LegendreGradientEvalSum

Evaluate finite sum of gradient of Legendre polynomials.

Interface 1

܀ Interface

INTERFACE
  MODULE PURE FUNCTION LegendreGradientEvalSum(n, x, coeff) RESULT(ans)
    INTEGER(I4B), INTENT(IN) :: n
    !! order of 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 Legendre polynomial of order n at point x
  END FUNCTION LegendreGradientEvalSum
END INTERFACE

️܀ See example

This example shows the usage of LegendreGradientEvalSum method.

This routine evaluates the gradient of sum of finite series of Legendre 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
  n = 3; coeff = [1, 0, 0, 0]
  x = -0.5_DFP
  ans = LegendreGradientEvalSum( n=n, x=x, coeff=coeff )
  exact = LegendreGradientEval( n=0_I4B, x=x )
  call ok( SOFTEQ(ans, exact, tol ))
  n = 3; coeff = [0, 1, 0, 0]
  x = -0.5_DFP
  ans = LegendreGradientEvalSum( n=n, x=x, coeff=coeff )
  exact = LegendreGradientEval( n=1_I4B, x=x )
  call ok( SOFTEQ(ans, exact, tol ))
  n = 3; coeff = [0, 0, 0, 1]
  x = 0.25_DFP
  ans = LegendreGradientEvalSum( n=n, x=x, coeff=coeff )
  exact = LegendreGradientEval( n=3_I4B, x=x )
  call ok( SOFTEQ(ans, exact, tol ))
  n = 3; coeff = [0, 0, 1, 2]
  x = -0.5_DFP
  ans = LegendreGradientEvalSum( n=n, x=x, coeff=coeff )
  exact = LegendreGradientEval( n=2_I4B, x=x) &
      & + 2.0_DFP * LegendreGradientEval( &
      & n=3_I4B, x=x)
  call ok( SOFTEQ(ans, exact, tol ))
end program main

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Interface 2

܀ Interface

INTERFACE
  MODULE PURE FUNCTION LegendreGradientEvalSum(n, x, coeff) &
    & RESULT(ans)
    INTEGER(I4B), INTENT(IN) :: n
    !! order of 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 Legendre polynomial of order n at point x
  END FUNCTION LegendreGradientEvalSum
END INTERFACE

️܀ See example

This example shows the usage of LegendreGradientEvalSum method.

This routine evaluates the gradient offinite sum of the Legendre 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
  type(string) :: astr
  n = 3
  x = [-0.5, 0.5, 0.25]
  coeff = [1, 0, 0, 0]
  ans = LegendreGradientEvalSum( n=n, x=x, coeff=coeff )
  exact = LegendreGradientEval( n=0_I4B, x=x )
  call ok( ALL(SOFTEQ(ans, exact, tol )))
  n = 3
  x = [-0.5, 0.5, 0.25]
  coeff = [0, 0, 0, 1]
  ans = LegendreGradientEvalSum( n=n, x=x, coeff=coeff )
  exact = LegendreGradientEval( n=3_I4B, x=x )
  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 = LegendreGradientEvalSum( n=n, x=x, coeff=coeff )
  exact = LegendreGradientEval( n=n, x=x )
  call ok( ALL(SOFTEQ(ans, exact, tol )))
end program main

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Interface 3

܀ Interface

INTERFACE
  MODULE PURE FUNCTION LegendreGradientEvalSum(n, x, coeff, k) RESULT(ans)
    INTEGER(I4B), INTENT(IN) :: n
    !! order of 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 Legendre polynomial of order n at point x
  END FUNCTION LegendreGradientEvalSum
END INTERFACE

️܀ See example

This example shows the usage of LegendreGradientEvalSum method.

This routine evaluates the kth gradient of sum of finite series of Legendre 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
  n = 3; coeff = [1, 0, 0, 0]
  x = -0.5_DFP
  ans = LegendreGradientEvalSum( n=n, x=x, coeff=coeff, k=1 )
  exact = LegendreGradientEval( n=0_I4B, x=x )
  call ok( SOFTEQ(ans, exact, tol ))
  n = 3; coeff = [0, 1, 0, 0]
  x = -0.5_DFP
  ans = LegendreGradientEvalSum( n=n, x=x, coeff=coeff, k=1 )
  exact = LegendreGradientEval( n=1_I4B, x=x )
  call ok( SOFTEQ(ans, exact, tol ))
  n = 3; coeff = [0, 0, 0, 1]
  x = 0.25_DFP
  ans = LegendreGradientEvalSum( n=n, x=x, coeff=coeff, k=1 )
  exact = LegendreGradientEval( n=3_I4B, x=x )
  call ok( SOFTEQ(ans, exact, tol ))
  n = 3; coeff = [0, 0, 1, 2]
  x = -0.5_DFP
  ans = LegendreGradientEvalSum( n=n, x=x, coeff=coeff, k = 1 )
  exact = LegendreGradientEval( n=2_I4B, x=x ) &
      & + 2.0_DFP * LegendreGradientEval( &
      & n=3_I4B, x=x )
  call ok( SOFTEQ(ans, exact, tol ))
end program main

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Interface 4

܀ Interface

INTERFACE
  MODULE PURE FUNCTION LegendreGradientEvalSum(n, x, coeff, k) RESULT(ans)
    INTEGER(I4B), INTENT(IN) :: n
    !! order of 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 Legendre polynomial of order n at point x
  END FUNCTION LegendreGradientEvalSum
END INTERFACE

️܀ See example

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