Package 'orthopolynom'

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

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

...

Author(s)

Frederick Novomestky [email protected]

References

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

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.inner.products

Examples

###
### generate the inner products vector for the
### C Chebyshev polynomials of orders 0 to 10
###
h <- chebyshev.c.inner.products( 10 )
print( h )

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

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

...

Author(s)

Frederick Novomestky [email protected]

References

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

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 )

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

Value

A data frame with the recurrence relation parameters.

Author(s)

Frederick Novomestky [email protected]

References

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

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.inner.products

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 )

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

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 [email protected]

References

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

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 )

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

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

...

Author(s)

Frederick Novomestky [email protected]

References

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

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 )

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

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

...

Author(s)

Frederick Novomestky [email protected]

References

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

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 )

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

Value

A data frame with the recurrence relation parameters.

Author(s)

Frederick Novomestky [email protected]

References

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

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.inner.products

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 )

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

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 [email protected]

References

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

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 )

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

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

...

Author(s)

Frederick Novomestky [email protected]

References

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

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 )

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

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

...

Author(s)

Frederick Novomestky [email protected]

References

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

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 )

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

Value

A data frame with the recurrence relation parameters.

Author(s)

Frederick Novomestky [email protected]

References

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

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.t.inner.products

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 )

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

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 [email protected]

References

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

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 )

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

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

...

Author(s)

Frederick Novomestky [email protected]

References

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

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 )

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

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

...

Author(s)

Frederick Novomestky [email protected]

References

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

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 )

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

Value

A data frame with the recurrence relation parameters.

Author(s)

Frederick Novomestky [email protected]

References

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

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.inner.products

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 )

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

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 [email protected]

References

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

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 )

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

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

...

Author(s)

Frederick Novomestky [email protected]

References

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

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

ultraspherical.inner.products

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 )

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

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

...

Author(s)

Frederick Novomestky [email protected]

References

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

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 )

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

Value

A data frame with the recurrence relation parameters.

Author(s)

Frederick Novomestky [email protected]

References

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

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.inner.products

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 )

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

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 [email protected]

References

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

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 )

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

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

...

Author(s)

Frederick Novomestky [email protected]

References

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

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 )

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

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

...

Author(s)

Frederick Novomestky [email protected]

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 )

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

Details

The parameter $\mu$ must be greater than -0.5.

Value

A data frame with the recurrence relation parameters.

Author(s)

Frederick Novomestky [email protected]

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.inner.products

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 )

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

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 [email protected]

References

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

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 )

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

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

...

Author(s)

Frederick Novomestky [email protected]

References

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

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 )

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

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

...

Author(s)

Frederick Novomestky [email protected]

References

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

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 )

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

Value

A data frame with the recurrence relation parameters.

Author(s)

Frederick Novomestky [email protected]

References

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

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.inner.products

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 )

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

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 [email protected]

References

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

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 )

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

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

...

Author(s)

Frederick Novomestky [email protected]

References

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

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 )

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

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

...

Author(s)

Frederick Novomestky [email protected]

References

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

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 )

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

Value

A data frame with the recurrence relation parameters.

Author(s)

Frederick Novomestky [email protected]

References

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

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.inner.products,

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 )

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

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 [email protected]

References

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

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 )

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

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

...

Author(s)

Frederick Novomestky [email protected]

References

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

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 )

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

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

...

Author(s)

Frederick Novomestky [email protected]

References

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

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 )

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

Value

A data frame with the recurrence relation parameters.

Author(s)

Frederick Novomestky [email protected]

References

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

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.inner.products

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 )

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

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 [email protected]

References

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

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 )

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

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

...

Author(s)

Frederick Novomestky [email protected]

References

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

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 )

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

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

...

Author(s)

Frederick Novomestky [email protected]

References

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

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 )

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

Value

A data frame with the recurrence relation parameters.

Author(s)

Frederick Novomestky [email protected]

References

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

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 )

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

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 [email protected]

References

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

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 )

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

Value

A list of symmetric, tri-diagnonal matrices

...

Author(s)

Frederick Novomestky [email protected]

References

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

Examples

r <- chebyshev.t.recurrences( 5 )
m.r <- monic.polynomial.recurrences( r )
j.m <- jacobi.matrices( m.r )

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

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

...

Author(s)

Frederick Novomestky [email protected]

References

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

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 )

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

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

...

Author(s)

Frederick Novomestky [email protected]

References

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

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 )

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

Value

A data frame with the recurrence relation parameters.

Author(s)

Frederick Novomestky [email protected]

References

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

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 )

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

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 [email protected]

References

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

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 )

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

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

...

Author(s)

Frederick Novomestky [email protected]

References

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

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 )

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

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

...

Author(s)

Frederick Novomestky [email protected]

References

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

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 )

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 $k = 0,\;1,\; \ldots ,\;n$.