1. Johann Friederich Carl Gauss Gauss Quadrature By: Derek Picklesimer

2. 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.

3. Background (cont.) ► Disquisitiones Arithmeticae - his greatest work on higher arithmetic and number theory.  Was written in 1798, but wasn’t 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.

4. 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.

5. 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.

6. 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.

7. Gauss Quadrature ► To find the unknown coefficients, you must solve 4 equations simultaneously.

8. Gauss Quadrature ► The result is:  c 0 =c 1 =1  x 0 =-1/√3=-0.5773503  x 1 = 1/√3 =0.5773503 ► Thus, the formula becomes: ► For any integral using the two-point Gauss- Legendre formula.

9. 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.

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

11. Table of c’s and x’s

12. Example of 2 pt Gauss-Legendre ► Using 2 pt Gauss-Legendre formula integrate the following.

13. Example of 2 pt Gauss-Legendre

15. 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.