Skip to main content

Methods

Chebyshev polynomial of first kind is defined. The nth order Chebyshev polynomial is denoted by Tn(x)T_{n}(x). Chebyshev polynomials are orthogonal polynomials.

It is a child of [[AbstractOrthopol1D_]].

Structure​

TYPE, EXTENDS( AbstractOrthoPol1D_ ) :: ChebyshevFirst1D_
  CONTAINS

ConstructorMethods​

Constructor function​

We can create an instance of TnT_{n} by calling ChebyshevFirst1D() function. Examples are included in

  • [[ChebyshevFirst1D_test_1]]
  • [[ChebyshevFirst1D_test_2]]

Interface is given below:

MODULE FUNCTION ChebyshevFirst1D( varname, n, isMonic, &
    & isOrthonormal ) RESULT( ans )
  CHARACTER( LEN = * ), INTENT( IN ) :: varname
    !! variable name
  INTEGER( I4B ), INTENT( IN ) :: n
    !! order of chebyshev polynomial
  LOGICAL( LGT ), OPTIONAL, INTENT( IN ) :: isMonic
    !! Default is .FALSE., if true then leading coeff of poly is 1
  LOGICAL( LGT ), OPTIONAL, INTENT( IN ) :: isOrthonormal
    !! Default is .FALSE., if true then the polynomials are orthonormal
  TYPE( ChebyshevFirst1D_ ) :: ans
    !! Chebyshev polynomial of first kind
END FUNCTION ChebyshevFirst1D

ChebyshevFirst1D_Pointer​

We can create an instance of new pointer by ChebyshevFirst1D_Pointer() function.

The interface is given below:

FUNCTION ChebyshevFirst1D_Pointer( varname, n, &
  & isMonic, isOrthonormal ) RESULT( ans )
  CHARACTER( LEN = * ), INTENT( IN ) :: varname
    !! variable name
  INTEGER( I4B ), INTENT( IN ) :: n
    !! order of chebyshev polynomial
  LOGICAL( LGT ), OPTIONAL, INTENT( IN ) :: isMonic
    !! Default is .FALSE., if true then leading coeff of poly is 1
  LOGICAL( LGT ), OPTIONAL, INTENT( IN ) :: isOrthonormal
    !! Default is .FALSE., if true then the polynomials are orthonormal
  CLASS( ChebyshevFirst1D_ ), POINTER :: ans
    !! Chebyshev polynomial of first kind
END FUNCTION ChebyshevFirst1D_Pointer

Deallocate​

You can deallocate (or destroy) the object by calling Deallocate() subroutine.

CALL obj%Deallocate()

GetMethods​

GetStringForUID​

Weight​

This function evaluate the weight of the Chebyshev polynomial at a given point x.

GetRecurrenceCoeff​

GetCoeffScale​

Zeros​

GaussQuadrature​

GaussRadauQuadratur​

GaussLobattoQuadrature​

Examples​

ChebyshevFirst1D_test_1

ChebyshevFirst1D example 1

This example shows the usage of [[ChebyshevFirst1D_]] class.

Modules and classes​

  • [[ChebyshevFirst1D_]]

Usage​

PROGRAM main
use easifemBase
use easifemClasses
implicit none
type(ChebyshevFirst1D_) :: obj
type(String) :: astr
integer(i4b) :: ii

n=1​

  obj = ChebyshevFirst1D(varname="x", n=1)
  call Display( "T(n=1) := " )
  call obj%Display( "=>" )
>result
J(n=1, alpha=0.0, beta=0.0) :=
=>J_1( x ) = ( x-.000 )* 1.000* J_0
=>J_0( x ) = 1

n=2​

  call blanklines( nol=5 )
  obj = ChebyshevFirst1D(varname="x", n=2)
  call Display( "J(n=2, alpha=0.0, beta=0.0) := " )
  call obj%Display( "=>" )
>result
J(n=2, alpha=0.0, beta=0.0) :=
=>J_2( x ) = ( x-.000 )* 1.500* J_1( x ) - .333* 1.500* J_0( x )
=>J_1( x ) = ( x-.000 )* 1.000* J_0
=>J_0( x ) = 1

n=3​

  call blanklines( nol=5 )
  obj = ChebyshevFirst1D(varname="x", n=3)
  call Display( "J(n=3, alpha=0.0, beta=0.0) := " )
  call obj%Display( "=>" )
>result
J(n=3, alpha=0.0, beta=0.0) :=
=>J_3( x ) = ( x-.000 )* 1.667* J_2( x ) - .267* 2.500* J_1( x )
=>J_2( x ) = ( x-.000 )* 1.500* J_1( x ) - .333* 1.500* J_0( x )
=>J_1( x ) = ( x-.000 )* 1.000* J_0
=>J_0( x ) = 1

n=4​

  call blanklines( nol=5 )
  obj = Jacobi1D(varname="x", n=4, alpha=0.0_DFP, beta=0.0_DFP)
  call Display( "J(n=4, alpha=0.0, beta=0.0) := " )
  call obj%Display( "=>" )
>result
J(n=4, alpha=0.0, beta=0.0) :=
=>J_4( x ) = ( x-.000 )* 1.750* J_3( x ) - .257* 2.917* J_2( x )
=>J_3( x ) = ( x-.000 )* 1.667* J_2( x ) - .267* 2.500* J_1( x )
=>J_2( x ) = ( x-.000 )* 1.500* J_1( x ) - .333* 1.500* J_0( x )
=>J_1( x ) = ( x-.000 )* 1.000* J_0
=>J_0( x ) = 1
END PROGRAM main
Checkout more examples

ChebyshevFirst1D_test_6

ChebyshevFirst1D example 6

This example shows the usage of [[ChebyshevFirst1D_]] class. We test Zeros function in this routine, which returns the zeros of Chebyshev1 polynomial.

Modules and classes​

  • [[ChebyshevFirst1D_]]

Usage​

PROGRAM main
use easifemBase
use easifemClasses
implicit none
type(ChebyshevFirst1D_) :: obj
real( dfp ), allocatable :: x( : )
integer( i4b ) :: n

n=1​

  n = 1
  obj = ChebyshevFirst1D(varname="x", n=n)
  x = obj%zeros()
  call display( x, "zeros for n="//tostring(n), orient="ROW" )

n=2​

  n = 2
  obj = ChebyshevFirst1D(varname="x", n=n)
  x = obj%zeros()
  call display( x, "zeros for n="//tostring(n), orient="ROW" )

n=3​

  n = 3
  obj = ChebyshevFirst1D(varname="x", n=n)
  x = obj%zeros()
  call display( x, "zeros for n="//tostring(n), orient="ROW" )

n=4​

  n = 4
  obj = ChebyshevFirst1D(varname="x", n=n)
  x = obj%zeros()
  call display( x, "zeros for n="//tostring(n), orient="ROW" )
END PROGRAM main
Checkout more examples