Johann Friederich Carl Johann Friederich Carl Gauss Gauss

Gauss QuadratureGauss Quadrature

By: Derek PicklesimerBy: Derek Picklesimer

BackgroundBackground

►A german mathematician born in 1777A german mathematician born in 1777►He was educated at the Caroline He was educated at the Caroline

College, Brunswick, and the Univ. of College, Brunswick, and the Univ. of GöttingenGöttingen His education and early research was His education and early research was

funded by the Duke of Brunswick.funded by the Duke of Brunswick.

► In 1807 he became the director of the In 1807 he became the director of the astronomical observatory in Göttingen.astronomical observatory in Göttingen.

Background (cont.)Background (cont.)

►Disquisitiones ArithmeticaeDisquisitiones Arithmeticae - his - his greatest work on higher arithmetic and greatest work on higher arithmetic and number theory.number theory. Was written in 1798, but wasn’t published Was written in 1798, but wasn’t published

until 1801.until 1801.► In 1809, he wrote In 1809, he wrote Theoria motus Theoria motus

corporum celestiumcorporum celestium a complete a complete treatment of the calculation of the treatment of the calculation of the orbits of planets and comets from orbits of planets and comets from observational data.observational data.

Background (cont)Background (cont)

► In 1821, he became involved in geodetic In 1821, he became involved in geodetic survey work and invented the heliotrope, survey work and invented the heliotrope, a device used to measure distances by a device used to measure distances by means of reflected sunlight.means of reflected sunlight.

►Later on in his life he becamed involved Later on in his life he becamed involved in various other topics ranging from in various other topics ranging from electromagnetism to topologyelectromagnetism to topology

►He died in 1855.He died in 1855.

Gauss QuadratureGauss Quadrature

►The main Type of Gauss Quadrature I The main Type of Gauss Quadrature I will discuss is the Gauss-Legendre will discuss is the Gauss-Legendre formula.formula.

►This method is a technique used to This method is a technique used to integrate functions when the function integrate functions when the function cannot be integrated analytically.cannot be integrated analytically.

►The Gauss-Legendre formulas are The Gauss-Legendre formulas are derived from the method of derived from the method of undetermined coefficients.undetermined coefficients.

Gauss QuadratureGauss Quadrature

►This is the generic form for the two This is the generic form for the two point Gauss-Legendre formula.point Gauss-Legendre formula.

►First to be able to integrate any given First to be able to integrate any given function we must solve for the function we must solve for the unknown coefficients c and x values.unknown coefficients c and x values.

)()( 1100 xfcxfcI

Gauss QuadratureGauss Quadrature

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

Gauss QuadratureGauss Quadrature

► The result is:The result is: cc00=c=c11=1=1 xx00=-1/√3=-0.5773503=-1/√3=-0.5773503 xx11= 1/√3 =0.5773503= 1/√3 =0.5773503

► Thus, the formula becomes:Thus, the formula becomes:

► For any integral using the two-point Gauss-For any integral using the two-point Gauss-Legendre formula.Legendre formula.

)3

1()

3

1( ffI

Gauss QuadratureGauss Quadrature

► In order to use the In order to use the Gauss-Legendre Gauss-Legendre formula, the formula, the integration limits integration limits need to be -1 to 1.need to be -1 to 1.

► A simple change of A simple change of variable can be used variable can be used to translate the to translate the limits of integration.limits of integration.

► Note: a and b are Note: a and b are the original limits.the original limits.

2

)()( dxababx

ddxab

dx2

)(

Gauss QuadratureGauss Quadrature

► The Gauss-Legendre formula is not limited The Gauss-Legendre formula is not limited to only two points.to only two points.

►Higher point versions can be developed in Higher point versions can be developed in the more general formthe more general form

)(...)()( 111100 nn xfcxfcxfcI

Table of c’s and x’sTable of c’s and x’s

Example of 2 pt Gauss-Example of 2 pt Gauss-LegendreLegendre

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

dxex x24

0

Example of 2 pt Gauss-Example of 2 pt Gauss-LegendreLegendre

dd xx

x 222

)04()04(

ddxdx 2

Example of 2 pt Gauss-Example of 2 pt Gauss-LegendreLegendre

1

1

)22(2 2])22[( dxex dxd

)3

1()

3

1( ffI

543936.3477

376279.3468167657324.9

I

I

ConclusionConclusion

► In comparison to the analytical In comparison to the analytical solution of 5216.926477, the Gauss-solution of 5216.926477, the Gauss-Legendre has a 33.3% error.Legendre has a 33.3% error.

►With an increase in the number of With an increase in the number of points used in the Gauss-Legendre points used in the Gauss-Legendre formula, there is a decrease in the formula, there is a decrease in the amount of error. amount of error.