| Version: | 1.0-6.1 |
| Date: | 2022-05-06 |
| Title: | Collection of Functions for Orthogonal and Orthonormal Polynomials |
| Author: | Frederick Novomestky <fnovomes@poly.edu> |
| Maintainer: | Frederick Novomestky <fnovomes@poly.edu> |
| Depends: | R (≥ 2.0.1) |
| Imports: | polynom, stats |
| Description: | A collection of functions to construct sets of orthogonal polynomials and their recurrence relations. Additional functions are provided to calculate the derivative, integral, value and roots of lists of polynomial objects. |
| License: | GPL-2 | GPL-3 [expanded from: GPL (≥ 2)] |
| NeedsCompilation: | no |
| Packaged: | 2022-10-03 09:51:57 UTC; hornik |
| Repository: | CRAN |
| Date/Publication: | 2022-10-03 10:11:45 UTC |
Inner products of Chebyshev polynomials
Description
This function returns a vector with n + 1 elements containing the inner product of
an order k Chebyshev polynomial of the first kind, C_k \left( x\right),
with itself (i.e. the norm squared) for orders k = 0,\;1,\; \ldots ,\;n .
Usage
chebyshev.c.inner.products(n)
Arguments
n |
integer value for the highest polynomial order |
Details
The formula used to compute the inner products is as follows.
h_n = \left\langle {C_n |C_n } \right\rangle = \left\{ {\begin{array}{cc}
{4\,\pi } & {n \ne 0} \\
{8\,\pi } & {n = 0} \\
\end{array} } \right.
Value
A vector with n + 1 elements
1 |
inner product of order 0 orthogonal polynomial |
2 |
inner product of order 1 orthogonal polynomial |
...
n+1 |
inner product of order |
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.
See Also
Examples
###
### generate the inner products vector for the
### C Chebyshev polynomials of orders 0 to 10
###
h <- chebyshev.c.inner.products( 10 )
print( h )
Create list of Chebyshev polynomials
Description
This function returns a list with n + 1 elements containing
the order k Chebyshev polynomials of the first kind, C_k \left( x\right),
for orders k = 0,\;1,\; \ldots ,\;n.
Usage
chebyshev.c.polynomials(n, normalized=FALSE)
Arguments
n |
integer value for the highest polynomial order |
normalized |
a boolean value which, if TRUE, returns a list of normalized orthogonal polynomials |
Details
The function chebyshev.c.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 Chebyshev polynomial |
2 |
order 1 Chebyshev polynomial |
...
n+1 |
order |
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.
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 <- chebyshev.c.polynomials( 10, normalized=TRUE )
print( normalized.p.list )
###
### gemerate a list of unnormalized C Chebyshev polynomials of orders 0 to 10
###
unnormalized.p.list <- chebyshev.c.polynomials( 10, normalized=FALSE )
print( unnormalized.p.list )
Recurrence relations for Chebyshev polynomials
Description
This function returns a data frame with n + 1 rows and four named columns containing
the coefficient vectors c, d, e and f of
the recurrence relations for the order k Chebyshev polynomial of the first kind, C_k \left( x \right),
and for orders k = 0,\;1,\; \ldots ,\;n.
Usage
chebyshev.c.recurrences(n, normalized=FALSE)
Arguments
n |
integer value for the highest polynomial order |
normalized |
boolean value which, if TRUE, returns recurrence relations for normalized polynomials |
Value
A data frame with the recurrence relation parameters.
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.
See Also
Examples
###
### generate the recurrences data frame for
### the normalized Chebyshev C polynomials
### of orders 0 to 10.
###
normalized.r <- chebyshev.c.recurrences( 10, normalized=TRUE )
print( normalized.r )
###
### generate the recurrences data frame for
### the normalized Chebyshev C polynomials
### of orders 0 to 10.
###
unnormalized.r <- chebyshev.c.recurrences( 10, normalized=FALSE )
print( unnormalized.r )
Weight function for the Chebyshev polynomial
Description
This function returns the value of the weight function for the order k
Chebyshev polynomial of the first kind, C_k \left( x \right) .
Usage
chebyshev.c.weight(x)
Arguments
x |
the function argument which can be a vector |
Details
The function takes on non-zero values in the interval \left( -2,2 \right) . The formula
used to compute the weight function is as follows.
w\left( x \right) = \frac{1}
{{\sqrt {1 - \frac{{x^2 }}
{4}} }}
Value
The value of the weight function
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.
Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992. Numerical Recipes in C, Cambridge University Press, Cambridge, U.K.
Szego, G., 1939. Orthogonal Polynomials, 23, American Mathematical Society Colloquium Publications, Providence, RI.
Examples
###
### compute the C Chebyshev weight function for arguments between -3 and 3
###
x <- seq( -3, 3, .01 )
y <- chebyshev.c.weight( x )
plot( x, y )
Inner products of Chebyshev polynomials
Description
This function returns a vector with n + 1 elements containing the inner product of
an order k Chebyshev polynomial of the second kind, S_k \left( x\right),
with itself (i.e. the norm squared) for orders k = 0,\;1,\; \ldots ,\;n .
Usage
chebyshev.s.inner.products(n)
Arguments
n |
integer value for the highest polynomial order |
Details
The formula used to compute the inner products is as follows.
h_n = \left\langle {S_n |S_n } \right\rangle = \pi .
Value
A vector with n + 1 elements
1 |
inner product of order 0 orthogonal polynomial |
2 |
inner product of order 1 orthogonal polynomial |
...
n+1 |
inner product of order |
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.
Examples
###
### generate the inner products vector for the
### S Chebyshev polynomials of orders 0 to 10
###
h <- chebyshev.s.inner.products( 10 )
print( h )
Create list of Chebyshev polynomials
Description
This function returns a list with n + 1 elements containing
the order k Chebyshev polynomials of the second kind, S_k \left( x\right),
for orders k = 0,\;1,\; \ldots ,\;n.
Usage
chebyshev.s.polynomials(n, normalized=FALSE)
Arguments
n |
integer value for the highest polynomial order |
normalized |
a boolean value which, if TRUE, returns a list of normalized orthogonal polynomials |
Details
The function chebyshev.s.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 Chebyshev polynomial |
2 |
order 1 Chebyshev polynomial |
...
n+1 |
order |
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.
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 <- chebyshev.s.polynomials( 10, normalized=TRUE )
print( normalized.p.list )
###
### gemerate a list of unnormalized S Chebyshev polynomials of orders 0 to 10
###
unnormalized.p.list <- chebyshev.s.polynomials( 10, normalized=FALSE )
print( unnormalized.p.list )
Recurrence relations for Chebyshev polynomials
Description
This function returns a data frame with n + 1 rows and four named columns containing
the coefficient vectors c, d, e and f of
the recurrence relations for the order k Chebyshev polynomial of the second kind, S_k \left( x \right),
and for orders k = 0,\;1,\; \ldots ,\;n.
Usage
chebyshev.s.recurrences(n, normalized=FALSE)
Arguments
n |
integer value for the highest polynomial order |
normalized |
boolean value which, if TRUE, returns recurrence relations for normalized polynomials |
Value
A data frame with the recurrence relation parameters.
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.
See Also
Examples
###
### generate the recurrences data frame for
### the normalized Chebyshev S polynomials
### of orders 0 to 10.
###
normalized.r <- chebyshev.s.recurrences( 10, normalized=TRUE )
print( normalized.r )
###
### generate the recurrences data frame for
### the normalized Chebyshev S polynomials
### of orders 0 to 10.
###
unnormalized.r <- chebyshev.s.recurrences( 10, normalized=FALSE )
print( unnormalized.r )
Weight function for the Chebyshev polynomial
Description
This function returns the value of the weight function for the order k Chebyshev polynomial of the second kind, S_k \left( x \right) .
Usage
chebyshev.s.weight(x)
Arguments
x |
the function argument which can be a vector |
Details
The function takes on non-zero values in the interval \left( -2,2 \right) . The formula
used to compute the weight function is as follows.
w\left( x \right) = \sqrt {1 - \frac{{x^2 }}{4}}
Value
The value of the weight function.
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.
Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992. Numerical Recipes in C, Cambridge University Press, Cambridge, U.K.
Szego, G., 1939. Orthogonal Polynomials, 23, American Mathematical Society Colloquium Publications, Providence, RI.
Examples
###
### compute the S Chebyshev weight function for arguments between -2 and 2
###
x <- seq( -2, 2, .01 )
y <- chebyshev.s.weight( x )
plot( x, y )
Inner products of Chebyshev polynomials
Description
This function returns a vector with n + 1 elements containing the inner product of
an order k Chebyshev polynomial of the first kind, T_k \left( x\right),
with itself (i.e. the norm squared) for orders k = 0,\;1,\; \ldots ,\;n .
Usage
chebyshev.t.inner.products(n)
Arguments
n |
integer value for the highest polynomial order |
Details
The formula used to compute the inner products is as follows.
h_n = \left\langle {T_n |T_n } \right\rangle = \left\{ {\begin{array}{cc}
{\frac{\pi }
{2}} & {n \ne 0} \\
\pi & {n = 0} \\
\end{array} } \right.
Value
A vector with n + 1 elements
1 |
inner product of order 0 orthogonal polynomial |
2 |
inner product of order 1 orthogonal polynomial |
...
n+1 |
inner product of order |
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.
Examples
###
### generate the inner products vector for the
### T Chybyshev polynomials of orders 0 to 10
###
h <- chebyshev.t.inner.products( 10 )
print( h )
Create list of Chebyshev polynomials
Description
This function returns a list with n + 1 elements containing
the order k Chebyshev polynomials of the first kind, T_k \left( x\right),
for orders k = 0,\;1,\; \ldots ,\;n.
Usage
chebyshev.t.polynomials(n, normalized=FALSE)
Arguments
n |
integer value for the highest polynomial order |
normalized |
a boolean value which, if TRUE, returns a list of normalized orthogonal polynomials |
Details
The function chebyshev.t.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 Chebyshev polynomial |
2 |
order 1 Chebyshev polynomial |
...
n+1 |
order |
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.
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 <- chebyshev.t.polynomials( 10, normalized=TRUE )
print( normalized.p.list )
###
### gemerate a list of unnormalized T Chebyshev polynomials of orders 0 to 10
###
unnormalized.p.list <- chebyshev.t.polynomials( 10, normalized=FALSE )
print( unnormalized.p.list )
Recurrence relations for Chebyshev polynomials
Description
This function returns a data frame with n + 1 rows and four named columns containing
the coefficient vectors c, d, e and f of
the recurrence relations for the order k Chebyshev polynomial of the first kind, T_k \left( x \right),
for orders k = 0,\;1,\; \ldots ,\;n.
Usage
chebyshev.t.recurrences(n, normalized=FALSE)
Arguments
n |
integer value for the highest polynomial order |
normalized |
boolean value which, if TRUE, returns recurrence relations for normalized polynomials |
Value
A data frame with the recurrence relation parameters.
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.
See Also
Examples
###
### generate the recurrence relations for
### the normalized T Chebyshev polynomials
### of orders 0 to 10
###
normalized.r <- chebyshev.t.recurrences( 10, normalized=TRUE )
print( normalized.r )
###
### generate the recurrence relations for
### the normalized T Chebyshev polynomials
### of orders 0 to 10
###
unnormalized.r <- chebyshev.t.recurrences( 10, normalized=FALSE )
print( unnormalized.r )
Weight function for the Chebyshev polynomial
Description
This function returns the value of the weight function for the order k Chebyshev polynomial
of the first kind, T_k \left( x \right) .
Usage
chebyshev.t.weight(x)
Arguments
x |
the function argument which can be a vector |
Details
The function takes on non-zero values in the interval \left( -1,1 \right) . The formula
used to compute the weight function is as follows.
w\left( x \right) = \frac{1}{{\sqrt {1 - x^2 } }}
Value
The value of the weight function.
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.
Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992. Numerical Recipes in C, Cambridge University Press, Cambridge, U.K.
Szego, G., 1939. Orthogonal Polynomials, 23, American Mathematical Society Colloquium Publications, Providence, RI.
Examples
###
### compute the T Chebyshev function for argument values between -2 and 2
x <- seq( -1, 1, .01 )
y <- chebyshev.t.weight( x )
plot( x, y )
Inner products of Chebyshev polynomials
Description
This function returns a vector with n + 1 elements containing the inner product of
an order k Chebyshev polynomial of the second kind, U_k \left( x\right),
with itself (i.e. the norm squared) for orders k = 0,\;1,\; \ldots ,\;n .
Usage
chebyshev.u.inner.products(n)
Arguments
n |
integer value for the highest polynomial order |
Details
The formula used to compute the inner products is as follows.
h_n = \left\langle {U_n |U_n } \right\rangle = \frac{\pi }{2}
Value
A vector with n + 1 elements
1 |
inner product of order 0 orthogonal polynomial |
2 |
inner product of order 1 orthogonal polynomial |
...
n+1 |
inner product of order |
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.
Examples
###
### generate the inner products vector for the
### U Chebyshev polynomials of orders 0 to 10
###
h <- chebyshev.u.inner.products( 10 )
print( h )
Create list of Chebyshev polynomials
Description
This function returns a list with n + 1 elements containing
the order k Chebyshev polynomials of the second kind, U_k \left( x\right),
for orders k = 0,\;1,\; \ldots ,\;n.
Usage
chebyshev.u.polynomials(n, normalized=FALSE)
Arguments
n |
integer value for the highest polynomial order |
normalized |
a boolean value which, if TRUE, returns a list of normalized orthogonal polynomials |
Details
The function chebyshev.u.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 Chebyshev polynomial |
2 |
order 1 Chebyshev polynomial |
...
n+1 |
order |
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.
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 <- chebyshev.u.polynomials( 10, normalized=TRUE )
print( normalized.p.list )
###
### gemerate a list of unnormalized T Chebyshev polynomials of orders 0 to 10
###
unnormalized.p.list <- chebyshev.u.polynomials( 10, normalized=FALSE )
print( unnormalized.p.list )
Recurrence relations for Chebyshev polynomials
Description
This function returns a data frame with n + 1 rows and four named columns containing
the coefficient vectors c, d, e and f of
the recurrence relations for the order k Chebyshev polynomial of the second kind, U_k \left( x \right),
for orders k = 0,\;1,\; \ldots ,\;n.
Usage
chebyshev.u.recurrences(n, normalized=FALSE)
Arguments
n |
integer value for the highest polynomial order |
normalized |
boolean value which, if TRUE, returns recurrence relations for normalized polynomials |
Value
A data frame with the recurrence relation parameters.
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.
See Also
Examples
###
### generate the recurrence relations for
### the normalized U Chebyshev polynomials
### of orders 0 to 10
###
normalized.r <- chebyshev.u.recurrences( 10, normalized=TRUE )
print( normalized.r )
###
### generate the recurrence relations for
### the unnormalized U Chebyshev polynomials
### of orders 0 to 10
###
unnormalized.r <- chebyshev.u.recurrences( 10, normalized=FALSE )
print( unnormalized.r )
Weight function for the Chebyshev polynomial
Description
This function returns the value of the weight function for the order k Chebyshev polynomial of the second kind, U_k \left( x \right) .
Usage
chebyshev.u.weight(x)
Arguments
x |
the function argument which can be a vector |
Details
The function takes on non-zero values in the interval \left( -1,1 \right) . The formula used
to compute the weight function is as follows.
w\left( x \right) = \sqrt {1 - x^2 }
Value
The value of the weight function.
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.
Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992. Numerical Recipes in C, Cambridge University Press, Cambridge, U.K.
Szego, G., 1939. Orthogonal Polynomials, 23, American Mathematical Society Colloquium Publications, Providence, RI.
Examples
###
### compute the U Chebyshev function for argument values between -2 and 2
###
x <- seq( -1, 1, .01 )
y <- chebyshev.u.weight( x )
plot( x, y )
Inner products of Gegenbauer polynomials
Description
This function returns a vector with n + 1 elements containing the inner product of
an order k Gegenbauer polynomial, C_k^{\left( \alpha \right)} \left( x \right),
with itself (i.e. the norm squared) for orders k = 0,\;1,\; \ldots ,\;n .
Usage
gegenbauer.inner.products(n,alpha)
Arguments
n |
integer value for the highest polynomial order |
alpha |
numeric value for the polynomial parameter |
Details
The formula used to compute the inner products is as follows.
h_n = \left\langle {C_n^{\left( \alpha \right)} |C_n^{\left( \alpha \right)} } \right\rangle = \left\{ {\begin{array}{cc}
{\frac{{\pi \;2^{1 - 2\,\alpha } \,\Gamma \left( {n + 2\,\alpha } \right)}}
{{n!\;\left( {n + \alpha } \right)\,\left[ {\Gamma \left( \alpha \right)} \right]^2 }}} & {\alpha \ne 0} \\
{\frac{{2\;\pi }}
{{n^2 }}} & {\alpha = 0} \\
\end{array} } \right..
Value
A vector with n + 1 elements
1 |
inner product of order 0 orthogonal polynomial |
2 |
inner product of order 1 orthogonal polynomial |
...
n+1 |
inner product of order |
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.
See Also
Examples
###
### generate the inner products vector for the
### Gegenbauer polynomials of orders 0 to 10
### the polynomial parameter is 1.0
###
h <- gegenbauer.inner.products( 10, 1 )
print( h )
Create list of Gegenbauer polynomials
Description
This function returns a list with n + 1 elements containing
the order k Gegenbauer polynomials, C_k^{\left( \alpha \right)} \left( x \right),
for orders k = 0,\;1,\; \ldots ,\;n.
Usage
gegenbauer.polynomials(n, alpha, normalized=FALSE)
Arguments
n |
integer value for the highest polynomial order |
alpha |
polynomial parameter |
normalized |
a boolean value which, if TRUE, returns a list of normalized orthogonal polynomials |
Details
The function gegenbauer.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 Gegenbauer polynomial |
2 |
order 1 Gegenbauer polynomial |
...
n+1 |
order |
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.
See Also
gegenbauer.recurrences,
orthogonal.polynomials,
orthonormal.polynomials
Examples
###
### gemerate a list of normalized Gegenbauer polynomials of orders 0 to 10
### polynomial parameter is 1.0
###
normalized.p.list <- gegenbauer.polynomials( 10, 1, normalized=TRUE )
print( normalized.p.list )
###
### gemerate a list of unnormalized Gegenbauer polynomials of orders 0 to 10
### polynomial parameter is 1.0
###
unnormalized.p.list <- gegenbauer.polynomials( 10, 1, normalized=FALSE )
print( unnormalized.p.list )
Recurrence relations for Gegenbauer polynomials
Description
This function returns a data frame with n + 1 rows and four named columns containing
the coefficient vectors c, d, e and f of
the recurrence relations for the order k Gegenbauer polynomial, C_k^{\left( \alpha \right)} \left( x \right),
and for orders k = 0,\;1,\; \ldots ,\;n.
Usage
gegenbauer.recurrences(n, alpha, normalized=FALSE)
Arguments
n |
integer value for the highest polynomial order |
alpha |
numeric value for polynomial parameter |
normalized |
boolean value which, if TRUE, returns recurrence relations for normalized polynomials |
Value
A data frame with the recurrence relation parameters.
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.
See Also
Examples
###
### generate the recurrences data frame for
### the normalized Gegenbauer polynomials
### of orders 0 to 10.
### polynomial parameter value is 1.0
###
normalized.r <- gegenbauer.recurrences( 10, 1, normalized=TRUE )
print( normalized.r )
###
### generate the recurrences data frame for
### the normalized Gegenbauer polynomials
### of orders 0 to 10.
### polynomial parameter value is 1.0
###
unnormalized.r <- gegenbauer.recurrences( 10, 1, normalized=FALSE )
print( unnormalized.r )
Weight function for the Gegenbauer polynomial
Description
This function returns the value of the weight function for the order k
Gegenbauer polynomial, C_k^{\left( \alpha \right)} \left( x \right).
Usage
gegenbauer.weight(x,alpha)
Arguments
x |
the function argument which can be a vector |
alpha |
polynomial parameter |
Details
The function takes on non-zero values in the interval \left( -1,1 \right) . The formula
used to compute the weight function is as follows.
w\left( x \right) = \left( {1 - x^2 } \right)^{\alpha - 0.5}
Value
The value of the weight function
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.
Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992. Numerical Recipes in C, Cambridge University Press, Cambridge, U.K.
Szego, G., 1939. Orthogonal Polynomials, 23, American Mathematical Society Colloquium Publications, Providence, RI.
Examples
###
### compute the Gegenbauer weight function for argument values between -1 and 1
###
x <- seq( -1, 1, .01 )
y <- gegenbauer.weight( x, 1 )
plot( x, y )
Inner products of generalized Hermite polynomials
Description
This function returns a vector with n + 1 elements containing the inner product of
an order k generalized Hermite polynomial, H_k^{\left( \mu \right)} \left( x \right),
with itself (i.e. the norm squared) for orders k = 0,\;1,\; \ldots ,\;n .
Usage
ghermite.h.inner.products(n, mu)
Arguments
n |
|
mu |
|
Details
The parameter \mu must be greater than -0.5. The formula used to compute the inner
products is as follows.
h_n \left( \mu \right) = \left\langle {H_m^{\left( \mu \right)} |H_n^{\left( \mu \right)} } \right\rangle = 2^{2\,n} \,\left[ {\frac{n}
{2}} \right]!\;\Gamma \left( {\left[ {\frac{{n + 1}}
{2}} \right] + \mu + \frac{1}
{2}} \right)
Value
A vector with n + 1 elements
1 |
inner product of order 0 orthogonal polynomial |
2 |
inner product of order 1 orthogonal polynomial |
...
n+1 |
inner product of order |
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.
Examples
###
### generate the inner products vector for the
### generalized Hermite polynomials of orders 0 to 10
### polynomial parameter is 1
###
h <- ghermite.h.inner.products( 10, 1 )
print( h )
Create list of generalized Hermite polynomials
Description
This function returns a list with n + 1 elements containing
the order k generalized Hermite polynomials, H_k^{\left( \mu \right)} \left( x \right),
for orders k = 0,\;1,\; \ldots ,\;n.
Usage
ghermite.h.polynomials(n, mu, normalized = FALSE)
Arguments
n |
integer value for the highest polynomial order |
mu |
numeric value for the polynomial parameter |
normalized |
boolean value which, if TRUE, returns recurrence relations for normalized polynomials |
Details
The parameter \mu must be greater than -0.5.
The function ghermite.h.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 generalized Hermite polynomial |
2 |
order 1 generalized Hermite polynomial |
...
n+1 |
order |
Author(s)
Frederick Novomestky fnovomes@poly.edu
References
Alvarez-Nordase, R., M. K. Atakishiyeva and N. M. Atakishiyeva, 2004. A q-extension of the generalized 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, 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.
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 <- ghermite.h.polynomials( 10, 1, normalized=TRUE )
print( normalized.p.list )
###
### gemerate a list of unnormalized generalized Hermite polynomials of orders 0 to 10
### polynomial parameter is 1.0
###
unnormalized.p.list <- ghermite.h.polynomials( 10, 1, normalized=FALSE )
print( unnormalized.p.list )
Recurrence relations for generalized Hermite polynomials
Description
This function returns a data frame with n + 1 rows and four named columns containing
the coefficient vectors c, d, e and f of
the recurrence relations for the order k generalized Hermite polynomial, H_k^{\left( \mu \right)} \left( x \right),
and for orders k = 0,\;1,\; \ldots ,\;n.
Usage
ghermite.h.recurrences(n, mu, normalized = FALSE)
Arguments
n |
integer value for the highest polynomial order |
mu |
numeric value for the polynomial parameter |
normalized |
|
Details
The parameter \mu must be greater than -0.5.
Value
A data frame with the recurrence relation parameters.
Author(s)
Frederick Novomestky fnovomes@poly.edu
References
Alvarez-Nordase, R., M. K. Atakishiyeva and N. M. Atakishiyeva, 2004. A q-extension of the generalized 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, 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.
See Also
Examples
###
### generate the recurrences data frame for
### the normalized generalized Hermite polynomials
### of orders 0 to 10.
### polynomial parameter value is 1.0
###
normalized.r <- ghermite.h.recurrences( 10, 1, normalized=TRUE )
print( normalized.r )
###
### generate the recurrences data frame for
### the unnormalized generalized Hermite polynomials
### of orders 0 to 10.
### polynomial parameter value is 1.0
###
unnormalized.r <- ghermite.h.recurrences( 10, 1, normalized=FALSE )
print( unnormalized.r )
Weight function for the generalized Hermite polynomial
Description
This function returns the value of the weight function for the order k
generalized Hermite polynomial, H_k^{\left( \mu \right)} \left( x \right) .
Usage
ghermite.h.weight(x, mu)
Arguments
x |
a numeric vector function argument |
mu |
polynomial parameter |
Details
The function takes on non-zero values in the interval \left( -\infty,\infty \right) .
The parameter \mu must be greater than -0.5. The formula used to compute the
generalized Hermite weight function is as follows.
w\left( {x,\mu } \right) = \left| x \right|^{2\;\mu } \;\exp \left( { - x^2 } \right)
Value
The value of the weight function
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.
Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992. Numerical Recipes in C, Cambridge University Press, Cambridge, U.K.
Szego, G., 1939. Orthogonal Polynomials, 23, American Mathematical Society Colloquium Publications, Providence, RI.
Examples
###
### compute the generalized Hermite weight function for argument values
### between -3 and 3
###
x <- seq( -3, 3, .01 )
y <- ghermite.h.weight( x, 1 )
Inner products of generalized Laguerre polynomials
Description
This function returns a vector with n + 1 elements containing the inner product of
an order k generalized Laguerre polynomial, L_n^{\left( \alpha \right)} \left( x \right),
with itself (i.e. the norm squared) for orders k = 0,\;1,\; \ldots ,\;n .
Usage
glaguerre.inner.products(n,alpha)
Arguments
n |
integer highest polynomial order |
alpha |
polynomial parameter |
Details
The formula used to compute the inner products is as follows.
h_n = \left\langle {L_n^{\left( \alpha \right)} |L_n^{\left( \alpha \right)} } \right\rangle = \frac{{\Gamma \left( {\alpha + n + 1} \right)}}
{{n!}}.
Value
A vector with n + 1 elements
1 |
inner product of order 0 orthogonal polynomial |
2 |
inner product of order 1 orthogonal polynomial |
...
n+1 |
inner product of order |
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.
Examples
###
### generate the inner products vector for the
### generalized Laguerre polynomial inner products of orders 0 to 10
### polynomial parameter is 1.
###
h <- glaguerre.inner.products( 10, 1 )
print( h )
Create list of generalized Laguerre polynomials
Description
This function returns a list with n + 1 elements containing
the order n generalized Laguerre polynomials, L_n^{\left( \alpha \right)} \left( x \right),
for orders k = 0,\;1,\; \ldots ,\;n.
Usage
glaguerre.polynomials(n, alpha, normalized=FALSE)
Arguments
n |
integer value for the highest polynomial order |
alpha |
numeric value for the polynomial parameter |
normalized |
a boolean value which, if TRUE, returns a list of normalized orthogonal polynomials |
Details
The function glaguerre.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 generalized Laguerre polynomial |
2 |
order 1 generalized Laguerre polynomial |
...
n+1 |
order |
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.
See Also
glaguerre.recurrences,
orthogonal.polynomials,
orthonormal.polynomials
Examples
###
### gemerate a list of normalized generalized Laguerre polynomials of orders 0 to 10
### polynomial parameter is 1.0
###
normalized.p.list <- glaguerre.polynomials( 10, 1, normalized=TRUE )
print( normalized.p.list )
###
### gemerate a list of unnormalized generalized Laguerre polynomials of orders 0 to 10
### polynomial parameter is 1.0
###
unnormalized.p.list <- glaguerre.polynomials( 10, 1, normalized=FALSE )
Recurrence relations for generalized Laguerre polynomials
Description
This function returns a data frame with n + 1 rows and four named columns containing
the coefficient vectors c, d, e and f of the recurrence relations for
the order k generalized Laguerre polynomial, L_n^{\left( \alpha \right)} \left( x \right)
and for orders k = 0,\;1,\; \ldots ,\;n.
Usage
glaguerre.recurrences(n, alpha, normalized=FALSE)
Arguments
n |
integer value for the highest polynomial order |
alpha |
numeric value for the polynomial parameter |
normalized |
boolean value which, if TRUE, returns recurrence relations for normalized polynomials |
Value
A data frame with the recurrence relation parameters.
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.
See Also
Examples
###
### generate the recurrences data frame for
### the normalized generalized Laguerre polynomials
### of orders 0 to 10. the polynomial parameter value is 1.0.
###
normalized.r <- glaguerre.recurrences( 10, 1, normalized=TRUE )
print( normalized.r )
###
### generate the recurrences data frame for
### the unnormalized generalized Laguerre polynomials
### of orders 0 to 10. the polynomial parameter value is 1.0.
###
unnormalized.r <- glaguerre.recurrences( 10, 1, normalized=FALSE )
print( unnormalized.r )
Weight function for the generalized Laguerre polynomial
Description
This function returns the value of the weight function for the order k
generalized Laguerre polynomial, L_n^{\left( \alpha \right)} \left( x \right).
Usage
glaguerre.weight(x,alpha)
Arguments
x |
the function argument which can be a vector |
alpha |
polynomial parameter |
Details
The function takes on non-zero values in the interval \left( 0,\infty \right) . The formula
used to compute the weight function is as follows.
w\left( x \right) = e^{ - x} \,x^\alpha
Value
The value of the weight function
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.
Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992. Numerical Recipes in C, Cambridge University Press, Cambridge, U.K.
Szego, G., 1939. Orthogonal Polynomials, 23, American Mathematical Society Colloquium Publications, Providence, RI.
Examples
###
### compute the generalized Laguerre weight function for argument values
### between -3 and 3
### polynomial parameter value is 1.0
###
x <- seq( -3, 3, .01 )
y <- glaguerre.weight( x, 1 )
Inner products of Hermite polynomials
Description
This function returns a vector with n + 1 elements containing the inner product of
an order k Hermite polynomial, H_k \left( x \right),
with itself (i.e. the norm squared) for orders k = 0,\;1,\; \ldots ,\;n .
Usage
hermite.h.inner.products(n)
Arguments
n |
integer value for highest polynomial order |
Details
The formula used to compute the innner product is as follows.
h_n = \left\langle {H_n |H_n } \right\rangle = \sqrt \pi \;2^n \;n!.
Value
A vector with n + 1 elements
1 |
inner product of order 0 orthogonal polynomial |
2 |
inner product of order 1 orthogonal polynomial |
...
n+1 |
inner product of order |
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.
Examples
###
### generate the inner products vector for the
### Hermite polynomials of orders 0 to 10
###
h <- hermite.h.inner.products( 10 )
print( h )
Create list of Hermite polynomials
Description
This function returns a list with n + 1 elements containing
the order k Hermite polynomials, H_k \left( x \right),
for orders k = 0,\;1,\; \ldots ,\;n.
Usage
hermite.h.polynomials(n, normalized=FALSE)
Arguments
n |
integer value for the highest polynomial order |
normalized |
a boolean value which, if TRUE, returns a list of normalized orthogonal polynomials |
Details
The function hermite.h.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
list of orthonormal polynomial objects.
Value
A list of n + 1 polynomial objects
1 |
order 0 Hermite polynomial |
2 |
order 1 Hermite polynomial |
...
n+1 |
order |
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.
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 <- hermite.h.polynomials( 10, normalized=TRUE )
print( normalized.p.list )
###
### gemerate a list of unnormalized Hermite polynomials of orders 0 to 10
###
unnormalized.p.list <- hermite.h.polynomials( 10, normalized=FALSE )
print( unnormalized.p.list )
Recurrence relations for Hermite polynomials
Description
This function returns a data frame with n + 1 rows and four named columns containing
the coefficient vectors c, d, e and f of
the recurrence relations for the order k Hermite polynomial, H_k \left( x \right),
and for orders k = 0,\;1,\; \ldots ,\;n.
Usage
hermite.h.recurrences(n, normalized=FALSE)
Arguments
n |
integer value for the highest polynomial order |
normalized |
boolean value which, if TRUE, returns recurrence relations for normalized polynomials |
Value
A data frame with the recurrence relation parameters.
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.
See Also
Examples
###
### generate the recurrences data frame for
### the normalized Hermite H polynomials
### of orders 0 to 10.
###
normalized.r <- hermite.h.recurrences( 10, normalized=TRUE )
print( normalized.r )
###
### generate the recurrences data frame for
### the unnormalized Hermite H polynomials
### of orders 0 to 10.
###
unnormalized.r <- hermite.h.recurrences( 10, normalized=FALSE )
print( unnormalized.r )
Weight function for the Hermite polynomial
Description
This function returns the value of the weight function for the order k
Hermite polynomial, H_k \left( x \right).
Usage
hermite.h.weight(x)
Arguments
x |
the function argument which can be a vector |
Details
The function takes on non-zero values in the interval \left( -\infty,\infty \right) . The
formula used to compute the weight function.
w\left( x \right) = \exp \left( { - x^2 } \right)
Value
The value of the weight function
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.
Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992. Numerical Recipes in C, Cambridge University Press, Cambridge, U.K.
Szego, G., 1939. Orthogonal Polynomials, 23, American Mathematical Society Colloquium Publications, Providence, RI.
Examples
###
### compute the Hermite weight function for argument values
### between -3 and 3
x <- seq( -3, 3, .01 )
y <- hermite.h.weight( x )
plot( x, y )
Inner products of Hermite polynomials
Description
This function returns a vector with n + 1 elements containing the inner product of
an order k Hermite polynomial, He_k \left( x \right),
with itself (i.e. the norm squared) for orders k = 0,\;1,\; \ldots ,\;n .
Usage
hermite.he.inner.products(n)
Arguments
n |
integer value for the highest polynomial order |
Details
The formula used to compute the inner products is as follows.
h_n = \left\langle {He_n |He_n } \right\rangle = \sqrt {2\,\pi } \;n!.
Value
A vector with n + 1 elements
1 |
inner product of order 0 orthogonal polynomial |
2 |
inner product of order 1 orthogonal polynomial |
...
n+1 |
inner product of order |
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.
Examples
###
### generate the inner products vector for the
### scaled Hermite polynomials of orders 0 to 10
###
h <- hermite.he.inner.products( 10 )
print( h )
Create list of Hermite polynomials
Description
This function returns a list with n + 1 elements containing
the order k Hermite polynomials, He_k \left( x \right),
for orders k = 0,\;1,\; \ldots ,\;n.
Usage
hermite.he.polynomials(n, normalized=FALSE)
Arguments
n |
integer value for thehighest polynomial order |
normalized |
a boolean value which, if TRUE, returns a list of normalized orthogonal polynomials |
Details
The function hermite.he.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 Hermite polynomial |
2 |
order 1 Hermite polynomial |
...
n+1 |
order |
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.
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 <- hermite.he.polynomials( 10, normalized=TRUE )
print( normalized.p.list )
###
### gemerate a list of unnormalized Hermite polynomials of orders 0 to 10
###
unnormalized.p.list <- hermite.he.polynomials( 10, normalized=FALSE )
print( unnormalized.p.list )
Recurrence relations for Hermite polynomials
Description
This function returns a data frame with n + 1 rows and four named columns containing
the coefficient vectors c, d, e and f of
the recurrence relations for the order k Hermite polynomial, He_k \left( x \right),
and for orders k = 0,\;1,\; \ldots ,\;n.
Usage
hermite.he.recurrences(n, normalized=FALSE)
Arguments
n |
integer value for the highest polynomial order |
normalized |
boolean value which, if TRUE, returns recurrence relations for normalized polynomials |
Value
A data frame with the recurrence relation parameters.
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.
See Also
Examples
###
### generate the recurrences data frame for
### the normalized Hermite H polynomials
### of orders 0 to 10.
###
normalized.r <- hermite.he.recurrences( 10, normalized=TRUE )
print( normalized.r )
###
### generate the recurrences data frame for
### the unnormalized Hermite H polynomials
### of orders 0 to 10.
###
unnormalized.r <- hermite.he.recurrences( 10, normalized=FALSE )
print( unnormalized.r )
Weight function for the Hermite polynomial
Description
This function returns the value of the weight function for the order k
Hermite polynomial, He_k \left( x \right).
Usage
hermite.he.weight(x)
Arguments
x |
the function argument which can be a vector |
Details
The function takes on non-zero values in the interval \left( -\infty,\infty \right) . The
formula used to compute the weight function is as follows.
w\left( x \right) = \exp \left( { - \frac{{x^2 }}{2}} \right)
Value
The value of the weight function
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.
Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992. Numerical Recipes in C, Cambridge University Press, Cambridge, U.K.
Szego, G., 1939. Orthogonal Polynomials, 23, American Mathematical Society Colloquium Publications, Providence, RI.
Examples
###
### compute the scaled Hermite weight function for argument values
### between -3 and 3
###
x <- seq( -3, 3, .01 )
y <- hermite.he.weight( x )
Inner products of Jacobi polynomials
Description
This function returns a vector with n + 1 elements containing the inner product of
an order k Jacobi polynomial, G_k \left( {p,q,x} \right),
with itself (i.e. the norm squared) for orders k = 0,\;1,\; \ldots ,\;n .
Usage
jacobi.g.inner.products(n,p,q)
Arguments
n |
integer value for the highest polynomial order |
p |
numeric value for the first polynomial parameter |
q |
numeric value for the first polynomial parameter |
Details
The formula used to compute the inner products is as follows.
h_n = \left\langle {G_n |G_n } \right\rangle = \frac{{n!\;\Gamma \left( {n + q} \right)\,\Gamma \left( {n + p} \right)\,\Gamma \left( {n + p - q + 1} \right)}}
{{\left( {2\,n + p} \right)\,\left[ {\Gamma \left( {2\,n + p} \right)} \right]^2 }}.
Value
A vector with n + 1 elements
1 |
inner product of order 0 orthogonal polynomial |
2 |
inner product of order 1 orthogonal polynomial |
...
n+1 |
inner product of order |
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.
Examples
###
### generate the inner products vector for the
### G Jacobi polynomials of orders 0 to 10
### parameter p is 3 and parameter q is 2
###
h <- jacobi.g.inner.products( 10, 3, 2 )
print( h )
Create list of Jacobi polynomials
Description
This function returns a list with n + 1 elements containing
the order k Jacobi polynomials, G_k \left( {p,q,x} \right),
for orders k = 0,\;1,\; \ldots ,\;n.
Usage
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 |
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.
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 <- 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 )
Recurrence relations for Jacobi polynomials
Description
This function returns a data frame with n + 1 rows and four named columns containing
the coefficient vectors c, d, e and f of
the recurrence relations for the order k Jacobi polynomial, G_k \left( {p,q,x} \right),
and for orders k = 0,\;1,\; \ldots ,\;n.
Usage
jacobi.g.recurrences(n, p, q, normalized=FALSE)
Arguments
n |
integer value for the highest polynomial order |
p |
numeric value for the first polynomial parameter |
q |
numeric value for the second polynomial parameter |
normalized |
boolean value which, if TRUE, returns recurrence relations for normalized polynomials |
Value
A data frame with the recurrence relation parameters.
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.
See Also
jacobi.g.inner.products,
pochhammer
Examples
###
### generate the recurrences data frame for
### the normalized Jacobi G polynomials
### of orders 0 to 10.
### parameter p is 3 and parameter q is 2
###
normalized.r <- jacobi.g.recurrences( 10, 3, 2, normalized=TRUE )
print( normalized.r )
###
### generate the recurrences data frame for
### the normalized Jacobi G polynomials
### of orders 0 to 10.
### parameter p is 3 and parameter q is 2
###
unnormalized.r <- jacobi.g.recurrences( 10, 3, 2, normalized=FALSE )
print( unnormalized.r )
Weight function for the Jacobi polynomial
Description
This function returns the value of the weight function for the order k
Jacobi polynomial, G_k \left( {p,q,x} \right).
Usage
jacobi.g.weight(x,p,q)
Arguments
x |
the function argument which can be a vector |
p |
the first polynomial parameter |
q |
the second polynomial parameter |
Details
The function takes on non-zero values in the interval \left( 0,1 \right) . The formula
used to compute the weight function is as follows.
w\left( x \right) = \left( {1 - x} \right)^{p - q} \;x^{q - 1}
Value
The value of the weight function
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.
Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992. Numerical Recipes in C, Cambridge University Press, Cambridge, U.K.
Szego, G., 1939. Orthogonal Polynomials, 23, American Mathematical Society Colloquium Publications, Providence, RI.
Examples
###
### compute the Jacobi G weight function for argument values
### between 0 and 1
### parameter p is 3 and q is 2
###
x <- seq( 0, 1, .01 )
y <- jacobi.g.weight( x, 3, 2 )
Create list of Jacobi matrices from monic recurrence parameters
Description
Return a list of $n$ real symmetric, tri-diagonal matrices which are
the principal minors of the n \times n Jacobi matrix derived from
the monic recurrence parameters, a and b, for orthogonal polynomials.
Usage
jacobi.matrices(r)
Arguments
r |
a data frame containing the parameters |
Value
A list of symmetric, tri-diagnonal matrices
1 |
a |
2 |
a |
...
n |
an |
Author(s)
Frederick Novomestky fnovomes@poly.edu
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
r <- chebyshev.t.recurrences( 5 )
m.r <- monic.polynomial.recurrences( r )
j.m <- jacobi.matrices( m.r )
Inner products of Jacobi polynomials
Description
This function returns a vector with n + 1 elements containing the inner product of
an order k Jacobi polynomial, P_k^{\left( {\alpha ,\beta } \right)} \left( x \right),
with itself (i.e. the norm squared) for orders k = 0,\;1,\; \ldots ,\;n .
Usage
jacobi.p.inner.products(n,alpha,beta)
Arguments
n |
integer value for the highest polynomial order |
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.
h_n = \left\langle {P_n^{\left( {\alpha ,\beta } \right)} |P_n^{\left( {\alpha ,\beta } \right)} } \right\rangle = \frac{{2^{\alpha + \beta + 1} }}
{{2\,n + \alpha + \beta + 1}}\frac{{\Gamma \left( {n + \alpha + 1} \right)\,\Gamma \left( {n + \beta + 1} \right)}}
{{n!\;\Gamma \left( {n + \alpha + \beta + 1} \right)}}.
Value
A vector with n + 1 elements
1 |
inner product of order 0 orthogonal polynomial |
2 |
inner product of order 1 orthogonal polynomial |
...
n+1 |
inner product of order |
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.
Examples
###
### generate the inner product vector for the P Jacobi polynomials of orders 0 to 10
###
h <- jacobi.p.inner.products( 10, 2, 2 )
print( h )
Create list of Jacobi polynomials
Description
This function returns a list with n + 1 elements containing
the order k Jacobi polynomials, P_k^{\left( {\alpha ,\beta } \right)} \left( x \right),
for orders k = 0,\;1,\; \ldots ,\;n.
Usage
jacobi.p.polynomials(n, alpha, beta, normalized=FALSE)
Arguments
n |
integer value for the highest polynomial order |
alpha |
numeric value for the first polynomial parameter |
beta |
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.p.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 |
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.
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 <- jacobi.p.polynomials( 10, 2, 2, normalized=TRUE )
print( normalized.p.list )
###
### gemerate a list of unnormalized Jacobi P polynomials of orders 0 to 10
### first parameter value a is 2 and second parameter value b is 2
###
unnormalized.p.list <- jacobi.p.polynomials( 10, 2, 2, normalized=FALSE )
print( unnormalized.p.list )
Recurrence relations for Jacobi polynomials
Description
This function returns a data frame with n + 1 rows and four named columns containing
the coefficient vectors c, d, e and f of
the recurrence relations for the order k Jacobi polynomial, P_k^{\left( {\alpha ,\beta } \right)} \left( x \right),
and for orders k = 0,\;1,\; \ldots ,\;n.
Usage
jacobi.p.recurrences(n, alpha, beta, normalized=FALSE)
Arguments
n |
integer value for the highest polynomial order |
alpha |
numeric value for the first polynomial parameter |
beta |
numeric value for the second polynomial parameter |
normalized |
boolean value which, if TRUE, returns recurrence relations for normalized polynomials |
Value
A data frame with the recurrence relation parameters.
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.
See Also
jacobi.p.inner.products,
pochhammer
Examples
###
### generate the recurrences data frame for
### the normalized Jacobi P polynomials
### of orders 0 to 10.
### parameter a is 2 and parameter b is 2
###
normalized.r <- jacobi.p.recurrences( 10, 2, 2, normalized=TRUE )
print( normalized.r )
###
### generate the recurrences data frame for
### the unnormalized Jacobi P polynomials
### of orders 0 to 10.
### parameter a is 2 and parameter b is 2
###
unnormalized.r <- jacobi.p.recurrences( 10, 2, 2, normalized=FALSE )
print( unnormalized.r )
Weight function for the Jacobi polynomial
Description
This function returns the value of the weight function for the order k
Jacobi polynomial, P_k^{\left( {\alpha ,\beta } \right)} \left( x \right).
Usage
jacobi.p.weight(x,alpha,beta)
Arguments
x |
the function argument which can be a vector |
alpha |
the first polynomial parameter |
beta |
the second polynomial parameter |
Details
The function takes on non-zero values in the interval \left( -1,1 \right) . The formula
used to compute the weight function is as follows.
w\left( x \right) = \left( {1 - x} \right)^\alpha \;\left( {1 + x} \right)^\beta
Value
The value of the weight function
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.
Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992. Numerical Recipes in C, Cambridge University Press, Cambridge, U.K.
Szego, G., 1939. Orthogonal Polynomials, 23, American Mathematical Society Colloquium Publications, Providence, RI.
Examples
###
### compute the Jacobi P weight function for argument values
### between -1 and 1
###
x <- seq( -1, 1, .01 )
y <- jacobi.p.weight( x, 2, 2 )
Inner products of Laguerre polynomials
Description
This function returns a vector with n + 1 elements containing the inner product of
an order k Laguerre polynomial, L_n \left( x \right),
with itself (i.e. the norm squared) for orders k = 0,\;1,\; \ldots ,\;n .
Usage
laguerre.inner.products(n)
Arguments
n |
integer value for the highest polynomial order |
Details
The formula used to compute the inner products is as follows.
h_n = \left\langle {L_n |L_n } \right\rangle = 1.
Value
A vector with n + 1 elements
1 |
inner product of order 0 orthogonal polynomial |
2 |
inner product of order 1 orthogonal polynomial |
...
n+1 |
inner product of order |
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.
Examples
###
### generate the inner products vector for the
### Laguerre polynomial inner products of orders 0 to 10
###
h <- laguerre.inner.products( 10 )
print( h )
Create list of Laguerre polynomials
Description
This function returns a list with n + 1 elements containing
the order k Laguerre polynomials, L_n \left( x \right),
for orders k = 0,\;1,\; \ldots ,\;n.
Usage
laguerre.polynomials(n, normalized=FALSE)
Arguments
n |
integer value for the highest polynomial order |
normalized |
a boolean value which, if TRUE, returns a list of normalized orthogonal polynomials |
Details
The function laguerre.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 Laguerre polynomial |
2 |
order 1 Laguerre polynomial |
...
n+1 |
order |
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.
See Also
laguerre.recurrences,
orthogonal.polynomials,
orthonormal.polynomials
Examples
###
### gemerate a list of normalized Laguerre polynomials of orders 0 to 10
###
normalized.p.list <- laguerre.polynomials( 10, normalized=TRUE )
print( normalized.p.list )
###
### gemerate a list of unnormalized Laguerre polynomials of orders 0 to 10
###
unnormalized.p.list <- laguerre.polynomials( 10, normalized=FALSE )
print( unnormalized.p.list )
Recurrence relations for Laguerre polynomials
Description
This function returns a data frame with n + 1 rows and four named columns containing
the coefficient vectors c, d, e and f of
the recurrence relations for the order k Laguerre polynomial, L_n \left( x \right),
and for orders k = 0,\;1,\; \ldots ,\;n.
Usage
laguerre.recurrences(n, normalized=FALSE)
Arguments
n |
integer value for the highest polynomial order |
normalized |
boolean value which, if TRUE, returns recurrence relations for normalized polynomials |
Value
A data frame with the recurrence relation parameters.
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.
See Also
Examples
###
### generate the recurrences data frame for
### the normalized Laguerre polynomials
### of orders 0 to 10.
###
normalized.r <- laguerre.recurrences( 10, normalized=TRUE )
print( normalized.r )
###
### generate the recurrences data frame for
### the normalized Laguerre polynomials
### of orders 0 to 10.
###
unnormalized.r <- laguerre.recurrences( 10, normalized=FALSE )
print( unnormalized.r )
Weight function for the Laguerre polynomial
Description
This function returns the value of the weight function for the order k
Laguerre polynomial, L_n \left( x \right).
Usage
laguerre.weight(x)
Arguments
x |
the function argument which can be a vector |
Details
The function takes on non-zero values in the interval \left( 0,\infty \right) . The formula
used to compute the weight function is as follows.
w\left( x \right) = e^{ - x}
Value
The value of the weight function
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.
Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992. Numerical Recipes in C, Cambridge University Press, Cambridge, U.K.
Szego, G., 1939. Orthogonal Polynomials, 23, American Mathematical Society Colloquium Publications, Providence, RI.
Examples
###
### compute the Laguerre weight function for argument values
### between 0 and 3
x <- seq( -0, 3, .01 )
y <- laguerre.weight( x )
plot( x, y )
Inner products of Legendre polynomials
Description
This function returns a vector with n + 1 elements containing the inner product
of an order k Legendre polynomial, P_k \left( x \right),
with itself (i.e. the norm squared) for orders k = 0,\;1,\; \ldots ,\;n .
Usage
legendre.inner.products(n)
Arguments
n |
integer value for the highest polynomial order |
Details
The formula used compute the inner products is as follows.
h_n = \left\langle {P_n |P_n } \right\rangle = \frac{2}
{{2\,n + 1}}.
Value
A vector with n + 1 elements
1 |
inner product of order 0 orthogonal polynomial |
2 |
inner product of order 1 orthogonal polynomial |
...
n+1 |
inner product of order |
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.
See Also
Examples
###
### compute the inner product for the
### Legendre polynomials of orders 0 to 1
###
h <- legendre.inner.products( 10 )
print( h )
Create list of Legendre polynomials
Description
This function returns a list with n + 1 elements containing
the order k Legendre polynomials, P_k \left( x \right),
for orders k = 0,\;1,\; \ldots ,\;n.
Usage
legendre.polynomials(n, normalized=FALSE)
Arguments
n |
integer value for the highest polynomial order |
normalized |
a boolean value which, if TRUE, returns a list of normalized orthogonal polynomials |
Details
The function legendre.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 Legendre polynomial |
2 |
order 1 Legendre polynomial |
...
n+1 |
order |
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.
See Also
legendre.recurrences,
orthogonal.polynomials,
orthonormal.polynomials
Examples
###
### gemerate a list of normalized Laguerre polynomials of orders 0 to 10
###
normalized.p.list <- legendre.polynomials( 10, normalized=TRUE )
print( normalized.p.list )
###
### gemerate a list of unnormalized Laguerre polynomials of orders 0 to 10
###
unnormalized.p.list <- legendre.polynomials( 10, normalized=FALSE )
print( unnormalized.p.list )
Recurrence relations for Legendre polynomials
Description
This function returns a data frame with n + 1 rows and four named columns containing
the coefficient vectors c, d, e and f of
the recurrence relations for the order k Legendre polynomial, P_k \left( x \right),
and for orders k = 0,\;1,\; \ldots ,\;n.
Usage
legendre.recurrences(n, normalized=FALSE)
Arguments
n |
integer value for the highest polynomial order |
normalized |
boolean value which, if TRUE, returns recurrence relations for normalized polynomials |
Value
A data frame with the recurrence relation parameters.
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.
See Also
Examples
###
### generate the recurrences data frame for
### the normalized Legendre polynomials
### of orders 0 to 10.
###
normalized.r <- legendre.recurrences( 10, normalized=TRUE )
print( normalized.r )
###
### generate the recurrences data frame for
### the normalized Legendre polynomials
### of orders 0 to 10.
###
unnormalized.r <- legendre.recurrences( 10, normalized=FALSE )
print( unnormalized.r )
Weight function for the Legendre polynomial
Description
This function returns the value of the weight function for the order k
Legendre polynomial, P_k \left( x \right).
Usage
legendre.weight(x)
Arguments
x |
the function argument which can be a vector |
Details
The function takes on non-zero values in the interval \left( -1,1 \right) . The formula used
to compute the weight function is as follows.
w\left( x \right) = 1
Value
The value of the weight function
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.
Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992. Numerical Recipes in C, Cambridge University Press, Cambridge, U.K.
Szego, G., 1939. Orthogonal Polynomials, 23, American Mathematical Society Colloquium Publications, Providence, RI.
Examples
###
### compute the Legendre weight function for argument values
### between -1 and 1
###
x <- seq( -1, 1, .01 )
y <- legendre.weight( x )
plot( x, y )
Calculate the logarithm of Pochhammer's symbol
Description
lpochhammer returns the value of the natural logarithm of Pochhammer's symbol
calculated as
\ln \left[ {\left( z \right)_n } \right] = \ln \Gamma \left( {z + n} \right) - \ln \Gamma \left( z \right)
where \Gamma \left( z \right) is the Gamma function
Usage
lpochhammer(z, n)
Arguments
z |
argument of the symbol |
n |
integer number of terms in the symbol |
Value
The value of the logarithm of the symbol
Author(s)
Frederick Novomestky fnovomes@poly.edu
See Also
Examples
lpochhammer( pi, 5 )
Create data frame of monic recurrences
Description
This function returns a data frame with parameters required to construct monic orthogonal polynomials based on the standard recurrence relation for the non-monic polynomials. The recurrence relation for monic orthogonal polynomials is as follows.
q_{k + 1} \left( x \right) = \left( {x - a_k } \right)\;q_k \left( x \right) - b_k \;q_{k - 1} \left( x \right)
We require that q_{-1} \left( x \right) = 0 and q_0 \left( x \right) = 1.
The recurrence for non-monic orthogonal polynomials is given by
c_k \;p_{k + 1} \left( x \right) = \left( {d_k + e_k \;x} \right)\;p_k \left( x \right) - f_k \;p_{k - 1} \left( x \right)
We require that p_{-1} \left( x \right) = 0 and p_0 \left( x \right) = 1.
The monic polynomial recurrence parameters, a and b, are related to
the non-monic polynomial parameter vectors c, d, e and f in the following manner.
a_k = - \frac{{d_k }}{{e_k }}
b_k = \frac{{c_{k - 1} \;f_k }}{{e_{k - 1} \;e_k }}
with b_0 = 0.
Usage
monic.polynomial.recurrences(recurrences)
Arguments
recurrences |
the data frame of recurrence parameter vectors c, d, e and f |
Value
A data frame with n + 1 rows and two named columns, a and b.
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.
Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992. Numerical Recipes in C, Cambridge University Press, Cambridge, U.K.
Szego, G., 1939. Orthogonal Polynomials, 23, American Mathematical Society Colloquium Publications, Providence, RI.
See Also
Examples
###
### construct a list of the recurrences for the T Chebyshev polynomials of
### orders 0 to 10
###
r <- chebyshev.t.recurrences( 10, normalized=TRUE )
###
### construct the monic polynomial recurrences from the above list
###
m.r <- monic.polynomial.recurrences( r )
Create list of monic orthogonal polynomials
Description
This function returns a list with n + 1 elements containing
the order k monic polynomials
for orders k = 0,\;1,\; \ldots ,\;n.
Usage
monic.polynomials(monic.recurrences)
Arguments
monic.recurrences |
a data frame containing the parameters a and b |
Value
A list with n + 1 polynomial objects
1 |
order 0 monic orthogonal polynomial |
2 |
order 1 monic orthogonal polynomial |
...
n+1 |
order |
Author(s)
Frederick Novomestky fnovomes@poly.edu
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.
See Also
Examples
###
### generate the recurrences for the T Chebyshev polynomials
### of orders 0 to 10
###
r <- chebyshev.t.recurrences( 10, normalized=TRUE )
###
### get the corresponding monic polynomial recurrences
###
m.r <- monic.polynomial.recurrences( r )
###
### obtain the list of monic polynomials
###
p.list <- monic.polynomials( m.r )
Create orthogonal polynomials
Description
Create list of orthogonal polynomials from the following recurrence relations for
k = 0,\;1,\; \ldots ,\;n.
c_k p_{k+1}\left( x \right) = \left( d_k + e_k x \right) p_k \left( x \right) - f_k p_{k-1} \left( x \right)
We require that p_{-1} \left( x \right) = 0 and p_0 \left( x \right) = 1.
The coefficients are the column vectors {\bf{c}}, {\bf{d}}, {\bf{e}} and {\bf{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 with n + 1 rows and four named columns.
The column names are c, d, e and f.
These columns correspond to the column vectors described above.
Value
A list of n + 1 polynomial objects
1 |
Order 0 orthogonal polynomial |
2 |
Order 1 orthogonal polynomial |
...
n+1 |
Order |
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.
Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992. Numerical Recipes in C, Cambridge University Press, Cambridge, U.K.
Szego, G., 1939. Orthogonal Polynomials, 23, American Mathematical Society Colloquium Publications, Providence, RI.
Examples
###
### generate the recurrence relations for T Chebyshev polynomials of orders 0 to 10
###
r <- chebyshev.t.recurrences( 10, normalized=FALSE )
print( r )
###
### generate the list of orthogonal polynomials
###
p.list <- orthogonal.polynomials( r )
print( p.list )
Create orthonormal polynomials
Description
Create list of orthonormal polynomials from the following recurrence relations for
k = 0,\;1,\; \ldots ,\;n
.
c_k p_{k+1}\left( x \right) = \left( d_k + e_k x \right) p_k \left( x \right) - f_k p_{k-1} \left( x \right)
We require that p_{-1} \left( x \right) = 0 and p_0 \left( x \right) = 1.
The coefficients are the column vectors {\bf{c}}, {\bf{d}}, {\bf{e}} and {\bf{f}}.
Usage
orthonormal.polynomials(recurrences, p.0)
Arguments
recurrences |
a data frame containing the parameters of the orthonormal polynomial recurrence relations |
p.0 |
a polynomial object for the order 0 orthonormal polynomial |
Details
The argument is a data frame with n + 1 rows and four named columns.
The column names are c, d, e and f.
These columns correspond to the column vectors described above.
Value
A list of n + 1polynomial objects
1 |
Order 0 orthonormal polynomial |
2 |
Order 1 orthonormal polynomial |
...
n+1 |
Order |
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.
Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992. Numerical Recipes in C, Cambridge University Press, Cambridge, U.K.
Szego, G., 1939. Orthogonal Polynomials, 23, American Mathematical Society Colloquium Publications, Providence, RI.
Examples
###
### generate a data frame with the recurrences parameters for normalized T Chebyshev
### polynomials of orders 0 to 10
###
r <- chebyshev.t.recurrences( 10, normalized=TRUE )
print( r )
norm <- sqrt( pi )
###
### create the order 0 orthonormal polynomial
###
library("polynom")
p.0 <- polynomial( c( 1 / norm ) )
###
### generate a list of orthonormal polynomial objects
###
p.list <- orthonormal.polynomials( r, p.0 )
print( p.list )
Calculate the value of Pochhammer's symbol
Description
pochhammer returns the value of Pochhammer's symbol calculated as
\left( z \right)_n = z\;\left( {z + 1} \right)\; \ldots \;\left( {z + n - 1} \right) = \frac{{\Gamma \left( {z + n} \right)}}{{\Gamma \left( z \right)}}
where \Gamma \left( z \right) is the Gamma function
Usage
pochhammer(z, n)
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 Pochhammer's symbol
Author(s)
Frederick Novomestky fnovomes@poly.edu
Examples
###
### compute the Pochhamer's symbol fo z equal to 1 and
### n equal to 5
###
pochhammer( 1, 5 )
Create list of polynomial coefficient vectors
Description
This function returns a list with n + 1 elements containing
the vector of coefficients of the order k polynomials
for orders k = 0,\;1,\; \ldots ,\;n.
Each element in the list is a vector.
Usage
polynomial.coefficients(polynomials)
Arguments
polynomials |
list of polynomial objects |
Value
A list of n + 1 polynomial 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 order |
Author(s)
Frederick Novomestky fnovomes@poly.edu
Examples
###
### generate a list of normalized T Chebyshev polynomials
### of orders 0 to 10
###
p.list <- chebyshev.t.polynomials( 10, normalized=TRUE )
###
### obtain the list of coefficients for these polynomials
###
p.coef <- polynomial.coefficients( p.list )
Create list of polynomial derivatives
Description
This function returns a list with n + 1 elements containing polynomial objects
which are the derivatives of the order k polynomials
for orders k = 0,\;1,\; \ldots ,\;n.
Usage
polynomial.derivatives(polynomials)
Arguments
polynomials |
list of polynomial objects |
Details
The polynomial objects in the argument polynomials are as follows
1order 0 polynomial
2order 1 polynomial ...
n+1order
npolynomial
Value
List of n + 1 polynomial objects
1 |
derivative of |
2 |
derivative of |
...
n+1 |
derivative of |
Author(s)
Frederick Novomestky fnovomes@poly.edu
Examples
###
### generate a list of normalized T Chebyshev polynomials of
### orders 0 to 10
###
p.list <- chebyshev.t.polynomials( 10, normalized=TRUE )
###
### generate the corresponding list of polynomial derivatives
###
p.deriv <- polynomial.derivatives( p.list )
Coerce polynomials to functions
Description
This function returns a list with n + 1 elements containing the functions of the
order $k$ polynomials for orders k = 0,\;1,\; \ldots ,\;n
and for the given argument x.
Usage
polynomial.functions(polynomials, ...)
Arguments
polynomials |
a list of polynomial objects |
... |
further arguments to be passed to or from methods |
Details
The function uses the method as.function.polynomial to coerce each polynomial object to a function object.
Value
A list of n + 1 polynomial objects where each element is the function for the polynomial.
1 |
Function for the order 0 polynomial |
2 |
Function for the order 1 polynomial |
...
n+1 |
Function for the order |
Author(s)
Frederick Novomestky fnovomes@poly.edu
Examples
###
### generate a list of T Chebyshev polynomials of
### orders 0 to 10
###
p.list <- chebyshev.t.polynomials( 10, normalized=FALSE )
###
### create the list of functions for each polynomial
###
f.list <- polynomial.functions( p.list )
Create list of polynomial integrals
Description
This function returns a list with n + 1 elements containing polynomial objects
which are the indefinite integrals of the order k polynomials
for orders k = 0,\;1,\; \ldots ,\;n.
Usage
polynomial.integrals(polynomials)
Arguments
polynomials |
list of polynomial objects |
Details
The polynomial objects in the argument polynomials are as follows
1order 0 polynomial
2order 1 polynomial ...
n+1order n polynomial
Value
List of n + 1 polynomial objects
1 |
integral of |
2 |
integral of |
...
n+1 |
integral of |
Author(s)
Frederick Novomestky fnovomes@poly.edu
Examples
###
### generate a list of normalized T Chebyshev polynomials
### of orders 0 to 10
###
p.list <- chebyshev.t.polynomials( 10, normalized=TRUE )
###
### generate the corresponding list of polynomial integrals
###
p.int <- polynomial.integrals( p.list )
Create vector of polynomial orders
Description
This function returns a vector with n elements containing the orders of the polynomials
Usage
polynomial.orders(polynomials)
Arguments
polynomials |
list of $n$ polynomial objects |
Value
A vector of n values
1 |
Order of polynomials[[1]] |
2 |
Order of polynomials[[2]] |
...
n |
Order of polynomials[[n]] |
Author(s)
Frederick Novomestky fnovomes@poly.edu
Examples
###
### generate a list of normalized T Chebyshev polynomials
### of orders 0 to 10
###
p.list <- chebyshev.t.polynomials( 10, normalized=TRUE )
###
### get the vector of polynomial orders
###
p.order <- polynomial.orders( p.list )
Create a list of polynomial linear combinations
Description
This function returns a list with n + 1 elements containing the vector of
linear combinations of the order k polynomials for orders k = 0,\;1,\; \ldots ,\;n.
Each element in the list is a vector.
Usage
polynomial.powers(polynomials)
Arguments
polynomials |
A list of polynomials |
Details
The j-th component in the list is a vector of linear combinations of
the order k polynomials for orders k = 0,\;1,\; \ldots ,\;j - 1 equal to
the monomial x raised to the power j - 1.
Value
A list of n + 1 elements where each element is a vector of linear combinations.
1 |
Linear combination(s) of polynomials up to order 0 |
2 |
Linear combination(s) of polynomials up to order 1 |
...
n+1 |
Linear combination(s) of polynomials up to order |
Author(s)
Frederick Novomestky fnovomes@poly.edu
Examples
###
### generate Legendre polynomials of orders 0 to 10
###
polynomials <- legendre.polynomials( 10 )
###
### generate list of linear combinations of these polynomials
###
alphas <- polynomial.powers( polynomials )
print( alphas )
Create a list of polynomial roots
Description
This function returns a list with n elements containing the roots of
the order $k$ monic orthogonal polynomials for orders k = 0,\;1,\; \ldots ,\;n
using a data frame with the monic polynomial recurrence parameter vectors
\bf{a} and \bf{b}
Usage
polynomial.roots(m.r)
Arguments
m.r |
monic recurrence data frame with parameters a and b |
Details
The parameter m.r is a data frame with $n$+1 rows and two names columns.
The columns which are names a and b correspond to the above referenced vectors.
Function jacobi.matrices is used to create a list of symmetric,
tridiagonal Jacobi matrices from these named columns. The eigenvalues of
the k \times k Jacobi matrix are the roots or zeros of the order $k$
monic orthogonal polynomial.
Value
A list with n elements each of which is a vector of polynomial roots
1 |
roots of the order 1 monic polynomial |
2 |
roots of the order 2 monic polynomial |
...
n |
roots of the order |
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.
See Also
monic.polynomial.recurrences,
jacobi.matrices
Examples
###
### generate the recurrences data frame for
### the normalized Chebyshev polynomials of
### orders 0 to 10
###
r <- chebyshev.t.recurrences( 10, normalized=TRUE )
###
### construct the corresponding monic polynomial
### recurrences
###
m.r <- monic.polynomial.recurrences( r )
###
### obtain the polynomial roots from the monic polynomial
### recurrences
p.roots <- polynomial.roots( m.r )
Create vector of polynomial values
Description
This function returns a list with n + 1 elements containing the values of the
order k polynomials for orders k = 0,\;1,\; \ldots ,\;n
and for the given argument x.
Usage
polynomial.values( polynomials, x )
Arguments
polynomials |
list of polynomial objects |
x |
the argument which can be any numeric object |
Value
A list of n + 1 polynomial objects where each element is the value of the polynomial.
1 |
Value(s) for the order 0 polynomial |
2 |
Value(s) for the order 1 polynomial |
...
n+1 |
Value(s) for the order |
Author(s)
Frederick Novomestky fnovomes@poly.edu
Examples
###
### generate a list of T Chebyshev polynomials of
### orders 0 to 10
###
p.list <- chebyshev.t.polynomials( 10, normalized=FALSE )
x <- seq( -2, 2, .01 )
###
### compute the value of the polynomials for the given range of values in x
###
y <- polynomial.values( p.list, x )
print( y )
Scale values from [a,b] to [u.v]
Description
This function returns a vector of values that have been mapped from the interval [a,b] to the interval [u.v].
Usage
scaleX(x, a = min(x, na.rm = TRUE), b = max(x, na.rm = TRUE), u, v)
Arguments
x |
A numerical vector of values to be mapped into a target interval |
a |
A numerical lower bound for the domain interval with min(x) as the default value |
b |
A numerical upper bound for the domain interval with max(x) as the default value |
u |
A numerical lower bound for the target interval |
v |
A numerical upper bound for the target interval |
Details
Target lower and/or upper bounds can be -\infty and \infty, respectively. This accomodates
finite target intervals, 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 fnovomes@poly.edu, Gregor Gorjanc gregor.gorjanc@bfro-uni-lj.si
References
Seber, G. A. F. (1997) Linear Regression Analysis, New York.
Examples
x <- rnorm( 1000, 0, 10 )
y0 <- scaleX( x, u=0 , v=1 )
y1 <- scaleX( x, u=-1, v=1 )
y2 <- scaleX( x, u=-Inf, v=0 )
y3 <- scaleX( x, u=0, v=Inf )
y4 <- scaleX( x, u=-Inf, v=Inf )
Inner products of shifted Chebyshev polynomials
Description
This function returns a vector with n + 1 elements containing the inner product of
an order k shifted Chebyshev polynomial of the first kind, T_k^* \left( x\right),
with itself (i.e. the norm squared) for orders k = 0,\;1,\; \ldots ,\;n .
Usage
schebyshev.t.inner.products(n)
Arguments
n |
integer value for the highest polynomial order |
Details
The formula used to compute the inner products is as follows.
h_n = \left\langle {T_n^* |T_n^* } \right\rangle = \left\{ {\begin{array}{cc}
{\frac{\pi }
{2}} & {n \ne 0} \\
\pi & {n = 0} \\
\end{array} } \right.
Value
A vector with n + 1 elements
1 |
inner product of order 0 orthogonal polynomial |
2 |
inner product of order 1 orthogonal polynomial |
...
n+1 |
inner product of order |
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., NY.
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.
Examples
###
### generate the inner products vector for the
### shifted T Chebyshev polynomials of orders 0 to 10
###
h <- schebyshev.t.inner.products( 10 )
print( h )
Create list of shifted Chebyshev polynomials
Description
This function returns a list with n + 1 elements containing
the order k shifted Chebyshev polynomials of the first kind, T_k^* \left( x\right),
for orders k = 0,\;1,\; \ldots ,\;n.
Usage
schebyshev.t.polynomials(n, normalized)
Arguments
n |
integer value for the highest polynomial order |
normalized |
a boolean value which, if TRUE, returns a list of normalized orthogonal polynomials |
Details
The function schebyshev.t.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 shifted Chebyshev polynomial |
2 |
order 1 shifted Chebyshev polynomial |
...
n+1 |
order |
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.
See Also
schebyshev.u.recurrences,
orthogonal.polynomials,
orthonormal.polynomials
Examples
###
### gemerate a list of normalized shifted T Chebyshev polynomials of orders 0 to 10
###
normalized.p.list <- schebyshev.t.polynomials( 10, normalized=TRUE )
print( normalized.p.list )
###
### gemerate a list of unnormalized shifted T Chebyshev polynomials of orders 0 to 10
###
unnormalized.p.list <- schebyshev.t.polynomials( 10, normalized=FALSE )
print( unnormalized.p.list )
Recurrence relations for shifted Chebyshev polynomials
Description
This function returns a data frame with n + 1 rows and four named columns containing
the coefficient vectors c, d, e and f of the recurrence relations
for the order k shifted Chebyshev polynomial of the first kind, T_k^* \left( x \right), and
for orders k = 0,\;1,\; \ldots ,\;n.
Usage
schebyshev.t.recurrences(n, normalized)
Arguments
n |
integer value for the highest polynomial order |
normalized |
boolean value which, if TRUE, returns recurrence relations for normalized polynomials |
Value
A data frame with the recurrence relation parameters.
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.
Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992. Numerical Recipes in C, Cambridge University Press, Cambridge, U.K.
Szego, G., 1939. Orthogonal Polynomials, 23, American Mathematical Society Colloquium Publications, Providence, RI.
See Also
Examples
###
### generate the recurrence relations for
### the normalized shifted T Chebyshev polynomials
### of orders 0 to 10
###
normalized.r <- schebyshev.t.recurrences( 10, normalized=TRUE )
print( normalized.r )
###
### generate the recurrence relations for
### the unnormalized shifted T Chebyshev polynomials
### of orders 0 to 10
###
unnormalized.r <- schebyshev.t.recurrences( 10, normalized=FALSE )
print( unnormalized.r )
Weight function for the shifted Chebyshev polynomial
Description
This function returns the value of the weight function for the order k
shifted Chebyshev polynomial of the first kind, T_k^* \left( x \right) .
Usage
schebyshev.t.weight(x)
Arguments
x |
the function argument which can be a vector |
Details
The function takes on non-zero values in the interval \left( 0,1 \right) . The formula
used to compute the weight function is as follows.
w\left( x \right) = \frac{1}{{\sqrt {x - x^2 } }}
Value
The value of the weight function
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.
Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992. Numerical Recipes in C, Cambridge University Press, Cambridge, U.K.
Szego, G., 1939. Orthogonal Polynomials, 23, American Mathematical Society Colloquium Publications, Providence, RI.
Examples
###
### compute the shifted T Chebyshev weight function for argument values
### between 0 and 1
x <- seq( 0, 1, .01 )
y <- schebyshev.t.weight( x )
plot( x, y )
Inner products of shifted Chebyshev polynomials
Description
This function returns a vector with n + 1 elements containing the inner product of
an order k shifted Chebyshev polynomial of the second kind, U_k^* \left( x\right),
with itself (i.e. the norm squared) for orders k = 0,\;1,\; \ldots ,\;n .
Usage
schebyshev.u.inner.products(n)
Arguments
n |
integer value for the highest polynomial order |
Details
The formula used to compute the inner products is as follows.
h_n = \left\langle {U_n^* |U_n^* } \right\rangle = \frac{\pi }{8}.
Value
A vector with n + 1 elements
1 |
inner product of order 0 orthogonal polynomial |
2 |
inner product of order 1 orthogonal polynomial |
...
n+1 |
inner product of order |
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.
Examples
h <- schebyshev.u.inner.products( 10 )
Create list of shifted Chebyshev polynomials
Description
This function returns a list with n + 1 elements containing
the order k shifted Chebyshev polynomials of the second kind, U_k^* \left( x\right),
for orders k = 0,\;1,\; \ldots ,\;n.
Usage
schebyshev.u.polynomials(n, normalized)
Arguments
n |
integer value for highest polynomial order |
normalized |
a boolean value which, if TRUE, returns a list of normalized orthogonal polynomials |
Details
The function schebyshev.u.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 shifted Chebyshev polynomial |
2 |
order 1 shifted Chebyshev polynomial |
...
n+1 |
order |
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.
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 <- schebyshev.u.polynomials( 10, normalized=TRUE )
print( normalized.p.list )
###
### gemerate a list of unnormalized shifted U Chebyshev polynomials of orders 0 to 10
###
unnormalized.p.list <- schebyshev.u.polynomials( 10, normalized=FALSE )
print( unnormalized.p.list )
Recurrence relations for shifted Chebyshev polynomials
Description
This function returns a data frame with n + 1 rows and four named columns containing
the coefficient vectors c, d, e and f of the recurrence relations
for the order k shifted Chebyshev polynomial of the second kind, U_k^* \left( x \right), and
for orders k = 0,\;1,\; \ldots ,\;n.
Usage
schebyshev.u.recurrences(n, normalized)
Arguments
n |
integer value for the highest polynomial order |
normalized |
boolean value which, if TRUE, returns recurrence relations for normalized polynomials |
Value
A data frame with the recurrence relation parameters.
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.
Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992. Numerical Recipes in C, Cambridge University Press, Cambridge, U.K.
Szego, G., 1939. Orthogonal Polynomials, 23, American Mathematical Society Colloquium Publications, Providence, RI.
See Also
Examples
###
### generate the recurrence relations for
### the normalized shifted U Chebyshev polynomials
### of orders 0 to 10
###
normalized.r <- schebyshev.u.recurrences( 10, normalized=TRUE )
print( normalized.r )
###
### generate the recurrence relations for
### the unnormalized shifted T Chebyshev polynomials
### of orders 0 to 10
unnormalized.r <- schebyshev.u.recurrences( 10, normalized=FALSE )
print( unnormalized.r )
Weight function for the shifted Chebyshev polynomial
Description
This function returns the value of the weight function for the order k
shifted Chebyshev polynomial of the second kind, U_k^* \left( x \right) .
Usage
schebyshev.u.weight(x)
Arguments
x |
the function argument which can be a vector |
Details
The function takes on non-zero values in the interval \left( 0,1 \right) . The formula
used to compute the weight function is as follows.
w\left( x \right) = \sqrt {x - x^2 }
Value
The value of the weight function.
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.
Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992. Numerical Recipes in C, Cambridge University Press, Cambridge, U.K.
Szego, G., 1939. Orthogonal Polynomials, 23, American Mathematical Society Colloquium Publications, Providence, RI.
Examples
###
### compute the shifted U Chebyshev weight function for argument values
### between 0 and 1
###
x <- seq( 0, 1, .01 )
y <- schebyshev.u.weight( x )
plot( x, y )
Inner products of shifted Legendre polynomials
Description
This function returns a vector with n + 1 elements containing the inner product of
an order k shifted Legendre polynomial, P_k^* \left( x \right),
with itself (i.e. the norm squared) for orders k = 0,\;1,\; \ldots ,\;n .
Usage
slegendre.inner.products(n)
Arguments
n |
integer value for the highest polynomial order |
Details
The formula used to compute the inner products is as follows.
h_n = \left\langle {P_n^* |P_n^* } \right\rangle = \frac{1}{{2\,n + 1}}.
Value
A vector with $n$+1 elements
1 |
inner product of order 0 orthogonal polynomial |
2 |
inner product of order 1 orthogonal polynomial |
...
n+1 |
inner product of order |
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.
Examples
###
### compute the inner products vector for the
### shifted Legendre polynomials of orders 0 to 10
###
h <- slegendre.inner.products( 10 )
print( h )
Create list of shifted Legendre polynomials
Description
This function returns a list with n + 1 elements containing
the order k shifted Legendre polynomials, P_k^* \left( x \right),
for orders k = 0,\;1,\; \ldots ,\;n.
Usage
slegendre.polynomials(n, normalized=FALSE)
Arguments
n |
integer value for the highest polynomial order |
normalized |
a boolean value which, if TRUE, returns a list of normalized orthogonal polynomials |
Details
The function slegendre.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 shifted Legendre polynomial |
2 |
order 1 shifted Legendre polynomial |
...
n+1 |
order |
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.
See Also
slegendre.recurrences,
orthogonal.polynomials,
orthonormal.polynomials
Examples
###
### gemerate a list of normalized shifted Legendre polynomials of orders 0 to 10
###
normalized.p.list <- slegendre.polynomials( 10, normalized=TRUE )
print( normalized.p.list )
###
### gemerate a list of unnormalized shifted Legendre polynomials of orders 0 to 10
###
unnormalized.p.list <- slegendre.polynomials( 10, normalized=FALSE )
print( unnormalized.p.list )
Recurrence relations for shifted Legendre polynomials
Description
This function returns a data frame with n + 1 rows and four named columns containing
the coefficient vectors c, d, e and f of the recurrence relations for
the order k shifted Legendre polynomial, P_k^* \left( x \right), and
for orders k = 0,\;1,\; \ldots ,\;n.
Usage
slegendre.recurrences(n, normalized=FALSE)
Arguments
n |
integer value for the highest polynomial order |
normalized |
boolean value which, if TRUE, returns recurrence relations for normalized polynomials |
Value
A data frame with the recurrence relation parameters.
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.
Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992. Numerical Recipes in C, Cambridge University Press, Cambridge, U.K.
Szego, G., 1939. Orthogonal Polynomials, 23, American Mathematical Society Colloquium Publications, Providence, RI.
See Also
Examples
###
### generate the recurrence relations for normalized shifted Legendre polynomials
### of orders 0 to 10
###
normalized.r <- slegendre.recurrences( 10, normalized=TRUE )
print( normalized.r )
###
### generate the recurrence relations for normalized shifted Legendre polynomials
### of orders 0 to 10
###
unnormalized.r <- slegendre.recurrences( 10, normalized=FALSE )
print( unnormalized.r )
Weight function for the shifted Legendre polynomial
Description
This function returns the value of the weight function for the order k
shifted Legendre polynomial, P_k^* \left( x \right).
Usage
slegendre.weight(x)
Arguments
x |
the function argument which can be a vector |
Details
The function takes on non-zero values in the interval \left( 0,1 \right) . The formula
used to compute the weight function is as follows.
w\left( x \right) = 1
Value
The value of the weight function
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.
Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992. Numerical Recipes in C, Cambridge University Press, Cambridge, U.K.
Szego, G., 1939. Orthogonal Polynomials, 23, American Mathematical Society Colloquium Publications, Providence, RI.
Examples
###
### compute the shifted Legendre weight function for argument values
### between 0 and 1
###
x <- seq( 0, 1, .01 )
y <- slegendre.weight( x )
Inner products of spherical polynomials
Description
This function returns a vector with n + 1 elements containing the inner product
of an order k spherical polynomial, P_k \left( x \right),
with itself (i.e. the norm squared) for orders k = 0,\;1,\; \ldots ,\;n.
Usage
spherical.inner.products(n)
Arguments
n |
integer value for the highest polynomial order |
Details
The formula used to compute the inner products of the spherical orthogonal polynomials is the same as that used for the Legendre orthogonal polynomials.
Value
A vector with n + 1 elements
1 |
inner product of order 0 orthogonal polynomial |
2 |
inner product of order 1 orthogonal polynomial |
...
n+1 |
inner product of order |
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.
See Also
Examples
###
### generate the inner products vector for the spherical polynomals
### of orders 0 to 10.
###
h <- spherical.inner.products( 10 )
print( h )
Create list of spherical polynomials
Description
This function returns a list with n + 1 elements containing
the order k spherical polynomials, P_k \left( x \right),
for orders k = 0,\;1,\; \ldots ,\;n.
Usage
spherical.polynomials(n, normalized=FALSE)
Arguments
n |
integer value for the highest polynomial order |
normalized |
a boolean value which, if TRUE, returns a list of normalized orthogonal polynomials |
Details
The function spherical.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 spherical polynomial |
2 |
order 1 spherical polynomial |
...
n+1 |
order |
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.
See Also
spherical.recurrences,
orthogonal.polynomials,
orthonormal.polynomials
Examples
###
### generate a list of spherical orthonormal polynomials of orders 0 to 10
###
normalized.p.list <- spherical.polynomials( 10, normalized=TRUE )
print( normalized.p.list )
###
### generate a list of spherical orthogonal polynomials of orders 0 to 10
###
unnormalized.p.list <- spherical.polynomials( 10, normalized=FALSE )
print( unnormalized.p.list )
Recurrence relations for spherical polynomials
Description
This function returns a data frame with n + 1 rows and four named columns containing
the coefficient vectors c, d, e and f of
the recurrence relations for the order k spherical polynomial, P_k \left( x \right),
and for orders k = 0,\;1,\; \ldots ,\;n.
Usage
spherical.recurrences(n, normalized=FALSE)
Arguments
n |
integer value for the highest polynomial order |
normalized |
boolean value which, if TRUE, returns recurrence relations for normalized polynomials |
Value
A data frame with the recurrence relation parameters.
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.
See Also
Examples
###
### generate the recurrence relations for
### the normalized spherical polynomials
### of orders 0 to 10
###
normalized.r <- spherical.recurrences( 10, normalized=TRUE )
print( normalized.r )
###
### generate the recurrence relations for
### the unnormalized spherical polynomials
### of orders 0 to 10
###
unnormalized.r <- spherical.recurrences( 10, normalized=FALSE )
print( unnormalized.r )
Weight function for the spherical polynomial
Description
This function returns the value of the weight function for the order k spherical polynomial, P_k \left( x \right).
Usage
spherical.weight(x)
Arguments
x |
the function argument which can be a vector or matrix |
Details
The function takes on non-zero values in the interval \left( -1,1 \right) . The formula used
to compute the weight function is as follows.
w\left( x \right) = 1
Value
The value of the weight function
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.
Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992. Numerical Recipes in C, Cambridge University Press, Cambridge, U.K.
Szego, G., 1939. Orthogonal Polynomials, 23, American Mathematical Society Colloquium Publications, Providence, RI.
Examples
###
### compute the spherical weight function for a sequence of values between -2 and 2
###
x <- seq( -2, 2, .01 )
y <- spherical.weight( x )
plot( x, y )
Inner products of ultraspherical polynomials
Description
This function returns a vector with n + 1 elements containing the inner product of
an order k ultraspherical polynomial, C_k^{\left( \alpha \right)} \left( x \right),
with itself (i.e. the norm squared) for orders k = 0,\;1,\; \ldots ,\;n .
Usage
ultraspherical.inner.products(n,alpha)
Arguments
n |
integer value for the highest polynomial order |
alpha |
numeric value for the polynomial parameter |
Details
This function uses the same formula as the function gegenbauer.inner.products.
Value
A vector with n + 1 elements
1 |
inner product of order 0 orthogonal polynomial |
2 |
inner product of order 1 orthogonal polynomial |
...
n+1 |
inner product of order |
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., NY.
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.
See Also
Examples
###
### generate the inner products vector for the
### ultraspherical polynomials of orders 0 to 10.
### the polynomial parameter is 1.0
###
h <- ultraspherical.inner.products( 10, 1 )
print( h )
Create list of ultraspherical polynomials
Description
This function returns a list with n + 1 elements containing
the order k ultraspherical polynomials, C_k^{\left( \alpha \right)} \left( x \right),
for orders k = 0,\;1,\; \ldots ,\;n.
Usage
ultraspherical.polynomials(n, alpha, normalized=FALSE)
Arguments
n |
integer value for the highest polynomial order |
alpha |
polynomial parameter |
normalized |
a boolean value which, if TRUE, returns a list of normalized orthogonal polynomials |
Details
The function ultraspherical.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 ultraspherical polynomial |
2 |
order 1 ultraspherical polynomial |
...
n+1 |
order |
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.
See Also
gegenbauer.recurrences,
orthogonal.polynomials,
orthonormal.polynomials
Examples
###
### gemerate a list of normalized ultra spherical polynomials
### of orders 0 to 10
###
normalized.p.list <- ultraspherical.polynomials( 10, 1, normalized=TRUE )
print( normalized.p.list )
###
### gemerate a list of unnormalized ultra spherical polynomials
### of orders 0 to 10
###
unnormalized.p.list <- ultraspherical.polynomials( 10, 1, normalized=FALSE )
print( unnormalized.p.list )
Recurrence relations for ultraspherical polynomials
Description
This function returns a data frame with n + 1 rows and four named columns containing
the coefficient vectors c, d, e and f of
the recurrence relations for the order k ultraspherical polynomial, C_k^{\left( \alpha \right)} \left( x \right),
and for orders k = 0,\;1,\; \ldots ,\;n.
Usage
ultraspherical.recurrences(n, alpha, normalized=FALSE)
Arguments
n |
integer value for the highest polynomial order |
alpha |
numeric value for the polynomial parameter |
normalized |
boolean value which, if TRUE, returns recurrence relations for normalized polynomials |
Value
A data frame with the recurrence relation parameters.
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.
See Also
Examples
###
### generate the recurrence relations for
### the normalized ultraspherical polynomials
### of orders 0 to 10
### polynomial parameter value is 1.0
###
normalized.r <- ultraspherical.recurrences( 10, 1, normalized=TRUE )
print( normalized.r )
###
### generate the recurrence relations for
### the normalized ultraspherical polynomials
### of orders 0 to 10
### polynomial parameter value is 1.0
###
unnormalized.r <- ultraspherical.recurrences( 10, 1, normalized=FALSE )
print( unnormalized.r )
Weight function for the ultraspherical polynomial
Description
This function returns the value of the weight function for the order k ultraspherical polynomial, C_k^{\left( \alpha \right)} \left( x \right).
Usage
ultraspherical.weight(x,alpha)
Arguments
x |
the function argument which can be a vector |
alpha |
polynomial parameter |
Details
The function takes on non-zero values in the interval \left( -1,1 \right) . The formula used
to compute the weight function is as follows.
w\left( x \right) = \left( {1 - x^2 } \right)^{\alpha - 0.5}
Value
The value of the weight function
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.
Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992. Numerical Recipes in C, Cambridge University Press, Cambridge, U.K.
Szego, G., 1939. Orthogonal Polynomials, 23, American Mathematical Society Colloquium Publications, Providence, RI.
Examples
###
### compute the ultraspherical weight function for arguments between -2 and 2
### polynomial parameter is 1.0
###
x <- seq( -2, 2, .01 )
y <- ultraspherical.weight( x, 1 )
plot( x, y )