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Diffstat (limited to 'src/GeneralizedPolynomial-Uniform/Lagrange-tensor-product/1d.log')
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diff --git a/src/GeneralizedPolynomial-Uniform/Lagrange-tensor-product/1d.log b/src/GeneralizedPolynomial-Uniform/Lagrange-tensor-product/1d.log deleted file mode 100644 index 4fbd840..0000000 --- a/src/GeneralizedPolynomial-Uniform/Lagrange-tensor-product/1d.log +++ /dev/null @@ -1,2264 +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 Lagrange interpolating functions/coords/coeffs -# $Header: /cactusdevcvs/CactusBase/LocalInterp/src/GeneralizedPolynomial-Uniform/Lagrange/fns.maple,v 1.2 2002/08/20 16:31:24 jthorn Exp $ -> -################################################################################ -> -# -# 1-D interpolating functions -# -> -> fn_1d_order1 := -> proc(x) -> + c0 + c1*x -> end proc; - 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_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_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_order4 := proc(x) c0 + c1*x + c2*x^2 + c3*x^3 + c4*x^4 end proc - -> -> fn_1d_order5 := -> proc(x) -> + c0 + c1*x + c2*x^2 + c3*x^3 + c4*x^4 + c5*x^5 -> end; - fn_1d_order5 := proc(x) c0 + c1*x + c2*x^2 + c3*x^3 + c4*x^4 + c5*x^5 end proc - -> -> fn_1d_order6 := -> proc(x) -> + c0 + c1*x + c2*x^2 + c3*x^3 + c4*x^4 + c5*x^5 + c6*x^6 -> 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 proc - -> -######################################## -> -# coordinates for 1-D interpolating functions -> coords_list_1d := [x]; - coords_list_1d := [x] - -> -######################################## -> -# -# coefficients in 1-D interpolating functions -# -> -> coeffs_list_1d_order1 := [c0, c1]; - coeffs_list_1d_order1 := [c0, c1] - -> coeffs_list_1d_order2 := [c0, c1, c2]; - coeffs_list_1d_order2 := [c0, c1, c2] - -> coeffs_list_1d_order3 := [c0, c1, c2, c3]; - coeffs_list_1d_order3 := [c0, c1, c2, c3] - -> coeffs_list_1d_order4 := [c0, c1, c2, c3, c4]; - coeffs_list_1d_order4 := [c0, c1, c2, c3, c4] - -> coeffs_list_1d_order5 := [c0, c1, c2, c3, c4, c5]; - coeffs_list_1d_order5 := [c0, c1, c2, c3, c4, c5] - -> coeffs_list_1d_order6 := [c0, c1, c2, c3, c4, c5, c6]; - coeffs_list_1d_order6 := [c0, c1, c2, c3, c4, c5, c6] - -> -################################################################################ -> -# -# 2-D interpolating functions -# -> -> fn_2d_order1 := -> proc(x,y) -> + c01*y -> + c00 + c10*x -> end proc; - fn_2d_order1 := proc(x, y) c01*y + c00 + c10*x end proc - -> -> fn_2d_order2 := -> proc(x,y) -> + c02*y^2 -> + c01*y + c11*x*y -> + c00 + c10*x + c20*x^2 -> end proc; -fn_2d_order2 := - - proc(x, y) c02*y^2 + c01*y + c11*x*y + c00 + c10*x + c20*x^2 end proc - -> -> fn_2d_order3 := -> proc(x,y) -> + c03*y^3 -> + c02*y^2 + c12*x*y^2 -> + c01*y + c11*x*y + c21*x^2*y -> + c00 + c10*x + c20*x^2 + c30*x^3 -> end proc; -fn_2d_order3 := proc(x, y) - c03*y^3 + c02*y^2 + c12*x*y^2 + c01*y + c11*x*y + c21*x^2*y + c00 - + c10*x + c20*x^2 + c30*x^3 -end proc - -> -> fn_2d_order4 := -> proc(x,y) -> + c04*y^4 -> + c03*y^3 + c13*x*y^3 -> + c02*y^2 + c12*x*y^2 + c22*x^2*y^2 -> + c01*y + c11*x*y + c21*x^2*y + c31*x^3*y -> + c00 + c10*x + c20*x^2 + c30*x^3 + c40*x^4 -> end; -fn_2d_order4 := proc(x, y) - c04*y^4 + c03*y^3 + c13*x*y^3 + c02*y^2 + c12*x*y^2 + c22*x^2*y^2 - + c01*y + c11*x*y + c21*x^2*y + c31*x^3*y + c00 + c10*x + c20*x^2 - + c30*x^3 + c40*x^4 -end proc - -> -######################################## -> -# coordinates for 2-D interpolating functions -> coords_list_2d := [x,y]; - coords_list_2d := [x, y] - -> -######################################## -> -# -# coefficients in 2-D interpolating functions -# -> -> coeffs_list_2d_order1 := [ -> c01, -> c00, c10 -> ]; - coeffs_list_2d_order1 := [c01, c00, c10] - -> coeffs_list_2d_order2 := [ -> c02, -> c01, c11, -> c00, c10, c20 -> ]; - coeffs_list_2d_order2 := [c02, c01, c11, c00, c10, c20] - -> coeffs_list_2d_order3 := [ -> c03, -> c02, c12, -> c01, c11, c21, -> c00, c10, c20, c30 -> ]; - coeffs_list_2d_order3 := [c03, c02, c12, c01, c11, c21, c00, c10, c20, c30] - -> coeffs_list_2d_order4 := [ -> c04, -> c03, c13, -> c02, c12, c22, -> c01, c11, c21, c31, -> c00, c10, c20, c30, c40 -> ]; -coeffs_list_2d_order4 := - - [c04, c03, c13, c02, c12, c22, c01, c11, c21, c31, c00, c10, c20, c30, c40] - -> -################################################################################ -> -# -# 3-D interpolating functions -# -> -> fn_3d_order1 := -> proc(x,y,z) -# z^0 ----------- -> + c010*y -> + c000 + c100*x -# z^1 ----------- -> + c001*z -> end proc; - fn_3d_order1 := proc(x, y, z) c010*y + c000 + c100*x + c001*z end proc - -> -> fn_3d_order2 := -> proc(x,y,z) -# z^0 -------------------------- -> + c020*y^2 -> + c010*y + c110*x*y -> + c000 + c100*x + c200*x^2 -# z^1 -------------------------- -> + c011*y*z -> + c001*z + c101*x*z -# z^2 -------------------------- -> + c002*z^2 -> end proc; -fn_3d_order2 := proc(x, y, z) - c020*y^2 + c010*y + c110*x*y + c000 + c100*x + c200*x^2 + c011*y*z - + c001*z + c101*x*z + c002*z^2 -end proc - -> -> fn_3d_order3 := -> proc(x,y,z) -# z^0 ------------------------------------------- -> + c030*y^3 -> + c020*y^2 + c120*x*y^2 -> + c010*y + c110*x*y + c210*x^2*y -> + c000 + c100*x + c200*x^2 + c300*x^3 -# z^1 ------------------------------------------- -> + c021*y^2*z -> + c011*y *z + c111*x*y*z -> + c001 *z + c101*x *z + c201*x^2*z -# z^2 ------------------------------------------- -> + c012*y*z^2 -> + c002 *z^2 + c102*x*z^2 -# z^3 ------------------------------------------- -> + c003 *z^3 -> end proc; -fn_3d_order3 := proc(x, y, z) - c030*y^3 + c020*y^2 + c120*x*y^2 + c010*y + c110*x*y + c210*x^2*y - + c000 + c100*x + c200*x^2 + c300*x^3 + c021*y^2*z + c011*y*z - + c111*x*y*z + c001*z + c101*x*z + c201*x^2*z + c012*y*z^2 + c002*z^2 - + c102*x*z^2 + c003*z^3 -end proc - -> -> fn_3d_order4 := -> proc(x,y,z) -# z^0 -------------------------------------------------------- -> + c040*y^4 -> + c030*y^3 + c130*x*y^3 -> + c020*y^2 + c120*x*y^2 + c220*x^2*y^2 -> + c010*y + c110*x*y + c210*x^2*y + c310*x^3*y -> + c000 + c100*x + c200*x^2 + c300*x^3 + c400*x^4 -# z^1 ------------------------------------------- -> + c031*y^3*z -> + c021*y^2*z + c121*x*y^2*z -> + c011*y *z + c111*x*y *z + c211*x^2*y*z -> + c001 *z + c101*x *z + c201*x^2 *z + c301*x^3*z -# z^2 ------------------------------------------- -> + c022*y^2*z^2 -> + c012*y *z^2 + c112*x*y*z^2 -> + c002 *z^2 + c102*x *z^2 + c202*x^2*z^2 -# z^3 ------------------------------------------- -> + c013*y *z^3 -> + c003 *z^3 + c103*x *z^3 -# z^4 ------------------------------------------- -> + c004 *z^4 -> end; -fn_3d_order4 := proc(x, y, z) - c102*x*z^2 + c012*y*z^2 + c111*x*y*z + c121*x*y^2*z + c211*x^2*y*z - + c112*x*y*z^2 + c010*y + c110*x*y + c011*y*z + c101*x*z + c120*x*y^2 - + c210*x^2*y + c021*y^2*z + c201*x^2*z + c130*x*y^3 + c220*x^2*y^2 - + c310*x^3*y + c031*y^3*z + c301*x^3*z + c022*y^2*z^2 + c202*x^2*z^2 - + c013*y*z^3 + c103*x*z^3 + c000 + c100*x + c001*z + c020*y^2 - + c200*x^2 + c002*z^2 + c030*y^3 + c300*x^3 + c003*z^3 + c040*y^4 - + c400*x^4 + c004*z^4 -end proc - -> -######################################## -> -# coordinates for 3-D interpolating functions -> coords_list_3d := [x,y,z]; - coords_list_3d := [x, y, z] - -> -######################################## -> -# -# coefficients in 3-D interpolating functions -# -> -> coeffs_list_3d_order1 := [ -> # z^0 ----- -> c010, -> c000, c100, -> # z^1 ----- -> c001 -> ]; - coeffs_list_3d_order1 := [c010, c000, c100, c001] - -> coeffs_list_3d_order2 := [ -> # z^0 ----------- -> c020, -> c010, c110, -> c000, c100, c200, -> # z^1 ----------- -> c011, -> c001, c101, -> # z^2 ----------- -> c002 -> ]; -coeffs_list_3d_order2 := - - [c020, c010, c110, c000, c100, c200, c011, c001, c101, c002] - -> coeffs_list_3d_order3 := [ -> # z^0 ---------------- -> c030, -> c020, c120, -> c010, c110, c210, -> c000, c100, c200, c300, -> # z^1 ---------------- -> c021, -> c011, c111, -> c001, c101, c201, -> # z^2 ---------------- -> c012, -> c002, c102, -> # z^3 ---------------- -> c003 -> ]; -coeffs_list_3d_order3 := [c030, c020, c120, c010, c110, c210, c000, c100, c200, - - c300, c021, c011, c111, c001, c101, c201, c012, c002, c102, c003] - -> coeffs_list_3d_order4 := [ -> # z^0 ----------------------- -> c040, -> c030, c130, -> c020, c120, c220, -> c010, c110, c210, c310, -> c000, c100, c200, c300, c400, -> # z^1 ----------------------- -> c031, -> c021, c121, -> c011, c111, c211, -> c001, c101, c201, c301, -> # z^2 ----------------------- -> c022, -> c012, c112, -> c002, c102, c202, -> # z^3 ----------------------- -> c013, -> c003, c103, -> # z^4 ----------------------- -> c004 -> ]; -coeffs_list_3d_order4 := [c040, c030, c130, c020, c120, c220, c010, c110, c210, - - c310, c000, c100, c200, c300, c400, c031, c021, c121, c011, c111, c211, - - c001, c101, c201, c301, c022, c012, c112, c002, c102, c202, c013, c003, - - c103, c004] - -> -################################################################################ -# 1d.maple -- compute Lagrange interpolation coefficients in 1-D -# $Header: /cactusdevcvs/CactusBase/LocalInterp/src/GeneralizedPolynomial-Uniform/Lagrange/1d.maple,v 1.2 2002/08/20 16:31:22 jthorn Exp $ -> -################################################################################ -> -# -# 1d, cube, order=1, smoothing=0 (size=2) -# -> -# interpolating polynomial -> interp_1d_cube_order1_smooth0 -> := Lagrange_polynomial_interpolant(fn_1d_order1, coeffs_list_1d_order1, -> coords_list_1d, posn_list_1d_size2); - interp_1d_cube_order1_smooth0 := DATA(0) + (DATA(1) - DATA(0)) x - -> -# I -> coeffs_as_lc_of_data(%, posn_list_1d_size2); - [COEFF(0) = 1 - x, COEFF(1) = x] - -> print_coeffs__lc_of_data(%, "coeffs_I->coeff_", "fp", -> "1d.coeffs/1d.cube.order1.smooth0/coeffs-I.compute.c"); -bytes used=1000428, alloc=917336, time=0.07 -> -# d/dx -> simplify( diff(interp_1d_cube_order1_smooth0,x) ); - DATA(1) - DATA(0) - -> coeffs_as_lc_of_data(%, posn_list_1d_size2); - [COEFF(0) = -1, COEFF(1) = 1] - -> print_coeffs__lc_of_data(%, "coeffs_dx->coeff_", "fp", -> "1d.coeffs/1d.cube.order1.smooth0/coeffs-dx.compute.c"); -> -################################################################################ -> -# -# 1d, cube, order=2, smoothing=0 (size=3) -# -> -# interpolating polynomial -> interp_1d_cube_order2_smooth0 -> := Lagrange_polynomial_interpolant(fn_1d_order2, coeffs_list_1d_order2, -> coords_list_1d, posn_list_1d_size3); -interp_1d_cube_order2_smooth0 := DATA(0) + (- 1/2 DATA(-1) + 1/2 DATA(1)) x - - 2 - + (1/2 DATA(-1) + 1/2 DATA(1) - DATA(0)) x - -> -# I -> coeffs_as_lc_of_data(%, posn_list_1d_size3); - 2 2 2 - [COEFF(-1) = - 1/2 x + 1/2 x , COEFF(0) = 1 - x , COEFF(1) = 1/2 x + 1/2 x ] - -> print_coeffs__lc_of_data(%, "coeffs_I->coeff_", "fp", -> "1d.coeffs/1d.cube.order2.smooth0/coeffs-I.compute.c"); -> -# d/dx -> simplify( diff(interp_1d_cube_order2_smooth0,x) ); - - 1/2 DATA(-1) + 1/2 DATA(1) + x DATA(-1) + DATA(1) x - 2 x DATA(0) - -> coeffs_as_lc_of_data(%, posn_list_1d_size3); - [COEFF(-1) = x - 1/2, COEFF(0) = -2 x, COEFF(1) = 1/2 + x] - -> print_coeffs__lc_of_data(%, "coeffs_dx->coeff_", "fp", -> "1d.coeffs/1d.cube.order2.smooth0/coeffs-dx.compute.c"); -> -# d^2/dx^2 -> simplify( diff(interp_1d_cube_order2_smooth0,x,x) ); -bytes used=2000692, alloc=1441528, time=0.11 - DATA(-1) + DATA(1) - 2 DATA(0) - -> coeffs_as_lc_of_data(%, posn_list_1d_size3); - [COEFF(-1) = 1, COEFF(0) = -2, COEFF(1) = 1] - -> print_coeffs__lc_of_data(%, "coeffs_dxx->coeff_", "fp", -> "1d.coeffs/1d.cube.order2.smooth0/coeffs-dxx.compute.c"); -> -################################################################################ -> -# -# 1d, cube, order=3, smoothing=0 (size=4) -# -> -# interpolating polynomial -> interp_1d_cube_order3_smooth0 -> := Lagrange_polynomial_interpolant(fn_1d_order3, coeffs_list_1d_order3, -> coords_list_1d, posn_list_1d_size4); -interp_1d_cube_order3_smooth0 := DATA(0) - - + (- 1/2 DATA(0) - 1/3 DATA(-1) + DATA(1) - 1/6 DATA(2)) x - - 2 - + (1/2 DATA(-1) + 1/2 DATA(1) - DATA(0)) x - - 3 - + (1/2 DATA(0) - 1/6 DATA(-1) - 1/2 DATA(1) + 1/6 DATA(2)) x - -> -# I -> coeffs_as_lc_of_data(%, posn_list_1d_size4); - 2 3 2 3 -[COEFF(-1) = - 1/3 x + 1/2 x - 1/6 x , COEFF(0) = 1 - 1/2 x - x + 1/2 x , - - 2 3 3 - COEFF(1) = x + 1/2 x - 1/2 x , COEFF(2) = - 1/6 x + 1/6 x ] - -> print_coeffs__lc_of_data(%, "coeffs_I->coeff_", "fp", -> "1d.coeffs/1d.cube.order3.smooth0/coeffs-I.compute.c"); -> -# d/dx -> simplify( diff(interp_1d_cube_order3_smooth0,x) ); -bytes used=3001280, alloc=1769148, time=0.17 -- 1/2 DATA(0) - 1/3 DATA(-1) + DATA(1) - 1/6 DATA(2) + x DATA(-1) + x DATA(1) - - 2 2 2 - - 2 x DATA(0) + 3/2 x DATA(0) - 1/2 x DATA(-1) - 3/2 x DATA(1) - - 2 - + 1/2 x DATA(2) - -> coeffs_as_lc_of_data(%, posn_list_1d_size4); - 2 2 -[COEFF(-1) = x - 1/2 x - 1/3, COEFF(0) = - 1/2 - 2 x + 3/2 x , - - 2 2 - COEFF(1) = x + 1 - 3/2 x , COEFF(2) = 1/2 x - 1/6] - -> print_coeffs__lc_of_data(%, "coeffs_dx->coeff_", "fp", -> "1d.coeffs/1d.cube.order3.smooth0/coeffs-dx.compute.c"); -> -# d^2/dx^2 -> simplify( diff(interp_1d_cube_order3_smooth0,x,x) ); -DATA(-1) + DATA(1) - 2 DATA(0) + 3 x DATA(0) - x DATA(-1) - 3 x DATA(1) - - + x DATA(2) - -> coeffs_as_lc_of_data(%, posn_list_1d_size4); - [COEFF(-1) = 1 - x, COEFF(0) = -2 + 3 x, COEFF(1) = 1 - 3 x, COEFF(2) = x] - -> print_coeffs__lc_of_data(%, "coeffs_dxx->coeff_", "fp", -> "1d.coeffs/1d.cube.order3.smooth0/coeffs-dxx.compute.c"); -> -################################################################################ -> -# -# 1d, cube, order=4, smoothing=0 (size=5) -# -> -# interpolating polynomial -> interp_1d_cube_order4_smooth0 -> := Lagrange_polynomial_interpolant(fn_1d_order4, coeffs_list_1d_order4, -> coords_list_1d, posn_list_1d_size5); -bytes used=4001452, alloc=1834672, time=0.27 -interp_1d_cube_order4_smooth0 := DATA(0) - - + (- 1/12 DATA(2) + 1/12 DATA(-2) - 2/3 DATA(-1) + 2/3 DATA(1)) x + - - (2/3 DATA(1) - 5/4 DATA(0) - 1/24 DATA(-2) + 2/3 DATA(-1) - 1/24 DATA(2)) - - 2 3 - x + (1/12 DATA(2) - 1/12 DATA(-2) + 1/6 DATA(-1) - 1/6 DATA(1)) x + - - (- 1/6 DATA(1) + 1/4 DATA(0) + 1/24 DATA(-2) - 1/6 DATA(-1) + 1/24 DATA(2)) - - 4 - x - -> -# I -> coeffs_as_lc_of_data(%, posn_list_1d_size5); - 2 3 4 -[COEFF(-2) = 1/12 x - 1/24 x - 1/12 x + 1/24 x , - - 2 3 4 - COEFF(-1) = - 2/3 x + 2/3 x + 1/6 x - 1/6 x , - - 2 4 - COEFF(0) = - 5/4 x + 1 + 1/4 x , - - 3 2 4 - COEFF(1) = - 1/6 x + 2/3 x + 2/3 x - 1/6 x , - - 3 2 4 - COEFF(2) = 1/12 x - 1/12 x - 1/24 x + 1/24 x ] - -> print_coeffs__lc_of_data(%, "coeffs_I->coeff_", "fp", -> "1d.coeffs/1d.cube.order4.smooth0/coeffs-I.compute.c"); -bytes used=5001648, alloc=1900196, time=0.34 -> -# d/dx -> simplify( diff(interp_1d_cube_order4_smooth0,x) ); -- 1/12 DATA(2) + 1/12 DATA(-2) - 2/3 DATA(-1) + 2/3 DATA(1) + 4/3 x DATA(1) - - - 5/2 x DATA(0) - 1/12 x DATA(-2) + 4/3 x DATA(-1) - 1/12 x DATA(2) - - 2 2 2 2 - + 1/4 x DATA(2) - 1/4 x DATA(-2) + 1/2 x DATA(-1) - 1/2 x DATA(1) - - 3 3 3 3 - - 2/3 x DATA(1) + x DATA(0) + 1/6 x DATA(-2) - 2/3 x DATA(-1) - - 3 - + 1/6 x DATA(2) - -> coeffs_as_lc_of_data(%, posn_list_1d_size5); - 2 3 -[COEFF(-2) = - 1/12 x - 1/4 x + 1/12 + 1/6 x , - - 3 2 3 - COEFF(-1) = - 2/3 + 4/3 x - 2/3 x + 1/2 x , COEFF(0) = x - 5/2 x, - - 3 2 - COEFF(1) = 4/3 x - 2/3 x - 1/2 x + 2/3, - - 2 3 - COEFF(2) = - 1/12 - 1/12 x + 1/4 x + 1/6 x ] - -> print_coeffs__lc_of_data(%, "coeffs_dx->coeff_", "fp", -> "1d.coeffs/1d.cube.order4.smooth0/coeffs-dx.compute.c"); -bytes used=6001824, alloc=1900196, time=0.42 -> -# d^2/dx^2 -> simplify( diff(interp_1d_cube_order4_smooth0,x,x) ); -4/3 DATA(1) - 5/2 DATA(0) - 1/12 DATA(-2) + 4/3 DATA(-1) - 1/12 DATA(2) - - 2 - + 1/2 x DATA(2) - 1/2 x DATA(-2) + x DATA(-1) - x DATA(1) - 2 x DATA(1) - - 2 2 2 2 - + 3 x DATA(0) + 1/2 x DATA(-2) - 2 x DATA(-1) + 1/2 x DATA(2) - -> coeffs_as_lc_of_data(%, posn_list_1d_size5); - 2 2 -[COEFF(-2) = - 1/12 - 1/2 x + 1/2 x , COEFF(-1) = 4/3 - 2 x + x, - - 2 2 - COEFF(0) = - 5/2 + 3 x , COEFF(1) = -x + 4/3 - 2 x , - - 2 - COEFF(2) = - 1/12 + 1/2 x + 1/2 x ] - -> print_coeffs__lc_of_data(%, "coeffs_dxx->coeff_", "fp", -> "1d.coeffs/1d.cube.order4.smooth0/coeffs-dxx.compute.c"); -bytes used=7002056, alloc=1900196, time=0.50 -> -################################################################################ -> -# -# 1d, cube, order=5, smoothing=0 (size=6) -# -> -# interpolating polynomial -> interp_1d_cube_order5_smooth0 -> := Lagrange_polynomial_interpolant(fn_1d_order5, coeffs_list_1d_order5, -> coords_list_1d, posn_list_1d_size6); -bytes used=8002252, alloc=1900196, time=0.55 -interp_1d_cube_order5_smooth0 := DATA(0) + (- 1/2 DATA(-1) - 1/4 DATA(2) - - - 1/3 DATA(0) + 1/20 DATA(-2) + DATA(1) + 1/30 DATA(3)) x + - - (2/3 DATA(1) - 5/4 DATA(0) - 1/24 DATA(-2) + 2/3 DATA(-1) - 1/24 DATA(2)) - - 2 - x + (- 1/24 DATA(-1) + 7/24 DATA(2) + 5/12 DATA(0) - 1/24 DATA(-2) - - 3 - - 7/12 DATA(1) - 1/24 DATA(3)) x + - - (- 1/6 DATA(1) + 1/4 DATA(0) + 1/24 DATA(-2) - 1/6 DATA(-1) + 1/24 DATA(2)) - - 4 - x + (1/24 DATA(-1) - 1/24 DATA(2) - 1/12 DATA(0) - 1/120 DATA(-2) - - 5 - + 1/12 DATA(1) + 1/120 DATA(3)) x - -> -# I -> coeffs_as_lc_of_data(%, posn_list_1d_size6); - 2 3 4 5 -[COEFF(-2) = 1/20 x - 1/24 x - 1/24 x + 1/24 x - 1/120 x , - - 2 3 4 5 - COEFF(-1) = - 1/2 x + 2/3 x - 1/24 x - 1/6 x + 1/24 x , - - 2 3 4 5 - COEFF(0) = 1 - 1/3 x - 5/4 x + 5/12 x + 1/4 x - 1/12 x , - - 2 3 4 5 - COEFF(1) = x + 2/3 x - 7/12 x - 1/6 x + 1/12 x , - - 3 4 2 5 - COEFF(2) = - 1/4 x + 7/24 x + 1/24 x - 1/24 x - 1/24 x , - - 5 3 - COEFF(3) = 1/30 x + 1/120 x - 1/24 x ] - -> print_coeffs__lc_of_data(%, "coeffs_I->coeff_", "fp", -> "1d.coeffs/1d.cube.order5.smooth0/coeffs-I.compute.c"); -bytes used=9004240, alloc=1965720, time=0.63 -> -# d/dx -> simplify( diff(interp_1d_cube_order5_smooth0,x) ); -- 1/3 DATA(0) + DATA(1) - 1/4 DATA(2) - 1/2 DATA(-1) + 1/30 DATA(3) - - 4 4 4 4 - + 5/24 x DATA(-1) - 5/24 x DATA(2) - 5/12 x DATA(0) - 1/24 x DATA(-2) - - 4 4 - + 5/12 x DATA(1) + 1/24 x DATA(3) + 4/3 x DATA(1) - 5/2 x DATA(0) - - 2 - - 1/12 x DATA(-2) + 4/3 x DATA(-1) - 1/12 x DATA(2) - 1/8 x DATA(-1) - - 2 2 2 2 - + 7/8 x DATA(2) + 5/4 x DATA(0) - 1/8 x DATA(-2) - 7/4 x DATA(1) - - 2 3 3 3 - - 1/8 x DATA(3) - 2/3 x DATA(1) + x DATA(0) + 1/6 x DATA(-2) - - 3 3 - - 2/3 x DATA(-1) + 1/6 x DATA(2) + 1/20 DATA(-2) - -> coeffs_as_lc_of_data(%, posn_list_1d_size6); - 4 3 2 -[COEFF(-2) = - 1/12 x - 1/24 x + 1/6 x + 1/20 - 1/8 x , - - 3 2 4 - COEFF(-1) = - 2/3 x - 1/8 x - 1/2 + 5/24 x + 4/3 x, - - 3 4 2 - COEFF(0) = - 5/2 x + x - 5/12 x - 1/3 + 5/4 x , - - 4 3 2 - COEFF(1) = 5/12 x - 2/3 x + 4/3 x - 7/4 x + 1, - - 3 2 4 - COEFF(2) = 1/6 x - 1/12 x + 7/8 x - 1/4 - 5/24 x , - - 4 2 - COEFF(3) = 1/24 x - 1/8 x + 1/30] - -> print_coeffs__lc_of_data(%, "coeffs_dx->coeff_", "fp", -> "1d.coeffs/1d.cube.order5.smooth0/coeffs-dx.compute.c"); -bytes used=10004620, alloc=1965720, time=0.72 -bytes used=11005012, alloc=1965720, time=0.81 -> -# d^2/dx^2 -> simplify( diff(interp_1d_cube_order5_smooth0,x,x) ); -4/3 DATA(1) - 5/2 DATA(0) - 1/12 DATA(-2) + 4/3 DATA(-1) - 1/12 DATA(2) - - - 1/4 x DATA(-1) + 7/4 x DATA(2) + 5/2 x DATA(0) - 1/4 x DATA(-2) - - 2 2 - - 7/2 x DATA(1) - 1/4 x DATA(3) - 2 x DATA(1) + 3 x DATA(0) - - 2 2 2 3 - + 1/2 x DATA(-2) - 2 x DATA(-1) + 1/2 x DATA(2) + 5/6 x DATA(-1) - - 3 3 3 3 - - 5/6 x DATA(2) - 5/3 x DATA(0) - 1/6 x DATA(-2) + 5/3 x DATA(1) - - 3 - + 1/6 x DATA(3) - -> coeffs_as_lc_of_data(%, posn_list_1d_size6); - 2 3 -[COEFF(-2) = - 1/12 + 1/2 x - 1/6 x - 1/4 x, - - 2 3 - COEFF(-1) = -2 x + 5/6 x + 4/3 - 1/4 x, - - 3 2 - COEFF(0) = - 5/3 x + 5/2 x + 3 x - 5/2, - - 2 3 - COEFF(1) = 4/3 - 2 x + 5/3 x - 7/2 x, - - 3 2 3 - COEFF(2) = - 5/6 x + 1/2 x - 1/12 + 7/4 x, COEFF(3) = - 1/4 x + 1/6 x ] - -> print_coeffs__lc_of_data(%, "coeffs_dxx->coeff_", "fp", -> "1d.coeffs/1d.cube.order5.smooth0/coeffs-dxx.compute.c"); -bytes used=12005228, alloc=1965720, time=0.88 -> -################################################################################ -> -# -# 1d, cube, order=6, smoothing=0 (size=7) -# -> -# interpolating polynomial -> interp_1d_cube_order6_smooth0 -> := Lagrange_polynomial_interpolant(fn_1d_order6, coeffs_list_1d_order6, -> coords_list_1d, posn_list_1d_size7); -bytes used=13005380, alloc=1965720, time=0.95 -interp_1d_cube_order6_smooth0 := DATA(0) + (3/20 DATA(-2) + 3/4 DATA(1) - - / - + 1/60 DATA(3) - 1/60 DATA(-3) - 3/4 DATA(-1) - 3/20 DATA(2)) x + | - \ - - 1/180 DATA(-3) + 3/4 DATA(1) - 3/40 DATA(-2) + 1/180 DATA(3) + 3/4 DATA(-1) - - 49 \ 2 / 13 - - -- DATA(0) - 3/40 DATA(2)| x + |- 1/6 DATA(-2) - -- DATA(1) - 36 / \ 48 - - 13 \ 3 / - - 1/48 DATA(3) + 1/48 DATA(-3) + -- DATA(-1) + 1/6 DATA(2)| x + | - 48 / \ - - 13 13 - - 1/144 DATA(-3) - -- DATA(1) + 1/12 DATA(-2) - 1/144 DATA(3) - -- DATA(-1) - 48 48 - - \ 4 - + 7/18 DATA(0) + 1/12 DATA(2)| x + (1/60 DATA(-2) + 1/48 DATA(1) - / - - 5 - + 1/240 DATA(3) - 1/240 DATA(-3) - 1/48 DATA(-1) - 1/60 DATA(2)) x + ( - - 1/720 DATA(-3) + 1/48 DATA(1) - 1/120 DATA(-2) + 1/720 DATA(3) - - 6 - + 1/48 DATA(-1) - 1/36 DATA(0) - 1/120 DATA(2)) x - -> -# I -> coeffs_as_lc_of_data(%, posn_list_1d_size7); - 2 3 4 5 6 -[COEFF(-3) = - 1/60 x + 1/180 x + 1/48 x - 1/144 x - 1/240 x + 1/720 x , - - 2 3 4 5 6 - COEFF(-2) = 3/20 x - 3/40 x - 1/6 x + 1/12 x + 1/60 x - 1/120 x , - - 2 13 3 13 4 5 6 - COEFF(-1) = - 3/4 x + 3/4 x + -- x - -- x - 1/48 x + 1/48 x , - 48 48 - - 4 49 2 6 - COEFF(0) = 7/18 x - -- x + 1 - 1/36 x , - 36 - - 13 3 2 5 6 13 4 - COEFF(1) = - -- x + 3/4 x + 3/4 x + 1/48 x + 1/48 x - -- x , - 48 48 - - 3 2 5 6 4 - COEFF(2) = 1/6 x - 3/20 x - 3/40 x - 1/60 x - 1/120 x + 1/12 x , - - 3 2 5 6 4 - COEFF(3) = - 1/48 x + 1/60 x + 1/180 x + 1/240 x + 1/720 x - 1/144 x ] - -> print_coeffs__lc_of_data(%, "coeffs_I->coeff_", "fp", -> "1d.coeffs/1d.cube.order6.smooth0/coeffs-I.compute.c"); -bytes used=14005568, alloc=1965720, time=1.03 -bytes used=15005724, alloc=1965720, time=1.12 -> -# d/dx -> simplify( diff(interp_1d_cube_order6_smooth0,x) ); -3/4 DATA(1) - 3/20 DATA(2) - 3/4 DATA(-1) - 1/60 DATA(-3) + 1/60 DATA(3) - - 13 3 3 3 3 - - -- x DATA(1) - 1/36 x DATA(-3) + 1/3 x DATA(-2) - 1/36 x DATA(3) - 12 - - 13 3 3 3 4 - - -- x DATA(-1) + 14/9 x DATA(0) + 1/3 x DATA(2) + 1/12 x DATA(-2) - 12 - - 4 4 4 4 - + 5/48 x DATA(1) + 1/48 x DATA(3) - 1/48 x DATA(-3) - 5/48 x DATA(-1) - - 4 5 5 5 - - 1/12 x DATA(2) + 1/120 x DATA(-3) + 1/8 x DATA(1) - 1/20 x DATA(-2) - - 5 5 5 5 - + 1/120 x DATA(3) + 1/8 x DATA(-1) - 1/6 x DATA(0) - 1/20 x DATA(2) - - + 1/90 x DATA(-3) + 3/2 x DATA(1) - 3/20 x DATA(-2) + 1/90 x DATA(3) - - 49 2 13 2 - + 3/2 x DATA(-1) - -- x DATA(0) - 1/2 x DATA(-2) - -- x DATA(1) - 18 16 - - 2 2 13 2 2 - - 1/16 x DATA(3) + 1/16 x DATA(-3) + -- x DATA(-1) + 1/2 x DATA(2) - 16 - - - 3/20 x DATA(2) + 3/20 DATA(-2) - -> coeffs_as_lc_of_data(%, posn_list_1d_size7); -bytes used=16005892, alloc=1965720, time=1.20 - 3 5 4 2 -[COEFF(-3) = - 1/36 x + 1/120 x - 1/48 x - 1/60 + 1/90 x + 1/16 x , - - 2 5 3 4 - COEFF(-2) = - 1/2 x - 1/20 x + 1/3 x + 3/20 - 3/20 x + 1/12 x , - - 4 13 2 13 3 5 - COEFF(-1) = 3/2 x - 5/48 x + -- x - -- x + 1/8 x - 3/4, - 16 12 - - 5 3 49 - COEFF(0) = - 1/6 x + 14/9 x - -- x, - 18 - - 4 5 13 2 13 3 - COEFF(1) = 5/48 x + 3/4 + 1/8 x + 3/2 x - -- x - -- x , - 16 12 - - 5 3 2 4 - COEFF(2) = - 1/20 x + 1/3 x + 1/2 x - 1/12 x - 3/20 - 3/20 x, - - 3 2 4 5 - COEFF(3) = - 1/36 x + 1/90 x - 1/16 x + 1/60 + 1/48 x + 1/120 x ] - -> print_coeffs__lc_of_data(%, "coeffs_dx->coeff_", "fp", -> "1d.coeffs/1d.cube.order6.smooth0/coeffs-dx.compute.c"); -bytes used=17006200, alloc=1965720, time=1.29 -> -# d^2/dx^2 -> simplify( diff(interp_1d_cube_order6_smooth0,x,x) ); -bytes used=18007928, alloc=1965720, time=1.39 - 49 -- -- DATA(0) + 3/2 DATA(1) - 3/20 DATA(2) + 3/2 DATA(-1) + 1/90 DATA(-3) - 18 - - 3 3 3 - + 1/90 DATA(3) + 5/12 x DATA(1) - 1/12 x DATA(-3) + 1/3 x DATA(-2) - - 3 3 3 4 - + 1/12 x DATA(3) - 5/12 x DATA(-1) - 1/3 x DATA(2) - 1/4 x DATA(-2) - - 4 4 4 4 - + 5/8 x DATA(1) + 1/24 x DATA(3) + 1/24 x DATA(-3) + 5/8 x DATA(-1) - - 4 - - 1/4 x DATA(2) + 1/8 x DATA(-3) - 13/8 x DATA(1) - x DATA(-2) - - 2 2 - - 1/8 x DATA(3) + 13/8 x DATA(-1) + x DATA(-2) - 13/4 x DATA(1) - - 2 2 2 2 - - 1/12 x DATA(3) - 1/12 x DATA(-3) - 13/4 x DATA(-1) + x DATA(2) - - 2 4 - + x DATA(2) - 3/20 DATA(-2) + 14/3 x DATA(0) - 5/6 x DATA(0) - -> coeffs_as_lc_of_data(%, posn_list_1d_size7); - 2 4 3 -[COEFF(-3) = - 1/12 x + 1/24 x + 1/90 + 1/8 x - 1/12 x , - - 4 2 3 - COEFF(-2) = -x - 1/4 x - 3/20 + x + 1/3 x , - - 4 3 2 - COEFF(-1) = 5/8 x + 3/2 - 5/12 x + 13/8 x - 13/4 x , - - 2 4 49 - COEFF(0) = 14/3 x - 5/6 x - --, - 18 - - 2 3 4 - COEFF(1) = - 13/4 x + 5/12 x - 13/8 x + 5/8 x + 3/2, - - 2 4 3 - COEFF(2) = - 3/20 + x + x - 1/4 x - 1/3 x , - - 4 2 3 - COEFF(3) = 1/90 + 1/24 x - 1/8 x - 1/12 x + 1/12 x ] - -> print_coeffs__lc_of_data(%, "coeffs_dxx->coeff_", "fp", -> "1d.coeffs/1d.cube.order6.smooth0/coeffs-dxx.compute.c"); -bytes used=19008116, alloc=1965720, time=1.47 -> -################################################################################ -> quit -bytes used=19802260, alloc=1965720, time=1.53 |