there a difference between 0.1 and 1/3?

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Representing Real Numbers

• In Base-10, we can express any number as a sum of
  weighted powers of ten.
• Hence

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Representing Real Numbers

• In Base-2 or binary we have a series in powers of
  two?

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Limiting Precision

• Truncate PI
• Truncate to what?
• To 8 significant digits
• Okay
• 3.1415926535897932384626433832795
• ?
• 3.1415926

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Truncation vs Rounding

• Round to nearest value
• 3.1415926535897932384626433832795
• ?
• 3.1415927
• Rules
• If value after last retained digit is gt 5, then
  round-up.
• If value after last retained digit is lt 5, then
  truncate.
• If value after last digit is 5, then
• If all remaining digits are not zero, round-up.
• Else, randomly round-up or truncate.