Computational Physics

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Computational Physics

• PS 587

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We are still waiting for the Ph D class to join
in

• Till then, refresh some concepts in programming
  (later).
• Discuss some general techniques which may be
  useful in any case
• Accuracy, and why it is important.

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Decimal System

• Foundation of our computer revolution.
• Imagine computing in the Roman system CCXXXII
  times XLVIII, i.e. 232 ? 48.
• Zero was invented by Indian mathematicians, who
  were inspired by the Babylonian and the Chinese
  number systems, particularly as used in abacuses.

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The Discovery of Decimal Fractions

• Persians and Arabs invented the representation of
  decimal fractions that we use today.
• They discovered the rules for basic arithmetic
  operations that we now learn in school.

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The Long Journey
Adelard 1080 AD
Khwarizmi 780 AD
Kashani 1380 AD
House of Wisdom 9thc. AD
Brahmagupta, 598 AD Sridhara, 850 AD
Diophantus 3rdc. AD
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Khwarizmi (780 850)

• Settled in the House of Wisdom (Baghdad).
• Wrote three books
• Hindu Arithmetic
• Al-jabr va Al-Moghabela
• Astronomical Tables
• The established wordsAlgorithm from
  Al-Khwarizmiand Algebra from
  Al-jabrtestify to his fundamentalcontribution
  to human thought.

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The Long Journey
Adelard 1080 AD
Khwarizmi 780 AD
Kashani 1380 AD
House of Wisdom 9thc. AD
Brahmagupta, 598 AD Sridhara, 850 AD
Diophantus 3rdc. AD
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Adelard of Bath (1080 1160)

• First English Scientist.
• Translated from Arabic to Latin Khwarizmis
  astronomical tables with their use of zero.

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The Long Journey
Adelard 1080 AD
Khwarizmi 780 AD
Kashani 1380 AD
House of Wisdom 9thc. AD
Brahmagupta, 598 AD Sridhara, 850 AD
Diophantus 3rdc. AD
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Kashani (1380 1429)

• Developed arithmetic algorithms for fractions,
  that we use today.

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Kashani (1380 1429)

• Kashani invented the first mechanical special
  purpose computers
• to find when the planets are closest,
• to calculate longitudes of planets,
• to predict lunar eclipses.

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Kashanis Planetarium
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Mechanical Computers in Europe
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Modern Computers Floating Point Numbers

• Any other number like ? is rounded or
  approximated to a close floating point number.

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Computers lie.

• One has to be alert.

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Floating Point Arithmetic is not sound

• Especially when adding BIG numbers

• But using IEEEs standard precision, we get
  three different results,

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What is 0 on the computer?

• 0 is the smallest number such that 011.
• Compute the 0 on your calculator. This is related
  to the number of bits used to represent a real
  number. Typically this will be something like
  10-8.

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Floating Point Arithmetic is not sound

• A simple calculation shows

• But using IEEEs standard precision, we get
  three different results, all wrong.

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Failure of Floating Point Computation
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Failure of Floating Point Computation

• Depending on the floating point format, the
  sequence tends to 1 or 2 or 3 or 4.
• In reality, it oscillates about 1.51 and 2.37.

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Failure of Floating Point Computation

• In any floating point format, the sequence
  converges to 100.
• In reality, it converges to 6.

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Failure of Floating Point Computation

• In any floating point format, the sequence
  converges to 100.
• In reality, it converges to 6.

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Failure of Floating Point Computation

• In any floating point format, the sequence
  converges to 100.
• In reality, it converges to 6.

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3.141592653589793238462643383279502884197169399375
10582974944592307816406286208998628034825342117067
98214808651328230664709384460955058223172535940812
84811174502841027019385211055596446229489549303819
64428810975665933446128475648233786783165271201909
14564856692346034861045432664821339360726024914127
37245870066063155881748815209209628292540917153643
67892590360011330530548820466521384146951941511609
...