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PPT – Johann Friederich Carl Gauss PowerPoint presentation | free to download - id: 3e6174-MDE2N

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Johann Friederich Carl Gauss

- Gauss Quadrature
- By Derek Picklesimer

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.

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.

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.

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.

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.

Gauss Quadrature

- To find the unknown coefficients, you must solve

4 equations simultaneously.

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.

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.

Gauss Quadrature

- The Gauss-Legendre formula is not limited to only

two points. - Higher point versions can be developed in the

more general form

Table of cs and xs

Example of 2 pt Gauss-Legendre

- Using 2 pt Gauss-Legendre formula integrate the

following.

Example of 2 pt Gauss-Legendre

Example of 2 pt Gauss-Legendre

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.

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