JacobiGradientEvalSum

Evaluate finite sum of gradient of Jacobi polynomials.

Interface 1

INTERFACE
  MODULE PURE FUNCTION JacobiGradientEvalSum(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 JacobiGradientEvalSum
END INTERFACE

Interface 2

INTERFACE
  MODULE PURE FUNCTION JacobiGradientEvalSum(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 JacobiGradientEvalSum
END INTERFACE

Interface 3

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

Interface 4

INTERFACE
  MODULE PURE FUNCTION JacobiGradientEvalSum(n, alpha, beta, x, coeff, k) &
    & 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
    INTEGER(I4B), INTENT(IN) :: k
    !! kth order derivative
    REAL(DFP) :: ans(SIZE(x))
    !! Evaluate Jacobi polynomial of order n at point x
  END FUNCTION JacobiGradientEvalSum
END INTERFACE

Examples

️܀ See example

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

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

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 = JacobiGradientEvalSum(n=n, x=x, alpha=alpha, &
    & beta=beta, coeff=coeff)
  exact = LegendreGradientEval(n=0_I4B, x=x)
  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 = JacobiGradientEvalSum(n=n, x=x, alpha=alpha, &
    & beta=beta, coeff=coeff)
  exact = LegendreGradientEval(n=1_I4B, x=x)
  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 = JacobiGradientEvalSum(n=n, x=x, alpha=alpha, &
    & beta=beta, coeff=coeff)
  exact = LegendreGradientEval(n=3_I4B, x=x)
  call ok( ALL(SOFTEQ(ans, exact, tol )))

end program main

↢

️܀ See example

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

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

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 = JacobiGradientEvalSum(n=n, x=x, alpha=alpha, &
    & beta=beta, coeff=coeff)
  exact = LegendreGradientEval(n=0_I4B, x=x)
  call ok( SOFTEQ(ans, exact, tol ))

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

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

end program main

↢

️܀ See example

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

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

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 = JacobiGradientEvalSum(n=n, x=x, alpha=alpha, &
    & beta=beta, coeff=coeff, k=1)
  exact = LegendreGradientEval(n=0_I4B, x=x)
  call ok( SOFTEQ(ans, exact, tol ))

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

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

end program main

↢