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|
# Maple code to define Lagrange interpolating functions/coords/coeffs
# $Header$
################################################################################
#
# 1-D interpolating functions
#
fn_1d_order1 :=
proc(x)
+ c0 + c1*x
end proc;
fn_1d_order2 :=
proc(x)
+ c0 + c1*x + c2*x^2
end proc;
fn_1d_order3 :=
proc(x)
+ c0 + c1*x + c2*x^2 + c3*x^3
end proc;
fn_1d_order4 :=
proc(x)
+ c0 + c1*x + c2*x^2 + c3*x^3 + c4*x^4
end;
fn_1d_order5 :=
proc(x)
+ c0 + c1*x + c2*x^2 + c3*x^3 + c4*x^4 + c5*x^5
end;
fn_1d_order6 :=
proc(x)
+ c0 + c1*x + c2*x^2 + c3*x^3 + c4*x^4 + c5*x^5 + c6*x^6
end;
########################################
# coordinates for 1-D interpolating functions
coords_list_1d := [x];
########################################
#
# coefficients in 1-D interpolating functions
#
coeffs_list_1d_order1 := [c0, c1];
coeffs_list_1d_order2 := [c0, c1, c2];
coeffs_list_1d_order3 := [c0, c1, c2, c3];
coeffs_list_1d_order4 := [c0, c1, c2, c3, c4];
coeffs_list_1d_order5 := [c0, c1, c2, c3, c4, c5];
coeffs_list_1d_order6 := [c0, c1, c2, c3, c4, c5, c6];
################################################################################
#
# 2-D interpolating functions (tensor-product)
#
fn_2d_order1 :=
proc(x,y)
+ c01*y + c11*x*y
+ c00 + c10*x
end proc;
fn_2d_order2 :=
proc(x,y)
+ c02*y^2 + c12*x*y^2 + c22*x^2*y^2
+ c01*y + c11*x*y + c21*x^2*y
+ c00 + c10*x + c20*x^2
end proc;
fn_2d_order3 :=
proc(x,y)
+ c03*y^3 + c13*x*y^3 + c23*x^2*y^3 + c33*x^3*y^3
+ c02*y^2 + c12*x*y^2 + c22*x^2*y^2 + c32*x^3*y^2
+ c01*y + c11*x*y + c21*x^2*y + c31*x^3*y
+ c00 + c10*x + c20*x^2 + c30*x^3
end proc;
fn_2d_order4 :=
proc(x,y)
+ c04*y^4 + c14*x*y^4 + c24*x^2*y^4 + c34*x^3*y^4 + c44*x^4*y^4
+ c03*y^3 + c13*x*y^3 + c23*x^2*y^3 + c33*x^3*y^3 + c43*x^4*y^3
+ c02*y^2 + c12*x*y^2 + c22*x^2*y^2 + c32*x^3*y^2 + c42*x^4*y^2
+ c01*y + c11*x*y + c21*x^2*y + c31*x^3*y + c41*x^4*y
+ c00 + c10*x + c20*x^2 + c30*x^3 + c40*x^4
end;
########################################
# coordinates for 2-D interpolating functions
coords_list_2d := [x,y];
########################################
#
# coefficients in 2-D interpolating functions (tensor-product)
#
coeffs_list_2d_order1 := [
c01, c11,
c00, c10
];
coeffs_list_2d_order2 := [
c02, c12, c22,
c01, c11, c21,
c00, c10, c20
];
coeffs_list_2d_order3 := [
c03, c13, c23, c33,
c02, c12, c22, c32,
c01, c11, c21, c31,
c00, c10, c20, c30
];
coeffs_list_2d_order4 := [
c04, c14, c24, c34, c44,
c03, c13, c23, c33, c43,
c02, c12, c22, c32, c42,
c01, c11, c21, c31, c41,
c00, c10, c20, c30, c40
];
################################################################################
#
# 3-D interpolating functions (tensor-product)
#
fn_3d_order1 :=
proc(x,y,z)
# z^1 -------------------
+ c011 *y*z + c111*x*y*z
+ c001 *z + c101*x *z
# z^0 -------------------
+ c010 *y + c110*x*y
+ c000 + c100*x
end proc;
fn_3d_order2 :=
proc(x,y,z)
# z^2 --------------------------------------------------
+ c022 *y^2*z^2 + c122*x *y^2*z^2 + c222*x^2*y^2*z^2
+ c012 *y *z^2 + c112*x *y *z^2 + c212*x^2*y *z^2
+ c002 *z^2 + c102*x *z^2 + c202*x^2 *z^2
# z --------------------------------------------------
+ c021 *y^2*z + c121*x *y^2*z + c221*x^2*y^2*z
+ c011 *y *z + c111*x *y *z + c211*x^2*y *z
+ c001 *z + c101*x *z + c201*x^2 *z
# z^0 ------------------------------------------------
+ c020 *y^2 + c120*x *y^2 + c220*x^2*y^2
+ c010 *y + c110*x *y + c210*x^2*y
+ c000 + c100*x + c200*x^2
end proc;
fn_3d_order3 :=
proc(x,y,z)
# z^3 ---------------------------------------------------------------------
+ c033 *y^3*z^3 + c133*x *y^3*z^3 + c233*x^2*y^3*z^3 + c333*x^3*y^3*z^3
+ c023 *y^2*z^3 + c123*x *y^2*z^3 + c223*x^2*y^2*z^3 + c323*x^3*y^2*z^3
+ c013 *y *z^3 + c113*x *y *z^3 + c213*x^2*y *z^3 + c313*x^3*y *z^3
+ c003 *z^3 + c103*x *z^3 + c203*x^2 *z^3 + c303*x^3 *z^3
# z^2 ---------------------------------------------------------------------
+ c032 *y^3*z^2 + c132*x *y^3*z^2 + c232*x^2*y^3*z^2 + c332*x^3*y^3*z^2
+ c022 *y^2*z^2 + c122*x *y^2*z^2 + c222*x^2*y^2*z^2 + c322*x^3*y^2*z^2
+ c012 *y *z^2 + c112*x *y *z^2 + c212*x^2*y *z^2 + c312*x^3*y *z^2
+ c002 *z^2 + c102*x *z^2 + c202*x^2 *z^2 + c302*x^3 *z^2
# z ---------------------------------------------------------------------
+ c031 *y^3*z + c131*x *y^3*z + c231*x^2*y^3*z + c331*x^3*y^3*z
+ c021 *y^2*z + c121*x *y^2*z + c221*x^2*y^2*z + c321*x^3*y^2*z
+ c011 *y *z + c111*x *y *z + c211*x^2*y *z + c311*x^3*y *z
+ c001 *z + c101*x *z + c201*x^2 *z + c301*x^3 *z
# z^0 ---------------------------------------------------------------------
+ c030 *y^3 + c130*x *y^3 + c230*x^2*y^3 + c330*x^3*y^3
+ c020 *y^2 + c120*x *y^2 + c220*x^2*y^2 + c320*x^3*y^2
+ c010 *y + c110*x *y + c210*x^2*y + c310*x^3*y
+ c000 + c100*x + c200*x^2 + c300*x^3
end proc;
fn_3d_order4 :=
proc(x,y,z)
# z^4 ----------------------------------------------------------------------------------------
+ c044 *y^4*z^4 + c144*x *y^4*z^4 + c244*x^2*y^4*z^4 + c344*x^3*y^4*z^4 + c444*x^4*y^4*z^4
+ c034 *y^3*z^4 + c134*x *y^3*z^4 + c234*x^2*y^3*z^4 + c334*x^3*y^3*z^4 + c434*x^4*y^3*z^4
+ c024 *y^2*z^4 + c124*x *y^2*z^4 + c224*x^2*y^2*z^4 + c324*x^3*y^2*z^4 + c424*x^4*y^2*z^4
+ c014 *y *z^4 + c114*x *y *z^4 + c214*x^2*y *z^4 + c314*x^3*y *z^4 + c414*x^4*y *z^4
+ c004 *z^4 + c104*x *z^4 + c204*x^2 *z^4 + c304*x^3 *z^4 + c404*x^4 *z^4
# z^3 ----------------------------------------------------------------------------------------
+ c043 *y^4*z^3 + c143*x *y^4*z^3 + c243*x^2*y^4*z^3 + c343*x^3*y^4*z^3 + c443*x^4*y^4*z^3
+ c033 *y^3*z^3 + c133*x *y^3*z^3 + c233*x^2*y^3*z^3 + c333*x^3*y^3*z^3 + c433*x^4*y^3*z^3
+ c023 *y^2*z^3 + c123*x *y^2*z^3 + c223*x^2*y^2*z^3 + c323*x^3*y^2*z^3 + c423*x^4*y^2*z^3
+ c013 *y *z^3 + c113*x *y *z^3 + c213*x^2*y *z^3 + c313*x^3*y *z^3 + c413*x^4*y *z^3
+ c003 *z^3 + c103*x *z^3 + c203*x^2 *z^3 + c303*x^3 *z^3 + c403*x^4 *z^3
# z^2 ----------------------------------------------------------------------------------------
+ c042 *y^4*z^2 + c142*x *y^4*z^2 + c242*x^2*y^4*z^2 + c342*x^3*y^4*z^2 + c442*x^4*y^4*z^2
+ c032 *y^3*z^2 + c132*x *y^3*z^2 + c232*x^2*y^3*z^2 + c332*x^3*y^3*z^2 + c432*x^4*y^3*z^2
+ c022 *y^2*z^2 + c122*x *y^2*z^2 + c222*x^2*y^2*z^2 + c322*x^3*y^2*z^2 + c422*x^4*y^2*z^2
+ c012 *y *z^2 + c112*x *y *z^2 + c212*x^2*y *z^2 + c312*x^3*y *z^2 + c412*x^4*y *z^2
+ c002 *z^2 + c102*x *z^2 + c202*x^2 *z^2 + c302*x^3 *z^2 + c402*x^4 *z^2
# z ----------------------------------------------------------------------------------------
+ c041 *y^4*z + c141*x *y^4*z + c241*x^2*y^4*z + c341*x^3*y^4*z + c441*x^4*y^4*z
+ c031 *y^3*z + c131*x *y^3*z + c231*x^2*y^3*z + c331*x^3*y^3*z + c431*x^4*y^3*z
+ c021 *y^2*z + c121*x *y^2*z + c221*x^2*y^2*z + c321*x^3*y^2*z + c421*x^4*y^2*z
+ c011 *y *z + c111*x *y *z + c211*x^2*y *z + c311*x^3*y *z + c411*x^4*y *z
+ c001 *z + c101*x *z + c201*x^2 *z + c301*x^3 *z + c401*x^4 *z
# z^0 ----------------------------------------------------------------------------------------
+ c040 *y^4 + c140*x *y^4 + c240*x^2*y^4 + c340*x^3*y^4 + c440*x^4*y^4
+ c030 *y^3 + c130*x *y^3 + c230*x^2*y^3 + c330*x^3*y^3 + c430*x^4*y^3
+ c020 *y^2 + c120*x *y^2 + c220*x^2*y^2 + c320*x^3*y^2 + c420*x^4*y^2
+ c010 *y + c110*x *y + c210*x^2*y + c310*x^3*y + c410*x^4*y
+ c000 + c100*x + c200*x^2 + c300*x^3 + c400*x^4
end;
fn_3d_order5 :=
proc(x,y,z)
# z^5 ----------------------------------------------------------------------------------------
+ c055 *y^5*z^5 + c155*x *y^5*z^5 + c255*x^2*y^5*z^5 + c355*x^3*y^5*z^5 + c455*x^4*y^5*z^5 + c555*x^5*y^5*z^5
+ c045 *y^4*z^5 + c145*x *y^4*z^5 + c245*x^2*y^4*z^5 + c345*x^3*y^4*z^5 + c445*x^4*y^4*z^5 + c545*x^5*y^4*z^5
+ c035 *y^3*z^5 + c135*x *y^3*z^5 + c235*x^2*y^3*z^5 + c335*x^3*y^3*z^5 + c435*x^4*y^3*z^5 + c535*x^5*y^3*z^5
+ c025 *y^2*z^5 + c125*x *y^2*z^5 + c225*x^2*y^2*z^5 + c325*x^3*y^2*z^5 + c425*x^4*y^2*z^5 + c525*x^5*y^2*z^5
+ c015 *y *z^5 + c115*x *y *z^5 + c215*x^2*y *z^5 + c315*x^3*y *z^5 + c415*x^4*y *z^5 + c515*x^5*y *z^5
+ c005 *z^5 + c105*x *z^5 + c205*x^2 *z^5 + c305*x^3 *z^5 + c405*x^4 *z^5 + c505*x^5 *z^5
# z^4 ----------------------------------------------------------------------------------------
+ c054 *y^5*z^4 + c154*x *y^5*z^4 + c254*x^2*y^5*z^4 + c354*x^3*y^5*z^4 + c454*x^4*y^5*z^4 + c554*x^5*y^5*z^4
+ c044 *y^4*z^4 + c144*x *y^4*z^4 + c244*x^2*y^4*z^4 + c344*x^3*y^4*z^4 + c444*x^4*y^4*z^4 + c544*x^5*y^4*z^4
+ c034 *y^3*z^4 + c134*x *y^3*z^4 + c234*x^2*y^3*z^4 + c334*x^3*y^3*z^4 + c434*x^4*y^3*z^4 + c534*x^5*y^3*z^4
+ c024 *y^2*z^4 + c124*x *y^2*z^4 + c224*x^2*y^2*z^4 + c324*x^3*y^2*z^4 + c424*x^4*y^2*z^4 + c524*x^5*y^2*z^4
+ c014 *y *z^4 + c114*x *y *z^4 + c214*x^2*y *z^4 + c314*x^3*y *z^4 + c414*x^4*y *z^4 + c514*x^5*y *z^4
+ c004 *z^4 + c104*x *z^4 + c204*x^2 *z^4 + c304*x^3 *z^4 + c404*x^4 *z^4 + c504*x^5 *z^4
# z^3 ----------------------------------------------------------------------------------------
+ c053 *y^5*z^3 + c153*x *y^5*z^3 + c253*x^2*y^5*z^3 + c353*x^3*y^5*z^3 + c453*x^4*y^5*z^3 + c553*x^5*y^5*z^3
+ c043 *y^4*z^3 + c143*x *y^4*z^3 + c243*x^2*y^4*z^3 + c343*x^3*y^4*z^3 + c443*x^4*y^4*z^3 + c543*x^5*y^4*z^3
+ c033 *y^3*z^3 + c133*x *y^3*z^3 + c233*x^2*y^3*z^3 + c333*x^3*y^3*z^3 + c433*x^4*y^3*z^3 + c533*x^5*y^3*z^3
+ c023 *y^2*z^3 + c123*x *y^2*z^3 + c223*x^2*y^2*z^3 + c323*x^3*y^2*z^3 + c423*x^4*y^2*z^3 + c523*x^5*y^2*z^3
+ c013 *y *z^3 + c113*x *y *z^3 + c213*x^2*y *z^3 + c313*x^3*y *z^3 + c413*x^4*y *z^3 + c513*x^5*y *z^3
+ c003 *z^3 + c103*x *z^3 + c203*x^2 *z^3 + c303*x^3 *z^3 + c403*x^4 *z^3 + c503*x^5 *z^3
# z^2 ----------------------------------------------------------------------------------------
+ c052 *y^5*z^2 + c152*x *y^5*z^2 + c252*x^2*y^5*z^2 + c352*x^3*y^5*z^2 + c452*x^4*y^5*z^2 + c552*x^5*y^5*z^2
+ c042 *y^4*z^2 + c142*x *y^4*z^2 + c242*x^2*y^4*z^2 + c342*x^3*y^4*z^2 + c442*x^4*y^4*z^2 + c542*x^5*y^4*z^2
+ c032 *y^3*z^2 + c132*x *y^3*z^2 + c232*x^2*y^3*z^2 + c332*x^3*y^3*z^2 + c432*x^4*y^3*z^2 + c532*x^5*y^3*z^2
+ c022 *y^2*z^2 + c122*x *y^2*z^2 + c222*x^2*y^2*z^2 + c322*x^3*y^2*z^2 + c422*x^4*y^2*z^2 + c522*x^5*y^2*z^2
+ c012 *y *z^2 + c112*x *y *z^2 + c212*x^2*y *z^2 + c312*x^3*y *z^2 + c412*x^4*y *z^2 + c512*x^5*y *z^2
+ c002 *z^2 + c102*x *z^2 + c202*x^2 *z^2 + c302*x^3 *z^2 + c402*x^4 *z^2 + c502*x^5 *z^2
# z ----------------------------------------------------------------------------------------
+ c051 *y^5*z + c151*x *y^5*z + c251*x^2*y^5*z + c351*x^3*y^5*z + c451*x^4*y^5*z + c551*x^5*y^5*z
+ c041 *y^4*z + c141*x *y^4*z + c241*x^2*y^4*z + c341*x^3*y^4*z + c441*x^4*y^4*z + c541*x^5*y^4*z
+ c031 *y^3*z + c131*x *y^3*z + c231*x^2*y^3*z + c331*x^3*y^3*z + c431*x^4*y^3*z + c531*x^5*y^3*z
+ c021 *y^2*z + c121*x *y^2*z + c221*x^2*y^2*z + c321*x^3*y^2*z + c421*x^4*y^2*z + c521*x^5*y^2*z
+ c011 *y *z + c111*x *y *z + c211*x^2*y *z + c311*x^3*y *z + c411*x^4*y *z + c511*x^5*y *z
+ c001 *z + c101*x *z + c201*x^2 *z + c301*x^3 *z + c401*x^4 *z + c501*x^5 *z
# z^0 ---------------------------------------------------------------------------------------------------
+ c050 *y^5 + c150*x *y^5 + c250*x^2*y^5 + c350*x^3*y^5 + c450*x^4*y^5 + c550*x^5*y^5
+ c040 *y^4 + c140*x *y^4 + c240*x^2*y^4 + c340*x^3*y^4 + c440*x^4*y^4 + c540*x^5*y^4
+ c030 *y^3 + c130*x *y^3 + c230*x^2*y^3 + c330*x^3*y^3 + c430*x^4*y^3 + c530*x^5*y^3
+ c020 *y^2 + c120*x *y^2 + c220*x^2*y^2 + c320*x^3*y^2 + c420*x^4*y^2 + c520*x^5*y^2
+ c010 *y + c110*x *y + c210*x^2*y + c310*x^3*y + c410*x^4*y + c510*x^5*y
+ c000 + c100*x + c200*x^2 + c300*x^3 + c400*x^4 + c500*x^5
end;
########################################
# coordinates for 3-D interpolating functions
coords_list_3d := [x,y,z];
########################################
#
# coefficients in 3-D interpolating functions
#
coeffs_list_3d_order1 := [
# z^1 -----
c011, c111,
c001, c101,
# z^0 -----
c010, c110,
c000, c100
];
coeffs_list_3d_order2 := [
# z^2 -----------
c022, c122, c222,
c012, c112, c212,
c002, c102, c202,
# z^1 -----------
c021, c121, c221,
c011, c111, c211,
c001, c101, c201,
# z^0 -----------
c020, c120, c220,
c010, c110, c210,
c000, c100, c200
];
coeffs_list_3d_order3 := [
# z^3 -----------------
c033, c133, c233, c333,
c023, c123, c223, c323,
c013, c113, c213, c313,
c003, c103, c203, c303,
# z^2 -----------------
c032, c132, c232, c332,
c022, c122, c222, c322,
c012, c112, c212, c312,
c002, c102, c202, c302,
# z^1 -----------------
c031, c131, c231, c331,
c021, c121, c221, c321,
c011, c111, c211, c311,
c001, c101, c201, c301,
# z^0 -----------------
c030, c130, c230, c330,
c020, c120, c220, c320,
c010, c110, c210, c310,
c000, c100, c200, c300
];
coeffs_list_3d_order4 := [
# z^4 -----------------------
c044, c144, c244, c344, c444,
c034, c134, c234, c334, c434,
c024, c124, c224, c324, c424,
c014, c114, c214, c314, c414,
c004, c104, c204, c304, c404,
# z^3 -----------------------
c043, c143, c243, c343, c443,
c033, c133, c233, c333, c433,
c023, c123, c223, c323, c423,
c013, c113, c213, c313, c413,
c003, c103, c203, c303, c403,
# z^2 -----------------------
c042, c142, c242, c342, c442,
c032, c132, c232, c332, c432,
c022, c122, c222, c322, c422,
c012, c112, c212, c312, c412,
c002, c102, c202, c302, c402,
# z^1 -----------------------
c041, c141, c241, c341, c441,
c031, c131, c231, c331, c431,
c021, c121, c221, c321, c421,
c011, c111, c211, c311, c411,
c001, c101, c201, c301, c401,
# z^0 -----------------------
c040, c140, c240, c340, c440,
c030, c130, c230, c330, c430,
c020, c120, c220, c320, c420,
c010, c110, c210, c310, c410,
c000, c100, c200, c300, c400
];
coeffs_list_3d_order5 := [
# z^5 -----------------------
c055, c155, c255, c355, c455, c555,
c045, c145, c245, c345, c445, c545,
c035, c135, c235, c335, c435, c535,
c025, c125, c225, c325, c425, c525,
c015, c115, c215, c315, c415, c515,
c005, c105, c205, c305, c405, c505,
# z^4 -----------------------
c054, c154, c254, c354, c454, c554,
c044, c144, c244, c344, c444, c544,
c034, c134, c234, c334, c434, c534,
c024, c124, c224, c324, c424, c524,
c014, c114, c214, c314, c414, c514,
c004, c104, c204, c304, c404, c504,
# z^3 -----------------------
c053, c153, c253, c353, c453, c553,
c043, c143, c243, c343, c443, c543,
c033, c133, c233, c333, c433, c533,
c023, c123, c223, c323, c423, c523,
c013, c113, c213, c313, c413, c513,
c003, c103, c203, c303, c403, c503,
# z^2 -----------------------
c052, c152, c252, c352, c452, c552,
c042, c142, c242, c342, c442, c542,
c032, c132, c232, c332, c432, c532,
c022, c122, c222, c322, c422, c522,
c012, c112, c212, c312, c412, c512,
c002, c102, c202, c302, c402, c502,
# z^1 -----------------------
c051, c151, c251, c351, c451, c551,
c041, c141, c241, c341, c441, c541,
c031, c131, c231, c331, c431, c531,
c021, c121, c221, c321, c421, c521,
c011, c111, c211, c311, c411, c511,
c001, c101, c201, c301, c401, c501,
# z^0 -----------------------
c050, c150, c250, c350, c450, c550,
c040, c140, c240, c340, c440, c540,
c030, c130, c230, c330, c430, c530,
c020, c120, c220, c320, c420, c520,
c010, c110, c210, c310, c410, c510,
c000, c100, c200, c300, c400, c500
];
################################################################################
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