LagrangeCoeff

Returns the coefficients of Lagrange polynomials for monomial basis.

A Lagrange polynomial in terms of monomials is given by

l(x,y,z) = \sum_{a,b,c}^{p,q,r} c_{a,b,c} x^{a} y^{b} z^{c}

A Lagrange polynomial in terms of general basis is given by

l(x,y,z) = \sum_{a,b,c}^{p,q,r} c_{a,b,c} \phi_{a,b,c}

note

This subroutine returns $c_{a,b,c}$ for Lagrange polynomial.

For example, if we select monomial basis:

1, x, y, xy, z, xz, yz, xyz

then for order = 1 (linear), we have following results:

coeff

basis	$l_1$	$l_2$	$l_3$	$l_4$	$l_5$	$l_6$	$l_7$	$l_8$
1	0.125	0.125	0.125	0.125	0.125	0.125	0.125	0.125
x	-0.125	0.125	0.125	-0.125	-0.125	0.125	0.125	-0.125
y	-0.125	-0.125	0.125	0.125	-0.125	-0.125	0.125	0.125
xy	0.125	-0.125	0.125	-0.125	0.125	-0.125	0.125	-0.125
z	-0.125	-0.125	-0.125	-0.125	0.125	0.125	0.125	0.125
xz	0.125	-0.125	-0.125	0.125	-0.125	0.125	0.125	-0.125
yz	0.125	0.125	-0.125	-0.125	-0.125	-0.125	0.125	0.125
xyz	-0.125	0.125	-0.125	0.125	0.125	-0.125	0.125	-0.125

Interface 1

INTERFACE LagrangeCoeff_Hexahedron
  MODULE FUNCTION LagrangeCoeff_Hexahedron1(order, i, xij) RESULT(ans)
    INTEGER(I4B), INTENT(IN) :: order
    !! order of polynomial
    INTEGER(I4B), INTENT(IN) :: i
    !! ith coefficients for lagrange polynomial
    REAL(DFP), INTENT(IN) :: xij(:, :)
    !! interpolation points in xij format
    !! number of rows in xij is 3
    !! number of columns should be equal to the number degree of freedom
    REAL(DFP) :: ans(SIZE(xij, 2))
    !! coefficients
  END FUNCTION LagrangeCoeff_Hexahedron1
END INTERFACE LagrangeCoeff_Hexahedron

Interface 2

INTERFACE LagrangeCoeff_Hexahedron
  MODULE FUNCTION LagrangeCoeff_Hexahedron2(order, i, v, isVandermonde) &
    & RESULT(ans)
    INTEGER(I4B), INTENT(IN) :: order
    !! order of polynomial, it should be SIZE(v,2)-1
    INTEGER(I4B), INTENT(IN) :: i
    !! coefficient for ith lagrange polynomial
    REAL(DFP), INTENT(IN) :: v(:, :)
    !! vandermonde matrix size should be (order+1,order+1)
    LOGICAL(LGT), INTENT(IN) :: isVandermonde
    !! This is just to resolve interface issue
    REAL(DFP) :: ans(SIZE(v, 1))
    !! coefficients
  END FUNCTION LagrangeCoeff_Hexahedron2
END INTERFACE LagrangeCoeff_Hexahedron

Interface 3

INTERFACE LagrangeCoeff_Hexahedron
  MODULE FUNCTION LagrangeCoeff_Hexahedron3(order, i, v, ipiv) RESULT(ans)
    INTEGER(I4B), INTENT(IN) :: order
    !! order of polynomial, it should be SIZE(x,2)-1
    INTEGER(I4B), INTENT(IN) :: i
    !! ith coefficients for lagrange polynomial
    REAL(DFP), INTENT(INOUT) :: v(:, :)
    !! LU decomposition of vandermonde matrix
    INTEGER(I4B), INTENT(IN) :: ipiv(:)
    !! inverse pivoting mapping, compes from LU decomposition
    REAL(DFP) :: ans(SIZE(v, 1))
    !! coefficients
  END FUNCTION LagrangeCoeff_Hexahedron3
END INTERFACE LagrangeCoeff_Hexahedron

Interface 4

܀ Interface
️܀ See example
↢

INTERFACE LagrangeCoeff_Hexahedron
  MODULE FUNCTION LagrangeCoeff_Hexahedron4(order, xij, basisType, &
    & refHexahedron, alpha, beta, lambda) RESULT(ans)
    INTEGER(I4B), INTENT(IN) :: order
    !! order of polynomial
    REAL(DFP), INTENT(IN) :: xij(:, :)
    !! points in xij format, size(xij,2)
    INTEGER(I4B), OPTIONAL, INTENT(IN) :: basisType
    !! Monomials
    !! Jacobi
    !! Legendre
    !! Chebyshev
    !! Ultraspherical
    !! Heirarchical
    CHARACTER(*), OPTIONAL, INTENT(IN) :: refHexahedron
    !! UNIT
    !! BIUNIT
    REAL(DFP), OPTIONAL, INTENT(IN) :: alpha
    !! This parameter is needed when basisType is Jacobi
    REAL(DFP), OPTIONAL, INTENT(IN) :: beta
    !! This parameter is needed when basisType is Jacobi
    REAL(DFP), OPTIONAL, INTENT(IN) :: lambda
    !! This parameter is needed when basisType is Ultraspherical
    REAL(DFP) :: ans(SIZE(xij, 2), SIZE(xij, 2))
    !! coefficients
  END FUNCTION LagrangeCoeff_Hexahedron4
END INTERFACE LagrangeCoeff_Hexahedron

order

order of polynomial

xij

interpolation points for Lagrange polynomials. The number of rows of xij should be 3. The number of columns of xij should be equal to the total number of degrees of freedom.

basisType

Monomials
Jacobi
Legendre
Chebyshev
Ultraspherical
Heirarchical

refHexahedron

Reference hexahedron can be UNIT or BIUNIT.

alpha, beta, lambda

alpha and beta are needed when basisType is Jacobi
lambda is needed when basisType is Ultraspherical

PROGRAM main
USE easifembase
IMPLICIT NONE
INTEGER(i4b) :: i1, i2, order, basisType
REAL(dfp), ALLOCATABLE :: xij(:, :), coeff(:, :)
TYPE(string) :: astr
REAL( DFP ) :: alpha, beta, lambda
CHARACTER(*), parameter :: refHexahedron="BIUNIT"
INTEGER( I4B ), ALLOCATABLE :: deg(:, :)


order = 1
basisType = Monomial

deg = LagrangeDegree_Hexahedron(order)

xij = InterpolationPoint_Hexahedron(  &
  & order=order,  &
  & ipType=Equidistance,  &
  & layout="VEFC")

coeff = LagrangeCoeff_Hexahedron(&
  & order=order, &
  & xij=xij, &
  & basisType=basisType, &
  & refHexahedron=refHexahedron, &
  & alpha=alpha, &
  & beta=beta, &
  & lambda=lambda)


CALL Display(Mdencode(deg), "monomial degrees: " )
CALL Display(Mdencode(coeff), "coeff : " )

END PROGRAM main

See results

degrees:

a	b	c
0	0	0
1	0	0
0	1	0
1	1	0
0	0	1
1	0	1
0	1	1
1	1	1

1, x, y, xy, z, xz, yz, xyz

coeff

basis	$l_1$	$l_2$	$l_3$	$l_4$	$l_5$	$l_6$	$l_7$	$l_8$
1	0.125	0.125	0.125	0.125	0.125	0.125	0.125	0.125
x	-0.125	0.125	0.125	-0.125	-0.125	0.125	0.125	-0.125
y	-0.125	-0.125	0.125	0.125	-0.125	-0.125	0.125	0.125
xy	0.125	-0.125	0.125	-0.125	0.125	-0.125	0.125	-0.125
z	-0.125	-0.125	-0.125	-0.125	0.125	0.125	0.125	0.125
xz	0.125	-0.125	-0.125	0.125	-0.125	0.125	0.125	-0.125
yz	0.125	0.125	-0.125	-0.125	-0.125	-0.125	0.125	0.125
xyz	-0.125	0.125	-0.125	0.125	0.125	-0.125	0.125	-0.125

Interface 5

܀ Interface
️܀ See example
↢

INTERFACE LagrangeCoeff_Hexahedron
  MODULE FUNCTION LagrangeCoeff_Hexahedron5(&
    & p, &
    & q, &
    & r, &
    & xij, &
    & basisType1, &
    & basisType2, &
    & basisType3, &
    & alpha1, &
    & beta1, &
    & lambda1, &
    & alpha2, &
    & beta2, &
    & lambda2, &
    & alpha3, &
    & beta3, &
    & lambda3, &
    & refHexahedron &
    & ) RESULT(ans)
    INTEGER(I4B), INTENT(IN) :: p
    !! order of polynomial
    INTEGER(I4B), INTENT(IN) :: q
    !! order of polynomial
    INTEGER(I4B), INTENT(IN) :: r
    !! order of polynomial
    REAL(DFP), INTENT(IN) :: xij(:, :)
    !! These are interpolation points in xij format, size(xij,2)
    INTEGER(I4B), INTENT(IN) :: basisType1
    !! basis type in x direction
    !! Monomials
    !! Jacobi
    !! Legendre
    !! Chebyshev
    !! Ultraspherical
    !! Heirarchical
    INTEGER(I4B), INTENT(IN) :: basisType2
    !! basis type in y direction
    !! Monomials
    !! Jacobi
    !! Legendre
    !! Chebyshev
    !! Ultraspherical
    !! Heirarchical
    INTEGER(I4B), INTENT(IN) :: basisType3
    !! basis type in z direction
    !! Monomials
    !! Jacobi
    !! Legendre
    !! Chebyshev
    !! Ultraspherical
    !! Heirarchical
    REAL(DFP), INTENT(IN) :: alpha1
    !! This parameter is needed when basisType1 is Jacobi
    REAL(DFP), INTENT(IN) :: beta1
    !! This parameter is needed when basisType1 is Jacobi
    REAL(DFP), INTENT(IN) :: lambda1
    !! This parameter is needed when basisType1 is Ultraspherical
    REAL(DFP), INTENT(IN) :: alpha2
    !! This parameter is needed when basisType2 is Jacobi
    REAL(DFP), INTENT(IN) :: beta2
    !! This parameter is needed when basisType2 is Jacobi
    REAL(DFP), INTENT(IN) :: lambda2
    !! This parameter is needed when basisType2 is Ultraspherical
    REAL(DFP), INTENT(IN) :: alpha3
    !! This parameter is needed when basisType3 is Jacobi
    REAL(DFP), INTENT(IN) :: beta3
    !! This parameter is needed when basisType3 is Jacobi
    REAL(DFP), INTENT(IN) :: lambda3
    !! This parameter is needed when basisType3 is Ultraspherical
    CHARACTER(*), OPTIONAL, INTENT(IN) :: refHexahedron
    !! UNIT
    !! BIUNIT
    REAL(DFP) :: ans(SIZE(xij, 2), SIZE(xij, 2))
    !! coefficients
  END FUNCTION LagrangeCoeff_Hexahedron5
END INTERFACE LagrangeCoeff_Hexahedron

p, q, r

p is order in x direction
q is order in y direction
r is order in z direction

xij

xij is the interpolation points. The number of rows in xij is 3, and the number of columns in xij should be equal to the total number of degrees of freedom.

basisType1, basisType2, basisType3

basisType in x, y, and z direction. It can take following values:

Monomials
Jacobi
Legendre
Chebyshev
Ultraspherical
Heirarchical

alpha1, beta1, lambda1

alpha1 and beta1 are needed when basisType1 is Jacobi
lambda1 is needed when basisType1 is Ultraspherical

alpha2, beta2, lambda2

alpha2 and beta2 are needed when basisType2 is Jacobi
lambda2 is needed when basisType2 is Ultraspherical

alpha3, beta, lambda3

alpha3 and beta3 are needed when basisType3 is Jacobi
lambda3 is needed when basisType3 is Ultraspherical

refHexahedron

Reference hexahedron can be UNIT or BIUNIT.

PROGRAM main
USE easifembase
IMPLICIT NONE
INTEGER(i4b) :: i1, i2, p, q, r, basisType1, basisType2, basisType3
REAL(dfp), ALLOCATABLE :: xij(:, :), coeff(:, :)
TYPE(string) :: astr
REAL( DFP ) :: alpha1, beta1, lambda1
REAL( DFP ) :: alpha2, beta2, lambda2
REAL( DFP ) :: alpha3, beta3, lambda3
CHARACTER(*), parameter :: refHexahedron="BIUNIT"
INTEGER( I4B ), ALLOCATABLE :: deg(:, :)


p = 1
q = 1
r = 1
basisType1 = Monomial
basisType2 = Monomial
basisType3 = Monomial

deg = LagrangeDegree_Hexahedron(p, q, r)

xij = InterpolationPoint_Hexahedron(  &
  & p=p,  &
  & q=q,  &
  & r=r,  &
  & ipType1=Equidistance,  &
  & ipType2=Equidistance,  &
  & ipType3=Equidistance,  &
  & layout="VEFC")

coeff = LagrangeCoeff_Hexahedron(&
  & p=p, &
  & q=q, &
  & r=r, &
  & xij=xij, &
  & basisType1=basisType1, &
  & basisType2=basisType2, &
  & basisType3=basisType3, &
  & refHexahedron=refHexahedron, &
  & alpha1=alpha1, &
  & beta1=beta1, &
  & lambda1=lambda1, &
  & alpha2=alpha2, &
  & beta2=beta2, &
  & lambda2=lambda2, &
  & alpha3=alpha3, &
  & beta3=beta3, &
  & lambda3=lambda3 &
  & )


CALL Display(Mdencode(deg), "monomial degrees: " )
CALL Display(Mdencode(coeff), "coeff : " )

END PROGRAM main

See results

degrees:

a	b	c
0	0	0
1	0	0
0	1	0
1	1	0
0	0	1
1	0	1
0	1	1
1	1	1

1, x, y, xy, z, xz, yz, xyz

coeff

basis	$l_1$	$l_2$	$l_3$	$l_4$	$l_5$	$l_6$	$l_7$	$l_8$
1	0.125	0.125	0.125	0.125	0.125	0.125	0.125	0.125
x	-0.125	0.125	0.125	-0.125	-0.125	0.125	0.125	-0.125
y	-0.125	-0.125	0.125	0.125	-0.125	-0.125	0.125	0.125
xy	0.125	-0.125	0.125	-0.125	0.125	-0.125	0.125	-0.125
z	-0.125	-0.125	-0.125	-0.125	0.125	0.125	0.125	0.125
xz	0.125	-0.125	-0.125	0.125	-0.125	0.125	0.125	-0.125
yz	0.125	0.125	-0.125	-0.125	-0.125	-0.125	0.125	0.125
xyz	-0.125	0.125	-0.125	0.125	0.125	-0.125	0.125	-0.125

LagrangeCoeff

Interface 1​

Interface 2​

Interface 3​

Interface 4​

Interface 5​

Interface 1

Interface 2

Interface 3

Interface 4

Interface 5