## Description

This function returns a list with n elements containing the order k quadrature rule data frame for the shperical polynomial for orders k = 1,\;2,\; … ,\;n.

## Arguments

 n integer value for the highest order normalized boolean value. if TRUE rules are for orthonormal polynomials, otherwise they are for orthgonal polynomials

## Details

An order k quadrature data frame is a named data frame that contains the roots and abscissa values of the corresponding order k orthogonal polynomial. The column with name x contains the roots or zeros and the column with name w contains the weights.

## Value

A list with n elements each of which is a data frame

 1 Quadrature rule data frame for the order 1 spherical polynomial 2 Quadrature rule data frame for the order 2 spherical polynomial

...

 n Quadrature rule data frame for the order n spherical polynomial

## Author(s)

Frederick Novomestky [email protected]

## References

Abramowitz, M. and I. A. Stegun, 1968. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc., New York.

Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992. Numerical Recipes in C, Cambridge University Press, Cambridge, U.K.

Stroud, A. H., and D. Secrest, 1966. Gaussian Quadrature Formulas, Prentice-Hall, Englewood Cliffs, NJ.

## Examples

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ### ### generate a list of quadrature rule data frames for ### the orthogonal spherical polynomials ### of orders 1 to 5 ### orthogonal.rules <- spherical.quadrature.rules( 5 ) print( orthogonal.rules ) ### ### generate a list of quadrature rule data frames for ### the orthonormal spherical polynomials ### of orders 1 to 5 ### orthonormal.rules <- spherical.quadrature.rules( 5, TRUE ) print( orthonormal.rules )

### Example output

[[1]]
x w
1 0 2

[[2]]
x w
1  0.5773503 1
2 -0.5773503 1

[[3]]
x         w
1  7.745967e-01 0.5555556
2  7.771561e-16 0.8888889
3 -7.745967e-01 0.5555556

[[4]]
x         w
1  0.8611363 0.3478548
2  0.3399810 0.6521452
3 -0.3399810 0.6521452
4 -0.8611363 0.3478548

[[5]]
x         w
1  9.061798e-01 0.2369269
2  5.384693e-01 0.4786287
3  6.661338e-16 0.5688889
4 -5.384693e-01 0.4786287
5 -9.061798e-01 0.2369269

[[1]]
x w
1 0 2

[[2]]
x w
1  0.5773503 1
2 -0.5773503 1

[[3]]
x         w
1  7.745967e-01 0.5555556
2  8.881784e-16 0.8888889
3 -7.745967e-01 0.5555556

[[4]]
x         w
1  0.8611363 0.3478548
2  0.3399810 0.6521452
3 -0.3399810 0.6521452
4 -0.8611363 0.3478548

[[5]]
x         w
1  9.061798e-01 0.2369269
2  5.384693e-01 0.4786287
3  4.440892e-16 0.5688889
4 -5.384693e-01 0.4786287
5 -9.061798e-01 0.2369269

gaussquad documentation built on May 30, 2017, 8:06 a.m.