Structure

Datatype for creating a space of Jacobi Polynomials.

TYPE, EXTENDS(AbstractOrthopolSpace1D_) :: JacobiSpace1D_
  PRIVATE
  REAL(DFP) :: alpha = 0.0_DFP
  REAL(DFP) :: beta = 0.0_DF

Methods

JacobiSpace1D function

Constructor for JacobiSpace1D

Interface:

INTERFACE
  MODULE FUNCTION JacobiSpace1D(alpha, beta) RESULT(ans)
    REAL(DFP), INTENT(IN) :: alpha
    !! alpha > -1.0
    REAL(DFP), INTENT(IN) :: beta
    !! beta > -1.0
    TYPE(JacobiSpace1D_) :: ans
    !! Chebyshev Basis of first kind
  END FUNCTION JacobiSpace1D
END INTERFACE

!!! example "Examples"

[[JacobiSpace1D_test_1]]

JacobiSpace1D_Pointer

Returns the pointer to the newly created instance of JacobiSpace1D.

Interface:

INTERFACE
  MODULE FUNCTION JacobiSpace1D_Pointer(alpha, beta) &
    & RESULT(ans)
    REAL(DFP), INTENT(IN) :: alpha
    !! alpha > -1.0
    REAL(DFP), INTENT(IN) :: beta
    !! beta > -1.0
    CLASS(JacobiSpace1D_), POINTER :: ans
    !! Jacobi polynomial of order = 0 to n
  END FUNCTION JacobiSpace1D_Pointer
END INTERFACE

!!! example "Examples"

[[JacobiSpace1D_test_1]]

Deallocate

Deallocate the object

INTERFACE
  MODULE SUBROUTINE Deallocate(obj)
    CLASS(JacobiSpace1D_), INTENT(INOUT) :: obj
  END SUBROUTINE Deallocate
END INTERFACE

!!! example "Examples"

[[JacobiSpace1D_test_1]]

Display

Display the contents

INTERFACE
  MODULE SUBROUTINE Display(obj, msg, unitno)
    CLASS(JacobiSpace1D_), INTENT(INOUT) :: obj
    CHARACTER(LEN=*), INTENT(IN) :: msg
    INTEGER(I4B), OPTIONAL, INTENT(IN) :: unitno
  END SUBROUTINE Display
END INTERFACE

!!! example "Examples"

[[JacobiSpace1D_test_1]]

GetParam

Returns $\alpha, \beta$ .

INTERFACE
  MODULE PURE FUNCTION Orthopol_GetParam(obj) RESULT(ans)
    CLASS(JacobiSpace1D_), INTENT(IN) :: obj
    REAL(DFP) :: ans(2)
      !! ans(1) = alpha
      !! ans(2) = beta
  END FUNCTION Orthopol_GetParam
END INTERFACE

!!! example "Examples"

[[JacobiSpace1D_test_1]]

GetRecurrenceCoeff

Returns the recurrence coefficient of monic polynomials

INTERFACE
  MODULE PURE FUNCTION GetRecurrenceCoeff(obj, n) RESULT(ans)
    CLASS(JacobiSpace1D_), INTENT(IN) :: obj
    INTEGER(I4B), INTENT(IN) :: n
    REAL(DFP) :: ans(0:n - 1, 1:2)
    !! ans(:,1) = alpha
    !! ans(:,2) = beta
  END FUNCTION GetRecurrenceCoeff
END INTERFACE

!!! example "Examples"

[[JacobiSpace1D_test_2]]

GetRecurrenceCoeff2

Returns the recurrence coefficient of standard polynomials.

p_{n+1} = (a_{n}x+b_{n})p_{n}-c_{n}p_{n-1}

INTERFACE
  MODULE PURE FUNCTION GetRecurrenceCoeff2(obj, n) RESULT(ans)
    CLASS(JacobiSpace1D_), INTENT(IN) :: obj
    INTEGER(I4B), INTENT(IN) :: n
    REAL(DFP) :: ans(0:n - 1, 1:3)
    !! ans(:,1) = a,
    !! ans(:,2) = b
    !! ans(:,3) = c
  END FUNCTION GetRecurrenceCoeff2
END INTERFACE

!!! example "Examples"

[[JacobiSpace1D_test_2]]

GetAlpha

Returns the recurrence coefficient $\alpha$ of monic polynomials

INTERFACE
  MODULE PURE FUNCTION GetAlpha(obj, n) RESULT(ans)
    CLASS(JacobiSpace1D_), INTENT(IN) :: obj
    INTEGER(I4B), INTENT(IN) :: n
    REAL(DFP) :: ans
  END FUNCTION GetAlpha
END INTERFACE

!!! example "Examples"

[[JacobiSpace1D_test_2]]

GetBeta

Returns the recurrence coefficient beta of monic polynomials

INTERFACE
  MODULE PURE FUNCTION GetBeta(obj, n) RESULT(ans)
    CLASS(JacobiSpace1D_), INTENT(IN) :: obj
    INTEGER(I4B), INTENT(IN) :: n
    REAL(DFP) :: ans
  END FUNCTION GetBeta
END INTERFACE

!!! example "Examples"

[[JacobiSpace1D_test_2]]

GetLeadingCoeff

Returns the leading coeff of orthopol.

INTERFACE
  MODULE PURE FUNCTION GetLeadingCoeff(obj, n) RESULT(ans)
    CLASS(JacobiSpace1D_), INTENT(IN) :: obj
    INTEGER(I4B), INTENT(IN) :: n
    REAL(DFP) :: ans
  END FUNCTION GetLeadingCoeff
END INTERFACE

!!! example "Examples"

[[JacobiSpace1D_test_3]]

GetLeadingCoeffRatio

Returns the ration of leading coeff of orthopol (n+1)/n

INTERFACE
  MODULE PURE FUNCTION GetLeadingCoeffRatio(obj, n) RESULT(ans)
    CLASS(JacobiSpace1D_), INTENT(IN) :: obj
    INTEGER(I4B), INTENT(IN) :: n
    REAL(DFP) :: ans
  END FUNCTION GetLeadingCoeffRatio
END INTERFACE

!!! example "Examples"

[[JacobiSpace1D_test_3]]

GetNormSqr

Returns the ration of leading coeff of orthopol (n+1)/n

INTERFACE
  MODULE PURE FUNCTION GetNormSqr(obj, n) RESULT(ans)
    CLASS(JacobiSpace1D_), INTENT(IN) :: obj
    INTEGER(I4B), INTENT(IN) :: n
    REAL(DFP) :: ans
  END FUNCTION GetNormSqr
END INTERFACE

!!! example "Examples"

[[JacobiSpace1D_test_3]]

GetNormSqr2

Returns the ratio of leading coeff of orthopol (n+1)/n

INTERFACE
  MODULE PURE FUNCTION GetNormSqr2(obj, n) RESULT(ans)
    CLASS(JacobiSpace1D_), INTENT(IN) :: obj
    INTEGER(I4B), INTENT(IN) :: n
    REAL(DFP) :: ans(0:n)
  END FUNCTION GetNormSqr2
END INTERFACE

!!! example "Examples"

[[JacobiSpace1D_test_3]]

GetNormSqrRatio

Returns the ratio of square of norm of orthopol (n+1)/n

INTERFACE
  MODULE PURE FUNCTION GetNormSqrRatio(obj, n) RESULT(ans)
    CLASS(JacobiSpace1D_), INTENT(IN) :: obj
    INTEGER(I4B), INTENT(IN) :: n
    REAL(DFP) :: ans
  END FUNCTION GetNormSqrRatio
END INTERFACE

!!! example "Examples"

[[JacobiSpace1D_test_3]]

GetZeros

Returns the zeros of orthopol

INTERFACE
  MODULE FUNCTION GetZeros(obj, n) RESULT(ans)
    CLASS(JacobiSpace1D_), INTENT(IN) :: obj
    INTEGER(I4B), INTENT(IN) :: n
    REAL(DFP) :: ans(n)
  END FUNCTION GetZeros
END INTERFACE

!!! example "Examples"

[[JacobiSpace1D_test_4]]

GetQuadrature

Returns the zeros of orthopol

INTERFACE
  MODULE FUNCTION GetQuadrature(obj, n, quadType) RESULT(ans)
    CLASS(JacobiSpace1D_), INTENT(IN) :: obj
    INTEGER(I4B), INTENT(IN) :: n
    !! number of points
    INTEGER(I4B), INTENT(IN) :: quadType
    !! Gauss, GaussRadauLeft, GaussRadauRight, GaussLobatto
    REAL(DFP) :: ans(n, 2)
    !! ans(:,1) = points
    !! ans(:,2) = weights
  END FUNCTION GetQuadrature
END INTERFACE

EvalAll

Evaluates the orthopol (0 to n) at single point.

INTERFACE
  MODULE PURE FUNCTION EvalAll(obj, n, x) RESULT(ans)
    CLASS(JacobiSpace1D_), INTENT(IN) :: obj
    INTEGER(I4B), INTENT(IN) :: n
    !! order of orthopol
    REAL(DFP), INTENT(IN) :: x
    !! point of evaluation
    REAL(DFP) :: ans(n + 1)
  END FUNCTION EvalAll
END INTERFACE

Evaluates the orthopol (0 to n) at several points.

INTERFACE
  MODULE PURE FUNCTION EvalAll(obj, n, x) RESULT(ans)
    CLASS(JacobiSpace1D_), INTENT(IN) :: obj
    INTEGER(I4B), INTENT(IN) :: n
    !! order of orthopol
    REAL(DFP), INTENT(IN) :: x(:)
    !! point of evaluation
    REAL(DFP) :: ans(SIZE(x), n + 1)
    !! Evaluate orthopol of order = 0 to n (total n+1) at point x
  END FUNCTION EvalAll
END INTERFACE

Eval

Evaluates the orthopol (order = n ) at single point.

INTERFACE
  MODULE PURE FUNCTION Eval(obj, n, x) RESULT(ans)
    CLASS(JacobiSpace1D_), INTENT(IN) :: obj
    INTEGER(I4B), INTENT(IN) :: n
    !! order of orthopol
    REAL(DFP), INTENT(IN) :: x
    !! point of evaluation
    REAL(DFP) :: ans
  END FUNCTION Eval
END INTERFACE

Evaluates the orthopol (order=n) at several points.

INTERFACE
  MODULE PURE FUNCTION Eval(obj, n, x) RESULT(ans)
    CLASS(JacobiSpace1D_), INTENT(IN) :: obj
    INTEGER(I4B), INTENT(IN) :: n
    !! order of orthopol
    REAL(DFP), INTENT(IN) :: x(:)
    !! point of evaluation
    REAL(DFP) :: ans(SIZE(x))
    !! Evaluate orthopol of order = n at several points
  END FUNCTION Eval
END INTERFACE

EvalSum

Evaluates the orthopol series at single point.

INTERFACE
  MODULE PURE FUNCTION EvalSum(obj, n, x, coeff) RESULT(ans)
    CLASS(JacobiSpace1D_), INTENT(IN) :: obj
    INTEGER(I4B), INTENT(IN) :: n
    !! order of orthopol
    REAL(DFP), INTENT(IN) :: x
    !! point of evaluation
    REAL(DFP), INTENT(IN) :: coeff(0:n)
    !! Coefficient of finite sum
    REAL(DFP) :: ans
  END FUNCTION EvalSum
END INTERFACE

Evaluates the orthopol series at several points.

INTERFACE
  MODULE PURE FUNCTION EvalSum(obj, n, x, coeff) RESULT(ans)
    CLASS(JacobiSpace1D_), INTENT(IN) :: obj
    INTEGER(I4B), INTENT(IN) :: n
    !! order of orthopol
    REAL(DFP), INTENT(IN) :: x(:)
    !! point of evaluation
    REAL(DFP), INTENT(IN) :: coeff(0:n)
    !! Coefficient of finite sum
    REAL(DFP) :: ans(SIZE(x))
  END FUNCTION EvalSum
END INTERFACE

GradientEvalAll

Evaluates the gradient of orthopol (0 to n) at single point.

INTERFACE
  MODULE PURE FUNCTION GradientEvalAll(obj, n, x) RESULT(ans)
    CLASS(JacobiSpace1D_), INTENT(IN) :: obj
    INTEGER(I4B), INTENT(IN) :: n
    !! order of orthopol
    REAL(DFP), INTENT(IN) :: x
    !! point of evaluation
    REAL(DFP) :: ans(n + 1)
  END FUNCTION GradientEvalAll
END INTERFACE

Evaluates the gradient of orthopol (0 to n) at several points.

INTERFACE
  MODULE PURE FUNCTION GradientEvalAll(obj, n, x) RESULT(ans)
    CLASS(JacobiSpace1D_), INTENT(IN) :: obj
    INTEGER(I4B), INTENT(IN) :: n
    !! order of orthopol
    REAL(DFP), INTENT(IN) :: x(:)
    !! point of evaluation
    REAL(DFP) :: ans(SIZE(x), n + 1)
    !! Evaluate orthopol of order = 0 to n (total n+1) at point x
  END FUNCTION GradientEvalAll
END INTERFACE

GradientEval

Evaluates the gradient orthopol (order = n ) at single point.

INTERFACE
  MODULE PURE FUNCTION GradientEval(obj, n, x) RESULT(ans)
    CLASS(JacobiSpace1D_), INTENT(IN) :: obj
    INTEGER(I4B), INTENT(IN) :: n
    !! order of orthopol
    REAL(DFP), INTENT(IN) :: x
    !! point of evaluation
    REAL(DFP) :: ans
  END FUNCTION GradientEval
END INTERFACE

Evaluates the gradient orthopol (order=n) at several points.

NTERFACE
  MODULE PURE FUNCTION GradientEval(obj, n, x) RESULT(ans)
    CLASS(JacobiSpace1D_), INTENT(IN) :: obj
    INTEGER(I4B), INTENT(IN) :: n
    !! order of orthopol
    REAL(DFP), INTENT(IN) :: x(:)
    !! point of evaluation
    REAL(DFP) :: ans(SIZE(x))
    !! Evaluate orthopol of order = n at several points
  END FUNCTION GradientEval
END INTERFACE

GradientEvalSum

Evaluates the gradient of orthopol series at single point.

INTERFACE
  MODULE PURE FUNCTION GradientEvalSum(obj, n, x, coeff) RESULT(ans)
    CLASS(JacobiSpace1D_), INTENT(IN) :: obj
    INTEGER(I4B), INTENT(IN) :: n
    !! order of orthopol
    REAL(DFP), INTENT(IN) :: x
    !! point of evaluation
    REAL(DFP), INTENT(IN) :: coeff(0:n)
    !! Coefficient of finite sum
    REAL(DFP) :: ans
  END FUNCTION GradientEvalSum
END INTERFACE

Evaluates the gradient of orthopol series at several points.

INTERFACE
  MODULE PURE FUNCTION GradientEvalSum(obj, n, x, coeff) RESULT(ans)
    CLASS(JacobiSpace1D_), INTENT(IN) :: obj
    INTEGER(I4B), INTENT(IN) :: n
    !! order of orthopol
    REAL(DFP), INTENT(IN) :: x(:)
    !! point of evaluation
    REAL(DFP), INTENT(IN) :: coeff(0:n)
    !! Coefficient of finite sum
    REAL(DFP) :: ans(SIZE(x))
  END FUNCTION GradientEvalSum
END INTERFACE

Evaluates the kth order gradient of orthopol series at single point.

INTERFACE
  MODULE PURE FUNCTION GradientEvalSum(obj, n, x, coeff, k) &
    & RESULT(ans)
    CLASS(JacobiSpace1D_), INTENT(IN) :: obj
    INTEGER(I4B), INTENT(IN) :: n
    !! order of orthopol
    REAL(DFP), INTENT(IN) :: x
    !! point of evaluation
    REAL(DFP), INTENT(IN) :: coeff(0:n)
    !! Coefficient of finite sum
    INTEGER(I4B), INTENT(IN) :: k
    !! order of derivative
    REAL(DFP) :: ans
  END FUNCTION GradientEvalSum
END INTERFACE

Evaluates the kth order gradient of orthopol series at several points.

INTERFACE
  MODULE PURE FUNCTION GradientEvalSum(obj, n, x, coeff, k) &
    & RESULT(ans)
    CLASS(JacobiSpace1D_), INTENT(IN) :: obj
    INTEGER(I4B), INTENT(IN) :: n
    !! order of orthopol
    REAL(DFP), INTENT(IN) :: x(:)
    !! point of evaluation
    REAL(DFP), INTENT(IN) :: coeff(0:n)
    !! Coefficient of finite sum
    INTEGER(I4B), INTENT(IN) :: k
    !! order of derivative
    REAL(DFP) :: ans(SIZE(x))
  END FUNCTION GradientEvalSum
END INTERFACE

Transform

Discrete Orthopol Transform

INTERFACE
  MODULE PURE FUNCTION Transform(obj, n, coeff, x, w, quadType) &
    & RESULT(ans)
    CLASS(JacobiSpace1D_), INTENT(IN) :: obj
    INTEGER(I4B), INTENT(IN) :: n
    !! order of jacobi polynomial
    REAL(DFP), INTENT(IN) :: coeff(0:n)
    !! nodal value (at quad points)
    REAL(DFP), INTENT(IN) :: x(0:n)
    !! quadrature points
    REAL(DFP), INTENT(IN) :: w(0:n)
    !! weights
    INTEGER(I4B), INTENT(IN) :: quadType
    !! Quadrature type
    !! Gauss, GaussLobatto, GaussRadauLeft
    !! GaussRadauRight
    REAL(DFP) :: ans(0:n)
    !! modal values  or coefficients
  END FUNCTION Transform
END INTERFACE

Discrete orthopol Transform of a function

INTERFACE
  MODULE FUNCTION Transform(obj, n, f, quadType) &
    & RESULT(ans)
    CLASS(JacobiSpace1D_), INTENT(IN) :: obj
    !! orthopol
    INTEGER(I4B), INTENT(IN) :: n
    !! order of jacobi polynomial
    PROCEDURE(iface_1DFunction), POINTER, INTENT(IN) :: f
    !! 1D space function
    INTEGER(I4B), INTENT(IN) :: quadType
    !! Quadrature type, Gauss, GaussLobatto, GaussRadau, GaussRadauLeft
    !! GaussRadauRight
    REAL(DFP) :: ans(0:n)
    !! modal values  or coefficients
  END FUNCTION Transform
END INTERFACE

InvTransform

Discrete Jacobi Inverse Transform

INTERFACE
  MODULE PURE FUNCTION InvTransform(obj, n, coeff, x) &
    & RESULT(ans)
    CLASS(JacobiSpace1D_), INTENT(IN) :: obj
    !! orthopol
    INTEGER(I4B), INTENT(IN) :: n
    !! order of Jacobi polynomial
    REAL(DFP), INTENT(IN) :: coeff(0:n)
    !! n+1  coefficient (modal values)
    REAL(DFP), INTENT(IN) :: x
    !! x point in physical space
    REAL(DFP) :: ans
    !! value in physical space
  END FUNCTION InvTransform
END INTERFACE

Discrete Jacobi Inverse Transform

INTERFACE
  MODULE PURE FUNCTION InvTransform(obj, n, coeff, x) &
    & RESULT(ans)
    CLASS(JacobiSpace1D_), INTENT(IN) :: obj
    !! orthopol
    INTEGER(I4B), INTENT(IN) :: n
    !! order of Jacobi polynomial
    REAL(DFP), INTENT(IN) :: coeff(0:n)
    !! n+1  coefficient (modal values)
    REAL(DFP), INTENT(IN) :: x(:)
    !! x point in physical space
    REAL(DFP) :: ans(SIZE(x))
    !! value in physical space
  END FUNCTION InvTransform
END INTERFACE

GradientCoeff

Returns coefficient for gradient of orthopol expansion

INTERFACE
  MODULE PURE FUNCTION GradientCoeff(obj, n, coeff) RESULT(ans)
    CLASS(JacobiSpace1D_), INTENT(IN) :: obj
    !! orthopol
    INTEGER(I4B), INTENT(IN) :: n
    !! order of Jacobi polynomial
    REAL(DFP), INTENT(IN) :: coeff(0:n)
    !! coefficients $\tilde{u}_{n}$ obtained from JacobiTransform
    REAL(DFP) :: ans(0:n)
    !! coefficient of gradient
  END FUNCTION GradientCoeff
END INTERFACE

DMatrix

Returns coefficient for gradient of Jacobi expansion

INTERFACE
  MODULE PURE FUNCTION DMatrix(obj, n, x, quadType) RESULT(ans)
    CLASS(JacobiSpace1D_), INTENT(IN) :: obj
    !! orthpol
    INTEGER(I4B), INTENT(IN) :: n
      !! order of Jacobi polynomial
    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 DMatrix
END INTERFACE

Structure

Methods​

JacobiSpace1D function​

JacobiSpace1D_Pointer​

Deallocate​

Display​

GetParam​

GetRecurrenceCoeff​

GetRecurrenceCoeff2​

GetAlpha​

GetBeta​

GetLeadingCoeff​

GetLeadingCoeffRatio​

GetNormSqr​

GetNormSqr2​

GetNormSqrRatio​

GetZeros​

GetQuadrature​

EvalAll​

Eval​

EvalSum​

GradientEvalAll​

GradientEval​

GradientEvalSum​

Transform​

InvTransform​

GradientCoeff​

DMatrix​

Methods

JacobiSpace1D function

JacobiSpace1D_Pointer

Deallocate

Display

GetParam

GetRecurrenceCoeff

GetRecurrenceCoeff2

GetAlpha

GetBeta

GetLeadingCoeff

GetLeadingCoeffRatio

GetNormSqr

GetNormSqr2

GetNormSqrRatio

GetZeros

GetQuadrature

EvalAll

Eval

EvalSum

GradientEvalAll

GradientEval

GradientEvalSum

Transform

InvTransform

GradientCoeff

DMatrix