UltrasphericalEvalSum

Evaluate the finite sum of Ultraspherical polynomials at point x.

Interface 1

܀ Interface

INTERFACE
  MODULE PURE FUNCTION UltrasphericalEvalSum(n, lambda, x, coeff) &
    & RESULT(ans)
    INTEGER(I4B), INTENT(IN) :: n
    !! order of polynomial
    REAL(DFP), INTENT(IN) :: lambda
    !! alpha of Ultraspherical polynomial
    REAL(DFP), INTENT(IN) :: x
    !! point
    REAL(DFP), INTENT(IN) :: coeff(0:n)
    !! Coefficient of finite sum, size = n+1
    REAL(DFP) :: ans
    !! Evaluate Ultraspherical polynomial of order n at point x
  END FUNCTION UltrasphericalEvalSum
END INTERFACE

️܀ See example

This example shows the usage of UltrasphericalEvalSum method.

This routine evaluates sum of finite series of Ultraspherical polynomials of order upto n, at a single point.

$$s(x) = \sum_{n=0}^{n=N}{c_n P_{n}^{(\lambda)}(x)}$$

program main
  use easifembase
  implicit none
  integer( i4b ) :: n
  real( dfp ) :: ans, x, exact
  real( dfp ), allocatable :: coeff(:)
  real( dfp ), parameter :: tol=1.0E-10, lambda=0.5_DFP

  n = 3; coeff = [1, 0, 0, 0]
  x = -0.5_DFP
  ans = UltrasphericalEvalSum( n=n, x=x, &
      & lambda=lambda, coeff=coeff )
  exact = UltrasphericalEval( n=0_I4B, x=x, &
      & lambda=lambda )
  call ok( SOFTEQ(ans, exact, tol ))

  n = 3; coeff = [0, 1, 0, 0]
  x = -0.5_DFP
  ans = UltrasphericalEvalSum( n=n, x=x, &
      & lambda=lambda, coeff=coeff )
  exact = UltrasphericalEval( n=1_I4B, x=x, &
      & lambda=lambda )
  call ok( SOFTEQ(ans, exact, tol ))

  n = 3; coeff = [0, 0, 0, 1]
  x = 0.25_DFP
  ans = UltrasphericalEvalSum( n=n, x=x, &
      & lambda=lambda, coeff=coeff )
  exact = UltrasphericalEval( n=3_I4B, x=x, &
      & lambda=lambda )
  call ok( SOFTEQ(ans, exact, tol ))

  n = 3; coeff = [0, 0, 1, 2]
  x = -0.5_DFP
  ans = UltrasphericalEvalSum( n=n, x=x, &
      & lambda=lambda, coeff=coeff )
  exact = UltrasphericalEval( n=2_I4B, x=x, &
      & lambda=lambda ) &
      & + 2.0_DFP * UltrasphericalEval( n=3_I4B, x=x, &
      & lambda=lambda )
  call ok( SOFTEQ(ans, exact, tol ))

end program main

↢

Interface 2

܀ Interface

INTERFACE
  MODULE PURE FUNCTION UltrasphericalEvalSum(n, lambda, x, coeff) RESULT(ans)
    INTEGER(I4B), INTENT(IN) :: n
    !! order of polynomial
    REAL(DFP), INTENT(IN) :: lambda
    !! alpha of Ultraspherical polynomial
    REAL(DFP), INTENT(IN) :: x(:)
    !! point
    REAL(DFP), INTENT(IN) :: coeff(0:n)
    !! Coefficient of finite sum, size = n+1
    REAL(DFP) :: ans(SIZE(x))
    !! Evaluate Ultraspherical polynomial of order n at point x
  END FUNCTION UltrasphericalEvalSum
END INTERFACE

️܀ See example

This example shows the usage of UltrasphericalEvalSum method.

This routine evaluates the finite sum of the Ultraspherical polynomials of order upto n, at several points.

program main
  use easifembase
  implicit none
  integer( i4b ) :: n
  real( dfp ), allocatable :: ans(:), x(:), exact(:), &
    & coeff(:)
  real( dfp ), parameter :: tol=1.0E-10, lambda=0.5_DFP
  type(string) :: astr

  n = 3
  x = [-0.5, 0.5, 0.25]
  coeff = [1, 0, 0, 0]
  ans = UltrasphericalEvalSum( n=n, x=x, &
      & lambda=lambda, coeff=coeff )
  exact = UltrasphericalEval( n=0_I4B, x=x, &
      & lambda=lambda )
  call ok( ALL(SOFTEQ(ans, exact, tol )))

  n = 3
  x = [-0.5, 0.5, 0.25]
  coeff = [0, 0, 0, 1]
  ans = UltrasphericalEvalSum( n=n, x=x, &
      & lambda=lambda, coeff=coeff )
  exact = UltrasphericalEval( n=3_I4B, x=x, &
      & lambda=lambda )
  call ok( ALL(SOFTEQ(ans, exact, tol )))

  n = 100
  x = [-0.5, 0.5, 0.25]
  coeff = zeros(n+1, 0.0_DFP)
  coeff( n+1 ) = 1.0_DFP
  ans = UltrasphericalEvalSum( n=n, x=x, &
      & lambda=lambda, coeff=coeff )
  exact = UltrasphericalEval( n=n, x=x, &
      & lambda=lambda )
  call ok( ALL(SOFTEQ(ans, exact, tol )))

end program main

↢