UltrasphericalGradientEvalSum

Evaluate finite sum of gradient of Ultraspherical polynomials.

Interface 1

܀ Interface

INTERFACE
  MODULE PURE FUNCTION UltrasphericalGradientEvalSum(n, lambda, x, &
    &  coeff) RESULT(ans)
    INTEGER(I4B), INTENT(IN) :: n
    !! order of polynomial
    REAL(DFP), INTENT(IN) :: lambda
    !! lambda 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 UltrasphericalGradientEvalSum
END INTERFACE

️܀ See example

This example shows the usage of UltrasphericalGradientEvalSum method.

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

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

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 = UltrasphericalGradientEvalSum( n=n, x=x, &
      & lambda=lambda, coeff=coeff )
  exact = UltrasphericalGradientEval( 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 = UltrasphericalGradientEvalSum( n=n, x=x, &
      & lambda=lambda, coeff=coeff )
  exact = UltrasphericalGradientEval( 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 = UltrasphericalGradientEvalSum( n=n, x=x, &
      & lambda=lambda, coeff=coeff )
  exact = UltrasphericalGradientEval( 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 = UltrasphericalGradientEvalSum( n=n, x=x, &
      & lambda=lambda, coeff=coeff )
  exact = UltrasphericalGradientEval( n=2_I4B, x=x, &
      & lambda=lambda ) &
      & + 2.0_DFP * UltrasphericalGradientEval( &
      & n=3_I4B, x=x, &
      & lambda=lambda )
  call ok( SOFTEQ(ans, exact, tol ))
end program main

↢

Interface 2

܀ Interface

INTERFACE
  MODULE PURE FUNCTION UltrasphericalGradientEvalSum(n, lambda, x, coeff) &
    & RESULT(ans)
    INTEGER(I4B), INTENT(IN) :: n
    !! order of polynomial
    REAL(DFP), INTENT(IN) :: lambda
    !! lambda 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 UltrasphericalGradientEvalSum
END INTERFACE

️܀ See example

This example shows the usage of UltrasphericalGradientEvalSum method.

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

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

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 = UltrasphericalGradientEvalSum( n=n, x=x, &
      & lambda=lambda, coeff=coeff )
  exact = UltrasphericalGradientEval( 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 = UltrasphericalGradientEvalSum( n=n, x=x, &
      & lambda=lambda, coeff=coeff )
  exact = UltrasphericalGradientEval( 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 = UltrasphericalGradientEvalSum( n=n, x=x, &
      & lambda=lambda, coeff=coeff )
  exact = UltrasphericalGradientEval( n=n, x=x, &
      & lambda=lambda )
  call ok( ALL(SOFTEQ(ans, exact, tol )))
end program main

↢

Interface 3

܀ Interface

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

️܀ See example

This example shows the usage of UltrasphericalGradientEvalSum method.

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

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

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 = UltrasphericalGradientEvalSum( n=n, x=x, &
      & lambda=lambda, coeff=coeff, k=1 )
  exact = UltrasphericalGradientEval( 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 = UltrasphericalGradientEvalSum( n=n, x=x, &
      & lambda=lambda, coeff=coeff, k=1 )
  exact = UltrasphericalGradientEval( 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 = UltrasphericalGradientEvalSum( n=n, x=x, &
      & lambda=lambda, coeff=coeff, k=1 )
  exact = UltrasphericalGradientEval( 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 = UltrasphericalGradientEvalSum( n=n, x=x, &
      & lambda=lambda, coeff=coeff, k = 1 )
  exact = UltrasphericalGradientEval( n=2_I4B, x=x, &
      & lambda=lambda ) &
      & + 2.0_DFP * UltrasphericalGradientEval( &
      & n=3_I4B, x=x, &
      & lambda=lambda )
  call ok( SOFTEQ(ans, exact, tol ))
end program main

↢

Interface 4

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