|\^/| 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 $ > # # <<>> # 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