JacobiEvalSum

Evaluate the finite sum of Jacobi polynomials at point x.

Interface 1

INTERFACE
  MODULE PURE FUNCTION JacobiEvalSum(n, alpha, beta, x, coeff) RESULT(ans)
    INTEGER(I4B), INTENT(IN) :: n
    !! order of polynomial
    REAL(DFP), INTENT(IN) :: alpha
    !! alpha of Jacobi polynomial
    REAL(DFP), INTENT(IN) :: beta
    !! beta of Jacobi 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 Jacobi polynomial of order n at point x
  END FUNCTION JacobiEvalSum
END INTERFACE

Interface 2

INTERFACE
  MODULE PURE FUNCTION JacobiEvalSum(n, alpha, beta, x, coeff) RESULT(ans)
    INTEGER(I4B), INTENT(IN) :: n
    !! order of polynomial
    REAL(DFP), INTENT(IN) :: alpha
    !! alpha of Jacobi polynomial
    REAL(DFP), INTENT(IN) :: beta
    !! beta of Jacobi 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 Jacobi polynomial of order n at point x
  END FUNCTION JacobiEvalSum
END INTERFACE

Example

️܀ See example

This example shows the usage of JacobiEvalSum method. This routine evaluates the finite sum of Jacobi polynomial of order upto n at a given point.

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

In this example $\alpha=\beta=0.0$ (that is, proportional to Legendre polynomial)

program main
  use easifembase
  implicit none
  integer( i4b ) :: n
  real( dfp ) :: ans, x, exact
  real(dfp), allocatable :: coeff(:)
  real( dfp ), parameter :: alpha=0.0_DFP, beta=0.0_DFP, tol=1.0E-10
  type(string) :: astr

  n = 3
  coeff = [1.0, 0.0, 0.0, 0.0]
  x = 0.5_DFP;
  ans = JacobiEvalSum(n=n, x=x, alpha=alpha, &
    & beta=beta, coeff=coeff)
  exact = JacobiEval(n=0_I4B, x=x, alpha=alpha,&
    & beta=beta)
  call ok( SOFTEQ(ans, exact, tol ))

  n = 3
  coeff = [0.0, 1.0, 0.0, 0.0]
  x = 0.5_DFP;
  ans = JacobiEvalSum(n=n, x=x, alpha=alpha, &
    & beta=beta, coeff=coeff)
  exact = JacobiEval(n=1_I4B, x=x, alpha=alpha,&
    & beta=beta)
  call ok( SOFTEQ(ans, exact, tol ))

  n = 3
  coeff = [0.0, 0.0, 0.0, 1.0]
  x = 0.5_DFP;
  ans = JacobiEvalSum(n=n, x=x, alpha=alpha, &
    & beta=beta, coeff=coeff)
  exact = JacobiEval(n=3_I4B, x=x, alpha=alpha,&
    & beta=beta)
  call ok( SOFTEQ(ans, exact, tol ))

  n = 3
  coeff = [1.0, 2.0, 3.0, 4.0]
  x = 0.5_DFP;
  ans = JacobiEvalSum(n=n, x=x, alpha=alpha, &
    & beta=beta, coeff=coeff)
  exact = &
  &  1.0*JacobiEval(n=0_I4B, x=x, alpha=alpha,beta=beta) &
  &+ 2.0*JacobiEval(n=1_I4B, x=x, alpha=alpha,beta=beta) &
  &+ 3.0*JacobiEval(n=2_I4B, x=x, alpha=alpha,beta=beta) &
  &+ 4.0*JacobiEval(n=3_I4B, x=x, alpha=alpha,beta=beta)
  call ok( SOFTEQ(ans, exact, tol ))

end program main

↢

️܀ See example

This example shows the usage of JacobiEvalSum method. This routine evaluates the finite sum of Jacobi polynomial of order upto n at several points.

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

In this example $\alpha=\beta=0.0$ (that is, proportional to Legendre polynomial)

program main
  use easifembase
  implicit none
  integer( i4b ) :: n
  real(dfp), allocatable :: coeff(:), ans(:), x(:), exact(:)
  real( dfp ), parameter :: alpha=0.0_DFP, beta=0.0_DFP, tol=1.0E-10
  type(string) :: astr

  n = 3
  coeff = [1.0, 0.0, 0.0, 0.0]
  x = [0.5_DFP, -0.5_DFP];
  ans = JacobiEvalSum(n=n, x=x, alpha=alpha, &
    & beta=beta, coeff=coeff)
  exact = JacobiEval(n=0_I4B, x=x, alpha=alpha,&
    & beta=beta)
  call ok( ALL(SOFTEQ(ans, exact, tol )))

  n = 3
  coeff = [0.0, 1.0, 0.0, 0.0]
  x = [0.5_DFP, -0.5_DFP];
  ans = JacobiEvalSum(n=n, x=x, alpha=alpha, &
    & beta=beta, coeff=coeff)
  exact = JacobiEval(n=1_I4B, x=x, alpha=alpha,&
    & beta=beta)
  call ok( ALL(SOFTEQ(ans, exact, tol )))

  n = 3
  coeff = [0.0, 0.0, 0.0, 1.0]
  x = [0.5_DFP, -0.5_DFP];
  ans = JacobiEvalSum(n=n, x=x, alpha=alpha, &
    & beta=beta, coeff=coeff)
  exact = JacobiEval(n=3_I4B, x=x, alpha=alpha,&
    & beta=beta)
  call ok( ALL(SOFTEQ(ans, exact, tol )))

end program main

↢