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Diffstat (limited to 'src/GeneralizedPolynomial-Uniform/Lagrange/1d.log')
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diff --git a/src/GeneralizedPolynomial-Uniform/Lagrange/1d.log b/src/GeneralizedPolynomial-Uniform/Lagrange/1d.log new file mode 100644 index 0000000..6ba0ee0 --- /dev/null +++ b/src/GeneralizedPolynomial-Uniform/Lagrange/1d.log @@ -0,0 +1,2022 @@ + |\^/| 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 +# $Id: util.maple,v 1.3 2002/05/19 13:12:18 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 a C declaration 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 C declaration for a list of names. +# +# Argument: +# name_list = A list of the names. +# name_type = 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}), +> name_type::string, +> file_name::string ) +> local blanks, separator_string; +> +> ftruncate(file_name); +> +# a sequence of blanks with the same length as name_type +> seq(" ", i=1..length(name_type)); +> +# string to separate names +> separator_string := cat(",\n", %, " "); +> +> map(convert, name_list, string); +> ListTools[Join](%, separator_string); +> cat(op(%)); +> +> fprintf(file_name, +> "%s %s;\n", +> name_type, %); +> +> fclose(file_name); +> NULL; +> end proc; +print_name_list_dcl := proc( +name_list::list({name, string}), name_type::string, file_name::string) +local blanks, separator_string; + ftruncate(file_name); + seq(" ", i = 1 .. length(name_type)); + separator_string := cat(",\n", %, " "); + map(convert, name_list, string); + ListTools[Join](%, separator_string); + cat(op(%)); + fprintf(file_name, "%s %s;\n", name_type, %); + 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 generalized interpolation formulas/coefficients +# $Id: interpolate.maple,v 1.4 2002/05/14 15:52:50 jthorn Exp $ +> +# +# <<<representation of numbers, data values, etc>>> +# Lagrange polynomial_interpolant - compute Lagrange polynomial interpolant +# coeff_as_lc_of_data - coefficients of ... (linear combination of data) +# +# print_coeff__lc_of_data - print C code to compute coefficients +# print_data_var_assign - print C code to assign data-value variables +# print_interp_coeff_var_store - print C code to store coeff vars "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 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 positions as was used to compute the +# interpolating polynomial. +# +# 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. +# +> coeff_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; +coeff_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 +# coeff_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_coeff__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_coeff__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_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_data_var_assign := +> 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_data_var_assign := 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 * coeff_dx_p1_m1; +# +# Arguments: +# posn_list = The same list of positions as was used to compute the +# interpolating polynomial. +# RHS_factor_name = If this string is non-empty, then the coefficient is +# multiplied by this factor before being stored, eg +# setting this to "factor" would give the example above. +# If this string is empty (""), the multiplication is +# omitted, eg +# COEFF(1,-1) = coeff_dx_p1_m1; +# coeff_name_prefix = A prefix string for the coefficient names. +# file_name = The file name to write the coefficients to. This is +# truncated before writing. +# +> print_interp_coeff_var_store := +> proc( +> posn_list::list(list(numeric)), +> RHS_factor_name::string, +> coeff_name_prefix::string, +> file_name::string +> ) +> +> ftruncate(file_name); +> map( +> proc(posn::list(numeric)) +> if (length(RHS_factor_name) > 0) +> then fprintf(file_name, +> "%a = %s * %s;\n", +> 'COEFF'(op(posn)), +> RHS_factor_name, +> coeff_name(posn,coeff_name_prefix)); +> else fprintf(file_name, +> "%a = %s;\n", +> 'COEFF'(op(posn)), +> coeff_name(posn,coeff_name_prefix)); +> end if; +> end proc +> , +> posn_list +> ); +> fclose(file_name); +> +> NULL; +> end proc; +print_interp_coeff_var_store := proc(posn_list::list(list(numeric)), +RHS_factor_name::string, coeff_name_prefix::string, file_name::string) + ftruncate(file_name); + map(proc(posn::list(numeric)) + if 0 < length(RHS_factor_name) then fprintf(file_name, + "%a = %s * %s;\n", 'COEFF'(op(posn)), RHS_factor_name, + coeff_name(posn, coeff_name_prefix)) + else fprintf(file_name, "%a = %s;\n", 'COEFF'(op(posn)), + coeff_name(posn, coeff_name_prefix)) + end if + end proc, posn_list); + fclose(file_name); + NULL +end proc + +> +################################################################################ +> +# +# This function prints a C expression to compute the interpolant, +# using the coefficients computed by print_coeff__lc_of_data() +# (i.e. expressing the interpolant as a linear combination of the +# data values). +# +# Arguments: +# posn_list = The same list of positions as was used to compute the +# interpolating polynomial. +# result_var_name = The (string) name of the variable to which the +# result is to be assigned. +# 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_interp_cmpt__lc_of_data := +> proc( +> posn_list::list(list(numeric)), +> result_var_name::string, +> coeff_name_prefix::string, +> data_var_name_prefix::string, +> file_name::string +> ) +> +> ftruncate(file_name); +> +> fprintf(file_name, "%s =\n", result_var_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\t+ "); +> cat(op(%)); +> fprintf(file_name, "\t%s;\n", %); +> +> fclose(file_name); +> +> NULL; +> end proc; +print_interp_cmpt__lc_of_data := proc(posn_list::list(list(numeric)), +result_var_name::string, coeff_name_prefix::string, +data_var_name_prefix::string, file_name::string) + ftruncate(file_name); + fprintf(file_name, "%s =\n", result_var_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\t+ "); + cat(op(%)); + fprintf(file_name, "\t%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 +# $Id: $ +> +################################################################################ +> +# +# 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]] + +> +################################################################################ +> +# +# 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]] + +# Maple code to define Lagrange interpolating functions/coords/coeffs +# $Id: 1d.maple,v 1.4 2002/05/14 15:54:01 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 +> coord_list_1d := [x]; + coord_list_1d := [x] + +> +######################################## +> +# +# coefficients in 1-D interpolating functions +# +> +> coeff_list_1d_order1 := [c0, c1]; + coeff_list_1d_order1 := [c0, c1] + +> coeff_list_1d_order2 := [c0, c1, c2]; + coeff_list_1d_order2 := [c0, c1, c2] + +> coeff_list_1d_order3 := [c0, c1, c2, c3]; + coeff_list_1d_order3 := [c0, c1, c2, c3] + +> coeff_list_1d_order4 := [c0, c1, c2, c3, c4]; + coeff_list_1d_order4 := [c0, c1, c2, c3, c4] + +> coeff_list_1d_order5 := [c0, c1, c2, c3, c4, c5]; + coeff_list_1d_order5 := [c0, c1, c2, c3, c4, c5] + +> coeff_list_1d_order6 := [c0, c1, c2, c3, c4, c5, c6]; + coeff_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 +> coord_list_2d := [x,y]; + coord_list_2d := [x, y] + +> +######################################## +> +# +# coefficients in 2-D interpolating functions +# +> +> coeff_list_2d_order1 := [ +> c01, +> c00, c10 +> ]; + coeff_list_2d_order1 := [c01, c00, c10] + +> coeff_list_2d_order2 := [ +> c02, +> c01, c11, +> c00, c10, c20 +> ]; + coeff_list_2d_order2 := [c02, c01, c11, c00, c10, c20] + +> coeff_list_2d_order3 := [ +> c03, +> c02, c12, +> c01, c11, c21, +> c00, c10, c20, c30 +> ]; + coeff_list_2d_order3 := [c03, c02, c12, c01, c11, c21, c00, c10, c20, c30] + +> coeff_list_2d_order4 := [ +> c04, +> c03, c13, +> c02, c12, c22, +> c01, c11, c21, c31, +> c00, c10, c20, c30, c40 +> ]; +coeff_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 + c000 + c100*x + c001*z + c020*y^2 + c110*x*y + + c011*y*z + c101*x*z + c200*x^2 + c002*z^2 + c030*y^3 + c300*x^3 + + c003*z^3 + c040*y^4 + c400*x^4 + c004*z^4 + 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 +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 +# +> +> coeff_list_3d_order1 := [ +> # z^0 ----- +> c010, +> c000, c100, +> # z^1 ----- +> c001 +> ]; + coeff_list_3d_order1 := [c010, c000, c100, c001] + +> coeff_list_3d_order2 := [ +> # z^0 ----------- +> c020, +> c010, c110, +> c000, c100, c200, +> # z^1 ----------- +> c011, +> c001, c101, +> # z^2 ----------- +> c002 +> ]; +coeff_list_3d_order2 := + + [c020, c010, c110, c000, c100, c200, c011, c001, c101, c002] + +> coeff_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 +> ]; +coeff_list_3d_order3 := [c030, c020, c120, c010, c110, c210, c000, c100, c200, + + c300, c021, c011, c111, c001, c101, c201, c012, c002, c102, c003] + +> coeff_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 +> ]; +coeff_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 +# $Id: $ +> +################################################################################ +> +# +# 1d, cube, order=1, smoothing=0 (size=2) +# +> +# interpolating polynomial +> interp_1d_cube_order1_smooth0 +> := Lagrange_polynomial_interpolant(fn_1d_order1, coeff_list_1d_order1, +> coord_list_1d, posn_list_1d_size2); + interp_1d_cube_order1_smooth0 := DATA(0) + (DATA(1) - DATA(0)) x + +> +# I +> coeff_as_lc_of_data(%, posn_list_1d_size2); + [COEFF(0) = 1 - x, COEFF(1) = x] + +> print_coeff__lc_of_data(%, "coeff_I_", "fp", +> "1d.coeffs/1d.cube.order1.smooth0/coeff-I.compute.c"); +> +# d/dx +> simplify( diff(interp_1d_cube_order1_smooth0,x) ); +bytes used=1005348, alloc=917336, time=0.13 + DATA(1) - DATA(0) + +> coeff_as_lc_of_data(%, posn_list_1d_size2); + [COEFF(0) = -1, COEFF(1) = 1] + +> print_coeff__lc_of_data(%, "coeff_dx_", "fp", +> "1d.coeffs/1d.cube.order1.smooth0/coeff-dx.compute.c"); +> +################################################################################ +> +# +# 1d, cube, order=2, smoothing=0 (size=3) +# +> +# interpolating polynomial +> interp_1d_cube_order2_smooth0 +> := Lagrange_polynomial_interpolant(fn_1d_order2, coeff_list_1d_order2, +> coord_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 +> coeff_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_coeff__lc_of_data(%, "coeff_I_", "fp", +> "1d.coeffs/1d.cube.order2.smooth0/coeff-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) + +> coeff_as_lc_of_data(%, posn_list_1d_size3); + [COEFF(-1) = x - 1/2, COEFF(0) = -2 x, COEFF(1) = 1/2 + x] + +> print_coeff__lc_of_data(%, "coeff_dx_", "fp", +> "1d.coeffs/1d.cube.order2.smooth0/coeff-dx.compute.c"); +> +# d^2/dx^2 +> simplify( diff(interp_1d_cube_order2_smooth0,x,x) ); + DATA(-1) + DATA(1) - 2 DATA(0) + +> coeff_as_lc_of_data(%, posn_list_1d_size3); + [COEFF(-1) = 1, COEFF(0) = -2, COEFF(1) = 1] + +> print_coeff__lc_of_data(%, "coeff_dxx_", "fp", +> "1d.coeffs/1d.cube.order2.smooth0/coeff-dxx.compute.c"); +bytes used=2005620, alloc=1507052, time=0.22 +> +################################################################################ +> +# +# 1d, cube, order=3, smoothing=0 (size=4) +# +> +# interpolating polynomial +> interp_1d_cube_order3_smooth0 +> := Lagrange_polynomial_interpolant(fn_1d_order3, coeff_list_1d_order3, +> coord_list_1d, posn_list_1d_size4); +interp_1d_cube_order3_smooth0 := DATA(0) + + + (- 1/2 DATA(0) + DATA(1) - 1/3 DATA(-1) - 1/6 DATA(2)) x + + 2 + + (1/2 DATA(-1) + 1/2 DATA(1) - DATA(0)) x + + 3 + + (1/2 DATA(0) - 1/2 DATA(1) - 1/6 DATA(-1) + 1/6 DATA(2)) x + +> +# I +> coeff_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_coeff__lc_of_data(%, "coeff_I_", "fp", +> "1d.coeffs/1d.cube.order3.smooth0/coeff-I.compute.c"); +> +# d/dx +> simplify( diff(interp_1d_cube_order3_smooth0,x) ); +bytes used=3006488, alloc=1703624, time=0.28 +- 1/2 DATA(0) + DATA(1) - 1/3 DATA(-1) - 1/6 DATA(2) + x DATA(-1) + x DATA(1) + + 2 2 2 + - 2 x DATA(0) + 3/2 x DATA(0) - 3/2 x DATA(1) - 1/2 x DATA(-1) + + 2 + + 1/2 x DATA(2) + +> coeff_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) = - 3/2 x + 1 + x, COEFF(2) = 1/2 x - 1/6] + +> print_coeff__lc_of_data(%, "coeff_dx_", "fp", +> "1d.coeffs/1d.cube.order3.smooth0/coeff-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) - 3 x DATA(1) - x DATA(-1) + + + x DATA(2) + +> coeff_as_lc_of_data(%, posn_list_1d_size4); + [COEFF(-1) = 1 - x, COEFF(0) = -2 + 3 x, COEFF(1) = -3 x + 1, COEFF(2) = x] + +> print_coeff__lc_of_data(%, "coeff_dxx_", "fp", +> "1d.coeffs/1d.cube.order3.smooth0/coeff-dxx.compute.c"); +> +################################################################################ +> +# +# 1d, cube, order=4, smoothing=0 (size=5) +# +> +# interpolating polynomial +> interp_1d_cube_order4_smooth0 +> := Lagrange_polynomial_interpolant(fn_1d_order4, coeff_list_1d_order4, +> coord_list_1d, posn_list_1d_size5); +bytes used=4006688, alloc=1834672, time=0.36 +interp_1d_cube_order4_smooth0 := DATA(0) + + + (1/12 DATA(-2) - 2/3 DATA(-1) + 2/3 DATA(1) - 1/12 DATA(2)) 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) + 1/6 DATA(-1) - 1/6 DATA(1) + 1/12 DATA(2)) x + + + (- 1/6 DATA(1) + 1/4 DATA(0) + 1/24 DATA(-2) - 1/6 DATA(-1) + 1/24 DATA(2)) + + 4 + x + +> +# I +> coeff_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_coeff__lc_of_data(%, "coeff_I_", "fp", +> "1d.coeffs/1d.cube.order4.smooth0/coeff-I.compute.c"); +bytes used=5006864, alloc=1900196, time=0.43 +> +# d/dx +> simplify( diff(interp_1d_cube_order4_smooth0,x) ); +1/12 DATA(-2) - 2/3 DATA(-1) + 2/3 DATA(1) - 1/12 DATA(2) + 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) + 1/2 x DATA(-1) - 1/2 x DATA(1) + 1/4 x DATA(2) + + 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) + +> coeff_as_lc_of_data(%, posn_list_1d_size5); + 2 3 +[COEFF(-2) = - 1/4 x - 1/12 x + 1/6 x + 1/12, + + 2 3 3 + COEFF(-1) = - 2/3 + 1/2 x + 4/3 x - 2/3 x , COEFF(0) = x - 5/2 x, + + 3 2 + COEFF(1) = - 2/3 x + 2/3 + 4/3 x - 1/2 x , + + 3 2 + COEFF(2) = 1/6 x - 1/12 + 1/4 x - 1/12 x] + +> print_coeff__lc_of_data(%, "coeff_dx_", "fp", +> "1d.coeffs/1d.cube.order4.smooth0/coeff-dx.compute.c"); +bytes used=6007068, alloc=1900196, time=0.53 +> +# 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) + x DATA(-1) - x DATA(1) + 1/2 x DATA(2) - 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) + +> coeff_as_lc_of_data(%, posn_list_1d_size5); + 2 2 +[COEFF(-2) = - 1/12 + 1/2 x - 1/2 x, COEFF(-1) = x + 4/3 - 2 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_coeff__lc_of_data(%, "coeff_dxx_", "fp", +> "1d.coeffs/1d.cube.order4.smooth0/coeff-dxx.compute.c"); +bytes used=7007288, alloc=1900196, time=0.60 +> +################################################################################ +> +# +# 1d, cube, order=5, smoothing=0 (size=6) +# +> +# interpolating polynomial +> interp_1d_cube_order5_smooth0 +> := Lagrange_polynomial_interpolant(fn_1d_order5, coeff_list_1d_order5, +> coord_list_1d, posn_list_1d_size6); +interp_1d_cube_order5_smooth0 := DATA(0) + (DATA(1) + 1/20 DATA(-2) + + - 1/3 DATA(0) - 1/4 DATA(2) - 1/2 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 + (- 7/12 DATA(1) - 1/24 DATA(-2) + 5/12 DATA(0) + 7/24 DATA(2) + + 3 + - 1/24 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 + (1/12 DATA(1) - 1/120 DATA(-2) - 1/12 DATA(0) - 1/24 DATA(2) + + 5 + + 1/24 DATA(-1) + 1/120 DATA(3)) x + +> +# I +> coeff_as_lc_of_data(%, posn_list_1d_size6); +bytes used=8007928, alloc=1900196, time=0.67 + 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_coeff__lc_of_data(%, "coeff_I_", "fp", +> "1d.coeffs/1d.cube.order5.smooth0/coeff-I.compute.c"); +bytes used=9008132, alloc=1900196, time=0.75 +> +# d/dx +> simplify( diff(interp_1d_cube_order5_smooth0,x) ); +- 1/3 DATA(0) + DATA(1) - 1/2 DATA(-1) - 1/4 DATA(2) + 1/30 DATA(3) + + 3 + - 2/3 x DATA(1) + 4/3 x DATA(1) - 5/2 x DATA(0) - 1/12 x DATA(-2) + + 2 2 + + 4/3 x DATA(-1) - 1/12 x DATA(2) - 7/4 x DATA(1) - 1/8 x DATA(-2) + + 2 2 2 2 + + 5/4 x DATA(0) + 7/8 x DATA(2) - 1/8 x DATA(-1) - 1/8 x DATA(3) + + 3 3 3 3 + + x DATA(0) + 1/6 x DATA(-2) - 2/3 x DATA(-1) + 1/6 x DATA(2) + + 4 4 4 4 + + 5/12 x DATA(1) - 1/24 x DATA(-2) - 5/12 x DATA(0) - 5/24 x DATA(2) + + 4 4 + + 5/24 x DATA(-1) + 1/24 x DATA(3) + 1/20 DATA(-2) + +> coeff_as_lc_of_data(%, posn_list_1d_size6); + 2 4 3 +[COEFF(-2) = - 1/12 x - 1/8 x - 1/24 x + 1/6 x + 1/20, + + 2 4 3 + COEFF(-1) = - 1/2 - 1/8 x + 5/24 x + 4/3 x - 2/3 x , + + 2 3 4 + COEFF(0) = 5/4 x + x - 5/12 x - 5/2 x - 1/3, + + 4 2 3 + COEFF(1) = 5/12 x + 4/3 x - 7/4 x + 1 - 2/3 x , + + 2 3 4 + COEFF(2) = - 1/4 + 7/8 x - 1/12 x + 1/6 x - 5/24 x , + + 4 2 + COEFF(3) = 1/24 x + 1/30 - 1/8 x ] + +> print_coeff__lc_of_data(%, "coeff_dx_", "fp", +> "1d.coeffs/1d.cube.order5.smooth0/coeff-dx.compute.c"); +bytes used=10008312, alloc=1965720, time=0.84 +bytes used=11008712, alloc=1965720, time=0.94 +> +# 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) + + - 7/2 x DATA(1) - 1/4 x DATA(-2) + 5/2 x DATA(0) + 7/4 x DATA(2) + + 2 2 + - 1/4 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) + 5/3 x DATA(1) + + 3 3 3 3 + - 1/6 x DATA(-2) - 5/3 x DATA(0) - 5/6 x DATA(2) + 5/6 x DATA(-1) + + 3 + + 1/6 x DATA(3) + +> coeff_as_lc_of_data(%, posn_list_1d_size6); + 2 3 +[COEFF(-2) = - 1/12 + 1/2 x - 1/4 x - 1/6 x , + + 2 3 + COEFF(-1) = - 1/4 x - 2 x + 4/3 + 5/6 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, + + 2 3 3 + COEFF(2) = 1/2 x + 7/4 x - 1/12 - 5/6 x , COEFF(3) = - 1/4 x + 1/6 x ] + +> print_coeff__lc_of_data(%, "coeff_dxx_", "fp", +> "1d.coeffs/1d.cube.order5.smooth0/coeff-dxx.compute.c"); +bytes used=12008884, alloc=1965720, time=1.03 +> +################################################################################ +> +# +# 1d, cube, order=6, smoothing=0 (size=7) +# +> +# interpolating polynomial +> interp_1d_cube_order6_smooth0 +> := Lagrange_polynomial_interpolant(fn_1d_order6, coeff_list_1d_order6, +> coord_list_1d, posn_list_1d_size7); +bytes used=13009076, alloc=1965720, time=1.11 +interp_1d_cube_order6_smooth0 := DATA(0) + (- 3/20 DATA(2) - 1/60 DATA(-3) + + / + - 3/4 DATA(-1) + 3/4 DATA(1) + 1/60 DATA(3) + 3/20 DATA(-2)) x + | + \ + + 3/4 DATA(1) + 1/180 DATA(-3) - 3/40 DATA(2) + 3/4 DATA(-1) + 1/180 DATA(3) + + 49 \ 2 / + - -- DATA(0) - 3/40 DATA(-2)| x + |1/6 DATA(2) + 1/48 DATA(-3) + 36 / \ + + 13 13 \ 3 / + + -- DATA(-1) - -- DATA(1) - 1/48 DATA(3) - 1/6 DATA(-2)| x + | + 48 48 / \ + + 13 13 + - -- DATA(1) - 1/144 DATA(-3) + 1/12 DATA(2) - -- DATA(-1) - 1/144 DATA(3) + 48 48 + + \ 4 + + 7/18 DATA(0) + 1/12 DATA(-2)| x + (- 1/60 DATA(2) - 1/240 DATA(-3) + / + + 5 + - 1/48 DATA(-1) + 1/48 DATA(1) + 1/240 DATA(3) + 1/60 DATA(-2)) x + ( + + 1/48 DATA(1) + 1/720 DATA(-3) - 1/120 DATA(2) + 1/48 DATA(-1) + + 6 + + 1/720 DATA(3) - 1/36 DATA(0) - 1/120 DATA(-2)) x + +> +# I +> coeff_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_coeff__lc_of_data(%, "coeff_I_", "fp", +> "1d.coeffs/1d.cube.order6.smooth0/coeff-I.compute.c"); +bytes used=14009280, alloc=1965720, time=1.19 +bytes used=15009668, alloc=1965720, time=1.28 +> +# d/dx +> simplify( diff(interp_1d_cube_order6_smooth0,x) ); + 13 2 +3/4 DATA(1) - 3/4 DATA(-1) - 1/60 DATA(-3) - 3/20 DATA(2) + -- x DATA(-1) + 16 + + 2 2 13 3 3 + - 1/16 x DATA(3) - 1/2 x DATA(-2) - -- x DATA(1) - 1/36 x DATA(-3) + 12 + + 3 13 3 3 3 + + 1/3 x DATA(2) - -- x DATA(-1) - 1/36 x DATA(3) + 14/9 x DATA(0) + 12 + + 3 4 4 13 2 + + 1/3 x DATA(-2) - 1/12 x DATA(2) - 1/48 x DATA(-3) - -- x DATA(1) + 16 + + + 3/2 x DATA(1) + 1/90 x DATA(-3) - 3/20 x DATA(2) + 3/2 x DATA(-1) + + 49 4 + + 1/90 x DATA(3) - -- x DATA(0) - 3/20 x DATA(-2) - 5/48 x DATA(-1) + 18 + + 4 4 4 5 + + 5/48 x DATA(1) + 1/48 x DATA(3) + 1/12 x DATA(-2) + 1/8 x DATA(1) + + 5 5 5 5 + + 1/120 x DATA(-3) - 1/20 x DATA(2) + 1/8 x DATA(-1) + 1/120 x DATA(3) + + 5 5 2 2 + - 1/6 x DATA(0) - 1/20 x DATA(-2) + 1/2 x DATA(2) + 1/16 x DATA(-3) + + + 1/60 DATA(3) + 3/20 DATA(-2) + +> coeff_as_lc_of_data(%, posn_list_1d_size7); + 5 3 2 4 +[COEFF(-3) = - 1/60 + 1/90 x + 1/120 x - 1/36 x + 1/16 x - 1/48 x , + + 4 2 3 5 + COEFF(-2) = 3/20 - 3/20 x + 1/12 x - 1/2 x + 1/3 x - 1/20 x , + + 5 13 2 13 3 4 + COEFF(-1) = 1/8 x - 3/4 + -- x - -- x + 3/2 x - 5/48 x , + 16 12 + + 5 3 49 + COEFF(0) = - 1/6 x + 14/9 x - -- x, + 18 + + 5 4 13 2 13 3 + COEFF(1) = 3/2 x + 1/8 x + 5/48 x + 3/4 - -- x - -- x , + 16 12 + + 3 5 2 4 + COEFF(2) = 1/3 x - 3/20 - 1/20 x - 3/20 x + 1/2 x - 1/12 x , + + 5 3 4 2 + COEFF(3) = 1/60 + 1/120 x - 1/36 x + 1/90 x + 1/48 x - 1/16 x ] + +> print_coeff__lc_of_data(%, "coeff_dx_", "fp", +> "1d.coeffs/1d.cube.order6.smooth0/coeff-dx.compute.c"); +bytes used=16010120, alloc=1965720, time=1.36 +bytes used=17010292, alloc=1965720, time=1.46 +> +# d^2/dx^2 +> simplify( diff(interp_1d_cube_order6_smooth0,x,x) ); +bytes used=18010476, alloc=1965720, time=1.55 + 49 +- -- DATA(0) + 3/2 DATA(1) + 3/2 DATA(-1) + 1/90 DATA(-3) - 3/20 DATA(2) + 18 + + 2 2 2 3 + - 13/4 x DATA(-1) - 1/12 x DATA(3) + x DATA(-2) + 5/12 x DATA(1) + + 3 3 3 3 + - 1/12 x DATA(-3) - 1/3 x DATA(2) - 5/12 x DATA(-1) + 1/12 x DATA(3) + + 3 4 4 2 + + 1/3 x DATA(-2) - 1/4 x DATA(2) + 1/24 x DATA(-3) - 13/4 x DATA(1) + + - 13/8 x DATA(1) + 1/8 x DATA(-3) + x DATA(2) + 13/8 x DATA(-1) + + 4 4 + - 1/8 x DATA(3) - x DATA(-2) + 5/8 x DATA(-1) + 5/8 x DATA(1) + + 4 4 2 2 + + 1/24 x DATA(3) - 1/4 x DATA(-2) + x DATA(2) - 1/12 x DATA(-3) + + 2 4 + + 1/90 DATA(3) - 3/20 DATA(-2) + 14/3 x DATA(0) - 5/6 x DATA(0) + +> coeff_as_lc_of_data(%, posn_list_1d_size7); + 3 4 2 +[COEFF(-3) = 1/8 x + 1/90 - 1/12 x + 1/24 x - 1/12 x , + + 4 2 3 + COEFF(-2) = - 1/4 x - 3/20 + x + 1/3 x - x, + + 3 4 2 + COEFF(-1) = - 5/12 x + 5/8 x - 13/4 x + 3/2 + 13/8 x, + + 2 4 49 + COEFF(0) = 14/3 x - 5/6 x - --, + 18 + + 4 3 2 + COEFF(1) = 5/8 x + 5/12 x - 13/8 x + 3/2 - 13/4 x , + + 4 3 2 + COEFF(2) = - 1/4 x - 1/3 x + x + x - 3/20, + + 2 3 4 + COEFF(3) = - 1/8 x + 1/90 - 1/12 x + 1/12 x + 1/24 x ] + +> print_coeff__lc_of_data(%, "coeff_dxx_", "fp", +> "1d.coeffs/1d.cube.order6.smooth0/coeff-dxx.compute.c"); +bytes used=19010860, alloc=1965720, time=1.62 +> +################################################################################ +> quit +bytes used=19691188, alloc=1965720, time=1.68 |