Johann Friederich Carl Gauss

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Johann Friederich Carl Gauss

• Gauss Quadrature
• By Derek Picklesimer

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Background

• A german mathematician born in 1777
• He was educated at the Caroline College,
  Brunswick, and the Univ. of Göttingen
• His education and early research was funded by
  the Duke of Brunswick.
• In 1807 he became the director of the
  astronomical observatory in Göttingen.

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Background (cont.)

• Disquisitiones Arithmeticae - his greatest work
  on higher arithmetic and number theory.
• Was written in 1798, but wasnt published until
  1801.
• In 1809, he wrote Theoria motus corporum
  celestium a complete treatment of the calculation
  of the orbits of planets and comets from
  observational data.

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Background (cont)

• In 1821, he became involved in geodetic survey
  work and invented the heliotrope, a device used
  to measure distances by means of reflected
  sunlight.
• Later on in his life he becamed involved in
  various other topics ranging from
  electromagnetism to topology
• He died in 1855.

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Gauss Quadrature

• The main Type of Gauss Quadrature I will discuss
  is the Gauss-Legendre formula.
• This method is a technique used to integrate
  functions when the function cannot be integrated
  analytically.
• The Gauss-Legendre formulas are derived from the
  method of undetermined coefficients.

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Gauss Quadrature

• This is the generic form for the two point
  Gauss-Legendre formula.
• First to be able to integrate any given function
  we must solve for the unknown coefficients c and
  x values.

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Gauss Quadrature

• To find the unknown coefficients, you must solve
  4 equations simultaneously.

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Gauss Quadrature

• The result is
• c0c11
• x0-1/v3-0.5773503
• x1 1/v3 0.5773503
• Thus, the formula becomes
• For any integral using the two-point
  Gauss-Legendre formula.

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Gauss Quadrature

• In order to use the Gauss-Legendre formula, the
  integration limits need to be -1 to 1.
• A simple change of variable can be used to
  translate the limits of integration.
• Note a and b are the original limits.

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Gauss Quadrature

• The Gauss-Legendre formula is not limited to only
  two points.
• Higher point versions can be developed in the
  more general form

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Table of cs and xs
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Example of 2 pt Gauss-Legendre

• Using 2 pt Gauss-Legendre formula integrate the
  following.

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Example of 2 pt Gauss-Legendre
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Example of 2 pt Gauss-Legendre
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Conclusion

• In comparison to the analytical solution of
  5216.926477, the Gauss-Legendre has a 33.3
  error.
• With an increase in the number of points used in
  the Gauss-Legendre formula, there is a decrease
  in the amount of error.