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+ |\^/| 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
generated by cgit v1.2.3 (git 2.39.1) at 2024-05-20 15:48:51 +0000