JacobiDMatrix

Interface

INTERFACE
  MODULE PURE FUNCTION JacobiDMatrix(n, alpha, beta, x, quadType) RESULT(ans)
    INTEGER(I4B), INTENT(IN) :: n
      !! order of Jacobi polynomial
    REAL(DFP), INTENT(IN) :: alpha
      !! alpha > -1.0
    REAL(DFP), INTENT(IN) :: beta
      !! beta > -1.0
    REAL(DFP), INTENT(IN) :: x(0:n)
      !! quadrature points
    INTEGER(I4B), INTENT(IN) :: quadType
      !! Gauss and GaussLobatto
    REAL(DFP) :: ans(0:n, 0:n)
      !! D matrix
  END FUNCTION JacobiDMatrix
END INTERFACE

Example

️܀ See example

• This example shows the usage of JacobiDMatrix method.
• This routine yields the differentiation matrix, $D$.

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

program main
  use easifembase
  use easifemclasses
  implicit none
  integer( i4b ) :: n, ii
  real(dfp), allocatable :: fval( : ), pt( : ), wt(:), &
    & f1val(:), error(:, :), D(:,:)
  real( dfp ), parameter :: alpha=0.0_DFP, beta=0.0_DFP, tol=1.0E-10
  type(string) :: astr
  integer( i4b ), parameter :: quadType = GaussLobatto

note

In this example we consider

$$f(x) = sin(4\pi x)$$

!!
error = zeros(30, 2, 1.0_DFP)
!!
DO n = 1, SIZE(error,1)
  !!
  call reallocate( pt, n+1, wt, n+1, fval, n+1 )
  !!
  call JacobiQuadrature( n=n+1, alpha=alpha, beta=beta, &
    & pt=pt, wt=wt, quadType=quadType )
  !!
  fval = func1(pt)
  !!
  D = JacobiDMatrix(n=n, alpha=alpha, &
    & beta=beta, x=pt, quadType=quadType)
  !!
  f1val = dfunc1(pt)
  !!
  error(n,1) = n
  error(n,2) = NORM2( ABS(f1val-MATMUL(D,fval)) )
  !!
END DO
!!
CALL display( MdEncode(error), "error=")

See results

error=

order(n)	MAX(err)
1	17.772
2	21.766
5	30.677
10	30.737
15	5.9239
20	8.60174E-02
25	1.11384E-04
30	1.93772E-07

contains
elemental function func1(x) result(ans)
  real(dfp), intent(in) :: x
  real(dfp) :: ans
  ans = SIN(4.0_DFP * pi * x)
end function func1
!!
elemental function dfunc1(x) result(ans)
  real(dfp), intent(in) :: x
  real(dfp) :: ans
  ans = 4.0_DFP * pi * COS(4.0_DFP * pi * x)
end function dfunc1

end program main

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