# jacobi.g.polynomials: Create list of Jacobi polynomials In orthopolynom: Collection of functions for orthogonal and orthonormal polynomials

## Description

This function returns a list with n + 1 elements containing the order k Jacobi polynomials, G_k ≤ft( {p,q,x} \right), for orders k = 0,\;1,\; … ,\;n.

## Usage

 1 jacobi.g.polynomials(n, p, q, normalized=FALSE) 

## Arguments

 n integer value for the highest polynomial order p numeic value for the first polynomial parameter q numeric value for the second polynomial parameter normalized a boolean value which, if TRUE, returns a list of normalized orthogonal polynomials

## Details

The function jacobi.g.recurrences produces a data frame with the recurrence relation parameters for the polynomials. If the normalized argument is FALSE, the function orthogonal.polynomials is used to construct the list of orthogonal polynomial objects. Otherwise, the function orthonormal.polynomials is used to construct the list of orthonormal polynomial objects.

## Value

A list of n + 1 polynomial objects

 1  order 0 Jacobi polynomial 2  order 1 Jacobi polynomial

...

 n+1  order n Jacobi polynomial

## Author(s)

Frederick Novomestky fnovomes@poly.edu

## References

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

Courant, R., and D. Hilbert, 1989. Methods of Mathematical Physics, John Wiley, New York, NY.

Szego, G., 1939. Orthogonal Polynomials, 23, American Mathematical Society Colloquium Publications, Providence, RI.

jacobi.g.recurrences, orthogonal.polynomials, orthonormal.polynomials

## Examples

  1 2 3 4 5 6 7 8 9 10 11 12 ### ### gemerate a list of normalized Jacobi G polynomials of orders 0 to 10 ### first parameter value p is 3 and second parameter value q is 2 ### normalized.p.list <- jacobi.g.polynomials( 10, 3, 2, normalized=TRUE ) print( normalized.p.list ) ### ### gemerate a list of normalized Jacobi G polynomials of orders 0 to 10 ### first parameter value p is 3 and second parameter value q is 2 ### unnormalized.p.list <- jacobi.g.polynomials( 10, 3, 2, normalized=FALSE ) print( unnormalized.p.list ) 

### Example output

Loading required package: polynom
[[1]]
2.44949

[[2]]
-5.477226 + 10.95445*x

[[3]]
9.165151 - 45.82576*x + 45.82576*x^2

[[4]]
-13.41641 + 120.7477*x - 281.7446*x^2 + 187.8297*x^3

[[5]]
18.1659 - 254.3226*x + 1017.291*x^2 - 1525.936*x^3 + 762.9679*x^4

[[6]]
-23.36664 + 467.3329*x - 2803.997*x^2 + 7009.993*x^3 - 7710.992*x^4 +
3084.397*x^5

[[7]]
28.98275 - 782.5343*x + 6521.12*x^2 - 23910.77*x^3 + 43039.39*x^4 - 37300.8*x^5
+ 12433.6*x^6

[[8]]
-34.98571 + 1224.5*x - 13469.5*x^2 + 67347.49*x^3 - 175103.5*x^4 + 245144.9*x^5
- 175103.5*x^6 + 50029.57*x^7

[[9]]
41.35215 - 1819.494*x + 25472.92*x^2 - 165574*x^3 + 579509*x^4 - 1159018*x^5 +
1324592*x^6 - 804216.5*x^7 + 201054.1*x^8

[[10]]
-48.06246 + 2595.373*x - 44986.46*x^2 + 367389.4*x^3 - 1653252*x^4 +
4408673*x^5 - 7137852*x^6 + 6882929*x^7 - 3632657*x^8 + 807257.1*x^9

[[11]]
55.09991 - 3581.494*x + 75211.38*x^2 - 752113.8*x^3 + 4211837*x^4 -
14320250*x^5 + 30686240*x^6 - 41645610*x^7 + 34704680*x^8 - 16195520*x^9 +
3239103*x^10

[[1]]
1

[[2]]
-0.5 + x

[[3]]
0.2 - x + x^2

[[4]]
-0.07142857 + 0.6428571*x - 1.5*x^2 + x^3

[[5]]
0.02380952 - 0.3333333*x + 1.333333*x^2 - 2*x^3 + x^4

[[6]]
-0.007575758 + 0.1515152*x - 0.9090909*x^2 + 2.272727*x^3 - 2.5*x^4 + x^5

[[7]]
0.002331002 - 0.06293706*x + 0.5244755*x^2 - 1.923077*x^3 + 3.461538*x^4 -
3*x^5 + x^6

[[8]]
-0.0006993007 + 0.02447552*x - 0.2692308*x^2 + 1.346154*x^3 - 3.5*x^4 + 4.9*x^5
- 3.5*x^6 + x^7

[[9]]
0.0002056767 - 0.009049774*x + 0.1266968*x^2 - 0.8235294*x^3 + 2.882353*x^4 -
5.764706*x^5 + 6.588235*x^6 - 4*x^7 + x^8

[[10]]
-5.953799e-05 + 0.003215051*x - 0.05572755*x^2 + 0.4551084*x^3 - 2.047988*x^4 +
5.4613*x^5 - 8.842105*x^6 + 8.526316*x^7 - 4.5*x^8 + x^9

[[11]]
1.701085e-05 - 0.001105705*x + 0.02321981*x^2 - 0.2321981*x^3 + 1.30031*x^4 -
4.421053*x^5 + 9.473684*x^6 - 12.85714*x^7 + 10.71429*x^8 - 5*x^9 + x^10


orthopolynom documentation built on May 2, 2019, 3:22 a.m.