orthopolynom

8/11/2019 orthopolynom

1/113

Package orthopolynom

July 2, 2014

Version 1.0-5

Date 2013-02-03

Title Collection of functions for orthogonal and orthonormal polynomials

Author Frederick Novomestky

Maintainer Frederick Novomestky

Depends R (>= 2.0.1), polynom

Description A collection of functions to construct sets of orthogonal

polynomials and their recurrence relations. Additional functions are provided to calcu-

late the derivative, integral,value and roots of lists of polynomial objects.

License GPL (>= 2)

Repository CRAN

Date/Publication 2013-02-04 07:42:24

NeedsCompilation no

R topics documented:

chebyshev.c.inner.products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

chebyshev.c.polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

chebyshev.c.recurrences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

chebyshev.c.weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

chebyshev.s.inner.products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

chebyshev.s.polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

c h e b y s h e v .s .r e c u r r e n c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

chebyshev.s.weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

chebyshev.t.inner.products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

chebyshev.t.polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

c h e b y s h e v .t .r e c u r r e n c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

chebyshev.t.weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

chebyshev.u.inner.products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1


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2 R topics documented:

chebyshev.u.polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

chebyshev.u.recurrences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

c h e b y s h e v . u . w e i g h t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

gegenbauer.inner.products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

gegenbauer.polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25g e g e n b a u e r .r e c u r r e n c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

gegenbauer.weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

g h e r m i t e .h .i n n e r .p r o d u c t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

g h e r m i t e .h .p o l y n o m i a l s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

ghermite.h.recurrences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

g h e r m i t e . h . w e i g h t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

glaguerre.inner.products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

glaguerre.polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

g l a g u e r r e .r e c u r r e n c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

glaguerre.weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

hermite.h.inner.products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

hermite.h.polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

h e r m i t e .h .r e c u r r e n c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

hermite.h.weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

h e r m i t e .h e .i n n e r .p r o d u c t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

h e r m i t e .h e .p o l y n o m i a l s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

hermite.he.recurrences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

h e r m i t e . h e . w e i g h t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

jacobi.g.inner.products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

jacobi.g.polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

jacobi.g.recurrences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

jacobi.g.weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

jacobi.matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

jacobi.p.inner.products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

jacobi.p.polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56jacobi.p.recurrences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

jacobi.p.weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

laguerre.inner.products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

laguerre.polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

laguerre.recurrences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

laguerre.weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

l e g e n d r e .i n n e r .p r o d u c t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

legendre.polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

legendre.recurrences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

legendre.weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

lpochhammer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

monic.polynomial.recurrences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

monic.polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

o r t h o g o n a l .p o l y n o m i a l s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

o r t h o n o r m a l .p o l y n o m i a l s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

pochhammer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

p o l y n o m i a l .c o e f fic i e n t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

polynomial.derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77


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polynomial.functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

polynomial.integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

p o l y n o m i a l . o r d e r s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

polynomial.powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

polynomial.roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82p o l y n o m i a l . v a l u e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

scaleX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

schebyshev.t .inner.products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

s c h e b y s h e v .t .p o l y n o m i a l s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

schebyshev.t.recurrences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

s c h e b y s h e v . t . w e i g h t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

schebyshev.u.inner.products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

schebyshev.u.polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

s c h e b y s h e v .u .r e c u r r e n c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

schebyshev.u.weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

slegendre.inner.products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

slegendre.polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

s l e g e n d r e .r e c u r r e n c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

slegendre.weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

s p h e r i c a l . i n n e r .p r o d u c t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

s p h e r i c a l .p o l y n o m i a l s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

spherical.recurrences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

s p h e r i c a l . w e i g h t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

ultraspherical.inner.products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

ultraspherical.polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

u l t r a s p h e r i c a l .r e c u r r e n c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

ultraspherical.weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Index 111

chebyshev.c.inner.products

Inner products of Chebyshev polynomials

Description

This function returns a vector with n + 1 elements containing the inner product of an order kChebyshev polynomial of the first kind, Ck(x), with itself (i.e. the norm squared) for ordersk =0, 1, . . . , n.

Usage

chebyshev.c.inner.products(n)

Arguments

n integer value for the highest polynomial order


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4 chebyshev.c.inner.products

Details

The formula used to compute the inner products is as follows.

hn=Cn|Cn= 4 n= 08 n= 0Value

A vector withn + 1elements

1 inner product of order 0 orthogonal polynomial


...

n+1 inner product of ordernorthogonal polynomial

Author(s)

Frederick Novomestky

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 Publi-

cations, Providence, RI.

See Also

gegenbauer.inner.products

Examples

###

### generate the inner products vector for the

### C Chebyshev polynomials of orders 0 to 10###

h


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chebyshev.c.polynomials 5

chebyshev.c.polynomials

Create list of Chebyshev polynomials

Description

This function returns a list with n + 1 elements containing the order k Chebyshev polynomials ofthe first kind,Ck(x), for ordersk = 0, 1, . . . , n.

Usage

chebyshev.c.polynomials(n, normalized=FALSE)

Arguments

n integer value for the highest polynomial ordernormalized a boolean value which, if TRUE, returns a list of normalized orthogonal poly-

nomials

Details

The function chebyshev.c.recurrencesproduces a data frame with the recurrence relation pa-

rameters 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 ofn + 1polynomial objects

1 order 0 Chebyshev polynomial


...

n+1 ordernChebyshev polynomial

Author(s)


References







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6 chebyshev.c.recurrences

See Also

chebyshev.c.recurrences,orthogonal.polynomials,orthonormal.polynomials

Examples

###

### gemerate a list of normalized C Chebyshev polynomials of orders 0 to 10

###

normalized.p.list


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chebyshev.c.weight 7

References






See Also

chebyshev.c.inner.products

Examples

###

### generate the recurrences data frame for

### the normalized Chebyshev C polynomials

### of orders 0 to 10.

###

normalized.r


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8 chebyshev.s.inner.products

Details

The function takes on non-zero values in the interval (2, 2). The formula used to compute theweight function is as follows.

w (x) = 1

1x24

Value

The value of the weight function

Author(s)

Frederick Novomestky< [email protected] >

References




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

Cambridge University Press, Cambridge, U.K.



Examples

###

### compute the C Chebyshev weight function for arguments between -3 and 3

###

x


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chebyshev.s.inner.products 9

Arguments


Details


hn=Sn|Sn= .

Value




...


Author(s)


References






Examples

###


### S Chebyshev polynomials of orders 0 to 10###

h


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10 chebyshev.s.polynomials

chebyshev.s.polynomials


Description

This function returns a list with n + 1 elements containing the order k Chebyshev polynomials ofthe second kind,Sk(x), for ordersk = 0, 1, . . . , n.

Usage

chebyshev.s.polynomials(n, normalized=FALSE)

Arguments


nomials

Details

The function chebyshev.s.recurrencesproduces a data frame with the recurrence relation pa-




Value




...


Author(s)


References







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chebyshev.s.recurrences 11

See Also

chebyshev.s.recurrences,orthogonal.polynomials,orthonormal.polynomials

Examples

###

### gemerate a list of normalized S Chebyshev polynomials of orders 0 to 10

###

normalized.p.list


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12 chebyshev.s.weight

References






See Also

chebyshev.s.inner.products

Examples

###


### the normalized Chebyshev S polynomials


###

normalized.r


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chebyshev.t.inner.products 13

Details


w (x) =

1 x24

Value

The value of the weight function.

Author(s)


References








Examples

###

### compute the S Chebyshev weight function for arguments between -2 and 2

###

x


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14 chebyshev.t.inner.products

Arguments


Details


hn=Tn|Tn=

2 n= 0 n= 0

Value




...


Author(s)


References






Examples

###


### T Chybyshev polynomials of orders 0 to 10###

h


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chebyshev.t.polynomials 15

chebyshev.t.polynomials


Description

This function returns a list with n + 1 elements containing the order k Chebyshev polynomials ofthe first kind,Tk(x), for ordersk = 0, 1, . . . , n.

Usage

chebyshev.t.polynomials(n, normalized=FALSE)

Arguments


nomials

Details

The function chebyshev.t.recurrencesproduces a data frame with the recurrence relation pa-




Value




...


Author(s)


References







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16 chebyshev.t.recurrences

See Also

chebyshev.u.recurrences,orthogonal.polynomials,orthonormal.polynomials

Examples

###

### gemerate a list of normalized T Chebyshev polynomials of orders 0 to 10

###

normalized.p.list


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chebyshev.t.weight 17

References






See Also

chebyshev.t.inner.products

Examples

###

### generate the recurrence relations for

### the normalized T Chebyshev polynomials

### of orders 0 to 10

###

normalized.r


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18 chebyshev.u.inner.products

Details


w (x) = 11x2

Value


Author(s)


References








Examples

###

### compute the T Chebyshev function for argument values between -2 and 2

x


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chebyshev.u.inner.products 19

Arguments


Details


hn=Un|Un= 2

Value




...


Author(s)


References






Examples

###


### U Chebyshev polynomials of orders 0 to 10###

h


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20 chebyshev.u.polynomials

chebyshev.u.polynomials


Description

This function returns a list with n + 1 elements containing the order k Chebyshev polynomials ofthe second kind,Uk(x), for ordersk = 0, 1, . . . , n.

Usage

chebyshev.u.polynomials(n, normalized=FALSE)

Arguments


nomials

Details

The function chebyshev.u.recurrencesproduces a data frame with the recurrence relation pa-




Value




...


Author(s)


References







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chebyshev.u.recurrences 21

See Also

chebyshev.u.recurrences,orthogonal.polynomials,orthonormal.polynomials

Examples

###

### gemerate a list of normalized U Chebyshev polynomials of orders 0 to 10

###

normalized.p.list


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22 chebyshev.u.weight

References



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 Publi-


See Also

chebyshev.u.inner.products

Examples

###


### the normalized U Chebyshev polynomials


###

normalized.r


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gegenbauer.inner.products 23

Value


Author(s)


References








Examples

###

### compute the U Chebyshev function for argument values between -2 and 2

###

x


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24 gegenbauer.inner.products

Details


hn=

C()n |C

()n

= 212 (n+2)

n! (n+) [()]2

= 0

2 n2

= 0 .

Value




...


Author(s)


References




Szego, G., 1939. Orthogonal Polynomials, 23, American Mathematical Society Colloquium Publi-cations, Providence, RI.

See Also

ultraspherical.inner.products

Examples

###


### Gegenbauer polynomials of orders 0 to 10

### the polynomial parameter is 1.0

###

h


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gegenbauer.polynomials 25

gegenbauer.polynomials

Create list of Gegenbauer polynomials

Description

This function returns a list with n + 1 elements containing the order k Gegenbauer polynomials,

C()k (x), for ordersk = 0, 1, . . . , n.

Usage

gegenbauer.polynomials(n, alpha, normalized=FALSE)

Arguments


alpha polynomial parameter

normalized a boolean value which, if TRUE, returns a list of normalized orthogonal poly-

nomials

Details

The functiongegenbauer.recurrencesproduces a data frame with the recurrence relation param-

eters for the polynomials. If the normalized argument is FALSE, the function orthogonal.polynomials



Value


1 order 0 Gegenbauer polynomial

2 order 1 Gegenbauer polynomial

...


Author(s)


References







26/113


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gegenbauer.weight 27

References






See Also

gegenbauer.inner.products

Examples

###


### the normalized Gegenbauer polynomials

### of orders 0 to 10.### polynomial parameter value is 1.0

###

normalized.r


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28 ghermite.h.inner.products

Details


w (x) =

1 x20.5Value


Author(s)


References








Examples

###

### compute the Gegenbauer weight function for argument values between -1 and 1

###

x


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ghermite.h.inner.products 29

Arguments

n ninteger value for the highest polynomial order

mu mupolynomial parameter

Details

The parameter must be greater than -0.5. The formula used to compute the inner products is as

follows.

hn() =

H()m|H()n

= 22n

n2

!

n+1

2

+ + 12

Value


1 inner product of order 0 orthogonal polynomial2 inner product of order 1 orthogonal polynomial

...


Author(s)


References






Examples

###


### generalized Hermite polynomials of orders 0 to 10

### polynomial parameter is 1

###

h


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30 ghermite.h.polynomials

ghermite.h.polynomials

Create list of generalized Hermite polynomials

Description

This function returns a list withn + 1elements containing the orderkgeneralized Hermite polyno-

mials,H()k (x), for ordersk = 0, 1, . . . , n.

Usage

ghermite.h.polynomials(n, mu, normalized = FALSE)

Arguments


mu numeric value for the polynomial parameter

normalized boolean value which, if TRUE, returns recurrence relations for normalized poly-

nomials

Details

The parameter must be greater than -0.5. The functionghermite.h.recurrences produces a

data frame with the recurrence relation parameters for the polynomials. If the normalizedargu-

ment is FALSE, the function orthogonal.polynomialsis used to construct the list of orthogonalpolynomial objects. Otherwise, the functionorthonormal.polynomialsis used to construct the

list of orthonormal polynomial objects.

Value


1 order 0 generalized Hermite polynomial

2 order 1 generalized Hermite polynomial

...

n+1 orderngeneralized Hermite polynomial

Author(s)



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ghermite.h.recurrences 31

References

Alvarez-Nordase, R., M. K. Atakishiyeva and N. M. Atakishiyeva, 2004. A q-extension of the gen-

eralized Hermite polynomials with continuous orthogonality property on R, International Journal

of Pure and Applied Mathematics, 10(3), 335-347.Abramowitz, M. and I. A. Stegun, 1968. Handbook of Mathematical Functions with Formulas,





See Also

ghermite.h.recurrences,orthogonal.polynomials,orthonormal.polynomials

Examples

###

### gemerate a list of normalized generalized Hermite polynomials of orders 0 to 10

### polynomial parameter is 1.0

###

normalized.p.list


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32 ghermite.h.recurrences

Arguments


mu numeric value for the polynomial parameter

normalized normalizedboolean value which, if TRUE, returns recurrence relations for nor-malized polynomials

Details

The parametermust be greater than -0.5.

Value

A data frame with the recurrence relation parameters.

Author(s)


References

Alvarez-Nordase, R., M. K. Atakishiyeva and N. M. Atakishiyeva, 2004. A q-extension of the gen-

eralized Hermite polynomials with continuous orthogonality property on R, International Journal

of Pure and Applied Mathematics, 10(3), 335-347.






See Also

ghermite.h.inner.products

Examples

###


### the normalized generalized Hermite polynomials


### polynomial parameter value is 1.0

###

normalized.r


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ghermite.h.weight 33

unnormalized.r

References









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34 glaguerre.inner.products

Examples

###

### compute the generalized Hermite weight function for argument values

### between -3 and 3

###

x


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glaguerre.polynomials 35

References






Examples

###


### generalized Laguerre polynomial inner products of orders 0 to 10

### polynomial parameter is 1.

###

h


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glaguerre.recurrences 37

glaguerre.recurrences Recurrence relations for generalized Laguerre polynomials

Description

This function returns a data frame withn+ 1 rows and four named columns containing the coeffi-cient vectorsc, d,e and fof the recurrence relations for the order k generalized Laguerre polyno-

mial,L()n (x)and for ordersk = 0, 1, . . . , n.

Usage

glaguerre.recurrences(n, alpha, normalized=FALSE)

Arguments


alpha numeric value for the polynomial parameter


nomials

Value


Author(s)


References





See Also

glaguerre.inner.products


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38 glaguerre.weight

Examples

###


### the normalized generalized Laguerre polynomials

### of orders 0 to 10. the polynomial parameter value is 1.0.

###

normalized.r


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References








Examples

###

### compute the generalized Laguerre weight function for argument values


### polynomial parameter value is 1.0###

x


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40 hermite.h.polynomials

Value




...


Author(s)


References






Examples

###


### Hermite polynomials of orders 0 to 10

###

h


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Details

The functionhermite.h.recurrencesproduces a data frame with the recurrence relation parame-

ters 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.polynomialsis used to construct list of orthonormal polynomial objects.

Value


1 order 0 Hermite polynomial


...

n+1 ordernHermite polynomial

Author(s)


References






See Also

hermite.h.recurrences,orthogonal.polynomials,orthonormal.polynomials

Examples

###

### gemerate a list of normalized Hermite polynomials of orders 0 to 10

###

normalized.p.list


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42 hermite.h.recurrences

hermite.h.recurrences Recurrence relations for Hermite polynomials

Description

This function returns a data frame withn+ 1 rows and four named columns containing the coeffi-cient vectors c, d, e and fof the recurrence relations for the orderk Hermite polynomial, Hk(x),and for ordersk = 0, 1, . . . , n.

Usage

hermite.h.recurrences(n, normalized=FALSE)

Arguments

n integer value for the highest polynomial ordernormalized boolean value which, if TRUE, returns recurrence relations for normalized poly-

nomials

Value


Author(s)


References






See Also

hermite.h.inner.products,

Examples

###### generate the recurrences data frame for

### the normalized Hermite H polynomials


###

normalized.r


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###


### the unnormalized Hermite H polynomials


###unnormalized.r

References









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44 hermite.he.inner.products

Examples

###

### compute the Hermite weight function for argument values


x


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hermite.he.polynomials 45

References






Examples

###


### scaled Hermite polynomials of orders 0 to 10

###

h


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46 hermite.he.recurrences

Value




...

n+1 ordernHermite polynomial

Author(s)


References


Graphs, and Mathematical Tables, Dover Publications, Inc., New York.Courant, R., and D. Hilbert, 1989. Methods of Mathematical Physics, John Wiley, New York, NY.



See Also

hermite.he.recurrences,orthogonal.polynomials,orthonormal.polynomials

Examples

###

### gemerate a list of normalized Hermite polynomials of orders 0 to 10

###normalized.p.list


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Usage

hermite.he.recurrences(n, normalized=FALSE)

Arguments



nomials

Value


Author(s)


References






See Also

hermite.he.inner.products

Examples

###


### the normalized Hermite H polynomials


###

normalized.r


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48 hermite.he.weight

hermite.he.weight Weight function for the Hermite polynomial

Description

This function returns the value of the weight function for the order kHermite polynomial,Hek(x).

Usage

hermite.he.weight(x)

Arguments

x the function argument which can be a vector

DetailsThe function takes on non-zero values in the interval (,). The formula used to compute theweight function is as follows.

w (x) = expx2

2

Value


Author(s)


References








Examples

###

### compute the scaled Hermite weight function for argument values


###

x


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jacobi.g.inner.products 49

jacobi.g.inner.products

Inner products of Jacobi polynomials

Description

This function returns a vector withn + 1elements containing the inner product of an order kJacobipolynomial,Gk(p, q, x), with itself (i.e. the norm squared) for ordersk = 0, 1, . . . , n.

Usage

jacobi.g.inner.products(n,p,q)

Arguments


p numeric value for the first polynomial parameter

q numeric value for the first polynomial parameter

Details


hn=Gn|Gn= n! (n+q) (n+p) (n+pq+1)(2n+p) [(2 n+p)]2 .

Value




...


Author(s)


References







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50 jacobi.g.polynomials

Examples

###


### G Jacobi polynomials of orders 0 to 10

### parameter p is 3 and parameter q is 2

###

h


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Author(s)


References






See Also

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

Examples

###

### 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


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Arguments


p numeric value for the first polynomial parameter

q numeric value for the second polynomial parameter


nomials

Value


Author(s)


References






See Also

jacobi.g.inner.products,pochhammer

Examples

###


### the normalized Jacobi G polynomials


### parameter p is 3 and parameter q is 2

###

normalized.r


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jacobi.g.weight 53

jacobi.g.weight Weight function for the Jacobi polynomial

Description

This function returns the value of the weight function for the order k Jacobi polynomial, Gk(p, q, x).

Usage

jacobi.g.weight(x,p,q)

Arguments


p the first polynomial parameter

q the second polynomial parameter

Details

The function takes on non-zero values in the interval (0, 1). The formula used to compute the weightfunction is as follows.

w (x) = (1 x)pq xq1

Value


Author(s)


References









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54 jacobi.matrices

Examples

###

### compute the Jacobi G weight function for argument values

### between 0 and 1

### parameter p is 3 and q is 2

###

x


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jacobi.p.inner.products

Inner products of Jacobi polynomials

Description

This function returns a vector withn + 1elements containing the inner product of an order kJacobi

polynomial,P(,)k (x), with itself (i.e. the norm squared) for ordersk = 0, 1, . . . , n.

Usage

jacobi.p.inner.products(n,alpha,beta)

Arguments


alpha numeric value for the first polynomial parameter

beta numeric value for the first polynomial parameter

Details

The formula used to compute the innser products is as follows.

hn=

P(,)n |P(,)n

= 2

++1

2 n+++1(n++1) (n++1)

n! (n+++1) .

Value




...


Author(s)


References







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56 jacobi.p.polynomials

Examples

###

### generate the inner product vector for the P Jacobi polynomials of orders 0 to 10

###

h


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References





See Also

jacobi.p.recurrences,orthogonal.polynomials,orthonormal.polynomials

Examples

###

### gemerate a list of normalized Jacobi P polynomials of orders 0 to 10

### first parameter value a is 2 and second parameter value b is 2

###normalized.p.list


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58 jacobi.p.weight

Value


Author(s)


References






See Also

jacobi.p.inner.products,pochhammer

Examples

###


### the normalized Jacobi P polynomials


### parameter a is 2 and parameter b is 2

###

normalized.r


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Arguments


alpha the first polynomial parameter

beta the second polynomial parameter

Details


w (x) = (1 x) (1 + x)

Value


Author(s)


References








Examples

###

### compute the Jacobi P weight function for argument values


###

x


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60 laguerre.inner.products

laguerre.inner.products

Inner products of Laguerre polynomials

Description

This function returns a vector with n + 1 elements containing the inner product of an order kLaguerre polynomial,Ln(x), with itself (i.e. the norm squared) for ordersk = 0, 1, . . . , n.

Usage

laguerre.inner.products(n)

Arguments


Details


hn=Ln|Ln= 1.

Value




...


Author(s)


References







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Examples

###


### Laguerre polynomial inner products of orders 0 to 10

###

h


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References





See Also

laguerre.recurrences,orthogonal.polynomials,orthonormal.polynomials

Examples

###

### gemerate a list of normalized Laguerre polynomials of orders 0 to 10

###

normalized.p.list


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Author(s)


References






See Also

glaguerre.recurrences

Examples

###


### the normalized Laguerre polynomials


###

normalized.r


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Details

The formula used compute the inner products is as follows.

hn=

Pn|Pn

= 2

2n+1

.

Value




...


Author(s)


References





See Also

spherical.inner.products

Examples

###

### compute the inner product for the

### Legendre polynomials of orders 0 to 1###

h


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66 legendre.polynomials

legendre.polynomials Create list of Legendre polynomials

Description

This function returns a list with n+1 elements containing the order k Legendre polynomials, Pk(x),for ordersk = 0, 1, . . . , n.

Usage

legendre.polynomials(n, normalized=FALSE)

Arguments



nomials

Details

The function legendre.recurrencesproduces a data frame with the recurrence relation parame-

ters for the polynomials. If the normalized argument is FALSE, the function orthogonal.polynomials



Value


1 order 0 Legendre polynomial

2 order 1 Legendre polynomial

...

n+1 ordernLegendre polynomial

Author(s)


References







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See Also

legendre.recurrences,orthogonal.polynomials,orthonormal.polynomials

Examples

###

### gemerate a list of normalized Laguerre polynomials of orders 0 to 10

###

normalized.p.list


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68 legendre.weight

References





See Also

legendre.inner.products

Examples

###


### the normalized Legendre polynomials


###

normalized.r


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lpochhammer 69

Value


Author(s)


References








Examples

###

### compute the Legendre weight function for argument values


###

x


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70 monic.polynomial.recurrences

Value

The value of the logarithm of the symbol

Author(s)


See Also

pochhammer

Examples

lpochhammer( pi, 5 )

monic.polynomial.recurrencesCreate data frame of monic recurrences

Description

This function returns a data frame with parameters required to construct monic orthogonal poly-

nomials based on the standard recurrence relation for the non-monic polynomials. The recurrence

relation for monic orthogonal polynomials is as follows.

qk+1(x) = (x ak) qk(x) bk qk1(x)

We require that q

1(x) = 0and q0(x) = 1. The recurrence for non-monic orthogonal polynomials

is given byck pk+1(x) = (dk+ ek x) pk(x) fk pk1(x)

We require thatp1(x) = 0and p0(x) = 1. The monic polynomial recurrence parameters, a andb, are related to the non-monic polynomial parameter vectors c,d,eand fin the following manner.

ak =dkek

bk = ck1 fkek1 ek

withb0 = 0.

Usage

monic.polynomial.recurrences(recurrences)

Arguments

recurrences the data frame of recurrence parameter vectorsc,d,e andf


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Value

A data frame withn + 1rows and two named columns, a and b.

Author(s)


References








See Also

orthogonal.polynomials,

Examples

###

### construct a list of the recurrences for the T Chebyshev polynomials of

### orders 0 to 10

###

r


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72 monic.polynomials

Arguments

monic.recurrences

a data frame containing the parameters a and b

Value

A list withn + 1polynomial objects

1 order 0 monic orthogonal polynomial

2 order 1 monic orthogonal polynomial

...

n+1 ordernmonic orthogonal polynomial

Author(s)


References



See Also

monic.polynomial.recurrences

Examples

###

### generate the recurrences for the T Chebyshev polynomials


###

r


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orthogonal.polynomials

Create orthogonal polynomials

Description

Create list of orthogonal polynomials from the following recurrence relations for k= 0, 1, . . . , n.

ckpk+1(x) = (dk+ ekx)pk(x) fkpk1(x)We require thatp1(x) = 0andp0(x) = 1. The coefficients are the column vectors c, d, e and f.

Usage

orthogonal.polynomials(recurrences)

Arguments

recurrences a data frame containing the parameters of the orthogonal polynomial recurrence

relations

Details

The argument is a data frame withn+ 1 rows and four named columns. The column names are c,d,eand f. These columns correspond to the column vectors described above.

Value


1 Order 0 orthogonal polynomial2 Order 1 orthogonal polynomial

...

n+1 Ordernorthogonal polynomial

Author(s)


References


Graphs, and Mathematical Tables, Dover Publications, Inc., New York.Courant, R., and D. Hilbert, 1989. Methods of Mathematical Physics, John Wiley, New York, NY.






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Author(s)


References








Examples

###

### generate a data frame with the recurrences parameters for normalized T Chebyshev

### polynomials of orders 0 to 10

###

r


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76 polynomial.coefficients

Arguments

z numeric value for the argument of the symbol

n integer value for the number of terms in the symbol

Value

The value of Pochhammers symbol

Author(s)


Examples

###

### compute the Pochhamers symbol fo z equal to 1 and

### n equal to 5

###

pochhammer( 1, 5 )

polynomial.coefficients

Create list of polynomial coefficient vectors

Description

This function returns a list withn+ 1 elements containing the vector of coefficients of the order k

polynomials for ordersk = 0, 1, . . . , n. Each element in the list is a vector.

Usage

polynomial.coefficients(polynomials)

Arguments

polynomials list of polynomial objects

Value

A list ofn + 1polynomial objects where each element is a vector of coefficients.

1 Coefficient(s) of the order 0 polynomial

2 Coefficient(s) of the order 1 polynomial

...

n+1 Coefficient(s) of the ordernpolynomial


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Author(s)


Examples###

### generate a list of normalized T Chebyshev polynomials


###

p.list


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Author(s)


Examples

###

### generate a list of normalized T Chebyshev polynomials of

### orders 0 to 10

###

p.list


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Author(s)


Examples###

### generate a list of T Chebyshev polynomials of

### orders 0 to 10

###

p.list


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Author(s)


Examples

###



###

p.list


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polynomial.powers 81

Examples

###



###

p.list


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82 polynomial.roots

Examples

###

### generate Legendre polynomials of orders 0 to 10

###

polynomials


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polynomial.values 83

References






See Also

monic.polynomial.recurrences,jacobi.matrices

Examples

###


### the normalized Chebyshev polynomials of

### orders 0 to 10

###

r


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Details

Target lower and/or upper bounds can beand, respectively. This accomodates finite targetintervals, semi-infinite target intervals and infinite target intervals.

Value

A vector of transformed values with four attributes. The first attribute is called "a" and it is the

domain interval lower bound. The second attribute is called "b" and it is the domain interval upper

bound. The third attribute is called "u" and it is the target interval lower bound. The fourth attribute

is called "v" and it is the target interval upper bound.

Author(s)

Frederick Novomestky , Gregor Gorjanc

References

Seber, G. A. F. (1997) Linear Regression Analysis, New York.

Examples

x


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Usage

schebyshev.t.polynomials(n, normalized)

Arguments



nomials

Details

The function schebyshev.t.recurrencesproduces a data frame with the recurrence relation pa-




Value


1 order 0 shifted Chebyshev polynomial


...

n+1 ordernshifted Chebyshev polynomial

Author(s)


References





See Also

schebyshev.u.recurrences,orthogonal.polynomials,orthonormal.polynomials


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88 schebyshev.t.recurrences

Examples

###

### gemerate a list of normalized shifted T Chebyshev polynomials of orders 0 to 10

###

normalized.p.list


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schebyshev.t.weight 89

See Also

schebyshev.t.inner.products

Examples###


### the normalized shifted T Chebyshev polynomials


###

normalized.r


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90 schebyshev.u.inner.products

References








Examples

###

### compute the shifted T Chebyshev weight function for argument values

### between 0 and 1

x


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Value




...


Author(s)


References

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




Examples

h


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92 schebyshev.u.polynomials

Details

The function schebyshev.u.recurrencesproduces a data frame with the recurrence relation pa-


is used to construct the list of orthogonal polynomial objects. Otherwise, the function orthonormal.polynomialsis used to construct the list of orthonormal polynomial objects.

Value




...

n+1 ordernshifted Chebyshev polynomial

Author(s)


References






See Also

schebyshev.u.recurrences,orthogonal.polynomials,orthonormal.polynomials

Examples

###

### gemerate a list of normalized shifted U Chebyshev polynomials of orders 0 to 10

###

normalized.p.list


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schebyshev.u.recurrences

Recurrence relations for shifted Chebyshev polynomials

Description

This function returns a data frame withn+ 1 rows and four named columns containing the coeffi-cient vectorsc,d,e and fof the recurrence relations for the orderk shifted Chebyshev polynomial

of the second kind,Uk(x), and for ordersk = 0, 1, . . . , n.

Usage

schebyshev.u.recurrences(n, normalized)

Arguments



nomials

Value


Author(s)


References







See Also

schebyshev.u.inner.products


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94 schebyshev.u.weight

Examples

###


### the normalized shifted U Chebyshev polynomials


###

normalized.r


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slegendre.inner.products 95

References



Courant, R., and D. Hilbert, 1989. Methods of Mathematical Physics,

orthopolynom

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Transcript of orthopolynom