Package 'gaussquad'

Perform Gauss Chebyshev quadrature

Description

This function evaluates the integral of the given function between the lower and upper limits using the weight and abscissa values specified in the rule data frame. The quadrature formula uses the weight function for Chebyshev C polynomials.

Usage

chebyshev.c.quadrature(functn, rule, lower = -2, upper = 2, 
weighted = TRUE, ...)

Arguments

functn	an R function which should take a numeric argument x and possibly some parameters. The function returns a numerical vector value for the given argument x.
rule	a data frame containing the order $n$ Chebyshev quadrature rule
lower	numeric value for the lower limit of the integral with a default value of -2
upper	numeric value for the upper limit of the integral with a default value of +2
weighted	boolean value which if true causes the Chebyshev weight function to be included in the integrand
...	other arguments passed to the give function

Details

The rule argument corresponds to an order $n$ Chebyshev polynomial, weight function and interval $\left[ { - 2,2} \right]$. The lower and upper limits of the integral must be finite.

Value

The value of definite integral evaluated using Gauss Chebyshev quadrature

Author(s)

Frederick Novomestky [email protected]

References

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

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

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

See Also

chebyshev.c.quadrature.rules

Examples

###
### this example evaluates the quadrature function for
### the Chebyshev C polynomials.  it computes the integral
### of the product for all pairs of orthogonal polynomials
### from order 0 to order 16.  the results are compared to
### the diagonal matrix of the inner products for the
### polynomials.  it also computes the integral of the product
### of all pairs of orthonormal polynomials from order 0
### to order 16.  the resultant matrix should be an identity matrix
###
###
### set the value for the maximum polynomial order
###
    n <- 16
###
### maximum order plus 1
###
    np1 <- n + 1
###
### function to construct the polynomial products by column
###
by.column.products <- function( c, p.list, p.p.list )
{
###
### function to construct the polynomial products by row
###
    by.row.products <- function( r, c, p.list )
    {
        row.column.product <- p.list[[r]] * p.list[[c]]
        return (row.column.product )
    }
    np1 <- length( p.list )
    row.list <- lapply( 1:np1, by.row.products, c, p.list )
    return( row.list )
}
###
### function construct the polynomial functions by column
###
by.column.functions <- function( c, p.p.products )
{
###
### function to construct the polynomial functions by row
###
    by.row.functions <- function( r, c, p.p.products )
    {
        row.column.function <- as.function( p.p.products[[r]][[c]] )
        return( row.column.function )
    }
    np1 <- length( p.p.products[[1]] )
    row.list <- lapply( 1:np1, by.row.functions, c, p.p.products )
    return( row.list )
}
###
### function to compute the integral of the polynomials by column
###
by.column.integrals <- function( c, p.p.functions )
{
###
### function to compute the integral of the polynomials by row
###
    by.row.integrals <- function( r, c, p.p.functions )
    {
        row.column.integral <- chebyshev.c.quadrature(
            p.p.functions[[r]][[c]], order.np1.rule )
        return( row.column.integral )
    }
    np1 <- length( p.p.functions[[1]] )
    row.vector <- sapply( 1:np1, by.row.integrals, c, p.p.functions )
    return( row.vector )
}
###
### construct a list of the Chebyshev C orthogonal polynomials
###
p.list <- chebyshev.c.polynomials( n )
###
### construct the two dimensional list of pair products
### of polynomials
###
p.p.products <- lapply( 1:np1, by.column.products, p.list )
###
### compute the two dimensional list of functions
### corresponding to the polynomial products in
### the two dimensional list p.p.products
###
p.p.functions <- lapply( 1:np1, by.column.functions, p.p.products )
###
### get the rule table for the order np1 polynomial
###
rules <- chebyshev.c.quadrature.rules( np1 )
order.np1.rule <- rules[[np1]]
###
### construct the square symmetric matrix containing
### the definite integrals over the default limits
### corresponding to the two dimensional list of
### polynomial functions
###
p.p.integrals <- sapply( 1:np1, by.column.integrals, p.p.functions )
###
### construct the diagonal matrix with the inner products
### of the orthogonal polynomials on the diagonal
###
p.p.inner.products <- diag( chebyshev.c.inner.products( n ) )
print( "Integral of cross products for the orthogonal polynomials " )
print( apply( p.p.integrals, 2, round, digits=5 ) )
print( apply( p.p.inner.products, 2, round, digits=5 ) )
###
### construct a list of the Chebyshev C orthonormal polynomials
###
p.list <- chebyshev.c.polynomials( n, TRUE )
###
### construct the two dimensional list of pair products
### of polynomials
###
p.p.products <- lapply( 1:np1, by.column.products, p.list )
###
### compute the two dimensional list of functions
### corresponding to the polynomial products in
### the two dimensional list p.p.products
###
p.p.functions <- lapply( 1:np1, by.column.functions, p.p.products )
###
### get the rule table for the order np1 polynomial
###
rules <- chebyshev.c.quadrature.rules( np1, TRUE )
order.np1.rule <- rules[[np1]]
###
### construct the square symmetric matrix containing
### the definite integrals over the default limits
### corresponding to the two dimensional list of
### polynomial functions
###
p.p.integrals <- sapply( 1:np1, by.column.integrals, p.p.functions )
###
### display the matrix of integrals
###
print( "Integral of cross products for the orthonormal polynomials " )
print(apply( p.p.integrals, 2, round, digits=5 ) )

---

Create list of Chebyshev quadrature rules

Description

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

Usage

chebyshev.c.quadrature.rules(n,normalized=FALSE)

Arguments

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

Details

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

Value

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

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

...

n	Quadrature rule data frame for the order $n$ Chebyshev polynomial

Author(s)

Frederick Novomestky [email protected]

References

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

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

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

See Also

quadrature.rules, chebyshev.c.quadrature

Examples

###
### construct the list of quadrature rules for
### the Chebyshev orthogonal polynomials
### of orders 1 to 5
###
orthogonal.rules <- chebyshev.c.quadrature.rules( 5 )
print( orthogonal.rules )
###
### construct the list of quadrature rules for
### the Chebyshev orthonormal polynomials
### of orders 1 to 5
###
orthonormal.rules <- chebyshev.c.quadrature.rules( 5, TRUE )
print( orthonormal.rules )

---

Perform Gauss Chebyshev quadrature

Description

This function evaluates the integral of the given function between the lower and upper limits using the weight and abscissa values specified in the rule data frame. The quadrature formula uses the weight function for Chebyshev S polynomials.

Usage

chebyshev.s.quadrature(functn, rule, lower = -2, upper = 2, 
weighted = TRUE, ...)

Arguments

functn	an R function which should take a numeric argument x and possibly some parameters. The function returns a numerical vector value for the given argument x.
rule	a data frame containing the order $n$ Chebyshev quadrature rule
lower	numeric value for the lower limit of the integral with a default value of -2
upper	numeric value for the upper limit of the integral with a default value of +2
weighted	boolean value which if true causes the Chebyshev weight function to be included in the integrand
...	other arguments passed to the give function

Details

The rule argument corresponds to an order $n$ Chebyshev polynomial, weight function and interval $\left[ { - 2,2} \right]$. The lower and upper limits of the integral must be finite.

Value

The value of definite integral evaluated using Gauss Chebyshev quadrature

Author(s)

Frederick Novomestky [email protected]

References

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

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

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

See Also

chebyshev.s.quadrature.rules

Examples

###
### this example evaluates the quadrature function for
### the Chebyshev S polynomials.  it computes the integral
### of the product for all pairs of orthogonal polynomials
### from order 0 to order 16.  the results are compared to
### the diagonal matrix of the inner products for the
### polynomials.  it also computes the integral of the product
### of all pairs of orthonormal polynomials from order 0
### to order 16.  the resultant matrix should be an identity matrix
###
###
### set the value for the maximum polynomial order
###
    n <- 16
###
### maximum order plus 1
###
    np1 <- n + 1
###
### function to construct the polynomial products by column
###
by.column.products <- function( c, p.list, p.p.list )
{
###
### function to construct the polynomial products by row
###
    by.row.products <- function( r, c, p.list )
    {
        row.column.product <- p.list[[r]] * p.list[[c]]
        return (row.column.product )
    }
    np1 <- length( p.list )
    row.list <- lapply( 1:np1, by.row.products, c, p.list )
    return( row.list )
}
###
### function construct the polynomial functions by column
###
by.column.functions <- function( c, p.p.products )
{
###
### function to construct the polynomial functions by row
###
    by.row.functions <- function( r, c, p.p.products )
    {
        row.column.function <- as.function( p.p.products[[r]][[c]] )
        return( row.column.function )
    }
    np1 <- length( p.p.products[[1]] )
    row.list <- lapply( 1:np1, by.row.functions, c, p.p.products )
    return( row.list )
}
###
### function to compute the integral of the polynomials by column
###
by.column.integrals <- function( c, p.p.functions )
{
###
### function to compute the integral of the polynomials by row
###
    by.row.integrals <- function( r, c, p.p.functions )
    {
        row.column.integral <- chebyshev.s.quadrature(
            p.p.functions[[r]][[c]], order.np1.rule )
        return( row.column.integral )
    }
    np1 <- length( p.p.functions[[1]] )
    row.vector <- sapply( 1:np1, by.row.integrals, c, p.p.functions )
    return( row.vector )
}
###
### construct a list of the Chebyshev S orthogonal polynomials
###
p.list <- chebyshev.s.polynomials( n )
###
### construct the two dimensional list of pair products
### of polynomials
###
p.p.products <- lapply( 1:np1, by.column.products, p.list )
###
### compute the two dimensional list of functions
### corresponding to the polynomial products in
### the two dimensional list p.p.products
###
p.p.functions <- lapply( 1:np1, by.column.functions, p.p.products )
###
### get the rule table for the order np1 polynomial
###
rules <- chebyshev.s.quadrature.rules( np1 )
order.np1.rule <- rules[[np1]]
###
### construct the square symmetric matrix containing
### the definite integrals over the default limits
### corresponding to the two dimensional list of
### polynomial functions
###
p.p.integrals <- sapply( 1:np1, by.column.integrals, p.p.functions )
###
### construct the diagonal matrix with the inner products
### of the orthogonal polynomials on the diagonal
###
p.p.inner.products <- diag( chebyshev.s.inner.products( n ) )
print( "Integral of cross products for the orthogonal polynomials " )
print( apply( p.p.integrals, 2, round, digits=5 ) )
print( apply( p.p.inner.products, 2, round, digits=5 ) )
###
### construct a list of the Chebyshev S orthonormal polynomials
###
p.list <- chebyshev.s.polynomials( n, TRUE )
###
### construct the two dimensional list of pair products
### of polynomials
###
p.p.products <- lapply( 1:np1, by.column.products, p.list )
###
### compute the two dimensional list of functions
### corresponding to the polynomial products in
### the two dimensional list p.p.products
###
p.p.functions <- lapply( 1:np1, by.column.functions, p.p.products )
###
### get the rule table for the order np1 polynomial
###
rules <- chebyshev.s.quadrature.rules( np1, TRUE )
order.np1.rule <- rules[[np1]]
###
### construct the square symmetric matrix containing
### the definite integrals over the default limits
### corresponding to the two dimensional list of
### polynomial functions
###
p.p.integrals <- sapply( 1:np1, by.column.integrals, p.p.functions )
###
### display the matrix of integrals
###
print( "Integral of cross products for the orthonormal polynomials " )
print(apply( p.p.integrals, 2, round, digits=5 ) )

---

Create list of Chebyshev quadrature rules

Description

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

Usage

chebyshev.s.quadrature.rules(n,normalized=FALSE)

Arguments

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

Details

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

Value

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

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

...

n	Quadrature rule data frame for the order $n$ Chebyshev polynomial

Author(s)

Frederick Novomestky [email protected]

References

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

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

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

See Also

quadrature.rules, chebyshev.s.quadrature

Examples

###
### generate a list of quadrature rules for
### the Chebyshev S orthogonal polynomials
### for orders 1 to 5
###
orthogonal.rules <- chebyshev.s.quadrature.rules( 5 )
print( orthogonal.rules )
###
### generate a list of quadrature rules for
### the Chebyshev S orthogonal polynomials
### for orders 1 to 5
###
orthonormal.rules <- chebyshev.s.quadrature.rules( 5, TRUE )
print( orthonormal.rules )

---

Perform Gauss Chebyshev quadrature

Description

This function evaluates the integral of the given function between the lower and upper limits using the weight and abscissa values specified in the rule data frame. The quadrature formula uses the weight function for Chebyshev T polynomials.

Usage

chebyshev.t.quadrature(functn, rule, lower = -1, upper = 1, 
weighted = TRUE, ...)

Arguments

functn	an R function which should take a numeric argument x and possibly some parameters. The function returns a numerical vector value for the given argument x.
rule	a data frame containing the order $n$ Chebyshev quadrature rule
lower	numeric value for the lower limit of the integral with a default value of -1
upper	numeric value for the upper limit of the integral with a default value of +1
weighted	boolean value which if true causes the Chebyshev weight function to be included in the integrand
...	other arguments passed to the given function

Details

The rule argument corresponds to an order $n$ Chebyshev polynomial, weight function and interval $\left[ { - 1,1} \right]$. The lower and upper limits of the integral must be finite.

Value

The value of definite integral evaluated using Gauss Chebyshev quadrature

Author(s)

Frederick Novomestky [email protected]

References

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

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

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

See Also

chebyshev.t.quadrature.rules

Examples

###
### this example evaluates the quadrature function for
### the Chebyshev T polynomials.  it computes the integral
### of the product for all pairs of orthogonal polynomials
### from order 0 to order 20.  the results are compared to
### the diagonal matrix of the inner products for the
### polynomials.  it also computes the integral of the product
### of all pairs of orthonormal polynomials from order 0
### to order 20.  the resultant matrix should be an identity matrix
###
###
### set the value for the maximum polynomial order
###
    n <- 20
###
### maximum order plus 1
###
    np1 <- n + 1
###
### function to construct the polynomial products by column
###
by.column.products <- function( c, p.list, p.p.list )
{
###
### function to construct the polynomial products by row
###
    by.row.products <- function( r, c, p.list )
    {
        row.column.product <- p.list[[r]] * p.list[[c]]
        return (row.column.product )
    }
    np1 <- length( p.list )
    row.list <- lapply( 1:np1, by.row.products, c, p.list )
    return( row.list )
}
###
### function construct the polynomial functions by column
###
by.column.functions <- function( c, p.p.products )
{
###
### function to construct the polynomial functions by row
###
    by.row.functions <- function( r, c, p.p.products )
    {
        row.column.function <- as.function( p.p.products[[r]][[c]] )
        return( row.column.function )
    }
    np1 <- length( p.p.products[[1]] )
    row.list <- lapply( 1:np1, by.row.functions, c, p.p.products )
    return( row.list )
}
###
### function to compute the integral of the polynomials by column
###
by.column.integrals <- function( c, p.p.functions )
{
###
### function to compute the integral of the polynomials by row
###
    by.row.integrals <- function( r, c, p.p.functions )
    {
        row.column.integral <- chebyshev.t.quadrature(
            p.p.functions[[r]][[c]], order.np1.rule )
        return( row.column.integral )
    }
    np1 <- length( p.p.functions[[1]] )
    row.vector <- sapply( 1:np1, by.row.integrals, c, p.p.functions )
    return( row.vector )
}
###
### construct a list of the Chebyshev T orthogonal polynomials
###
p.list <- chebyshev.t.polynomials( n )
###
### construct the two dimensional list of pair products
### of polynomials
###
p.p.products <- lapply( 1:np1, by.column.products, p.list )
###
### compute the two dimensional list of functions
### corresponding to the polynomial products in
### the two dimensional list p.p.products
###
p.p.functions <- lapply( 1:np1, by.column.functions, p.p.products )
###
### get the rule table for the order np1 polynomial
###
rules <- chebyshev.t.quadrature.rules( np1 )
order.np1.rule <- rules[[np1]]
###
### construct the square symmetric matrix containing
### the definite integrals over the default limits
### corresponding to the two dimensional list of
### polynomial functions
###
p.p.integrals <- sapply( 1:np1, by.column.integrals, p.p.functions )
###
### construct the diagonal matrix with the inner products
### of the orthogonal polynomials on the diagonal
###
p.p.inner.products <- diag( chebyshev.t.inner.products( n ) )
print( "Integral of cross products for the orthogonal polynomials " )
print( apply( p.p.integrals, 2, round, digits=5 ) )
print( apply( p.p.inner.products, 2, round, digits=5 ) )
###
### construct a list of the Chebyshev T orthonormal polynomials
###
p.list <- chebyshev.t.polynomials( n, TRUE )
###
### construct the two dimensional list of pair products
### of polynomials
###
p.p.products <- lapply( 1:np1, by.column.products, p.list )
###
### compute the two dimensional list of functions
### corresponding to the polynomial products in
### the two dimensional list p.p.products
###
p.p.functions <- lapply( 1:np1, by.column.functions, p.p.products )
###
### get the rule table for the order np1 polynomial
###
rules <- chebyshev.t.quadrature.rules( np1, TRUE )
order.np1.rule <- rules[[np1]]
###
### construct the square symmetric matrix containing
### the definite integrals over the default limits
### corresponding to the two dimensional list of
### polynomial functions
###
p.p.integrals <- sapply( 1:np1, by.column.integrals, p.p.functions )
###
### display the matrix of integrals
###
print( "Integral of cross products for the orthonormal polynomials " )
print(apply( p.p.integrals, 2, round, digits=5 ) )

---

Create list of Chebyshev quadrature rules

Description

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

Usage

chebyshev.t.quadrature.rules(n,normalized=FALSE)

Arguments

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

Details

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

Value

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

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

...

n	Quadrature rule data frame for the order $n$ Chebyshev polynomial

Author(s)

Frederick Novomestky [email protected]

References

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

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

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

See Also

quadrature.rules, chebyshev.t.quadrature

Examples

###
### generate the list of quadrature rules for
### the orthogonal Chebyshev polynomials
### for orders 1 to 5
###
orthogonal.rules <- chebyshev.t.quadrature.rules( 5 )
print( orthogonal.rules )
###
### generate the list of quadrature rules for
### the orthonormal Chebyshev polynomials
### for orders 1 to 5
###
orthonormal.rules <- chebyshev.t.quadrature.rules( 5, normalized=TRUE )
print( orthonormal.rules )

---

Perform Gauss Chebyshev quadrature

Description

This function evaluates the integral of the given function between the lower and upper limits using the weight and abscissa values specified in the rule data frame. The quadrature formula uses the weight function for Chebyshev U polynomials.

Usage

chebyshev.u.quadrature(functn, rule, lower = -1, upper = 1, 
weighted = TRUE, ...)

Arguments

functn	an R function which should take a numeric argument x and possibly some parameters. The function returns a numerical vector value for the given argument x.
rule	a data frame containing the order $n$ Chebyshev quadrature rule
lower	numeric value for the lower limit of the integral with a default value of -1
upper	numeric value for the upper limit of the integral with a default value of +1
weighted	a boolean value which if true causes the Chebyshev weight function to be included in the integrand
...	other arguments passed to the give function

Details

The rule argument corresponds to an order $n$ Chebyshev polynomial, weight function and interval $\left[ { - 1,1} \right]$. The lower and upper limits of the integral must be finite.

Value

The value of definite integral evaluated using Gauss Chebyshev quadrature

Author(s)

Frederick Novomestky [email protected]

References

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

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

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

See Also

chebyshev.u.quadrature.rules

Examples

###
### this example evaluates the quadrature function for
### the Chebyshev U polynomials.  it computes the integral
### of the product for all pairs of orthogonal polynomials
### from order 0 to order 20.  the results are compared to
### the diagonal matrix of the inner products for the
### polynomials.  it also computes the integral of the product
### of all pairs of orthonormal polynomials from order 0
### to order 20.  the resultant matrix should be an identity matrix
###
###
### set the value for the maximum polynomial order
###
    n <- 20
###
### maximum order plus 1
###
    np1 <- n + 1
###
### function to construct the polynomial products by column
###
by.column.products <- function( c, p.list, p.p.list )
{
###
### function to construct the polynomial products by row
###
    by.row.products <- function( r, c, p.list )
    {
        row.column.product <- p.list[[r]] * p.list[[c]]
        return (row.column.product )
    }
    np1 <- length( p.list )
    row.list <- lapply( 1:np1, by.row.products, c, p.list )
    return( row.list )
}
###
### function construct the polynomial functions by column
###
by.column.functions <- function( c, p.p.products )
{
###
### function to construct the polynomial functions by row
###
    by.row.functions <- function( r, c, p.p.products )
    {
        row.column.function <- as.function( p.p.products[[r]][[c]] )
        return( row.column.function )
    }
    np1 <- length( p.p.products[[1]] )
    row.list <- lapply( 1:np1, by.row.functions, c, p.p.products )
    return( row.list )
}
###
### function to compute the integral of the polynomials by column
###
by.column.integrals <- function( c, p.p.functions )
{
###
### function to compute the integral of the polynomials by row
###
    by.row.integrals <- function( r, c, p.p.functions )
    {
        row.column.integral <- chebyshev.u.quadrature(
            p.p.functions[[r]][[c]], order.np1.rule )
        return( row.column.integral )
    }
    np1 <- length( p.p.functions[[1]] )
    row.vector <- sapply( 1:np1, by.row.integrals, c, p.p.functions )
    return( row.vector )
}
###
### construct a list of the Chebyshev U orthogonal polynomials
###
p.list <- chebyshev.u.polynomials( n )
###
### construct the two dimensional list of pair products
### of polynomials
###
p.p.products <- lapply( 1:np1, by.column.products, p.list )
###
### compute the two dimensional list of functions
### corresponding to the polynomial products in
### the two dimensional list p.p.products
###
p.p.functions <- lapply( 1:np1, by.column.functions, p.p.products )
###
### get the rule table for the order np1 polynomial
###
rules <- chebyshev.u.quadrature.rules( np1 )
order.np1.rule <- rules[[np1]]
###
### construct the square symmetric matrix containing
### the definite integrals over the default limits
### corresponding to the two dimensional list of
### polynomial functions
###
p.p.integrals <- sapply( 1:np1, by.column.integrals, p.p.functions )
###
### construct the diagonal matrix with the inner products
### of the orthogonal polynomials on the diagonal
###
p.p.inner.products <- diag( chebyshev.u.inner.products( n ) )
print( "Integral of cross products for the orthogonal polynomials " )
print( apply( p.p.integrals, 2, round, digits=5 ) )
print( apply( p.p.inner.products, 2, round, digits=5 ) )
###
### construct a list of the Chebyshev U orthonormal polynomials
###
p.list <- chebyshev.u.polynomials( n, TRUE )
###
### construct the two dimensional list of pair products
### of polynomials
###
p.p.products <- lapply( 1:np1, by.column.products, p.list )
###
### compute the two dimensional list of functions
### corresponding to the polynomial products in
### the two dimensional list p.p.products
###
p.p.functions <- lapply( 1:np1, by.column.functions, p.p.products )
###
### get the rule table for the order np1 polynomial
###
rules <- chebyshev.u.quadrature.rules( np1, TRUE )
order.np1.rule <- rules[[np1]]
###
### construct the square symmetric matrix containing
### the definite integrals over the default limits
### corresponding to the two dimensional list of
### polynomial functions
###
p.p.integrals <- sapply( 1:np1, by.column.integrals, p.p.functions )
###
### display the matrix of integrals
###
print( "Integral of cross products for the orthonormal polynomials " )
print(apply( p.p.integrals, 2, round, digits=5 ) )

---

Create list of Chebyshev quadrature rules

Description

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

Usage

chebyshev.u.quadrature.rules(n,normalized=FALSE)

Arguments

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

Details

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

Value

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

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

...

n	Quadrature rule data frame for the order $n$ Chebyshev polynomial

Author(s)

Frederick Novomestky [email protected]

References

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

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

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

See Also

quadrature.rules, chebyshev.u.quadrature

Examples

###
### generate the list of quadrature rules for
### the Chebyshev U orthogonal polynomials for
### orders 1 to 5
###
orthogonal.rules <- chebyshev.u.quadrature.rules( 5 )
print( orthogonal.rules )
###
### generate the list of quadrature rules for
### the Chebyshev U orthonormal polynomials for
### orders 1 to 5
###
orthonormal.rules <- chebyshev.u.quadrature.rules( 5, TRUE )
print( orthonormal.rules )

---

Perform Gauss Gegenbauer quadrature

Description

This function evaluates the integral of the given function between the lower and upper limits using the weight and abscissa values specified in the rule data frame. The quadrature formula uses the weight function for Gegenbauer polynomials.

Usage

gegenbauer.quadrature(functn, rule, alpha = 0, lower = -1, upper = 1, 
weighted = TRUE, ...)

Arguments

functn	an R function which should take a numeric argument x and possibly some parameters. The function returns a numerical vector value for the given argument x.
rule	a data frame containing the order $n$ Gegenbauer quadrature rule
alpha	numeric value for the Gegenbauer polynomial parameter
lower	numeric value for the lower limit of the integral with a default value of -1
upper	numeric value the upper limit of the integral with a default value of 1
weighted	boolean value which if true causes the Gegenbauer weight function to be included in the integrand
...	other arguments passed to the give function

Details

The rule argument corresponds to an order $n$ Gegenbauer polynomial, weight function and interval $\left[ { - 1,1} \right]$. The lower and upper limits of the integral must be finite.

Value

The value of definite integral evaluated using Gauss Gegenbauer quadrature

Author(s)

Frederick Novomestky [email protected]

References

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

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

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

See Also

gegenbauer.quadrature.rules, ultraspherical.quadrature

Examples

###
### this example evaluates the quadrature function for
### the Gegenbauer polynomials.  it computes the integral
### of the product for all pairs of orthogonal polynomials
### from order 0 to order 16.  the results are compared to
### the diagonal matrix of the inner products for the
### polynomials.  it also computes the integral of the product
### of all pairs of orthonormal polynomials from order 0
### to order 16.  the resultant matrix should be an identity matrix
###
###
### set the polynomial parameter
###
alpha <- 0.25
###
### set the value for the maximum polynomial order
###
    n <- 16
###
### maximum order plus 1
###
    np1 <- n + 1
###
### function to construct the polynomial products by column
###
by.column.products <- function( c, p.list, p.p.list )
{
###
### function to construct the polynomial products by row
###
    by.row.products <- function( r, c, p.list )
    {
        row.column.product <- p.list[[r]] * p.list[[c]]
        return (row.column.product )
    }
    np1 <- length( p.list )
    row.list <- lapply( 1:np1, by.row.products, c, p.list )
    return( row.list )
}
###
### function construct the polynomial functions by column
###
by.column.functions <- function( c, p.p.products )
{
###
### function to construct the polynomial functions by row
###
    by.row.functions <- function( r, c, p.p.products )
    {
        row.column.function <- as.function( p.p.products[[r]][[c]] )
        return( row.column.function )
    }
    np1 <- length( p.p.products[[1]] )
    row.list <- lapply( 1:np1, by.row.functions, c, p.p.products )
    return( row.list )
}
###
### function to compute the integral of the polynomials by column
###
by.column.integrals <- function( c, p.p.functions )
{
###
### function to compute the integral of the polynomials by row
###
    by.row.integrals <- function( r, c, p.p.functions )
    {
        row.column.integral <- gegenbauer.quadrature(
            p.p.functions[[r]][[c]], order.np1.rule, alpha )
        return( row.column.integral )
    }
    np1 <- length( p.p.functions[[1]] )
    row.vector <- sapply( 1:np1, by.row.integrals, c, p.p.functions )
    return( row.vector )
}
###
### construct a list of the Gegenbauer orthogonal polynomials
###
p.list <- gegenbauer.polynomials( n, alpha )
###
### construct the two dimensional list of pair products
### of polynomials
###
p.p.products <- lapply( 1:np1, by.column.products, p.list )
###
### compute the two dimensional list of functions
### corresponding to the polynomial products in
### the two dimensional list p.p.products
###
p.p.functions <- lapply( 1:np1, by.column.functions, p.p.products )
###
### get the rule table for the order np1 polynomial
###
rules <- gegenbauer.quadrature.rules( np1, alpha )
order.np1.rule <- rules[[np1]]
###
### construct the square symmetric matrix containing
### the definite integrals over the default limits
### corresponding to the two dimensional list of
### polynomial functions
###
p.p.integrals <- sapply( 1:np1, by.column.integrals, p.p.functions )
###
### construct the diagonal matrix with the inner products
### of the orthogonal polynomials on the diagonal
###
p.p.inner.products <- diag( gegenbauer.inner.products( n,alpha ) )
print( "Integral of cross products for the orthogonal polynomials " )
print( apply( p.p.integrals, 2, round, digits=6 ) )
print( apply( p.p.inner.products, 2, round, digits=6 ) )
###
### construct a list of the Gegenbauer orthonormal polynomials
###
p.list <- gegenbauer.polynomials( n, alpha, TRUE )
###
### construct the two dimensional list of pair products
### of polynomials
###
p.p.products <- lapply( 1:np1, by.column.products, p.list )
###
### compute the two dimensional list of functions
### corresponding to the polynomial products in
### the two dimensional list p.p.products
###
p.p.functions <- lapply( 1:np1, by.column.functions, p.p.products )
###
### get the rule table for the order np1 polynomial
###
rules <- gegenbauer.quadrature.rules( np1, alpha, TRUE )
order.np1.rule <- rules[[np1]]
###
### construct the square symmetric matrix containing
### the definite integrals over the default limits
### corresponding to the two dimensional list of
### polynomial functions
###
p.p.integrals <- sapply( 1:np1, by.column.integrals, p.p.functions )
###
### display the matrix of integrals
###
print( "Integral of cross products for the orthonormal polynomials " )
print(apply( p.p.integrals, 2, round, digits=6 ) )

---

Create list of Gegenbauer quadrature rules

Description

This function returns a list with $n$ elements containing the order $k$ quadrature rule data frame for the Gegenbauer polynomials for orders $k = 1,\;2,\; \ldots ,\;n$.

Usage

gegenbauer.quadrature.rules(n,alpha,normalized=FALSE)

Arguments

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

Details

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

Value

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

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

...

n	Quadrature rule data frame for the order $n$ Gegenbauer polynomial

Author(s)

Frederick Novomestky [email protected]

References

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

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

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

See Also

quadrature.rules, gegenbauer.quadrature

Examples

###
### generate the list of quadrature rule data frames for
### the orthogonal Gegenbauer polynomials
### of orders 1 to 5
### polynomial parameter alpha is 1.0
###
orthogonal.rules <- gegenbauer.quadrature.rules( 5, 1 )
print( orthogonal.rules )
###
### generate the list of quadrature rule data frames for
### the orthonormal Gegenbauer polynomials
### of orders 1 to 5
### polynomial parameter alpha is 1.0
###
orthonormal.rules <- gegenbauer.quadrature.rules( 5, 1, TRUE )
print( orthonormal.rules )

---

Perform generalized Gauss Hermite quadrature

Description

This function evaluates the integral of the given function between the lower and upper limits using the weight and abscissa values specified in the rule data frame. The quadrature formula uses the weight function for generalized Hermite polynomials.

Usage

ghermite.h.quadrature(functn, rule, mu = 0, lower = -Inf, upper = Inf, 
weighted = TRUE, ...)

Arguments

functn	an R function which should take a numeric argument x and possibly some parameters. The function returns a numerical vector value for the given argument x.
rule	a data frame containing the order $n$ generalized Hermite quadrature rule
mu	numeric value for the parameter for the generalized Hermite polynomials
lower	numeric value for the lower limit of the integral with a default value of $- \infty$
upper	numeric value for the upper limit of the integral with a default value of $+ \infty$
weighted	a boolean value which if true causes the Hermite weight function to be included in the integrand
...	other arguments passed to the give function

Details

The rule argument corresponds to an order $n$ generalized Hermite polynomial, weight function and interval $\left( { - \infty ,\infty } \right)$. The lower and upper limits of the integral must be infinite.

Value

The value of definite integral evaluated using Gauss Hermite quadrature

Author(s)

Frederick Novomestky [email protected]

References

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

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

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

See Also

hermite.h.quadrature.rules

Examples

###
### this example evaluates the quadrature function for
### the generalized Hermite H polynomials.  it computes the integral
### of the product for all pairs of orthogonal polynomials
### from order 0 to order 10.  the results are compared to
### the diagonal matrix of the inner products for the
### polynomials.  it also computes the integral of the product
### of all pairs of orthonormal polynomials from order 0
### to order 10.  the resultant matrix should be an identity matrix
###
###
### set the polynomial parameter
###
    mu <- 1
###
### set the value for the maximum polynomial order
###
    n <- 10
###
### maximum order plus 1
###
    np1 <- n + 1
###
### function to construct the polynomial products by column
###
by.column.products <- function( c, p.list, p.p.list )
{
###
### function to construct the polynomial products by row
###
    by.row.products <- function( r, c, p.list )
    {
        row.column.product <- p.list[[r]] * p.list[[c]]
        return (row.column.product )
    }
    np1 <- length( p.list )
    row.list <- lapply( 1:np1, by.row.products, c, p.list )
    return( row.list )
}
###
### function construct the polynomial functions by column
###
by.column.functions <- function( c, p.p.products )
{
###
### function to construct the polynomial functions by row
###
    by.row.functions <- function( r, c, p.p.products )
    {
        row.column.function <- as.function( p.p.products[[r]][[c]] )
        return( row.column.function )
    }
    np1 <- length( p.p.products[[1]] )
    row.list <- lapply( 1:np1, by.row.functions, c, p.p.products )
    return( row.list )
}
###
### function to compute the integral of the polynomials by column
###
by.column.integrals <- function( c, p.p.functions )
{
###
### function to compute the integral of the polynomials by row
###
    by.row.integrals <- function( r, c, p.p.functions )
    {
        row.column.integral <- ghermite.h.quadrature(
            p.p.functions[[r]][[c]], order.np1.rule, mu )
        return( row.column.integral )
    }
    np1 <- length( p.p.functions[[1]] )
    row.vector <- sapply( 1:np1, by.row.integrals, c, p.p.functions )
    return( row.vector )
}
###
### construct a list of the generalized Hermite H orthogonal polynomials
###
p.list <- ghermite.h.polynomials( n, mu )
###
### construct the two dimensional list of pair products
### of polynomials
###
p.p.products <- lapply( 1:np1, by.column.products, p.list )
###
### compute the two dimensional list of functions
### corresponding to the polynomial products in
### the two dimensional list p.p.products
###
p.p.functions <- lapply( 1:np1, by.column.functions, p.p.products )
###
### get the rule table for the order np1 polynomial
###
rules <- ghermite.h.quadrature.rules( np1, mu )
order.np1.rule <- rules[[np1]]
###
### construct the square symmetric matrix containing
### the definite integrals over the default limits
### corresponding to the two dimensional list of
### polynomial functions
###
p.p.integrals <- sapply( 1:np1, by.column.integrals, p.p.functions )
###
### construct the diagonal matrix with the inner products
### of the orthogonal polynomials on the diagonal
###
p.p.inner.products <- diag( ghermite.h.inner.products( n, mu ) )
print( "Integral of cross products for the orthogonal polynomials " )
print( apply( p.p.integrals, 2, round, digits=5 ) )
print( apply( p.p.inner.products, 2, round, digits=5 ) )
###
### construct a list of the Hermite H orthonormal polynomials
###
p.list <- ghermite.h.polynomials( n, mu, TRUE )
###
### construct the two dimensional list of pair products
### of polynomials
###
p.p.products <- lapply( 1:np1, by.column.products, p.list )
###
### compute the two dimensional list of functions
### corresponding to the polynomial products in
### the two dimensional list p.p.products
###
p.p.functions <- lapply( 1:np1, by.column.functions, p.p.products )
###
### get the rule table for the order np1 polynomial
###
rules <- ghermite.h.quadrature.rules( np1, mu, TRUE )
order.np1.rule <- rules[[np1]]
###
### construct the square symmetric matrix containing
### the definite integrals over the default limits
### corresponding to the two dimensional list of
### polynomial functions
###
p.p.integrals <- sapply( 1:np1, by.column.integrals, p.p.functions )
###
### display the matrix of integrals
###
print( "Integral of cross products for the orthonormal polynomials " )
print(apply( p.p.integrals, 2, round, digits=5 ) )

---

Create list of generalized Hermite quadrature rules

Description

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

Usage

ghermite.h.quadrature.rules(n, mu, normalized=FALSE)

Arguments

n	integer value for the highest integer order
mu	numeric value for the parameter of the generalized Hermite polynomial
normalized	boolean value. if TRUE rules are for orthonormal polynomials, otherwise they are for orthgonal polynomials

Details

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

Value

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

1	Quadrature rule data frame for the order 1 generalized Hermite polynomial
2	Quadrature rule data frame for the order 2 generalized Hermite polynomial

...

n	Quadrature rule data frame for the order $n$ generalized Hermite polynomial

Author(s)

Frederick Novomestky [email protected]

References

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

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

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

See Also

quadrature.rules, schebyshev.t.quadrature

Examples

###
### generate a list of quadrature rule data frames for
### the generalized orthogonal Hermite polynomial
### of orders 1 to 5.
### polynomial parameter mu is 1.0
###
orthogonal.rules <- ghermite.h.quadrature.rules( 5, 1 )
print( orthogonal.rules )
###
### generate a list of quadrature rule data frames for
### the generalized orthonormal Hermite polynomial
### of orders 1 to 5.
### polynomial parameter mu is 1.0
###
orthonormal.rules <- ghermite.h.quadrature.rules( 5, 1, TRUE )
print( orthonormal.rules )

---

Perform Gauss Laguerre quadrature

Description

This function evaluates the integral of the given function between the lower and upper limits using the weight and abscissa values specified in the rule data frame. The quadrature formula uses the weight function for generalized Laguerre polynomials.

Usage

glaguerre.quadrature(functn, rule, alpha = 0, lower = 0, upper = Inf, 
weighted = TRUE, ...)

Arguments

functn	an R function which should take a numeric argument x and possibly some parameters. The function returns a numerical vector value for the given argument x.
rule	a data frame containing the order $n$ generalized Laguerre quadrature rule
alpha	numeric value for the generalized Laguerre polynomial parameter
lower	numeric value for the lower limit of the integral with a default value of 0
upper	numeric value for the upper limit of the integral with a default value of Inf
weighted	a boolean value which if true causes the generalized Laguerre weight function to be included in the integrand
...	other arguments passed to the give function

Details

The rule argument corresponds to an order $n$ generalized Laguerre polynomial, weight function and interval $\left[ {0,\infty } \right)$. The one limit of the integral must be finite and the other must be infinite.

Value

The value of definite integral evaluated using Gauss Laguerre quadrature

Author(s)

Frederick Novomestky [email protected]

References

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

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

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

See Also

glaguerre.quadrature.rules

Examples

###
### this example evaluates the quadrature function for
### the generalized Laguerre polynomials.  it computes the integral
### of the product for all pairs of orthogonal polynomials
### from order 0 to order 12.  the results are compared to
### the diagonal matrix of the inner products for the
### polynomials.  it also computes the integral of the product
### of all pairs of orthonormal polynomials from order 0
### to order 12.  the resultant matrix should be an identity matrix
###
###
### set the polynomial parameter
###
alpha <- 1
###
### set the value for the maximum polynomial order
###
    n <- 12
###
### maximum order plus 1
###
    np1 <- n + 1
###
### function to construct the polynomial products by column
###
by.column.products <- function( c, p.list, p.p.list )
{
###
### function to construct the polynomial products by row
###
    by.row.products <- function( r, c, p.list )
    {
        row.column.product <- p.list[[r]] * p.list[[c]]
        return (row.column.product )
    }
    np1 <- length( p.list )
    row.list <- lapply( 1:np1, by.row.products, c, p.list )
    return( row.list )
}
###
### function construct the polynomial functions by column
###
by.column.functions <- function( c, p.p.products )
{
###
### function to construct the polynomial functions by row
###
    by.row.functions <- function( r, c, p.p.products )
    {
        row.column.function <- as.function( p.p.products[[r]][[c]] )
        return( row.column.function )
    }
    np1 <- length( p.p.products[[1]] )
    row.list <- lapply( 1:np1, by.row.functions, c, p.p.products )
    return( row.list )
}
###
### function to compute the integral of the polynomials by column
###
by.column.integrals <- function( c, p.p.functions )
{
###
### function to compute the integral of the polynomials by row
###
    by.row.integrals <- function( r, c, p.p.functions )
    {
        row.column.integral <- glaguerre.quadrature(
            p.p.functions[[r]][[c]], order.np1.rule, alpha )
        return( row.column.integral )
    }
    np1 <- length( p.p.functions[[1]] )
    row.vector <- sapply( 1:np1, by.row.integrals, c, p.p.functions )
    return( row.vector )
}
###
### construct a list of the generalized Laguerre orthogonal polynomials
###
p.list <- glaguerre.polynomials( n, alpha )
###
### construct the two dimensional list of pair products
### of polynomials
###
p.p.products <- lapply( 1:np1, by.column.products, p.list )
###
### compute the two dimensional list of functions
### corresponding to the polynomial products in
### the two dimensional list p.p.products
###
p.p.functions <- lapply( 1:np1, by.column.functions, p.p.products )
###
### get the rule table for the order np1 polynomial
###
rules <- glaguerre.quadrature.rules( np1, alpha )
order.np1.rule <- rules[[np1]]
###
### construct the square symmetric matrix containing
### the definite integrals over the default limits
### corresponding to the two dimensional list of
### polynomial functions
###
p.p.integrals <- sapply( 1:np1, by.column.integrals, p.p.functions )
###
### construct the diagonal matrix with the inner products
### of the orthogonal polynomials on the diagonal
###
p.p.inner.products <- diag( glaguerre.inner.products( n,alpha ) )
print( "Integral of cross products for the orthogonal polynomials " )
print( apply( p.p.integrals, 2, round, digits=6 ) )
print( apply( p.p.inner.products, 2, round, digits=6 ) )
###
### construct a list of theg generalized Laguerre orthonormal polynomials
###
p.list <- glaguerre.polynomials( n, alpha, TRUE )
###
### construct the two dimensional list of pair products
### of polynomials
###
p.p.products <- lapply( 1:np1, by.column.products, p.list )
###
### compute the two dimensional list of functions
### corresponding to the polynomial products in
### the two dimensional list p.p.products
###
p.p.functions <- lapply( 1:np1, by.column.functions, p.p.products )
###
### get the rule table for the order np1 polynomial
###
rules <- glaguerre.quadrature.rules( np1, alpha, TRUE )
order.np1.rule <- rules[[np1]]
###
### construct the square symmetric matrix containing
### the definite integrals over the default limits
### corresponding to the two dimensional list of
### polynomial functions
###
p.p.integrals <- sapply( 1:np1, by.column.integrals, p.p.functions )
###
### display the matrix of integrals
###
print( "Integral of cross products for the orthonormal polynomials " )
print(apply( p.p.integrals, 2, round, digits=6 ) )

---

Create list of generalized Laguerre quadrature rules

Description

This function returns a list with $n$ elements containing the order $k$ quadrature rule data frame for the generalized Laguerre polynomials for orders $k = 1,\;2,\; \ldots ,\;n$.

Usage

glaguerre.quadrature.rules(n,alpha,normalized=FALSE)

Arguments

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

Details

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

Value

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

1	Quadrature rule for the order 1 generalized Laguerre polynomial
2	Quadrature rule for the order 2 generalized Laguerre polynomial

...

n	Quadrature rule for the order $n$ generalized Laguerre polynomial

Author(s)

Frederick Novomestky [email protected]

References

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

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

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

See Also

quadrature.rules, glaguerre.quadrature

Examples

###
### generate a list of quadrature rule data frames for
### the generalized orthogonal Laguerre polynomials
### of orders 1 to 5
### polynomial parameter is 1.0
###
orthogonal.rules <- glaguerre.quadrature.rules( 5, 1 )
print( orthogonal.rules )
###
### generate a list of quadrature rule data frames for
### the generalized orthonormal Laguerre polynomials
### of orders 1 to 5
### polynomial parameter is 1.0
###
orthonormal.rules <- glaguerre.quadrature.rules( 5, 1, TRUE )
print( orthonormal.rules )

---

Perform Gauss Hermite quadrature

Description

This function evaluates the integral of the given function between the lower and upper limits using the weight and abscissa values specified in the rule data frame. The quadrature formula uses the weight function for Hermite H polynomials.

Usage

hermite.h.quadrature(functn, rule, lower = -Inf, upper = Inf, 
weighted = TRUE, ...)

Arguments

functn	an R function which should take a numeric argument x and possibly some parameters. The function returns a numerical vector value for the given argument x.
rule	a data frame containing the order $n$ Hermite quadrature rule
lower	numeric value for the lower limit of the integral with a default value of $- \infty$
upper	numeric value for the upper limit of the integral with a default value of $+ \infty$
weighted	boolean value which if true causes the Hermite weight function to be included in the integrand
...	other arguments passed to the give function

Details

The rule argument corresponds to an order $n$ Hermite polynomial, weight function and interval $\left( { - \infty ,\infty } \right)$ The lower and upper limits of the integral must be infinite.

Value

The value of definite integral evaluated using Gauss Hermite quadrature

Author(s)

Frederick Novomestky [email protected]

References

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

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

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

See Also

hermite.h.quadrature.rules

Examples

###
### this example evaluates the quadrature function for
### the Hermite H polynomials.  it computes the integral
### of the product for all pairs of orthogonal polynomials
### from order 0 to order 8.  the results are compared to
### the diagonal matrix of the inner products for the
### polynomials.  it also computes the integral of the product
### of all pairs of orthonormal polynomials from order 0
### to order 8.  the resultant matrix should be an identity matrix
###
###
### set the value for the maximum polynomial order
###
    n <- 8
###
### maximum order plus 1
###
    np1 <- n + 1
###
### function to construct the polynomial products by column
###
by.column.products <- function( c, p.list, p.p.list )
{
###
### function to construct the polynomial products by row
###
    by.row.products <- function( r, c, p.list )
    {
        row.column.product <- p.list[[r]] * p.list[[c]]
        return (row.column.product )
    }
    np1 <- length( p.list )
    row.list <- lapply( 1:np1, by.row.products, c, p.list )
    return( row.list )
}
###
### function construct the polynomial functions by column
###
by.column.functions <- function( c, p.p.products )
{
###
### function to construct the polynomial functions by row
###
    by.row.functions <- function( r, c, p.p.products )
    {
        row.column.function <- as.function( p.p.products[[r]][[c]] )
        return( row.column.function )
    }
    np1 <- length( p.p.products[[1]] )
    row.list <- lapply( 1:np1, by.row.functions, c, p.p.products )
    return( row.list )
}
###
### function to compute the integral of the polynomials by column
###
by.column.integrals <- function( c, p.p.functions )
{
###
### function to compute the integral of the polynomials by row
###
    by.row.integrals <- function( r, c, p.p.functions )
    {
        row.column.integral <- hermite.h.quadrature(
            p.p.functions[[r]][[c]], order.np1.rule )
        return( row.column.integral )
    }
    np1 <- length( p.p.functions[[1]] )
    row.vector <- sapply( 1:np1, by.row.integrals, c, p.p.functions )
    return( row.vector )
}
###
### construct a list of the Hermite H orthogonal polynomials
###
p.list <- hermite.h.polynomials( n )
###
### construct the two dimensional list of pair products
### of polynomials
###
p.p.products <- lapply( 1:np1, by.column.products, p.list )
###
### compute the two dimensional list of functions
### corresponding to the polynomial products in
### the two dimensional list p.p.products
###
p.p.functions <- lapply( 1:np1, by.column.functions, p.p.products )
###
### get the rule table for the order np1 polynomial
###
rules <- hermite.h.quadrature.rules( np1 )
order.np1.rule <- rules[[np1]]
###
### construct the square symmetric matrix containing
### the definite integrals over the default limits
### corresponding to the two dimensional list of
### polynomial functions
###
p.p.integrals <- sapply( 1:np1, by.column.integrals, p.p.functions )
###
### construct the diagonal matrix with the inner products
### of the orthogonal polynomials on the diagonal
###
p.p.inner.products <- diag( hermite.h.inner.products( n ) )
print( "Integral of cross products for the orthogonal polynomials " )
print( apply( p.p.integrals, 2, round, digits=5 ) )
print( apply( p.p.inner.products, 2, round, digits=5 ) )
###
### construct a list of the Hermite H orthonormal polynomials
###
p.list <- hermite.h.polynomials( n, TRUE )
###
### construct the two dimensional list of pair products
### of polynomials
###
p.p.products <- lapply( 1:np1, by.column.products, p.list )
###
### compute the two dimensional list of functions
### corresponding to the polynomial products in
### the two dimensional list p.p.products
###
p.p.functions <- lapply( 1:np1, by.column.functions, p.p.products )
###
### get the rule table for the order np1 polynomial
###
rules <- hermite.h.quadrature.rules( np1, TRUE )
order.np1.rule <- rules[[np1]]
###
### construct the square symmetric matrix containing
### the definite integrals over the default limits
### corresponding to the two dimensional list of
### polynomial functions
###
p.p.integrals <- sapply( 1:np1, by.column.integrals, p.p.functions )
###
### display the matrix of integrals
###
print( "Integral of cross products for the orthonormal polynomials " )
print(apply( p.p.integrals, 2, round, digits=5 ) )

---

Create list of Hermite quadrature rules

Description

This function returns a list with $n$ elements containing the order $n$ quadrature rule data frame for the Hermite polynomials for orders $k = 1,\;2,\; \ldots ,\;n$.

Usage

hermite.h.quadrature.rules(n,normalized=FALSE)

Arguments

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

Details

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

Value

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

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

...

n	Quadrature rule data frame for the order $n$ Hermite polynomial

Author(s)

Frederick Novomestky [email protected]

References

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

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

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

See Also

quadrature.rules, hermite.h.quadrature

Examples

###
### generate the list of quadrature rules for
### the Hermite orthogonal polynomials
### of orders 1 to 5
###
orthogonal.rules <- hermite.h.quadrature.rules( 5 )
print( orthogonal.rules )
###
### generate the list of quadrature rules for
### the Hermite orthonormal polynomials
### of orders 1 to 5
###
orthonormal.rules <- hermite.h.quadrature.rules( 5, TRUE )
print( orthonormal.rules )

---

Perform Gauss Hermite quadrature

Description

This function evaluates the integral of the given function between the lower and upper limits using the weight and abscissa values specified in the rule data frame. The quadrature formula uses the weight function for hermite He polynomials.

Usage

hermite.he.quadrature(functn, rule, lower = -Inf, upper = Inf, 
weighted = TRUE, ...)

Arguments

functn	an R function which should take a numeric argument x and possibly some parameters. The function returns a numerical vector value for the given argument x.
rule	a data frame containing the order $n$ Hermite quadrature rule
lower	the lower limit of the integral with a default value of $- \infty$
upper	the upper limit of the integral with a default value of $+ \infty$
weighted	a boolean value which if true causes the Hermite weight function to be included in the integrand
...	other arguments passed to the give function

Details

The rule argument corresponds to an order $n$ Hermite polynomial, weight function and interval $\left( { - \infty ,\infty } \right)$ The lower and upper limits of the integral must be infinite.

Value

The value of definite integral evaluated using Gauss Hermite quadrature

Author(s)

Frederick Novomestky [email protected]

References

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

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

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

See Also

hermite.h.quadrature.rules

Examples

###
### this example evaluates the quadrature function for
### the scaled Hermite He polynomials.  it computes the integral
### of the product for all pairs of orthogonal polynomials
### from order 0 to order 16.  the results are compared to
### the diagonal matrix of the inner products for the
### polynomials.  it also computes the integral of the product
### of all pairs of orthonormal polynomials from order 0
### to order 16.  the resultant matrix should be an identity matrix
###
###
### set the value for the maximum polynomial order
###
    n <- 12
###
### maximum order plus 1
###
    np1 <- n + 1
###
### function to construct the polynomial products by column
###
by.column.products <- function( c, p.list, p.p.list )
{
###
### function to construct the polynomial products by row
###
    by.row.products <- function( r, c, p.list )
    {
        row.column.product <- p.list[[r]] * p.list[[c]]
        return (row.column.product )
    }
    np1 <- length( p.list )
    row.list <- lapply( 1:np1, by.row.products, c, p.list )
    return( row.list )
}
###
### function construct the polynomial functions by column
###
by.column.functions <- function( c, p.p.products )
{
###
### function to construct the polynomial functions by row
###
    by.row.functions <- function( r, c, p.p.products )
    {
        row.column.function <- as.function( p.p.products[[r]][[c]] )
        return( row.column.function )
    }
    np1 <- length( p.p.products[[1]] )
    row.list <- lapply( 1:np1, by.row.functions, c, p.p.products )
    return( row.list )
}
###
### function to compute the integral of the polynomials by column
###
by.column.integrals <- function( c, p.p.functions )
{
###
### function to compute the integral of the polynomials by row
###
    by.row.integrals <- function( r, c, p.p.functions )
    {
        row.column.integral <- hermite.h.quadrature(
            p.p.functions[[r]][[c]], order.np1.rule )
        return( row.column.integral )
    }
    np1 <- length( p.p.functions[[1]] )
    row.vector <- sapply( 1:np1, by.row.integrals, c, p.p.functions )
    return( row.vector )
}
###
### construct a list of the scaled Hermite He orthogonal polynomials
###
p.list <- hermite.he.polynomials( n )
###
### construct the two dimensional list of pair products
### of polynomials
###
p.p.products <- lapply( 1:np1, by.column.products, p.list )
###
### compute the two dimensional list of functions
### corresponding to the polynomial products in
### the two dimensional list p.p.products
###
p.p.functions <- lapply( 1:np1, by.column.functions, p.p.products )
###
### get the rule table for the order np1 polynomial
###
rules <- hermite.he.quadrature.rules( np1 )
order.np1.rule <- rules[[np1]]
###
### construct the square symmetric matrix containing
### the definite integrals over the default limits
### corresponding to the two dimensional list of
### polynomial functions
###
p.p.integrals <- sapply( 1:np1, by.column.integrals, p.p.functions )
###
### construct the diagonal matrix with the inner products
### of the orthogonal polynomials on the diagonal
###
p.p.inner.products <- diag( hermite.he.inner.products( n ) )
print( "Integral of cross products for the orthogonal polynomials " )
print( apply( p.p.integrals, 2, round, digits=5 ) )
print( apply( p.p.inner.products, 2, round, digits=5 ) )
###
### construct a list of the scaled Hermite H orthonormal polynomials
###
p.list <- hermite.he.polynomials( n, TRUE )
###
### construct the two dimensional list of pair products
### of polynomials
###
p.p.products <- lapply( 1:np1, by.column.products, p.list )
###
### compute the two dimensional list of functions
### corresponding to the polynomial products in
### the two dimensional list p.p.products
###
p.p.functions <- lapply( 1:np1, by.column.functions, p.p.products )
###
### get the rule table for the order np1 polynomial
###
rules <- hermite.he.quadrature.rules( np1, TRUE )
order.np1.rule <- rules[[np1]]
###
### construct the square symmetric matrix containing
### the definite integrals over the default limits
### corresponding to the two dimensional list of
### polynomial functions
###
p.p.integrals <- sapply( 1:np1, by.column.integrals, p.p.functions )
###
### display the matrix of integrals
###
print( "Integral of cross products for the orthonormal polynomials " )
print(apply( p.p.integrals, 2, round, digits=5 ) )

---

Create list of Hermite quadrature rules

Description

This function returns a list with $n$ elements containing the order $k$ quadrature rule data frame for the Hermite polynomials for orders $k = 1,\;2,\; \ldots ,\;n$.

Usage

hermite.he.quadrature.rules(n,normalized=FALSE)

Arguments

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

Details

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

Value

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

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

...

n	Quadrature rule data frame for the order $n$ Hermite polynomial

Author(s)

Frederick Novomestky [email protected]

References

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

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

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

See Also

quadrature.rules, hermite.he.quadrature

Examples

###
### generate a list of quadrature rule frames for
### the Hermite orthogonal polynomials
### of orders 1 to 5
###
orthogonal.rules <- hermite.he.quadrature.rules( 5 )
print( orthogonal.rules )
###
### generate a list of quadrature rule frames for
### the Hermite orthonormal polynomials
### of orders 1 to 5
###
orthonormal.rules <- hermite.he.quadrature.rules( 5, TRUE )
print( orthonormal.rules )

---

Perform Gauss Jacobi quadrature

Description

This function evaluates the integral of the given function between the lower and upper limits using the weight and abscissa values specified in the rule data frame. The quadrature formula uses the weight function for Jacobi G polynomials.

Usage

jacobi.g.quadrature(functn, rule,  p = 1, q = 1, lower = 0, upper = 1, 
weighted = TRUE, ...)

Arguments

functn	an R function which should take a numeric argument x and possibly some parameters. The function returns a numerical vector value for the given argument x.
rule	a data frame containing the order $n$ ultraspherical quadrature rule
p	numeric value for the first Jacobi polynomial parameter
q	numeric value for the second Jacobi polynomial parameter
lower	numeric value for the lower limit of the integral with a default value of 0
upper	numeric value for the upper limit of the integral with a default value of 1
weighted	boolean value which if true causes the ultraspherical weight function to be included in the integrand
...	other arguments passed to the give function

Details

The rule argument corresponds to an order $n$ Jacobi polynomial, weight function and interval $\left[ {0,1} \right]$. The lower and upper limits of the integral must be finite.

Value

The value of definite integral evaluated using Gauss Jacobi quadrature

Author(s)

Frederick Novomestky [email protected]

References

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

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

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

See Also

jacobi.p.quadrature.rules

Examples

###
### this example evaluates the quadrature function for
### the Jacobi G polynomials.  it computes the integral
### of the product for all pairs of orthogonal polynomials
### from order 0 to order 10.  the results are compared to
### the diagonal matrix of the inner products for the
### polynomials.  it also computes the integral of the product
### of all pairs of orthonormal polynomials from order 0
### to order 10.  the resultant matrix should be an identity matrix
###
###
### set the polynomial parameter
###
p <- 3
q <- 2
###
### set the value for the maximum polynomial order
###
    n <- 10
###
### maximum order plus 1
###
    np1 <- n + 1
###
### function to construct the polynomial products by column
###
by.column.products <- function( c, p.list, p.p.list )
{
###
### function to construct the polynomial products by row
###
    by.row.products <- function( r, c, p.list )
    {
        row.column.product <- p.list[[r]] * p.list[[c]]
        return (row.column.product )
    }
    np1 <- length( p.list )
    row.list <- lapply( 1:np1, by.row.products, c, p.list )
    return( row.list )
}
###
### function construct the polynomial functions by column
###
by.column.functions <- function( c, p.p.products )
{
###
### function to construct the polynomial functions by row
###
    by.row.functions <- function( r, c, p.p.products )
    {
        row.column.function <- as.function( p.p.products[[r]][[c]] )
        return( row.column.function )
    }
    np1 <- length( p.p.products[[1]] )
    row.list <- lapply( 1:np1, by.row.functions, c, p.p.products )
    return( row.list )
}
###
### function to compute the integral of the polynomials by column
###
by.column.integrals <- function( c, p.p.functions )
{
###
### function to compute the integral of the polynomials by row
###
    by.row.integrals <- function( r, c, p.p.functions )
    {
        row.column.integral <- jacobi.g.quadrature(
            p.p.functions[[r]][[c]], order.np1.rule, p, q )
        return( row.column.integral )
    }
    np1 <- length( p.p.functions[[1]] )
    row.vector <- sapply( 1:np1, by.row.integrals, c, p.p.functions )
    return( row.vector )
}
###
### construct a list of the Jacobi G orthogonal polynomials
###
p.list <- jacobi.g.polynomials( n, p, q )
###
### construct the two dimensional list of pair products
### of polynomials
###
p.p.products <- lapply( 1:np1, by.column.products, p.list )
###
### compute the two dimensional list of functions
### corresponding to the polynomial products in
### the two dimensional list p.p.products
###
p.p.functions <- lapply( 1:np1, by.column.functions, p.p.products )
###
### get the rule table for the order np1 polynomial
###
rules <- jacobi.g.quadrature.rules( np1, p, q )
order.np1.rule <- rules[[np1]]
###
### construct the square symmetric matrix containing
### the definite integrals over the default limits
### corresponding to the two dimensional list of
### polynomial functions
###
p.p.integrals <- sapply( 1:np1, by.column.integrals, p.p.functions )
###
### construct the diagonal matrix with the inner products
### of the orthogonal polynomials on the diagonal
###
p.p.inner.products <- diag( jacobi.g.inner.products( n,p, q ) )
print( "Integral of cross products for the orthogonal polynomials " )
print( apply( p.p.integrals, 2, round, digits=5 ) )
print( apply( p.p.inner.products, 2, round, digits=5 ) )
###
### construct a list of the Jacobi G orthonormal polynomials
###
p.list <- jacobi.g.polynomials( n, p, q, TRUE )
###
### construct the two dimensional list of pair products
### of polynomials
###
p.p.products <- lapply( 1:np1, by.column.products, p.list )
###
### compute the two dimensional list of functions
### corresponding to the polynomial products in
### the two dimensional list p.p.products
###
p.p.functions <- lapply( 1:np1, by.column.functions, p.p.products )
###
### get the rule table for the order np1 polynomial
###
rules <- jacobi.g.quadrature.rules( np1, p, q, TRUE )
order.np1.rule <- rules[[np1]]
###
### construct the square symmetric matrix containing
### the definite integrals over the default limits
### corresponding to the two dimensional list of
### polynomial functions
###
p.p.integrals <- sapply( 1:np1, by.column.integrals, p.p.functions )
###
### display the matrix of integrals
###
print( "Integral of cross products for the orthonormal polynomials " )
print(apply( p.p.integrals, 2, round, digits=5 ) )

---

Create list of Jacobi quadrature rules

Description

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

Usage

jacobi.g.quadrature.rules(n,p,q,normalized=FALSE)

Arguments

n	integer value for the highest order
p	numeric value for the first polynomial parameter
q	numeric value for the second polynomial parameter
normalized	boolean value. if TRUE rules are for orthonormal polynomials, otherwise they are for orthgonal polynomials

Details

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

Value

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

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

...

n	Quadrature rule data frame for the order $n$ Jacobi polynomial

Author(s)

Frederick Novomestky [email protected]

References

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

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

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

See Also

quadrature.rules

Examples

###
### generate the list of quadrature rule data frames for
### the orthogonal Jacobi G polynomial
### of orders 1 to 5
### parameter p is 3 and parameter q is 2
###
orthogonal.rules <- jacobi.g.quadrature.rules( 5, 3, 2 )
print( orthogonal.rules )
###
### generate the list of quadrature rule data frames for
### the orthonormal Jacobi G polynomial
### of orders 1 to 5
### parameter p is 3 and parameter q is 2
###
orthonormal.rules <- jacobi.g.quadrature.rules( 5, 3, 2, TRUE )
print( orthonormal.rules )

---

Perform Gauss Jacobi quadrature

Description

This function evaluates the integral of the given function between the lower and upper limits using the weight and abscissa values specified in the rule data frame. The quadrature formula uses the weight function for Jacobi P polynomials.

Usage

jacobi.p.quadrature(functn, rule, alpha = 0, beta = 0, lower = -1, upper = 1, 
weighted = TRUE,  ...)

Arguments

functn	an R function which should take a numeric argument x and possibly some parameters. The function returns a numerical vector value for the given argument x.
rule	a data frame containing the order $n$ ultraspherical quadrature rule
alpha	numeric value for the first Jacobi polynomial parameter
beta	numeric value for the second Jacobi polynomial parameter
lower	numeric value for the lower limit of the integral with a default value of -1
upper	numeric value for the upper limit of the integral with a default value of 1
weighted	a boolean value which if true causes the ultraspherical weight function to be included in the integrand
...	other arguments passed to the give function

Details

The rule argument corresponds to an order $n$ Jacobi polynomial, weight function and interval $\left[ { - 1,1} \right]$. The lower and upper limits of the integral must be finite.

Value

The value of definite integral evaluated using Gauss Jacobi quadrature

Author(s)

Frederick Novomestky [email protected]

References

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

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

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

See Also

jacobi.p.quadrature.rules

Examples

###
### this example evaluates the quadrature function for
### the Jacobi P polynomials.  it computes the integral
### of the product for all pairs of orthogonal polynomials
### from order 0 to order 12.  the results are compared to
### the diagonal matrix of the inner products for the
### polynomials.  it also computes the integral of the product
### of all pairs of orthonormal polynomials from order 0
### to order 12.  the resultant matrix should be an identity matrix
###
###
### set the polynomial parameter
###
alpha <- 1
beta <- -0.6
###
### set the value for the maximum polynomial order
###
    n <- 12
###
### maximum order plus 1
###
    np1 <- n + 1
###
### function to construct the polynomial products by column
###
by.column.products <- function( c, p.list, p.p.list )
{
###
### function to construct the polynomial products by row
###
    by.row.products <- function( r, c, p.list )
    {
        row.column.product <- p.list[[r]] * p.list[[c]]
        return (row.column.product )
    }
    np1 <- length( p.list )
    row.list <- lapply( 1:np1, by.row.products, c, p.list )
    return( row.list )
}
###
### function construct the polynomial functions by column
###
by.column.functions <- function( c, p.p.products )
{
###
### function to construct the polynomial functions by row
###
    by.row.functions <- function( r, c, p.p.products )
    {
        row.column.function <- as.function( p.p.products[[r]][[c]] )
        return( row.column.function )
    }
    np1 <- length( p.p.products[[1]] )
    row.list <- lapply( 1:np1, by.row.functions, c, p.p.products )
    return( row.list )
}
###
### function to compute the integral of the polynomials by column
###
by.column.integrals <- function( c, p.p.functions )
{
###
### function to compute the integral of the polynomials by row
###
    by.row.integrals <- function( r, c, p.p.functions )
    {
        row.column.integral <- jacobi.p.quadrature(
            p.p.functions[[r]][[c]], order.np1.rule, alpha, beta )
        return( row.column.integral )
    }
    np1 <- length( p.p.functions[[1]] )
    row.vector <- sapply( 1:np1, by.row.integrals, c, p.p.functions )
    return( row.vector )
}
###
### construct a list of the Jacobi P orthogonal polynomials
###
p.list <- jacobi.p.polynomials( n, alpha, beta )
###
### construct the two dimensional list of pair products
### of polynomials
###
p.p.products <- lapply( 1:np1, by.column.products, p.list )
###
### compute the two dimensional list of functions
### corresponding to the polynomial products in
### the two dimensional list p.p.products
###
p.p.functions <- lapply( 1:np1, by.column.functions, p.p.products )
###
### get the rule table for the order np1 polynomial
###
rules <- jacobi.p.quadrature.rules( np1, alpha, beta )
order.np1.rule <- rules[[np1]]
###
### construct the square symmetric matrix containing
### the definite integrals over the default limits
### corresponding to the two dimensional list of
### polynomial functions
###
p.p.integrals <- sapply( 1:np1, by.column.integrals, p.p.functions )
###
### construct the diagonal matrix with the inner products
### of the orthogonal polynomials on the diagonal
###
p.p.inner.products <- diag( jacobi.p.inner.products( n,alpha, beta ) )
print( "Integral of cross products for the orthogonal polynomials " )
print( apply( p.p.integrals, 2, round, digits=6 ) )
print( apply( p.p.inner.products, 2, round, digits=6 ) )
###
### construct a list of the Jacobi P orthonormal polynomials
###
p.list <- jacobi.p.polynomials( n, alpha, beta, TRUE )
###
### construct the two dimensional list of pair products
### of polynomials
###
p.p.products <- lapply( 1:np1, by.column.products, p.list )
###
### compute the two dimensional list of functions
### corresponding to the polynomial products in
### the two dimensional list p.p.products
###
p.p.functions <- lapply( 1:np1, by.column.functions, p.p.products )
###
### get the rule table for the order np1 polynomial
###
rules <- jacobi.p.quadrature.rules( np1, alpha, beta, TRUE )
order.np1.rule <- rules[[np1]]
###
### construct the square symmetric matrix containing
### the definite integrals over the default limits
### corresponding to the two dimensional list of
### polynomial functions
###
p.p.integrals <- sapply( 1:np1, by.column.integrals, p.p.functions )
###
### display the matrix of integrals
###
print( "Integral of cross products for the orthonormal polynomials " )
print(apply( p.p.integrals, 2, round, digits=6 ) )

---

Create list of Jacobi quadrature rules

Description

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

Usage

jacobi.p.quadrature.rules(n,alpha,beta,normalized=FALSE)

Arguments

n	integer value for the highest order
alpha	numeric value for the first polynomial parameter
beta	numeric value for the second polynomial parameter
normalized	boolean value. if TRUE rules are for orthonormal polynomials, otherwise they are for orthgonal polynomials

Details

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

Value

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

1	Quadrature rule for the order 1 Jacobi polynomial
2	Quadrature rule for the order 2 Jacobi polynomial

...

n	Quadrature rule for the order $n$ Jacobi polynomial

Author(s)

Frederick Novomestky [email protected]

References

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

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

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

See Also

quadrature.rules

Examples

###
### construct the list of quadrature rules for
### the Jacobi orthogonal polynomials
### of orders 1 to 5
### alpha = 3 and beta = 2
###
orthogonal.rules <- jacobi.p.quadrature.rules( 5, 3, 2 )
print( orthogonal.rules )
###
### construct the list of quadrature rules for
### the Jacobi orthonormal polynomials
### of orders 1 to 5
### alpha = 3 and beta = 2
###
orthonormal.rules <- jacobi.p.quadrature.rules( 5, 3, 2, TRUE )
print( orthonormal.rules )

---

Perform Gauss Laguerre quadrature

Description

This function evaluates the integral of the given function between the lower and upper limits using the weight and abscissa values specified in the rule data frame. The quadrature formula uses the weight function for Laguerre polynomials.

Usage

laguerre.quadrature(functn, rule, lower = 0, upper = Inf, weighted = TRUE, ...)

Arguments

functn	an R function which should take a numeric argument x and possibly some parameters. The function returns a numerical vector value for the given argument x.
rule	a data frame containing the order $n$ Laguerre quadrature rule
lower	numeric value for the lower limit of the integral with a default value of 0
upper	numeric value for the upper limit of the integral with a default value of Inf
weighted	boolean value which if true causes the Laguerre weight function to be included in the integrand
...	other arguments passed to the give function

Details

The rule argument corresponds to an order $n$ Laguerre polynomial, weight function and interval $\left[ {0,\infty } \right]$. The one limit of the integral must be finite and the other must be infinite.

Value

The value of definite integral evaluated using Gauss Laguerre quadrature

Author(s)

Frederick Novomestky [email protected]

References

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

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

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

See Also

laguerre.quadrature.rules

Examples

###
### this example evaluates the quadrature function for
### the Laguerre polynomials.  it computes the integral
### of the product for all pairs of orthogonal polynomials
### from order 0 to order 12.  the results are compared to
### the diagonal matrix of the inner products for the
### polynomials.  it also computes the integral of the product
### of all pairs of orthonormal polynomials from order 0
### to order 12.  the resultant matrix should be an identity matrix
###
###
### set the value for the maximum polynomial order
###
    n <- 12
###
### maximum order plus 1
###
    np1 <- n + 1
###
### function to construct the polynomial products by column
###
by.column.products <- function( c, p.list, p.p.list )
{
###
### function to construct the polynomial products by row
###
    by.row.products <- function( r, c, p.list )
    {
        row.column.product <- p.list[[r]] * p.list[[c]]
        return (row.column.product )
    }
    np1 <- length( p.list )
    row.list <- lapply( 1:np1, by.row.products, c, p.list )
    return( row.list )
}
###
### function construct the polynomial functions by column
###
by.column.functions <- function( c, p.p.products )
{
###
### function to construct the polynomial functions by row
###
    by.row.functions <- function( r, c, p.p.products )
    {
        row.column.function <- as.function( p.p.products[[r]][[c]] )
        return( row.column.function )
    }
    np1 <- length( p.p.products[[1]] )
    row.list <- lapply( 1:np1, by.row.functions, c, p.p.products )
    return( row.list )
}
###
### function to compute the integral of the polynomials by column
###
by.column.integrals <- function( c, p.p.functions )
{
###
### function to compute the integral of the polynomials by row
###
    by.row.integrals <- function( r, c, p.p.functions )
    {
        row.column.integral <- laguerre.quadrature(
            p.p.functions[[r]][[c]], order.np1.rule )
        return( row.column.integral )
    }
    np1 <- length( p.p.functions[[1]] )
    row.vector <- sapply( 1:np1, by.row.integrals, c, p.p.functions )
    return( row.vector )
}
###
### construct a list of the Laguerre orthogonal polynomials
###
p.list <- laguerre.polynomials( n )
###
### construct the two dimensional list of pair products
### of polynomials
###
p.p.products <- lapply( 1:np1, by.column.products, p.list )
###
### compute the two dimensional list of functions
### corresponding to the polynomial products in
### the two dimensional list p.p.products
###
p.p.functions <- lapply( 1:np1, by.column.functions, p.p.products )
###
### get the rule table for the order np1 polynomial
###
rules <- laguerre.quadrature.rules( np1 )
order.np1.rule <- rules[[np1]]
###
### construct the square symmetric matrix containing
### the definite integrals over the default limits
### corresponding to the two dimensional list of
### polynomial functions
###
p.p.integrals <- sapply( 1:np1, by.column.integrals, p.p.functions )
###
### construct the diagonal matrix with the inner products
### of the orthogonal polynomials on the diagonal
###
p.p.inner.products <- diag( laguerre.inner.products( n ) )
print( "Integral of cross products for the orthogonal polynomials " )
print( apply( p.p.integrals, 2, round, digits=6 ) )
print( apply( p.p.inner.products, 2, round, digits=6 ) )
###
### construct a list of the Laguerre orthonormal polynomials
###
p.list <- laguerre.polynomials( n, TRUE )
###
### construct the two dimensional list of pair products
### of polynomials
###
p.p.products <- lapply( 1:np1, by.column.products, p.list )
###
### compute the two dimensional list of functions
### corresponding to the polynomial products in
### the two dimensional list p.p.products
###
p.p.functions <- lapply( 1:np1, by.column.functions, p.p.products )
###
### get the rule table for the order np1 polynomial
###
rules <- laguerre.quadrature.rules( np1, TRUE )
order.np1.rule <- rules[[np1]]
###
### construct the square symmetric matrix containing
### the definite integrals over the default limits
### corresponding to the two dimensional list of
### polynomial functions
###
p.p.integrals <- sapply( 1:np1, by.column.integrals, p.p.functions )
###
### display the matrix of integrals
###
print( "Integral of cross products for the orthonormal polynomials " )
print(apply( p.p.integrals, 2, round, digits=6 ) )

---

Create list of Laguerre quadrature rules

Description

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

Usage

laguerre.quadrature.rules(n,normalized=FALSE)

Arguments

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

Details

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

Value

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

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

...

n	Quadrature rule data frame for the order $n$ Laguerre polynomial

Author(s)

Frederick Novomestky [email protected]

References

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

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

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

See Also

quadrature.rules, laguerre.quadrature

Examples

###
### generate a list of the quadrature rule frames for
### the orthogonal Laguerre polynomials
### of orders 1 to 5
###
orthogonal.rules <- laguerre.quadrature.rules( 5 )
print( orthogonal.rules )
###
### generate a list of the quadrature rule frames for
### the orthonormal Laguerre polynomials
### of orders 1 to 5
###
orthonormal.rules <- laguerre.quadrature.rules( 5, TRUE )
print( orthonormal.rules )

---

Perform Gauss Legendre quadrature

Description

This function evaluates the integral of the given function between the lower and upper limits using the weight and abscissa values specified in the rule data frame. The quadrature formula uses the weight function for Legendre polynomials.

Usage

legendre.quadrature(functn, rule, lower=-1, upper=1, 
weighted = TRUE, ...)

Arguments

functn	an R function which should take a numeric argument x and possibly some parameters. The function returns a numerical vector value for the given argument x.
rule	a data frame containing the order $n$ Legendre quadrature rule
lower	numeric value for the lower limit of the integral
upper	numeric value for the upper limit of the integral
weighted	boolean value which if true causes the Legendre weight function to be included in the integrand
...	other arguments passed to the give function

Details

The rule argument corresponds to an order $n$ Legendre polynomial, weight function and interval $\left[ { - 1,1} \right]$. The lower and upper limits of the integral must be finite.

Value

The value of definite integral evaluated using Gauss Legendre quadrature

Author(s)

Frederick Novomestky [email protected]

References

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

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

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

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

See Also

legendre.quadrature.rules

Examples

###
### this example evaluates the quadrature function for
### the Legendre polynomials.  it computes the integral
### of the product for all pairs of orthogonal polynomials
### from order 0 to order 20.  the results are compared to
### the diagonal matrix of the inner products for the
### polynomials.  it also computes the integral of the product
### of all pairs of orthonormal polynomials from order 0
### to order 20.  the resultant matrix should be an identity matrix
###
###
### set the value for the maximum polynomial order
###
    n <- 20
###
### maximum order plus 1
###
    np1 <- n + 1
###
### function to construct the polynomial products by column
###
by.column.products <- function( c, p.list, p.p.list )
{
###
### function to construct the polynomial products by row
###
    by.row.products <- function( r, c, p.list )
    {
        row.column.product <- p.list[[r]] * p.list[[c]]
        return (row.column.product )
    }
    np1 <- length( p.list )
    row.list <- lapply( 1:np1, by.row.products, c, p.list )
    return( row.list )
}
###
### function construct the polynomial functions by column
###
by.column.functions <- function( c, p.p.products )
{
###
### function to construct the polynomial functions by row
###
    by.row.functions <- function( r, c, p.p.products )
    {
        row.column.function <- as.function( p.p.products[[r]][[c]] )
        return( row.column.function )
    }
    np1 <- length( p.p.products[[1]] )
    row.list <- lapply( 1:np1, by.row.functions, c, p.p.products )
    return( row.list )
}
###
### function to compute the integral of the polynomials by column
###
by.column.integrals <- function( c, p.p.functions )
{
###
### function to compute the integral of the polynomials by row
###
    by.row.integrals <- function( r, c, p.p.functions )
    {
        row.column.integral <- legendre.quadrature(
            p.p.functions[[r]][[c]], order.np1.rule )
        return( row.column.integral )
    }
    np1 <- length( p.p.functions[[1]] )
    row.vector <- sapply( 1:np1, by.row.integrals, c, p.p.functions )
    return( row.vector )
}
###
### construct a list of the Legendre orthogonal polynomials
###
p.list <- legendre.polynomials( n )
###
### construct the two dimensional list of pair products
### of polynomials
###
p.p.products <- lapply( 1:np1, by.column.products, p.list )
###
### compute the two dimensional list of functions
### corresponding to the polynomial products in
### the two dimensional list p.p.products
###
p.p.functions <- lapply( 1:np1, by.column.functions, p.p.products )
###
### get the rule table for the order np1 polynomial
###
rules <- legendre.quadrature.rules( np1 )
order.np1.rule <- rules[[np1]]
###
### construct the square symmetric matrix containing
### the definite integrals over the default limits
### corresponding to the two dimensional list of
### polynomial functions
###
p.p.integrals <- sapply( 1:np1, by.column.integrals, p.p.functions )
###
### construct the diagonal matrix with the inner products
### of the orthogonal polynomials on the diagonal
###
p.p.inner.products <- diag( legendre.inner.products( n ) )
print( "Integral of cross products for the orthogonal polynomials " )
print( apply( p.p.integrals, 2, round, digits=5 ) )
print( apply( p.p.inner.products, 2, round, digits=5 ) )
###
### construct a list of the Legendre orthonormal polynomials
###
p.list <- legendre.polynomials( n, TRUE )
###
### construct the two dimensional list of pair products
### of polynomials
###
p.p.products <- lapply( 1:np1, by.column.products, p.list )
###
### compute the two dimensional list of functions
### corresponding to the polynomial products in
### the two dimensional list p.p.products
###
p.p.functions <- lapply( 1:np1, by.column.functions, p.p.products )
###
### get the rule table for the order np1 polynomial
###
rules <- legendre.quadrature.rules( np1, TRUE )
order.np1.rule <- rules[[np1]]
###
### construct the square symmetric matrix containing
### the definite integrals over the default limits
### corresponding to the two dimensional list of
### polynomial functions
###
p.p.integrals <- sapply( 1:np1, by.column.integrals, p.p.functions )
###
### display the matrix of integrals
###
print( "Integral of cross products for the orthonormal polynomials " )
print(apply( p.p.integrals, 2, round, digits=5 ) )

---