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Diffstat (limited to 'src/GeneralizedPolynomial-Uniform/Hermite/1d.log')
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diff --git a/src/GeneralizedPolynomial-Uniform/Hermite/1d.log b/src/GeneralizedPolynomial-Uniform/Hermite/1d.log deleted file mode 100644 index f855fa5..0000000 --- a/src/GeneralizedPolynomial-Uniform/Hermite/1d.log +++ /dev/null @@ -1,2385 +0,0 @@ - |\^/| Maple 7 (IBM INTEL LINUX) -._|\| |/|_. Copyright (c) 2001 by Waterloo Maple Inc. - \ MAPLE / All rights reserved. Maple is a registered trademark of - <____ ____> Waterloo Maple Inc. - | Type ? for help. -# util.maple -- misc utility routines -# $Header: /cactusdevcvs/CactusBase/LocalInterp/src/GeneralizedPolynomial-Uniform/util.maple,v 1.4 2002/08/20 16:46:06 jthorn Exp $ -> -# -# fix_rationals - convert numbers to RATIONAL() calls -# nonmatching_names - find names in a list which *don't* have a specified prefix -# sprint_numeric_list - convert a numeric list to a valid C identifier suffix -# print_name_list_dcl - print C declarations for a list of names -# -# hypercube_points - compute all (integer) points in an N-dimensional hypercube -# -# ftruncate - truncate a file to zero length -# -> -################################################################################ -################################################################################ -################################################################################ -> -# -# This function converts all {integer, rational} subexpressions of its -# input except integer exponents and -1 factors in products, into function -# calls -# RATIONAL(num,den) -# This is useful in conjunction with the C() library function, since -# -# C( (1/3) * foo * bar ) -# t0 = foo*bar/3; -# -# generates a (slow) division (and runs the risk of mixed-mode-arithmetic -# problems), while -# -# C((1.0/3.0) * foo * bar); -# t0 = 0.3333333333*foo*bar; -# -# suffers from roundoff error. With this function, -# -# fix_rationals((1/3) * foo * bar); -# RATIONAL(1,3) foo bar -# C(%); -# t0 = RATIONAL(1.0,3.0)*foo*bar; -# -# which a C preprocessor macro can easily convert to the desired -# -# t0 = (1.0/3.0)*foo*bar; -# -# Additionally, this function can be told to leave certain types of -# subexpressions unconverged. For example, -# fix_rationals(expr, type, specfunc(integer, DATA)); -# will leave all subexpressions of the form DATA(integer arguments) -# unconverted. -# -# Arguments: -# expr = (in) The expression to be converted. -# inert_fn = (optional in) -# If specified, this argument should be a Boolean procedure -# or the name of a Boolean procedure. This procedure should -# take one or more argument, and return true if and only if -# the first argument should *not* be converted, i.e. if we -# should leave this expression unchanged. See the last -# example above. -# ... = (optional in) -# Any further arguments are passed as additional arguments to -# the inert_fn procedure. -# -> fix_rationals := -> proc( -> expr::{ -> algebraic, name = algebraic, -> list({algebraic, name = algebraic}), -> set ({algebraic, name = algebraic}) -> }, -> inert_fn::{name, procedure} -> ) -> local nn, k, -> base, power, fbase, fpower, -> fn, fn_args_list, -> num, den, mult; -> -# do we want to convert this expression? -> if ((nargs >= 2) and inert_fn(expr, args[3..nargs])) -> then return expr; -> end if; -> -# recurse over lists and sets -> if (type(expr, {list,set})) -> then return map(fix_rationals, expr, args[2..nargs]); -> end if; -> -# recurse over equation right hand sides -> if (type(expr, name = algebraic)) -> then return ( lhs(expr) = fix_rationals(rhs(expr), args[2..nargs]) ); -> end if; -> -# recurse over functions other than RATIONAL() -> if (type(expr, function)) -> then -> fn := op(0, expr); -> if (fn <> 'RATIONAL') -> then -> fn_args_list := [op(expr)]; -> fn_args_list := map(fix_rationals, fn_args_list, args[2..nargs]); -> fn; return '%'( op(fn_args_list) ); -> end if; -> end if; -> -> nn := nops(expr); -> -# recurse over sums -> if (type(expr, `+`)) -> then return sum('fix_rationals(op(k,expr), args[2..nargs])', 'k'=1..nn); -> end if; -> -# recurse over products -# ... leaving leading -1 factors intact, i.e. not converted to RATIONAL(-1,1) -> if (type(expr, `*`)) -> then -> if (op(1, expr) = -1) -> then return -1*fix_rationals(remove(type, expr, 'identical(-1)'), -> args[2..nargs]); -> else return product('fix_rationals(op(k,expr), args[2..nargs])', -> 'k'=1..nn); -> end if; -> end if; -> -# recurse over powers -# ... leaving integer exponents intact -> if (type(expr, `^`)) -> then -> base := op(1, expr); -> power := op(2, expr); -> -> fbase := fix_rationals(base, args[2..nargs]); -> if (type(power, integer)) -> then fpower := power; -> else fpower := fix_rationals(power, args[2..nargs]); -> end if; -> return fbase ^ fpower; -> end if; -> -# fix integers and fractions -> if (type(expr, integer)) -> then return 'RATIONAL'(expr, 1); -> end if; -> if (type(expr, fraction)) -> then -> num := op(1, expr); -> den := op(2, expr); -> -> return 'RATIONAL'(num, den); -> end if; -> -# turn Maple floating-point into integer fraction, then recursively fix that -> if (type(expr, float)) -> then -> mult := op(1, expr); -> power := op(2, expr); -> return fix_rationals(mult * 10^power, args[2..nargs]); -> end if; -> -# identity op on names -> if (type(expr, name)) -> then return expr; -> end if; -> -# unknown type -> error "%0", -> "unknown type for expr!", -> " whattype(expr) = ", whattype(expr), -> " expr = ", expr; -> end proc; -fix_rationals := proc(expr::{algebraic, name = algebraic, -list({algebraic, name = algebraic}), set({algebraic, name = algebraic})}, -inert_fn::{procedure, name}) -local nn, k, base, power, fbase, fpower, fn, fn_args_list, num, den, mult; - if 2 <= nargs and inert_fn(expr, args[3 .. nargs]) then return expr - end if; - if type(expr, {set, list}) then - return map(fix_rationals, expr, args[2 .. nargs]) - end if; - if type(expr, name = algebraic) then - return lhs(expr) = fix_rationals(rhs(expr), args[2 .. nargs]) - end if; - if type(expr, function) then - fn := op(0, expr); - if fn <> 'RATIONAL' then - fn_args_list := [op(expr)]; - fn_args_list := - map(fix_rationals, fn_args_list, args[2 .. nargs]); - fn; - return '%'(op(fn_args_list)) - end if - end if; - nn := nops(expr); - if type(expr, `+`) then return - sum('fix_rationals(op(k, expr), args[2 .. nargs])', 'k' = 1 .. nn) - end if; - if type(expr, `*`) then - if op(1, expr) = -1 then return -fix_rationals( - remove(type, expr, 'identical(-1)'), args[2 .. nargs]) - else return product('fix_rationals(op(k, expr), args[2 .. nargs])', - 'k' = 1 .. nn) - end if - end if; - if type(expr, `^`) then - base := op(1, expr); - power := op(2, expr); - fbase := fix_rationals(base, args[2 .. nargs]); - if type(power, integer) then fpower := power - else fpower := fix_rationals(power, args[2 .. nargs]) - end if; - return fbase^fpower - end if; - if type(expr, integer) then return 'RATIONAL'(expr, 1) end if; - if type(expr, fraction) then - num := op(1, expr); den := op(2, expr); return 'RATIONAL'(num, den) - end if; - if type(expr, float) then - mult := op(1, expr); - power := op(2, expr); - return fix_rationals(mult*10^power, args[2 .. nargs]) - end if; - if type(expr, name) then return expr end if; - error "%0", "unknown type for expr!", " whattype(expr) = ", - whattype(expr), " expr = ", expr -end proc - -> -################################################################################ -> -# -# This function finds names in a list which *don't* have a specified prefix. -# -# Arguments: -# name_list = A list of the names. -# prefix = The prefix we want to filter out. -# -# Results: -# This function returns the subset list of names which don't have the -# specified prefix. -# -> nonmatching_names := -> proc( name_list::list({name,string}), prefix::{name,string} ) -> -> select( proc(n) -> evalb(not StringTools[IsPrefix](prefix,n)); -> end proc -> , -> name_list -> ); -> end proc; -nonmatching_names := proc( -name_list::list({name, string}), prefix::{name, string}) - select(proc(n) evalb(not StringTools[IsPrefix](prefix, n)) end proc, - name_list) -end proc - -> -################################################################################ -> -# -# This function converts a numeric list to a string which is a valid -# C identifier suffix: elements are separated by "_", decimal points are -# replaced by "x", and all nonzero values have explicit +/- signs, which -# are replaced by "p"/"m". -# -# For example, [0,-3.5,+4] --> "0_m3x5_p4". -# -> sprint_numeric_list := -> proc(nlist::list(numeric)) -> -# generate preliminary string, eg "+0_-3.5_+4" -> map2(sprintf, "%+a", nlist); -> ListTools[Join](%, "_"); -> cat(op(%)); -> -# fixup bad characters -> StringTools[SubstituteAll](%, "+0", "0"); -> StringTools[CharacterMap](".+-", "xpm", %); -> -> return %; -> end proc; -sprint_numeric_list := proc(nlist::list(numeric)) - map2(sprintf, "%+a", nlist); - ListTools[Join](%, "_"); - cat(op(%)); - StringTools[SubstituteAll](%, "+0", "0"); - StringTools[CharacterMap](".+-", "xpm", %); - return % -end proc - -> -################################################################################ -> -# -# This function prints a sequence of C declarations for a list of names. -# -# Argument: -# name_list = A list of the names. -# type_name = The C type of the names, eg. "double". -# file_name = The file name to write the declaration to. This is -# truncated before writing. -# -> print_name_list_dcl := -> proc( name_list::list({name,string}), -> type_name::string, -> file_name::string ) -> local blanks, separator_string; -> -> ftruncate(file_name); -> -> map( -> proc(var::{name,string}) -> fprintf(file_name, -> "%s %s;\n", -> type_name, var); -> end proc -> , -> name_list -> ); -> -> fclose(file_name); -> NULL; -> end proc; -print_name_list_dcl := proc( -name_list::list({name, string}), type_name::string, file_name::string) -local blanks, separator_string; - ftruncate(file_name); - map(proc(var::{name, string}) - fprintf(file_name, "%s %s;\n", type_name, var) - end proc, name_list); - fclose(file_name); - NULL -end proc - -> -################################################################################ -################################################################################ -################################################################################ -> -# -# This function computes a list of all the (integer) points in an -# N-dimensional hypercube, in lexicographic order. The present -# implementation requires N <= 4. -# -# Arguments: -# cmin,cmax = N-element lists of cube minimum/maximum coordinates. -# -# Results: -# The function returns a set of d-element lists giving the coordinates. -# For example, -# hypercube([0,0], [2,1] -# returns -# { [0,0], [0,1], [1,0], [1,1], [2,0], [2,1] } -> hypercube_points := -> proc(cmin::list(integer), cmax::list(integer)) -> local N, i,j,k,l; -> -> N := nops(cmin); -> if (nops(cmax) <> N) -> then error -> "must have same number of dimensions for min and max coordinates!"; -> fi; -> -> if (N = 1) -> then return [seq([i], i=cmin[1]..cmax[1])]; -> elif (N = 2) -> then return [ -> seq( -> seq([i,j], j=cmin[2]..cmax[2]), -> i=cmin[1]..cmax[1]) -> ]; -> elif (N = 3) -> then return [ -> seq( -> seq( -> seq([i,j,k], k=cmin[3]..cmax[3]), -> j=cmin[2]..cmax[2] ), -> i=cmin[1]..cmax[1]) -> ]; -> elif (N = 4) -> then return [ -> seq( -> seq( -> seq( -> seq([i,j,k,l], l=cmin[4]..cmax[4]), -> k=cmin[3]..cmax[3] ), -> j=cmin[2]..cmax[2]), -> i=cmin[1]..cmax[1]) -> ]; -> else -> error "implementation restriction: must have N <= 4, got %1!", N; -> fi; -> end proc; -hypercube_points := proc(cmin::list(integer), cmax::list(integer)) -local N, i, j, k, l; - N := nops(cmin); - if nops(cmax) <> N then error - "must have same number of dimensions for min and max coordinates!" - end if; - if N = 1 then return [seq([i], i = cmin[1] .. cmax[1])] - elif N = 2 then return - [seq(seq([i, j], j = cmin[2] .. cmax[2]), i = cmin[1] .. cmax[1])] - elif N = 3 then return [seq( - seq(seq([i, j, k], k = cmin[3] .. cmax[3]), j = cmin[2] .. cmax[2]) - , i = cmin[1] .. cmax[1])] - elif N = 4 then return [seq(seq(seq( - seq([i, j, k, l], l = cmin[4] .. cmax[4]), k = cmin[3] .. cmax[3]), - j = cmin[2] .. cmax[2]), i = cmin[1] .. cmax[1])] - else error "implementation restriction: must have N <= 4, got %1!", N - end if -end proc - -> -################################################################################ -################################################################################ -################################################################################ -> -# -# This function truncates a file to 0 length if it exists, or creates -# it at that length if it doesn't exist. -# -# Arguments: -# file_name = (in) The name of the file. -# -> ftruncate := -> proc(file_name::string) -> fopen(file_name, 'WRITE'); -> fclose(%); -> NULL; -> end proc; -ftruncate := - - proc(file_name::string) fopen(file_name, 'WRITE'); fclose(%); NULL end proc - -# interpolate.maple -- compute interpolation formulas/coefficients -# $Header: /cactusdevcvs/CactusBase/LocalInterp/src/GeneralizedPolynomial-Uniform/interpolate.maple,v 1.10 2002/08/28 11:31:09 jthorn Exp $ -> -# -# <<<representation of numbers, data values, etc>>> -# Lagrange_polynomial_interpolant - compute Lagrange polynomial interpolant -# Hermite_polynomial_interpolant - compute Hermite polynomial interpolant -# coeffs_as_lc_of_data - coefficients of ... (linear combination of data) -# -# print_coeffs__lc_of_data - print C code to compute coefficients -# print_fetch_data - print C code to fetch input array chunk into struct data -# print_store_coeffs - print C code to store struct coeffs "somewhere" -# print_interp_cmpt__lc_of_data - print C code for computation of interpolant -# -# coeff_name - name of coefficient of data at a given [m] coordinate -# data_var_name - name of variable storing data value at a given [m] coordinate -# -> -################################################################################ -> -# -# ***** representation of numbers, data values, etc ***** -# -# We use RATIONAL(p.0,q.0) to denote the rational number p/q. -# -# We use DATA(...) to represent the data values being interpolated at a -# specified [m] coordinate, where the arguments are the [m] coordinates. -# -# We use COEFF(...) to represent the molecule coefficient at a specified -# [m] coordinate, where the arguments are the [m] coordinates. -# -# For example, the usual 1-D centered 2nd order 1st derivative molecule -# would be written -# RATIONAL(-1.0,2.0)*DATA(-1) + RATIONA(1.0,2.0)*DATA(1) -# and its coefficients as -# COEFF(-1) = RATIONAL(-1.0,2.0) -# COEFF(1) = RATIONAL(1.0,2.0) -# -> -################################################################################ -################################################################################ -################################################################################ -> -# -# This function computes a Lagrange polynomial interpolant in any -# number of dimensions. -# -# Arguments: -# fn = The interpolation function. This should be a procedure in the -# coordinates, having the coefficients as global variables. For -# example, -# proc(x,y) c00 + c10*x + c01*y end proc -# coeff_list = A set of the interpolation coefficients (coefficients in -# the interpolation function), for example [c00, c10, c01]. -# coord_list = A list of the coordinates (independent variables in the -# interpolation function), for example [x,y]. -# posn_list = A list of positions (each a list of numeric values) where the -# interpolant is to use data, for example hypercube([0,0], [1,1]). -# Any positions may be used; if they're redundant (as in the -# example) the least-squares interpolant is computed. -# -# Results: -# This function returns the interpolating polynomial, in the form of -# an algebraic expression in the coordinates and the data values. -# -> Lagrange_polynomial_interpolant := -> proc( -> fn::procedure, coeff_list::list(name), -> coord_list::list(name), posn_list::list(list(numeric)) -> ) -> local posn, data_eqns, coeff_eqns; -> -# coefficients of interpolating polynomial -> data_eqns := { seq( fn(op(posn))='DATA'(op(posn)) , posn=posn_list ) }; -> coeff_eqns := linalg[leastsqrs](data_eqns, {op(coeff_list)}); -> if (has(coeff_eqns, '_t')) -> then error "interpolation coefficients aren't uniquely determined!"; -> end if; -> -# interpolant as a polynomial in the coordinates -> return subs(coeff_eqns, eval(fn))(op(coord_list)); -> end proc; -Lagrange_polynomial_interpolant := proc(fn::procedure, coeff_list::list(name), -coord_list::list(name), posn_list::list(list(numeric))) -local posn, data_eqns, coeff_eqns; - data_eqns := {seq(fn(op(posn)) = 'DATA'(op(posn)), posn = posn_list)}; - coeff_eqns := linalg[leastsqrs](data_eqns, {op(coeff_list)}); - if has(coeff_eqns, '_t') then - error "interpolation coefficients aren't uniquely determined!" - end if; - return subs(coeff_eqns, eval(fn))(op(coord_list)) -end proc - -> -################################################################################ -> -# -# This function computes a Hermite polynomial interpolant in any -# number of dimensions. This is a polynomial which -# * has values which match the given data DATA() at a specified set of -# points, and -# * has derivatives which match the specified finite-difference derivatives -# of the given data DATA() at a specified set of points -# -# For the derivative matching, we actually match all possible products -# of 1st derivatives, i.e. in 2-D we match dx, dy, and dxy, in 3-D we -# match dx, dy, dz, dxy, dxz, dyz, and dxyz, etc etc. -# -# Arguments: -# fn = The interpolation function. This should be a procedure in the -# coordinates, having the coefficients as global variables. For -# example, -# 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; -# coeff_set = A set of the interpolation coefficients (coefficients in -# the interpolation function), for example -# { -# c03, c13, c23, c33, -# c02, c12, c22, c32, -# c01, c11, c21, c31, -# c00, c10, c20, c30 -# } -# coord_list = A list of the coordinates (independent variables in the -# interpolation function), for example [x,y]. -# deriv_set = A set of equations of the form -# {coords} = proc -# giving the derivatives which are to be matched, and the -# procedures to compute their finite-difference approximations. -# Each procedure should take N_dims integer arguments specifying -# an evaluation point, and return a suitable linear combination -# of the DATA() for the derivative at that point. For example -# { -# {x} = proc(i::integer, j::integer) -# - 1/2*DATA(i-1,j) + 1/2*DATA(i+1,j) -# end proc -# , -# {y} = proc(i::integer, j::integer) -# - 1/2*DATA(i,j-1) + 1/2*DATA(i,j+1) -# end proc -# , -# {x,y} = proc(i::integer, j::integer) -# - 1/4*DATA(i-1,j+1) + 1/4*DATA(i+1,j+1) -# + 1/4*DATA(i-1,j-1) - 1/4*DATA(i+1,j-1) -# end proc -# } -# fn_posn_set = A set of positions (each a list of numeric values) -# where the interpolant is to match the given data DATA(), -# for example -# {[0,0], [0,1], [1,0], [1,1]} -# deriv_posn_set = A list of positions (each a list of numeric values) -# where the interpolant is to match the derivatives -# specified by deriv_set , for example -# {[0,0], [0,1], [1,0], [1,1]} -# -# Results: -# This function returns the interpolating polynomial, in the form of -# an algebraic expression in the coordinates and the data values. -# -> Hermite_polynomial_interpolant := -> proc( -> fn::procedure, -> coeff_set::set(name), -> coord_list::list(name), -> deriv_set::set(set(name) = procedure), -> fn_posn_set::set(list(numeric)), -> deriv_posn_set::set(list(numeric)) -> ) -> local fn_eqnset, deriv_eqnset, coeff_eqns, subs_eqnset; -> -> -# -# compute a set of equations -# {fn(posn) = DATA(posn)} -# giving the function values to be matched -# -> fn_eqnset := map( -> # return equation that fn(posn) = DATA(posn) -> proc(posn::list(integer)) -> fn(op(posn)) = 'DATA'(op(posn)); -> end proc -> , -> fn_posn_set -> ); -> -> -# -# compute a set of equations -# { diff(fn,coords)(posn) = DERIV(coords)(posn) } -# giving the derivative values to be matched, where DERIV(coords) -# is a placeholder for the appropriate derivative -# -> map( -> # return set of equations for this particular derivative -> proc(deriv_coords::set(name)) -> local deriv_fn; -> fn(op(coord_list)); -> diff(%, op(deriv_coords)); -> deriv_fn := unapply(%, op(coord_list)); -> map( -> proc(posn::list(integer)) -> deriv_fn(op(posn)) = 'DERIV'(op(deriv_coords))(op(posn)); -> end proc -> , -> deriv_posn_set -> ); -> end proc -> , -> map(lhs, deriv_set) -> ); -> deriv_eqnset := `union`(op(%)); -> -> -# -# solve overall set of equations for coefficients -# in terms of DATA() and DERIV() values -# -> coeff_eqns := solve[linear](fn_eqnset union deriv_eqnset, coeff_set); -> if (indets(map(rhs,%)) <> {}) -> then error "no unique solution for coefficients -- %1 eqns for %2 coeffs", -> nops(fn_eqnset union deriv_eqnset), -> nops(coeff_set); -> fi; -> -> -# -# compute a set of substitution equations -# {'DERIV'(coords) = procedure} -# -> subs_eqnset := map( -> proc(eqn::set(name) = procedure) -> 'DERIV'(op(lhs(eqn))) = rhs(eqn); -> end proc -> , -> deriv_set -> ); -> -> -# -# compute the coefficients in terms of the DATA() values -# -> subs(subs_eqnset, coeff_eqns); -> eval(%); -> -# -# compute the interpolant as a polynomial in the coordinates -# -> subs(%, fn(op(coord_list))); -> end proc; -Hermite_polynomial_interpolant := proc(fn::procedure, coeff_set::set(name), -coord_list::list(name), deriv_set::set(set(name) = procedure), -fn_posn_set::set(list(numeric)), deriv_posn_set::set(list(numeric))) -local fn_eqnset, deriv_eqnset, coeff_eqns, subs_eqnset; - fn_eqnset := map( - proc(posn::list(integer)) fn(op(posn)) = 'DATA'(op(posn)) end proc, - fn_posn_set); - map(proc(deriv_coords::set(name)) - local deriv_fn; - fn(op(coord_list)); - diff(%, op(deriv_coords)); - deriv_fn := unapply(%, op(coord_list)); - map(proc(posn::list(integer)) - deriv_fn(op(posn)) = - 'DERIV'(op(deriv_coords))(op(posn)) - end proc, deriv_posn_set) - end proc, map(lhs, deriv_set)); - deriv_eqnset := `union`(op(%)); - coeff_eqns := solve[linear](fn_eqnset union deriv_eqnset, coeff_set); - if indets(map(rhs, %)) <> {} then error - "no unique solution for coefficients -- %1 eqns for %2 coeffs", - nops(fn_eqnset union deriv_eqnset), nops(coeff_set) - end if; - subs_eqnset := map(proc(eqn::(set(name) = procedure)) - 'DERIV'(op(lhs(eqn))) = rhs(eqn) - end proc, deriv_set); - subs(subs_eqnset, coeff_eqns); - eval(%); - subs(%, fn(op(coord_list))) -end proc - -> -################################################################################ -> -# -# This function takes as input an interpolating polynomial, expresses -# it as a linear combination of the data values, and returns the coefficeints -# of that form. -# -# Arguments: -# interpolant = The interpolating polynomial (an algebraic expression -# in the coordinates and the data values). -# posn_list = The same list of data positions used in the interpolant. -# -# Results: -# This function returns the coefficients, as a list of equations of the -# form COEFF(...) = value , where each value is a polynomial in the -# coordinates. The order of the list matches that of posn_list. -# -> coeffs_as_lc_of_data := -> proc( -> interpolant::algebraic, -> posn_list::list(list(numeric)) -> ) -> local data_list, interpolant_as_lc_of_data; -> -# interpolant as a linear combination of the data values -> data_list := [ seq( 'DATA'(op(posn)) , posn=posn_list ) ]; -> interpolant_as_lc_of_data := collect(interpolant, data_list); -> -# coefficients of the data values in the linear combination -> return map( -> proc(posn::list(numeric)) -> coeff(interpolant_as_lc_of_data, DATA(op(posn))); -> 'COEFF'(op(posn)) = %; -> end proc -> , -> posn_list -> ); -> end proc; -coeffs_as_lc_of_data := proc( -interpolant::algebraic, posn_list::list(list(numeric))) -local data_list, interpolant_as_lc_of_data; - data_list := [seq('DATA'(op(posn)), posn = posn_list)]; - interpolant_as_lc_of_data := collect(interpolant, data_list); - return map(proc(posn::list(numeric)) - coeff(interpolant_as_lc_of_data, DATA(op(posn))); - 'COEFF'(op(posn)) = % - end proc, posn_list) -end proc - -> -################################################################################ -################################################################################ -################################################################################ -> -# -# This function prints C expressions for the coefficients of an -# interpolating polynomial. (The polynomial is expressed as linear -# combinations of the data values with coefficients which are -# RATIONAL(p,q) calls.) -# -# Arguments: -# coeff_list = A list of the coefficients, as returned from -# coeffs_as_lc_of_data() . -# coeff_name_prefix = A prefix string for the coefficient names. -# temp_name_type = The C type to be used for Maple-introduced temporary -# names, eg. "double". -# file_name = The file name to write the coefficients to. This is -# truncated before writing. -# -> print_coeffs__lc_of_data := -> proc( coeff_list::list(specfunc(numeric,COEFF) = algebraic), -> coeff_name_prefix::string, -> temp_name_type::string, -> file_name::string ) -> global `codegen/C/function/informed`; -> local coeff_list2, cmpt_list, temp_name_list; -> -# convert LHS of each equation from a COEFF() call (eg COEFF(-1,+1)) -# to a Maple/C variable name (eg coeff_I_m1_p1) -> coeff_list2 := map( -> proc(coeff_eqn::specfunc(numeric,COEFF) = algebraic) -> local posn; -> posn := [op(lhs(coeff_eqn))]; -> coeff_name(posn,coeff_name_prefix); -> convert(%, name); # codegen[C] wants LHS -> # to be an actual Maple *name* -> % = fix_rationals(rhs(coeff_eqn)); -> end proc -> , -> coeff_list -> ); -> -# -# generate the C code -# -> -# tell codegen[C] not to warn about unknown RATIONAL() and DATA() "fn calls" -# via undocumented :( global table -> `codegen/C/function/informed`['RATIONAL'] := true; -> `codegen/C/function/informed`['DATA'] := true; -> -> ftruncate(file_name); -> -# optimized computation sequence for all the coefficients -# (may use local variables t0,t1,t2,...) -> cmpt_list := [codegen[optimize](coeff_list2, tryhard)]; -> -# list of the t0,t1,t2,... local variables -> temp_name_list := nonmatching_names(map(lhs,cmpt_list), coeff_name_prefix); -> -# declare the t0,t1,t2,... local variables (if there are any) -> if (nops(temp_name_list) > 0) -> then print_name_list_dcl(%, temp_name_type, file_name); -> fi; -> -# now print the optimized computation sequence -> codegen[C](cmpt_list, filename=file_name); -> -> fclose(file_name); -> -> NULL; -> end proc; -print_coeffs__lc_of_data := proc( -coeff_list::list(specfunc(numeric, COEFF) = algebraic), -coeff_name_prefix::string, temp_name_type::string, file_name::string) -local coeff_list2, cmpt_list, temp_name_list; -global `codegen/C/function/informed`; - coeff_list2 := map(proc( - coeff_eqn::(specfunc(numeric, COEFF) = algebraic)) - local posn; - posn := [op(lhs(coeff_eqn))]; - coeff_name(posn, coeff_name_prefix); - convert(%, name); - % = fix_rationals(rhs(coeff_eqn)) - end proc, coeff_list); - `codegen/C/function/informed`['RATIONAL'] := true; - `codegen/C/function/informed`['DATA'] := true; - ftruncate(file_name); - cmpt_list := [codegen[optimize](coeff_list2, tryhard)]; - temp_name_list := - nonmatching_names(map(lhs, cmpt_list), coeff_name_prefix); - if 0 < nops(temp_name_list) then - print_name_list_dcl(%, temp_name_type, file_name) - end if; - codegen[C](cmpt_list, filename = file_name); - fclose(file_name); - NULL -end proc - -> -################################################################################ -> -# -# This function prints a sequence of C expression to assign the data-value -# variables, eg -# data->data_m1_p1 = DATA(-1,1); -# -# Arguments: -# posn_list = The same list of positions as was used to compute the -# interpolating polynomial. -# data_var_name_prefix = A prefix string for the data variable names. -# file_name = The file name to write the coefficients to. This is -# truncated before writing. -# -> print_fetch_data := -> proc( -> posn_list::list(list(numeric)), -> data_var_name_prefix::string, -> file_name::string -> ) -> -> ftruncate(file_name); -> map( -> proc(posn::list(numeric)) -> fprintf(file_name, -> "%s = %a;\n", -> data_var_name(posn,data_var_name_prefix), -> DATA(op(posn))); -> end proc -> , -> posn_list -> ); -> fclose(file_name); -> -> NULL; -> end proc; -print_fetch_data := proc(posn_list::list(list(numeric)), -data_var_name_prefix::string, file_name::string) - ftruncate(file_name); - map(proc(posn::list(numeric)) - fprintf(file_name, "%s = %a;\n", - data_var_name(posn, data_var_name_prefix), DATA(op(posn))) - end proc, posn_list); - fclose(file_name); - NULL -end proc - -> -################################################################################ -> -# -# This function prints a sequence of C expression to store the interpolation -# coefficients in COEFF(...) expressions, eg -# COEFF(1,-1) = factor * coeffs->coeff_p1_m1; -# -# Arguments: -# posn_list = The list of positions in the molecule. -# coeff_name_prefix = A prefix string for the coefficient names, -# eg "factor * coeffs->coeff_" -# file_name = The file name to write the coefficients to. This is -# truncated before writing. -# -> print_store_coeffs := -> proc( -> posn_list::list(list(numeric)), -> coeff_name_prefix::string, -> file_name::string -> ) -> -> ftruncate(file_name); -> map( -> proc(posn::list(numeric)) -> fprintf(file_name, -> "%a = %s;\n", -> 'COEFF'(op(posn)), -> coeff_name(posn,coeff_name_prefix)); -> end proc -> , -> posn_list -> ); -> fclose(file_name); -> -> NULL; -> end proc; -print_store_coeffs := proc(posn_list::list(list(numeric)), -coeff_name_prefix::string, file_name::string) - ftruncate(file_name); - map(proc(posn::list(numeric)) - fprintf(file_name, "%a = %s;\n", 'COEFF'(op(posn)), - coeff_name(posn, coeff_name_prefix)) - end proc, posn_list); - fclose(file_name); - NULL -end proc - -> -################################################################################ -> -# -# This function prints a C expression to evaluate a molecule, i.e. -# to compute the molecule as a linear combination of the data values. -# -# Arguments: -# posn_list = The list of positions in the molecule. -# coeff_name_prefix = A prefix string for the coefficient names. -# data_var_name_prefix = A prefix string for the data variable names. -# file_name = The file name to write the coefficients to. This is -# truncated before writing. -# -> print_evaluate_molecule := -> proc( -> posn_list::list(list(numeric)), -> coeff_name_prefix::string, -> data_var_name_prefix::string, -> file_name::string -> ) -> -> ftruncate(file_name); -> -# list of "coeff*data_var" terms -> map( -> proc(posn::list(numeric)) -> sprintf("%s*%s", -> coeff_name(posn,coeff_name_prefix), -> data_var_name(posn,data_var_name_prefix)); -> end proc -> , -> posn_list -> ); -> -> ListTools[Join](%, "\n + "); -> cat(op(%)); -> fprintf(file_name, " %s;\n", %); -> -> fclose(file_name); -> -> NULL; -> end proc; -print_evaluate_molecule := proc(posn_list::list(list(numeric)), -coeff_name_prefix::string, data_var_name_prefix::string, file_name::string) - ftruncate(file_name); - map(proc(posn::list(numeric)) - sprintf("%s*%s", coeff_name(posn, coeff_name_prefix), - data_var_name(posn, data_var_name_prefix)) - end proc, posn_list); - ListTools[Join](%, "\n + "); - cat(op(%)); - fprintf(file_name, " %s;\n", %); - fclose(file_name); - NULL -end proc - -> -################################################################################ -################################################################################ -################################################################################ -> -# -# This function computes the name of the coefficient of the data at a -# given [m] position, i.e. it encapsulates our naming convention for this. -# -# Arguments: -# posn = (in) The [m] coordinates. -# name_prefix = A prefix string for the coefficient name. -# -# Results: -# The function returns the coefficient, as a Maple string. -# -> coeff_name := -> proc(posn::list(numeric), name_prefix::string) -> cat(name_prefix, sprint_numeric_list(posn)); -> end proc; -coeff_name := proc(posn::list(numeric), name_prefix::string) - cat(name_prefix, sprint_numeric_list(posn)) -end proc - -> -################################################################################ -> -# -# This function computes the name of the variable in which the C code -# will store the input data at a given [m] position, i.e. it encapsulates -# our naming convention for this. -# -# Arguments: -# posn = (in) The [m] coordinates. -# name_prefix = A prefix string for the variable name. -# -# Results: -# The function returns the variable name, as a Maple string. -# -> data_var_name := -> proc(posn::list(numeric), name_prefix::string) -> cat(name_prefix, sprint_numeric_list(posn)); -> end proc; -data_var_name := proc(posn::list(numeric), name_prefix::string) - cat(name_prefix, sprint_numeric_list(posn)) -end proc - -# Maple code to compute lists of point positions in hypercube-shaped molecules -# $Header: /cactusdevcvs/CactusBase/LocalInterp/src/GeneralizedPolynomial-Uniform/common/cube_posns.maple,v 1.3 2002/08/20 16:56:41 jthorn Exp $ -> -################################################################################ -> -# -# 1D interpolation points -# -> posn_list_1d_size2 := hypercube_points([ 0], [+1]); - posn_list_1d_size2 := [[0], [1]] - -> posn_list_1d_size3 := hypercube_points([-1], [+1]); - posn_list_1d_size3 := [[-1], [0], [1]] - -> posn_list_1d_size4 := hypercube_points([-1], [+2]); - posn_list_1d_size4 := [[-1], [0], [1], [2]] - -> posn_list_1d_size5 := hypercube_points([-2], [+2]); - posn_list_1d_size5 := [[-2], [-1], [0], [1], [2]] - -> posn_list_1d_size6 := hypercube_points([-2], [+3]); - posn_list_1d_size6 := [[-2], [-1], [0], [1], [2], [3]] - -> posn_list_1d_size7 := hypercube_points([-3], [+3]); - posn_list_1d_size7 := [[-3], [-2], [-1], [0], [1], [2], [3]] - -> -################################################################################ -> -# -# 2D interpolation points (Fortran ordering) -# -> posn_list_2d_size2 := map(ListTools[Reverse], -> hypercube_points([ 0, 0], [+1,+1])); - posn_list_2d_size2 := [[0, 0], [1, 0], [0, 1], [1, 1]] - -> posn_list_2d_size3 := map(ListTools[Reverse], -> hypercube_points([-1,-1], [+1,+1])); -posn_list_2d_size3 := [[-1, -1], [0, -1], [1, -1], [-1, 0], [0, 0], [1, 0], - - [-1, 1], [0, 1], [1, 1]] - -> posn_list_2d_size4 := map(ListTools[Reverse], -> hypercube_points([-1,-1], [+2,+2])); -posn_list_2d_size4 := [[-1, -1], [0, -1], [1, -1], [2, -1], [-1, 0], [0, 0], - - [1, 0], [2, 0], [-1, 1], [0, 1], [1, 1], [2, 1], [-1, 2], [0, 2], [1, 2], - - [2, 2]] - -> posn_list_2d_size5 := map(ListTools[Reverse], -> hypercube_points([-2,-2], [+2,+2])); -posn_list_2d_size5 := [[-2, -2], [-1, -2], [0, -2], [1, -2], [2, -2], [-2, -1], - - [-1, -1], [0, -1], [1, -1], [2, -1], [-2, 0], [-1, 0], [0, 0], [1, 0], - - [2, 0], [-2, 1], [-1, 1], [0, 1], [1, 1], [2, 1], [-2, 2], [-1, 2], [0, 2], - - [1, 2], [2, 2]] - -> posn_list_2d_size6 := map(ListTools[Reverse], -> hypercube_points([-2,-2], [+3,+3])); -posn_list_2d_size6 := [[-2, -2], [-1, -2], [0, -2], [1, -2], [2, -2], [3, -2], - - [-2, -1], [-1, -1], [0, -1], [1, -1], [2, -1], [3, -1], [-2, 0], [-1, 0], - - [0, 0], [1, 0], [2, 0], [3, 0], [-2, 1], [-1, 1], [0, 1], [1, 1], [2, 1], - - [3, 1], [-2, 2], [-1, 2], [0, 2], [1, 2], [2, 2], [3, 2], [-2, 3], [-1, 3], - - [0, 3], [1, 3], [2, 3], [3, 3]] - -> -################################################################################ -> -# -# 3D interpolation points (Fortran ordering) -# -> posn_list_3d_size2 := map(ListTools[Reverse], -> hypercube_points([ 0, 0, 0], [+1,+1,+1])); -posn_list_3d_size2 := [[0, 0, 0], [1, 0, 0], [0, 1, 0], [1, 1, 0], [0, 0, 1], - - [1, 0, 1], [0, 1, 1], [1, 1, 1]] - -> posn_list_3d_size3 := map(ListTools[Reverse], -> hypercube_points([-1,-1,-1], [+1,+1,+1])); -posn_list_3d_size3 := [[-1, -1, -1], [0, -1, -1], [1, -1, -1], [-1, 0, -1], - - [0, 0, -1], [1, 0, -1], [-1, 1, -1], [0, 1, -1], [1, 1, -1], [-1, -1, 0], - - [0, -1, 0], [1, -1, 0], [-1, 0, 0], [0, 0, 0], [1, 0, 0], [-1, 1, 0], - - [0, 1, 0], [1, 1, 0], [-1, -1, 1], [0, -1, 1], [1, -1, 1], [-1, 0, 1], - - [0, 0, 1], [1, 0, 1], [-1, 1, 1], [0, 1, 1], [1, 1, 1]] - -> posn_list_3d_size4 := map(ListTools[Reverse], -> hypercube_points([-1,-1,-1], [+2,+2,+2])); -posn_list_3d_size4 := [[-1, -1, -1], [0, -1, -1], [1, -1, -1], [2, -1, -1], - - [-1, 0, -1], [0, 0, -1], [1, 0, -1], [2, 0, -1], [-1, 1, -1], [0, 1, -1], - - [1, 1, -1], [2, 1, -1], [-1, 2, -1], [0, 2, -1], [1, 2, -1], [2, 2, -1], - - [-1, -1, 0], [0, -1, 0], [1, -1, 0], [2, -1, 0], [-1, 0, 0], [0, 0, 0], - - [1, 0, 0], [2, 0, 0], [-1, 1, 0], [0, 1, 0], [1, 1, 0], [2, 1, 0], - - [-1, 2, 0], [0, 2, 0], [1, 2, 0], [2, 2, 0], [-1, -1, 1], [0, -1, 1], - - [1, -1, 1], [2, -1, 1], [-1, 0, 1], [0, 0, 1], [1, 0, 1], [2, 0, 1], - - [-1, 1, 1], [0, 1, 1], [1, 1, 1], [2, 1, 1], [-1, 2, 1], [0, 2, 1], - - [1, 2, 1], [2, 2, 1], [-1, -1, 2], [0, -1, 2], [1, -1, 2], [2, -1, 2], - - [-1, 0, 2], [0, 0, 2], [1, 0, 2], [2, 0, 2], [-1, 1, 2], [0, 1, 2], - - [1, 1, 2], [2, 1, 2], [-1, 2, 2], [0, 2, 2], [1, 2, 2], [2, 2, 2]] - -> posn_list_3d_size5 := map(ListTools[Reverse], -> hypercube_points([-2,-2,-2], [+2,+2,+2])); -posn_list_3d_size5 := [[-2, -2, -2], [-1, -2, -2], [0, -2, -2], [1, -2, -2], - - [2, -2, -2], [-2, -1, -2], [-1, -1, -2], [0, -1, -2], [1, -1, -2], - - [2, -1, -2], [-2, 0, -2], [-1, 0, -2], [0, 0, -2], [1, 0, -2], [2, 0, -2], - - [-2, 1, -2], [-1, 1, -2], [0, 1, -2], [1, 1, -2], [2, 1, -2], [-2, 2, -2], - - [-1, 2, -2], [0, 2, -2], [1, 2, -2], [2, 2, -2], [-2, -2, -1], [-1, -2, -1], - - [0, -2, -1], [1, -2, -1], [2, -2, -1], [-2, -1, -1], [-1, -1, -1], - - [0, -1, -1], [1, -1, -1], [2, -1, -1], [-2, 0, -1], [-1, 0, -1], [0, 0, -1], - - [1, 0, -1], [2, 0, -1], [-2, 1, -1], [-1, 1, -1], [0, 1, -1], [1, 1, -1], - - [2, 1, -1], [-2, 2, -1], [-1, 2, -1], [0, 2, -1], [1, 2, -1], [2, 2, -1], - - [-2, -2, 0], [-1, -2, 0], [0, -2, 0], [1, -2, 0], [2, -2, 0], [-2, -1, 0], - - [-1, -1, 0], [0, -1, 0], [1, -1, 0], [2, -1, 0], [-2, 0, 0], [-1, 0, 0], - - [0, 0, 0], [1, 0, 0], [2, 0, 0], [-2, 1, 0], [-1, 1, 0], [0, 1, 0], - - [1, 1, 0], [2, 1, 0], [-2, 2, 0], [-1, 2, 0], [0, 2, 0], [1, 2, 0], - - [2, 2, 0], [-2, -2, 1], [-1, -2, 1], [0, -2, 1], [1, -2, 1], [2, -2, 1], - - [-2, -1, 1], [-1, -1, 1], [0, -1, 1], [1, -1, 1], [2, -1, 1], [-2, 0, 1], - - [-1, 0, 1], [0, 0, 1], [1, 0, 1], [2, 0, 1], [-2, 1, 1], [-1, 1, 1], - - [0, 1, 1], [1, 1, 1], [2, 1, 1], [-2, 2, 1], [-1, 2, 1], [0, 2, 1], - - [1, 2, 1], [2, 2, 1], [-2, -2, 2], [-1, -2, 2], [0, -2, 2], [1, -2, 2], - - [2, -2, 2], [-2, -1, 2], [-1, -1, 2], [0, -1, 2], [1, -1, 2], [2, -1, 2], - - [-2, 0, 2], [-1, 0, 2], [0, 0, 2], [1, 0, 2], [2, 0, 2], [-2, 1, 2], - - [-1, 1, 2], [0, 1, 2], [1, 1, 2], [2, 1, 2], [-2, 2, 2], [-1, 2, 2], - - [0, 2, 2], [1, 2, 2], [2, 2, 2]] - -> posn_list_3d_size6 := map(ListTools[Reverse], -> hypercube_points([-2,-2,-2], [+3,+3,+3])); -posn_list_3d_size6 := [[-2, -2, -2], [-1, -2, -2], [0, -2, -2], [1, -2, -2], - - [2, -2, -2], [3, -2, -2], [-2, -1, -2], [-1, -1, -2], [0, -1, -2], - - [1, -1, -2], [2, -1, -2], [3, -1, -2], [-2, 0, -2], [-1, 0, -2], [0, 0, -2], - - [1, 0, -2], [2, 0, -2], [3, 0, -2], [-2, 1, -2], [-1, 1, -2], [0, 1, -2], - - [1, 1, -2], [2, 1, -2], [3, 1, -2], [-2, 2, -2], [-1, 2, -2], [0, 2, -2], - - [1, 2, -2], [2, 2, -2], [3, 2, -2], [-2, 3, -2], [-1, 3, -2], [0, 3, -2], - - [1, 3, -2], [2, 3, -2], [3, 3, -2], [-2, -2, -1], [-1, -2, -1], [0, -2, -1], - - [1, -2, -1], [2, -2, -1], [3, -2, -1], [-2, -1, -1], [-1, -1, -1], - - [0, -1, -1], [1, -1, -1], [2, -1, -1], [3, -1, -1], [-2, 0, -1], - - [-1, 0, -1], [0, 0, -1], [1, 0, -1], [2, 0, -1], [3, 0, -1], [-2, 1, -1], - - [-1, 1, -1], [0, 1, -1], [1, 1, -1], [2, 1, -1], [3, 1, -1], [-2, 2, -1], - - [-1, 2, -1], [0, 2, -1], [1, 2, -1], [2, 2, -1], [3, 2, -1], [-2, 3, -1], - - [-1, 3, -1], [0, 3, -1], [1, 3, -1], [2, 3, -1], [3, 3, -1], [-2, -2, 0], - - [-1, -2, 0], [0, -2, 0], [1, -2, 0], [2, -2, 0], [3, -2, 0], [-2, -1, 0], - - [-1, -1, 0], [0, -1, 0], [1, -1, 0], [2, -1, 0], [3, -1, 0], [-2, 0, 0], - - [-1, 0, 0], [0, 0, 0], [1, 0, 0], [2, 0, 0], [3, 0, 0], [-2, 1, 0], - - [-1, 1, 0], [0, 1, 0], [1, 1, 0], [2, 1, 0], [3, 1, 0], [-2, 2, 0], - - [-1, 2, 0], [0, 2, 0], [1, 2, 0], [2, 2, 0], [3, 2, 0], [-2, 3, 0], - - [-1, 3, 0], [0, 3, 0], [1, 3, 0], [2, 3, 0], [3, 3, 0], [-2, -2, 1], - - [-1, -2, 1], [0, -2, 1], [1, -2, 1], [2, -2, 1], [3, -2, 1], [-2, -1, 1], - - [-1, -1, 1], [0, -1, 1], [1, -1, 1], [2, -1, 1], [3, -1, 1], [-2, 0, 1], - - [-1, 0, 1], [0, 0, 1], [1, 0, 1], [2, 0, 1], [3, 0, 1], [-2, 1, 1], - - [-1, 1, 1], [0, 1, 1], [1, 1, 1], [2, 1, 1], [3, 1, 1], [-2, 2, 1], - - [-1, 2, 1], [0, 2, 1], [1, 2, 1], [2, 2, 1], [3, 2, 1], [-2, 3, 1], - - [-1, 3, 1], [0, 3, 1], [1, 3, 1], [2, 3, 1], [3, 3, 1], [-2, -2, 2], - - [-1, -2, 2], [0, -2, 2], [1, -2, 2], [2, -2, 2], [3, -2, 2], [-2, -1, 2], - - [-1, -1, 2], [0, -1, 2], [1, -1, 2], [2, -1, 2], [3, -1, 2], [-2, 0, 2], - - [-1, 0, 2], [0, 0, 2], [1, 0, 2], [2, 0, 2], [3, 0, 2], [-2, 1, 2], - - [-1, 1, 2], [0, 1, 2], [1, 1, 2], [2, 1, 2], [3, 1, 2], [-2, 2, 2], - - [-1, 2, 2], [0, 2, 2], [1, 2, 2], [2, 2, 2], [3, 2, 2], [-2, 3, 2], - - [-1, 3, 2], [0, 3, 2], [1, 3, 2], [2, 3, 2], [3, 3, 2], [-2, -2, 3], - - [-1, -2, 3], [0, -2, 3], [1, -2, 3], [2, -2, 3], [3, -2, 3], [-2, -1, 3], - - [-1, -1, 3], [0, -1, 3], [1, -1, 3], [2, -1, 3], [3, -1, 3], [-2, 0, 3], - - [-1, 0, 3], [0, 0, 3], [1, 0, 3], [2, 0, 3], [3, 0, 3], [-2, 1, 3], - - [-1, 1, 3], [0, 1, 3], [1, 1, 3], [2, 1, 3], [3, 1, 3], [-2, 2, 3], - - [-1, 2, 3], [0, 2, 3], [1, 2, 3], [2, 2, 3], [3, 2, 3], [-2, 3, 3], - - [-1, 3, 3], [0, 3, 3], [1, 3, 3], [2, 3, 3], [3, 3, 3]] - -# Maple code to define Hermite interpolating functions/coords/coeffs/mols -# $Header: /cactusdevcvs/CactusBase/LocalInterp/src/GeneralizedPolynomial-Uniform/Hermite/fns.maple,v 1.2 2002/09/01 18:33:34 jthorn Exp $ -> -# -# Note: -# interpolation order 2 <==> fn order 3, 3-point (2nd order) derivative mols -# interpolation order 3 <==> fn order 3, 5-point (4th order) derivative mols -# interpolation order 4 <==> fn order 5, 5-point (4th order) derivative mols -# -> -################################################################################ -################################################################################ -################################################################################ -> -# -# derivative primitives -# (argument is a procedure which should map m into the DATA() reference) -# -> -> dx_3point := -> proc(f::procedure(integer)) -> (1/2) * (-f(-1) + f(+1)) -> end proc; - dx_3point := proc(f::procedure(integer)) -1/2*f(-1) + 1/2*f(1) end proc - -> -> dx_5point := -> proc(f::procedure(integer)) -> (1/12) * (f(-2) - 8*f(-1) + 8*f(+1) - f(+2)) -> end proc; -dx_5point := proc(f::procedure(integer)) - 1/12*f(-2) - 2/3*f(-1) + 2/3*f(1) - 1/12*f(2) -end proc - -> -################################################################################ -################################################################################ -################################################################################ -> -# -# 1-D interpolating functions -# -> -> fn_1d_order3 := -> proc(x) -> + c0 + c1*x + c2*x^2 + c3*x^3 -> end proc; - fn_1d_order3 := proc(x) c0 + c1*x + c2*x^2 + c3*x^3 end proc - -> -> fn_1d_order5 := -> proc(x) -> + c0 + c1*x + c2*x^2 + c3*x^3 + c4*x^4 + c5*x^5 -> end proc; - fn_1d_order5 := proc(x) c0 + c1*x + c2*x^2 + c3*x^3 + c4*x^4 + c5*x^5 end proc - -> -################################################################################ -> -# coordinates for 1-D interpolating functions -> coord_list_1d := [x]; - coord_list_1d := [x] - -> -################################################################################ -> -# -# coefficients in 1-D interpolating functions -# -> -> coeffs_set_1d_order3 := {c0, c1, c2, c3}; - coeffs_set_1d_order3 := {c0, c1, c2, c3} - -> coeffs_set_1d_order5 := {c0, c1, c2, c3, c4, c5}; - coeffs_set_1d_order5 := {c0, c1, c2, c3, c4, c5} - -> -################################################################################ -> -# -# 1-D derivative molecules (argument = molecule center) -# -> -> deriv_1d_dx_3point := proc(i::integer) -> dx_3point(proc(mi::integer) DATA(i+mi) end proc) -> end proc; -deriv_1d_dx_3point := proc(i::integer) - dx_3point(proc(mi::integer) DATA(i + mi) end proc) -end proc - -> deriv_1d_dx_5point := proc(i::integer) -> dx_5point(proc(mi::integer) DATA(i+mi) end proc) -> end proc; -deriv_1d_dx_5point := proc(i::integer) - dx_5point(proc(mi::integer) DATA(i + mi) end proc) -end proc - -> -################################################################################ -################################################################################ -################################################################################ -> -# -# 2-D interpolating functions -# -> -> 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_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_order5 := -> proc(x,y) -> + c05*y^5 + c15*x*y^5 + c25*x^2*y^5 + c35*x^3*y^5 + c45*x^4*y^5 + c55*x^5*y^5 -> + c04*y^4 + c14*x*y^4 + c24*x^2*y^4 + c34*x^3*y^4 + c44*x^4*y^4 + c54*x^5*y^4 -> + c03*y^3 + c13*x*y^3 + c23*x^2*y^3 + c33*x^3*y^3 + c43*x^4*y^3 + c53*x^5*y^3 -> + c02*y^2 + c12*x*y^2 + c22*x^2*y^2 + c32*x^3*y^2 + c42*x^4*y^2 + c52*x^5*y^2 -> + c01*y + c11*x*y + c21*x^2*y + c31*x^3*y + c41*x^4*y + c51*x^5*y -> + c00 + c10*x + c20*x^2 + c30*x^3 + c40*x^4 + c50*x^5 -> end proc; -fn_2d_order5 := proc(x, y) - c34*x^3*y^4 + c14*x*y^4 + c03*y^3 + c02*y^2 + c01*y + c10*x + c20*x^2 - + c30*x^3 + c05*y^5 + c04*y^4 + c40*x^4 + c50*x^5 + c13*x*y^3 - + c23*x^2*y^3 + c33*x^3*y^3 + c12*x*y^2 + c22*x^2*y^2 + c32*x^3*y^2 - + c11*x*y + c21*x^2*y + c31*x^3*y + c15*x*y^5 + c25*x^2*y^5 - + c35*x^3*y^5 + c45*x^4*y^5 + c55*x^5*y^5 + c24*x^2*y^4 + c44*x^4*y^4 - + c54*x^5*y^4 + c43*x^4*y^3 + c53*x^5*y^3 + c42*x^4*y^2 + c52*x^5*y^2 - + c00 + c41*x^4*y + c51*x^5*y -end proc - -> -################################################################################ -> -# coordinates for 2-D interpolating functions -> coord_list_2d := [x,y]; - coord_list_2d := [x, y] - -> -################################################################################ -> -# -# coefficients in 2-D interpolating functions -# -> -> coeffs_set_2d_order3 := { -> c03, c13, c23, c33, -> c02, c12, c22, c32, -> c01, c11, c21, c31, -> c00, c10, c20, c30 -> }; -coeffs_set_2d_order3 := {c03, c13, c23, c33, c02, c12, c22, c32, c01, c11, c21, - - c31, c00, c10, c20, c30} - -> -> coeffs_set_2d_order5 := { -> c05, c15, c25, c35, c45, c55, -> c04, c14, c24, c34, c44, c54, -> c03, c13, c23, c33, c43, c53, -> c02, c12, c22, c32, c42, c52, -> c01, c11, c21, c31, c41, c51, -> c00, c10, c20, c30, c40, c50 -> }; -coeffs_set_2d_order5 := {c03, c13, c23, c33, c02, c12, c22, c32, c01, c11, c21, - - c31, c00, c10, c20, c30, c05, c15, c25, c35, c45, c55, c04, c14, c24, c34, - - c44, c54, c43, c53, c42, c52, c41, c51, c40, c50} - -> -################################################################################ -> -# -# 2-D derivative molecules (arguments = molecule center) -# -> -> deriv_2d_dx_3point := proc(i::integer, j::integer) -> dx_3point( -> proc(mi::integer) DATA(i+mi,j) end proc -> ) -> end proc; -deriv_2d_dx_3point := proc(i::integer, j::integer) - dx_3point(proc(mi::integer) DATA(i + mi, j) end proc) -end proc - -> deriv_2d_dy_3point := proc(i::integer, j::integer) -> dx_3point( -> proc(mj::integer) DATA(i,j+mj) end proc -> ) -> end proc; -deriv_2d_dy_3point := proc(i::integer, j::integer) - dx_3point(proc(mj::integer) DATA(i, j + mj) end proc) -end proc - -> deriv_2d_dxy_3point := proc(i::integer, j::integer) -> dx_3point( -> proc(mi::integer) -> dx_3point(proc(mj::integer) DATA(i+mi,j+mj) end proc) -> end proc -> ) -> end proc; -deriv_2d_dxy_3point := proc(i::integer, j::integer) - dx_3point(proc(mi::integer) - dx_3point(proc(mj::integer) DATA(i + mi, j + mj) end proc) - end proc) -end proc - -> -> deriv_2d_dx_5point := proc(i::integer, j::integer) -> dx_5point( -> proc(mi::integer) DATA(i+mi,j) end proc -> ) -> end proc; -deriv_2d_dx_5point := proc(i::integer, j::integer) - dx_5point(proc(mi::integer) DATA(i + mi, j) end proc) -end proc - -> deriv_2d_dy_5point := proc(i::integer, j::integer) -> dx_5point( -> proc(mj::integer) DATA(i,j+mj) end proc -> ) -> end proc; -deriv_2d_dy_5point := proc(i::integer, j::integer) - dx_5point(proc(mj::integer) DATA(i, j + mj) end proc) -end proc - -> deriv_2d_dxy_5point := proc(i::integer, j::integer) -> dx_5point( -> proc(mi::integer) -> dx_5point(proc(mj::integer) DATA(i+mi,j+mj) end proc) -> end proc -> ) -> end proc; -deriv_2d_dxy_5point := proc(i::integer, j::integer) - dx_5point(proc(mi::integer) - dx_5point(proc(mj::integer) DATA(i + mi, j + mj) end proc) - end proc) -end proc - -> -################################################################################ -################################################################################ -################################################################################ -> -# -# 3-D interpolating functions -# -> -> 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^1 --------------------------------------------------------------- -> + 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_order3 := proc(x, y, z) - c330*x^3*y^3 + c031*y^3*z + c103*x*z^3 + c022*y^2*z^2 + c301*x^3*z - + c133*x*y^3*z^3 + c233*x^2*y^3*z^3 + c333*x^3*y^3*z^3 - + c123*x*y^2*z^3 + c223*x^2*y^2*z^3 + c323*x^3*y^2*z^3 + c113*x*y*z^3 - + c213*x^2*y*z^3 + c313*x^3*y*z^3 + c132*x*y^3*z^2 + c232*x^2*y^3*z^2 - + c332*x^3*y^3*z^2 + c122*x*y^2*z^2 + c222*x^2*y^2*z^2 - + c322*x^3*y^2*z^2 + c112*x*y*z^2 + c212*x^2*y*z^2 + c312*x^3*y*z^2 - + c131*x*y^3*z + c231*x^2*y^3*z + c331*x^3*y^3*z + c121*x*y^2*z - + c221*x^2*y^2*z + c321*x^3*y^2*z + c111*x*y*z + c211*x^2*y*z - + c311*x^3*y*z + c033*y^3*z^3 + c023*y^2*z^3 + c013*y*z^3 - + c203*x^2*z^3 + c303*x^3*z^3 + c032*y^3*z^2 + c012*y*z^2 + c102*x*z^2 - + c202*x^2*z^2 + c302*x^3*z^2 + c021*y^2*z + c011*y*z + c101*x*z - + c201*x^2*z + c130*x*y^3 + c230*x^2*y^3 + c120*x*y^2 + c220*x^2*y^2 - + c320*x^3*y^2 + c110*x*y + c210*x^2*y + c310*x^3*y + c003*z^3 - + c002*z^2 + c001*z + c030*y^3 + c020*y^2 + c010*y + c000 + c100*x - + c200*x^2 + c300*x^3 -end proc - -> -> 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^1 -> + 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 proc; -fn_3d_order5 := proc(x, y, z) - c043*y^4*z^3 + c104*x*z^4 + c330*x^3*y^3 + c503*x^5*z^3 + c250*x^2*y^5 - + c031*y^3*z + c103*x*z^3 + c540*x^5*y^4 + c052*y^5*z^2 + c051*y^5*z - + c550*x^5*y^5 + c204*x^2*z^4 + c340*x^3*y^4 + c304*x^3*z^4 - + c042*y^4*z^2 + c140*x*y^4 + c022*y^2*z^2 + c205*x^2*z^5 + c150*x*y^5 - + c301*x^3*z + c133*x*y^3*z^3 + c233*x^2*y^3*z^3 + c333*x^3*y^3*z^3 - + c123*x*y^2*z^3 + c223*x^2*y^2*z^3 + c323*x^3*y^2*z^3 + c113*x*y*z^3 - + c213*x^2*y*z^3 + c313*x^3*y*z^3 + c132*x*y^3*z^2 + c232*x^2*y^3*z^2 - + c332*x^3*y^3*z^2 + c122*x*y^2*z^2 + c222*x^2*y^2*z^2 - + c322*x^3*y^2*z^2 + c112*x*y*z^2 + c212*x^2*y*z^2 + c312*x^3*y*z^2 - + c131*x*y^3*z + c231*x^2*y^3*z + c331*x^3*y^3*z + c121*x*y^2*z - + c221*x^2*y^2*z + c321*x^3*y^2*z + c111*x*y*z + c211*x^2*y*z - + c311*x^3*y*z + 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 + 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 + 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 - + 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 + 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 + 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 + 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 - + c134*x*y^3*z^4 + c035*y^3*z^5 + c033*y^3*z^3 + c023*y^2*z^3 - + c013*y*z^3 + c203*x^2*z^3 + c303*x^3*z^3 + c032*y^3*z^2 + c012*y*z^2 - + c102*x*z^2 + c202*x^2*z^2 + c302*x^3*z^2 + c021*y^2*z + c011*y*z - + c101*x*z + c201*x^2*z + c130*x*y^3 + c230*x^2*y^3 + c120*x*y^2 - + c220*x^2*y^2 + c320*x^3*y^2 + c110*x*y + c210*x^2*y + c310*x^3*y - + c003*z^3 + c002*z^2 + c001*z + c030*y^3 + c020*y^2 + c010*y + c000 - + c100*x + c200*x^2 + c300*x^3 + c005*z^5 + c055*y^5*z^5 - + c045*y^4*z^5 + c025*y^2*z^5 + c015*y*z^5 + c105*x*z^5 + c305*x^3*z^5 - + c405*x^4*z^5 + c505*x^5*z^5 + c054*y^5*z^4 + c044*y^4*z^4 - + c034*y^3*z^4 + c024*y^2*z^4 + c014*y*z^4 + c404*x^4*z^4 - + c504*x^5*z^4 + c053*y^5*z^3 + c403*x^4*z^3 + c402*x^4*z^2 - + c502*x^5*z^2 + c041*y^4*z + c401*x^4*z + c501*x^5*z + c350*x^3*y^5 - + c450*x^4*y^5 + c240*x^2*y^4 + c440*x^4*y^4 + c430*x^4*y^3 - + c530*x^5*y^3 + c420*x^4*y^2 + c520*x^5*y^2 + 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 - + 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 + 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 + 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 + 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 - + c433*x^4*y^3*z^3 + c533*x^5*y^3*z^3 + c004*z^4 + c050*y^5 + c040*y^4 - + c400*x^4 + c500*x^5 + c423*x^4*y^2*z^3 + c523*x^5*y^2*z^3 - + c413*x^4*y*z^3 + c513*x^5*y*z^3 + 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 - + 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 + c432*x^4*y^3*z^2 - + c532*x^5*y^3*z^2 + c422*x^4*y^2*z^2 + c522*x^5*y^2*z^2 - + c412*x^4*y*z^2 + c512*x^5*y*z^2 + 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 + 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 - + c431*x^4*y^3*z + c531*x^5*y^3*z + c421*x^4*y^2*z + c521*x^5*y^2*z - + c411*x^4*y*z + c511*x^5*y*z + c410*x^4*y + c510*x^5*y -end proc - -> -################################################################################ -> -# coordinates for 3-D interpolating functions -> coord_list_3d := [x,y,z]; - coord_list_3d := [x, y, z] - -> -################################################################################ -> -# -# coefficients in 3-D interpolating functions -# -> -> coeffs_set_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_set_3d_order3 := {c033, c133, c233, c333, c023, c123, c223, c323, c013, - - c113, c213, c313, c003, c103, c203, c303, c032, c132, c232, c332, c022, - - c122, c222, c322, c012, c112, c212, c312, c002, c102, c202, c302, c031, - - c131, c231, c331, c021, c121, c221, c321, c011, c111, c211, c311, c001, - - c101, c201, c301, c030, c130, c230, c330, c020, c120, c220, c320, c010, - - c110, c210, c310, c000, c100, c200, c300} - -> -> coeffs_set_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 -> }; -coeffs_set_3d_order5 := {c033, c133, c233, c333, c023, c123, c223, c323, c013, - - c113, c213, c313, c003, c103, c203, c303, c032, c132, c232, c332, c022, - - c122, c222, c322, c012, c112, c212, c312, c002, c102, c202, c302, c031, - - c131, c231, c331, c021, c121, c221, c321, c011, c111, c211, c311, c001, - - c101, c201, c301, c030, c130, c230, c330, c020, c120, c220, c320, c010, - - c110, c210, c310, c000, c100, c200, c300, 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, 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, c053, c153, c253, c353, c453, - - c553, c043, c143, c243, c343, c443, c543, c433, c533, c423, c523, c413, - - c513, c403, c503, c052, c152, c252, c352, c452, c552, c042, c142, c242, - - c342, c442, c542, c432, c532, c422, c522, c412, c512, c402, c502, c051, - - c151, c251, c351, c451, c551, c041, c141, c241, c341, c441, c541, c431, - - c531, c421, c521, c411, c511, c401, c501, c050, c150, c250, c350, c450, - - c550, c040, c140, c240, c340, c440, c540, c430, c530, c420, c520, c410, - - c510, c400, c500} - -> -################################################################################ -> -# -# 3-D derivative molecules (arguments = molecule center) -# -> -> deriv_3d_dx_3point := proc(i::integer, j::integer, k::integer) -> dx_3point( -> proc(mi::integer) DATA(i+mi,j,k) end proc -> ) -> end proc; -deriv_3d_dx_3point := proc(i::integer, j::integer, k::integer) - dx_3point(proc(mi::integer) DATA(i + mi, j, k) end proc) -end proc - -> deriv_3d_dy_3point := proc(i::integer, j::integer, k::integer) -> dx_3point( -> proc(mj::integer) DATA(i,j+mj,k) end proc -> ) -> end proc; -deriv_3d_dy_3point := proc(i::integer, j::integer, k::integer) - dx_3point(proc(mj::integer) DATA(i, j + mj, k) end proc) -end proc - -> deriv_3d_dz_3point := proc(i::integer, j::integer, k::integer) -> dx_3point( -> proc(mk::integer) DATA(i,j,k+mk) end proc -> ) -> end proc; -deriv_3d_dz_3point := proc(i::integer, j::integer, k::integer) - dx_3point(proc(mk::integer) DATA(i, j, k + mk) end proc) -end proc - -> deriv_3d_dxy_3point := proc(i::integer, j::integer, k::integer) -> dx_3point( -> proc(mi::integer) -> dx_3point( -> proc(mj::integer) DATA(i+mi,j+mj,k) end proc -> ) -> end proc -> ) -> end proc; -deriv_3d_dxy_3point := proc(i::integer, j::integer, k::integer) - dx_3point(proc(mi::integer) - dx_3point(proc(mj::integer) DATA(i + mi, j + mj, k) end proc) - end proc) -end proc - -> deriv_3d_dxz_3point := proc(i::integer, j::integer, k::integer) -> dx_3point( -> proc(mi::integer) -> dx_3point( -> proc(mk::integer) DATA(i+mi,j,k+mk) end proc -> ) -> end proc -> ) -> end proc; -deriv_3d_dxz_3point := proc(i::integer, j::integer, k::integer) - dx_3point(proc(mi::integer) - dx_3point(proc(mk::integer) DATA(i + mi, j, k + mk) end proc) - end proc) -end proc - -> deriv_3d_dyz_3point := proc(i::integer, j::integer, k::integer) -> dx_3point( -> proc(mj::integer) -> dx_3point( -> proc(mk::integer) DATA(i,j+mj,k+mk) end proc -> ) -> end proc -> ) -> end proc; -deriv_3d_dyz_3point := proc(i::integer, j::integer, k::integer) - dx_3point(proc(mj::integer) - dx_3point(proc(mk::integer) DATA(i, j + mj, k + mk) end proc) - end proc) -end proc - -> deriv_3d_dxyz_3point := proc(i::integer, j::integer, k::integer) -> dx_3point( -> proc(mi::integer) -> dx_3point( -> proc(mj::integer) -> dx_3point( -> proc(mk::integer) -> DATA(i+mi,j+mj,k+mk) -> end proc -> ) -> end proc -> ) -> end proc -> ) -> end proc; -deriv_3d_dxyz_3point := proc(i::integer, j::integer, k::integer) - dx_3point(proc(mi::integer) - dx_3point(proc(mj::integer) - dx_3point( - proc(mk::integer) DATA(i + mi, j + mj, k + mk) end proc) - end proc) - end proc) -end proc - -> -> deriv_3d_dx_5point := proc(i::integer, j::integer, k::integer) -> dx_5point( -> proc(mi::integer) DATA(i+mi,j,k) end proc -> ) -> end proc; -deriv_3d_dx_5point := proc(i::integer, j::integer, k::integer) - dx_5point(proc(mi::integer) DATA(i + mi, j, k) end proc) -end proc - -> deriv_3d_dy_5point := proc(i::integer, j::integer, k::integer) -> dx_5point( -> proc(mj::integer) DATA(i,j+mj,k) end proc -> ) -> end proc; -deriv_3d_dy_5point := proc(i::integer, j::integer, k::integer) - dx_5point(proc(mj::integer) DATA(i, j + mj, k) end proc) -end proc - -> deriv_3d_dz_5point := proc(i::integer, j::integer, k::integer) -> dx_5point( -> proc(mk::integer) DATA(i,j,k+mk) end proc -> ) -> end proc; -deriv_3d_dz_5point := proc(i::integer, j::integer, k::integer) - dx_5point(proc(mk::integer) DATA(i, j, k + mk) end proc) -end proc - -> deriv_3d_dxy_5point := proc(i::integer, j::integer, k::integer) -> dx_5point( -> proc(mi::integer) -> dx_5point( -> proc(mj::integer) DATA(i+mi,j+mj,k) end proc -> ) -> end proc -> ) -> end proc; -deriv_3d_dxy_5point := proc(i::integer, j::integer, k::integer) - dx_5point(proc(mi::integer) - dx_5point(proc(mj::integer) DATA(i + mi, j + mj, k) end proc) - end proc) -end proc - -> deriv_3d_dxz_5point := proc(i::integer, j::integer, k::integer) -> dx_5point( -> proc(mi::integer) -> dx_5point( -> proc(mk::integer) DATA(i+mi,j,k+mk) end proc -> ) -> end proc -> ) -> end proc; -deriv_3d_dxz_5point := proc(i::integer, j::integer, k::integer) - dx_5point(proc(mi::integer) - dx_5point(proc(mk::integer) DATA(i + mi, j, k + mk) end proc) - end proc) -end proc - -> deriv_3d_dyz_5point := proc(i::integer, j::integer, k::integer) -> dx_5point( -> proc(mj::integer) -> dx_5point( -> proc(mk::integer) DATA(i,j+mj,k+mk) end proc -> ) -> end proc -> ) -> end proc; -deriv_3d_dyz_5point := proc(i::integer, j::integer, k::integer) - dx_5point(proc(mj::integer) - dx_5point(proc(mk::integer) DATA(i, j + mj, k + mk) end proc) - end proc) -end proc - -> deriv_3d_dxyz_5point := proc(i::integer, j::integer, k::integer) -> dx_5point( -> proc(mi::integer) -> dx_5point( -> proc(mj::integer) -> dx_5point( -> proc(mk::integer) -> DATA(i+mi,j+mj,k+mk) -> end proc -> ) -> end proc -> ) -> end proc -> ) -> end proc; -deriv_3d_dxyz_5point := proc(i::integer, j::integer, k::integer) - dx_5point(proc(mi::integer) - dx_5point(proc(mj::integer) - dx_5point( - proc(mk::integer) DATA(i + mi, j + mj, k + mk) end proc) - end proc) - end proc) -end proc - -> -################################################################################ -################################################################################ -################################################################################ -# 1d.maple -- compute Hermite interpolation coefficients in 1-D -# $Header: /cactusdevcvs/CactusBase/LocalInterp/src/GeneralizedPolynomial-Uniform/Hermite/1d.maple,v 1.2 2002/09/01 18:33:33 jthorn Exp $ -> -################################################################################ -> -# -# 1d, cube, polynomial order=3, derivatives via 3-point order=2 formula -# ==> overall order=2, 4-point molecule -# -> -# interpolating polynomial -> interp_1d_cube_order2 -> := Hermite_polynomial_interpolant(fn_1d_order3, -> coeffs_set_1d_order3, -> [x], -> { {x} = deriv_1d_dx_3point }, -> {op(posn_list_1d_size2)}, -> {op(posn_list_1d_size2)}); -interp_1d_cube_order2 := DATA(0) + (- 1/2 DATA(-1) + 1/2 DATA(1)) x - - 2 - + (DATA(-1) + 2 DATA(1) - 5/2 DATA(0) - 1/2 DATA(2)) x - - 3 - + (3/2 DATA(0) - 1/2 DATA(-1) - 3/2 DATA(1) + 1/2 DATA(2)) x - -> -# I -> coeffs_as_lc_of_data(%, posn_list_1d_size4); - 2 3 2 3 -[COEFF(-1) = - 1/2 x + x - 1/2 x , COEFF(0) = - 5/2 x + 1 + 3/2 x , - - 3 2 3 2 - COEFF(1) = - 3/2 x + 1/2 x + 2 x , COEFF(2) = 1/2 x - 1/2 x ] - -> print_coeffs__lc_of_data(%, "coeffs_I->coeff_", "fp", -> "1d.coeffs/1d.cube.order2/coeffs-I.compute.c"); -bytes used=1001192, alloc=917336, time=0.08 -> -# d/dx -> simplify( diff(interp_1d_cube_order2,x) ); -- 1/2 DATA(-1) + 1/2 DATA(1) + 2 x DATA(-1) + 4 x DATA(1) - 5 x DATA(0) - - 2 2 2 - - x DATA(2) + 9/2 x DATA(0) - 3/2 x DATA(-1) - 9/2 x DATA(1) - - 2 - + 3/2 x DATA(2) - -> coeffs_as_lc_of_data(%, posn_list_1d_size4); - 2 2 -[COEFF(-1) = 2 x - 3/2 x - 1/2, COEFF(0) = -5 x + 9/2 x , - - 2 2 - COEFF(1) = 1/2 + 4 x - 9/2 x , COEFF(2) = -x + 3/2 x ] - -> print_coeffs__lc_of_data(%, "coeffs_dx->coeff_", "fp", -> "1d.coeffs/1d.cube.order2/coeffs-dx.compute.c"); -bytes used=2001768, alloc=1441528, time=0.15 -> -# d^2/dx^2 -> simplify( diff(interp_1d_cube_order2,x,x) ); -2 DATA(-1) + 4 DATA(1) - 5 DATA(0) - DATA(2) + 9 x DATA(0) - 3 x DATA(-1) - - - 9 x DATA(1) + 3 x DATA(2) - -> coeffs_as_lc_of_data(%, posn_list_1d_size4); -[COEFF(-1) = 2 - 3 x, COEFF(0) = -5 + 9 x, COEFF(1) = -9 x + 4, - - COEFF(2) = -1 + 3 x] - -> print_coeffs__lc_of_data(%, "coeffs_dxx->coeff_", "fp", -> "1d.coeffs/1d.cube.order2/coeffs-dxx.compute.c"); -> -################################################################################ -> -# -# 1d, cube, polynomial order=3, derivatives via 5-point order=4 formula -# ==> overall order=3, 6-point molecule -# -> -# interpolating polynomial -> interp_1d_cube_order3 -> := Hermite_polynomial_interpolant(fn_1d_order3, -> coeffs_set_1d_order3, -> [x], -> { {x} = deriv_1d_dx_5point }, -> {op(posn_list_1d_size2)}, -> {op(posn_list_1d_size2)}); -interp_1d_cube_order3 := DATA(0) - - + (1/12 DATA(-2) - 2/3 DATA(-1) + 2/3 DATA(1) - 1/12 DATA(2)) x + ( - - - 1/6 DATA(-2) + 5/4 DATA(-1) + 5/3 DATA(1) - 1/2 DATA(2) - 7/3 DATA(0) - - 2 - + 1/12 DATA(3)) x + (4/3 DATA(0) + 1/12 DATA(-2) - 7/12 DATA(-1) - - 3 - - 4/3 DATA(1) + 7/12 DATA(2) - 1/12 DATA(3)) x - -> -# I -> coeffs_as_lc_of_data(%, posn_list_1d_size6); - 2 3 2 3 -[COEFF(-2) = 1/12 x - 1/6 x + 1/12 x , COEFF(-1) = - 2/3 x + 5/4 x - 7/12 x , - - 2 3 3 2 - COEFF(0) = - 7/3 x + 1 + 4/3 x , COEFF(1) = - 4/3 x + 2/3 x + 5/3 x , - - 3 2 3 2 - COEFF(2) = 7/12 x - 1/12 x - 1/2 x , COEFF(3) = - 1/12 x + 1/12 x ] - -> print_coeffs__lc_of_data(%, "coeffs_I->coeff_", "fp", -> "1d.coeffs/1d.cube.order3/coeffs-I.compute.c"); -bytes used=3001984, alloc=1703624, time=0.23 -> -# d/dx -> simplify( diff(interp_1d_cube_order3,x) ); -1/12 DATA(-2) - 2/3 DATA(-1) + 2/3 DATA(1) - 1/12 DATA(2) - 1/3 x DATA(-2) - - + 5/2 x DATA(-1) + 10/3 x DATA(1) - x DATA(2) - 14/3 x DATA(0) - - 2 2 2 - + 1/6 x DATA(3) + 4 x DATA(0) + 1/4 x DATA(-2) - 7/4 x DATA(-1) - - 2 2 2 - - 4 x DATA(1) + 7/4 x DATA(2) - 1/4 x DATA(3) - -> coeffs_as_lc_of_data(%, posn_list_1d_size6); - 2 2 -[COEFF(-2) = - 1/3 x + 1/12 + 1/4 x , COEFF(-1) = - 2/3 + 5/2 x - 7/4 x , - - 2 2 - COEFF(0) = - 14/3 x + 4 x , COEFF(1) = -4 x + 10/3 x + 2/3, - - 2 2 - COEFF(2) = 7/4 x - 1/12 - x, COEFF(3) = 1/6 x - 1/4 x ] - -> print_coeffs__lc_of_data(%, "coeffs_dx->coeff_", "fp", -> "1d.coeffs/1d.cube.order3/coeffs-dx.compute.c"); -bytes used=4002684, alloc=1769148, time=0.30 -> -# d^2/dx^2 -> simplify( diff(interp_1d_cube_order3,x,x) ); -- 1/3 DATA(-2) + 5/2 DATA(-1) + 10/3 DATA(1) - DATA(2) - 14/3 DATA(0) - - + 1/6 DATA(3) + 8 x DATA(0) + 1/2 x DATA(-2) - 7/2 x DATA(-1) - - - 8 x DATA(1) + 7/2 x DATA(2) - 1/2 x DATA(3) - -> coeffs_as_lc_of_data(%, posn_list_1d_size6); -[COEFF(-2) = - 1/3 + 1/2 x, COEFF(-1) = - 7/2 x + 5/2, COEFF(0) = - 14/3 + 8 x, - - COEFF(1) = 10/3 - 8 x, COEFF(2) = 7/2 x - 1, COEFF(3) = 1/6 - 1/2 x] - -> print_coeffs__lc_of_data(%, "coeffs_dxx->coeff_", "fp", -> "1d.coeffs/1d.cube.order3/coeffs-dxx.compute.c"); -> -################################################################################ -> -# -# 1d, cube, polynomial order=5, derivatives via 5-point order=4 formula -# ==> overall order=4, 6-point molecule -# -# n.b. in higher dimensions this doesn't work -- there aren't enough -# equations to determine all the coefficients :( :( -# -> -# interpolating polynomial -> interp_1d_cube_order4 -> := Hermite_polynomial_interpolant(fn_1d_order5, -> coeffs_set_1d_order5, -> [x], -> { {x} = deriv_1d_dx_5point }, -> {op(posn_list_1d_size4)}, -> {op(posn_list_1d_size2)}); -bytes used=5003340, alloc=1769148, time=0.40 -interp_1d_cube_order4 := DATA(0) - - / - + (1/12 DATA(-2) - 2/3 DATA(-1) + 2/3 DATA(1) - 1/12 DATA(2)) x + | - \ - - 13 11 25 - - 1/8 DATA(-2) + -- DATA(-1) + 3/2 DATA(1) - -- DATA(2) - -- DATA(0) - 12 24 12 - - \ 2 - + 1/12 DATA(3)| x + (5/12 DATA(0) - 1/24 DATA(-2) - 1/24 DATA(-1) - / - - 3 /13 - - 7/12 DATA(1) + 7/24 DATA(2) - 1/24 DATA(3)) x + |-- DATA(0) - \12 - - 11 \ 4 - + 1/8 DATA(-2) - 7/12 DATA(-1) - DATA(1) + -- DATA(2) - 1/12 DATA(3)| x - 24 / - - + (- 5/12 DATA(0) - 1/24 DATA(-2) + 5/24 DATA(-1) + 5/12 DATA(1) - - 5 - - 5/24 DATA(2) + 1/24 DATA(3)) x - -> -# I -> coeffs_as_lc_of_data(%, posn_list_1d_size6); - 2 3 4 5 -[COEFF(-2) = 1/12 x - 1/8 x - 1/24 x + 1/8 x - 1/24 x , - - 13 2 3 4 5 - COEFF(-1) = - 2/3 x + -- x - 1/24 x - 7/12 x + 5/24 x , - 12 - - 25 2 3 13 4 5 - COEFF(0) = - -- x + 5/12 x + -- x + 1 - 5/12 x , - 12 12 - - 3 2 5 4 - COEFF(1) = - 7/12 x + 2/3 x + 3/2 x + 5/12 x - x , - - 11 2 3 5 11 4 - COEFF(2) = - 1/12 x - -- x + 7/24 x - 5/24 x + -- x , - 24 24 - - 3 5 2 4 - COEFF(3) = - 1/24 x + 1/24 x + 1/12 x - 1/12 x ] - -> print_coeffs__lc_of_data(%, "coeffs_I->coeff_", "fp", -> "1d.coeffs/1d.cube.order4/coeffs-I.compute.c"); -bytes used=6003504, alloc=1834672, time=0.48 -> -# d/dx -> simplify( diff(interp_1d_cube_order4,x) ); -bytes used=7003716, alloc=1834672, time=0.56 - 3 -2/3 DATA(1) - 2/3 DATA(-1) - 1/12 DATA(2) + 1/12 DATA(-2) - 4 x DATA(1) - - 3 3 25 4 4 - + 11/6 x DATA(2) - 1/3 x DATA(3) - -- x DATA(0) - 5/24 x DATA(-2) - 12 - - 25 4 25 4 25 4 4 - + -- x DATA(-1) + -- x DATA(1) - -- x DATA(2) + 5/24 x DATA(3) - 24 12 24 - - 11 - - -- x DATA(2) - 1/4 x DATA(-2) + 13/6 x DATA(-1) + 3 x DATA(1) - 12 - - 2 2 - - 25/6 x DATA(0) + 1/6 x DATA(3) + 5/4 x DATA(0) - 1/8 x DATA(-2) - - 2 2 2 2 - - 1/8 x DATA(-1) - 7/4 x DATA(1) + 7/8 x DATA(2) - 1/8 x DATA(3) - - 3 3 3 - + 13/3 x DATA(0) + 1/2 x DATA(-2) - 7/3 x DATA(-1) - -> coeffs_as_lc_of_data(%, posn_list_1d_size6); - 4 3 2 -[COEFF(-2) = - 5/24 x + 1/2 x + 1/12 - 1/8 x - 1/4 x, - - 25 4 2 3 - COEFF(-1) = 13/6 x + -- x - 1/8 x - 2/3 - 7/3 x , - 24 - - 2 3 25 4 - COEFF(0) = 5/4 x + 13/3 x - 25/6 x - -- x , - 12 - - 25 4 2 3 - COEFF(1) = 2/3 + -- x - 7/4 x - 4 x + 3 x, - 12 - - 25 4 3 11 2 - COEFF(2) = - -- x - 1/12 + 11/6 x - -- x + 7/8 x , - 24 12 - - 3 4 2 - COEFF(3) = - 1/3 x + 1/6 x + 5/24 x - 1/8 x ] - -> print_coeffs__lc_of_data(%, "coeffs_dx->coeff_", "fp", -> "1d.coeffs/1d.cube.order4/coeffs-dx.compute.c"); -bytes used=8003876, alloc=1834672, time=0.63 -> -# d^2/dx^2 -> simplify( diff(interp_1d_cube_order4,x,x) ); - 11 -- 1/4 DATA(-2) + 13/6 DATA(-1) + 3 DATA(1) - -- DATA(2) - 25/6 DATA(0) - 12 - - + 1/6 DATA(3) + 5/2 x DATA(0) - 1/4 x DATA(-2) - 1/4 x DATA(-1) - - 2 - - 7/2 x DATA(1) + 7/4 x DATA(2) - 1/4 x DATA(3) + 13 x DATA(0) - - 2 2 2 2 - + 3/2 x DATA(-2) - 7 x DATA(-1) - 12 x DATA(1) + 11/2 x DATA(2) - - 2 3 3 3 - - x DATA(3) - 25/3 x DATA(0) - 5/6 x DATA(-2) + 25/6 x DATA(-1) - - 3 3 3 - + 25/3 x DATA(1) - 25/6 x DATA(2) + 5/6 x DATA(3) - -> coeffs_as_lc_of_data(%, posn_list_1d_size6); - 2 3 -[COEFF(-2) = - 1/4 - 1/4 x + 3/2 x - 5/6 x , - - 2 3 - COEFF(-1) = 13/6 - 1/4 x - 7 x + 25/6 x , - - 3 2 - COEFF(0) = - 25/3 x + 13 x - 25/6 + 5/2 x, - - 2 3 - COEFF(1) = 3 - 7/2 x - 12 x + 25/3 x , - - 2 3 11 - COEFF(2) = 11/2 x - 25/6 x + 7/4 x - --, - 12 - - 3 2 - COEFF(3) = 5/6 x + 1/6 - 1/4 x - x ] - -> print_coeffs__lc_of_data(%, "coeffs_dxx->coeff_", "fp", -> "1d.coeffs/1d.cube.order4/coeffs-dxx.compute.c"); -bytes used=9004032, alloc=1900196, time=0.70 -> -################################################################################ -> quit -bytes used=9134616, alloc=1900196, time=0.73 |