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"N ormal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 " \+ " }{TEXT 256 13 "Help for poly" }}{PARA 0 "" 0 "" {TEXT -1 51 " " } }{PARA 0 "" 0 "" {TEXT -1 161 "Main commands: monomial_part, monomial_ terms, enum_of_monomial, monomial_of_enum, coeff_part, poly2vector, ve ctor2poly, is_in, is_in_poly, solve_poly, diff_inner" }}{PARA 0 "" 0 " " {TEXT -1 82 "CALLING SEQUENCE : monomial_part();" }}{PARA 0 "" 0 "" {TEXT -1 64 "CALLING SE QUENCE : monomial_terms();" }}{PARA 0 "" 0 "" {TEXT -1 64 "CALLING SEQUENCE : enum_of_monomial();" }}{PARA 0 "" 0 "" {TEXT -1 56 "CALLING SEQUENCE : monomi al_of_enum();" }}{PARA 0 "" 0 "" {TEXT -1 60 "CALLIN G SEQUENCE : coeff_part();" }}{PARA 0 "" 0 "" {TEXT -1 61 "CALLING SEQUENCE : poly2vector();" }}{PARA 0 "" 0 "" {TEXT -1 55 "CALLING SEQUENCE : vector2po ly();" }}{PARA 0 "" 0 "" {TEXT -1 81 "CALLING SE QUENCE : is_in(,) ;" }}{PARA 0 "" 0 "" {TEXT -1 98 "CALLING SEQUENCE : is_in_poly(,);" }} {PARA 0 "" 0 "" {TEXT -1 98 "CALLING SEQUENCE : solve_poly(,);" }}{PARA 0 " " 0 "" {TEXT -1 89 "CALLING SEQUENCE : solve_poly(,);" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{TEXT 298 4 "poly" }{TEXT -1 351 " contains routines for typ e checking of polynomials in several variables, converting polynomials into vectors, solving linear equations between polynomials, determini ng if a polynomial belongs to the vector space spanned by other polyno mials, defining a \"differentiation inner product\" between two polyno mials, and enumerating the monomials in a list." }}{PARA 0 "" 0 "" {TEXT -1 13 " The package " }{TEXT 258 8 "poly.mpl" }{TEXT -1 65 " mus t be loaded into MAPLE. Some commands also require the MAPLE " }{TEXT 257 6 "linalg" }{TEXT -1 9 " package." }}{PARA 0 "" 0 "" {TEXT -1 28 " Further details are in the " }{TEXT 259 9 "poly0.mws" }{TEXT -1 11 " \+ worksheet." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 16 "global variables" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 " " }{TEXT 284 3 " " }{TEXT 285 26 "There is a global variable" }{TEXT -1 2 " " }{TEXT 281 7 "glo bal_" }{TEXT 283 1 "n" }{TEXT -1 1 " " }{TEXT 286 65 "which denotes th e number of (algebraically independent) variables" }{TEXT -1 1 " " } {TEXT 282 11 "x1, x2, ..." }{TEXT -1 1 " " }{TEXT 287 118 "which the p olynomials will be written in. This must be set in the beginning befor e running any of the procedures below" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 " For example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "global_n:=3:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 5 "types" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "The re are three MAPLE \"types\":\n\n " }{TEXT 260 19 "`type/MY_VARIABLE S`" }{TEXT -1 9 " returns " }{TEXT 261 6 "`true`" }{TEXT -1 41 " if th e argument is one of the variables " }{TEXT 262 19 "x1, ..., x.global_ n" }{TEXT -1 5 ";\n " }{TEXT 263 15 "`type/MONOMIAL`" }{TEXT -1 9 " \+ returns " }{TEXT 264 6 "`true`" }{TEXT -1 34 " if the argument is a mo nomial in " }{TEXT 265 12 "MY_VARIABLES" }{TEXT -1 93 ". (Caution: Thi s does not seem to work correctly in every case but I can't find the p roblem.)" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{TEXT 266 17 "`type/POLY NOMIAL`" }{TEXT -1 9 " returns " }{TEXT 267 6 "`true`" }{TEXT -1 36 " \+ if the argument is a polynomial in " }{TEXT 268 12 "MY_VARIABLES" } {TEXT -1 51 ". (This does not always give ideal results either.)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "type(x1^2+x2^2,POLYNOMIAL([ x1,x2]));\ntype(1/x^2,POLYNOMIAL(x));\ntype(-x2,MY_VARIABLES);\ntype(x 3,MY_VARIABLES);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 "monomial_part, monomial_terms" }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 21 "There is a procedure " }{TEXT 269 14 "monomial_par t " }{TEXT -1 25 "which takes a polynomial " }{TEXT 271 1 "f" }{TEXT -1 4 " in " }{TEXT 270 12 "MY_VARIABLES" }{TEXT -1 52 " and returns th e coefficientless monomial factor if " }{TEXT 272 1 "f" }{TEXT -1 35 " has only one term. The procedure " }{TEXT 273 14 "monomial_terms" } {TEXT -1 20 " takes a polynomial " }{TEXT 275 1 "f" }{TEXT -1 4 " in \+ " }{TEXT 274 12 "MY_VARIABLES" }{TEXT -1 67 " and returns the list of \+ monomial terms (without coefficients) in " }{TEXT 276 1 "f" }{TEXT -1 25 ". The analogous procedure" }{TEXT 277 12 " coeff_part " }{TEXT -1 20 " takes a polynomial " }{TEXT 279 1 "f" }{TEXT -1 4 " in " } {TEXT 278 12 "MY_VARIABLES" }{TEXT -1 42 " and returns the list of co efficients in " }{TEXT 280 1 "f" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 71 "monomial_part(x1*x2);\nmonomial_part(2*x1*x2); \nmonomial_part(3*x1*x2^3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "monomial_terms(x3*x2^3+2*x1*x3);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 34 "enum_of_monomial, monomial_o f_enum" }}{PARA 0 "" 0 "" {TEXT -1 21 "There is a procedure " }{TEXT 288 17 " enum_of_monomial" }{TEXT -1 27 " which takes in a monomial " }{TEXT 289 1 "f" }{TEXT -1 37 " and returns the integer k such that " }{TEXT 290 6 "f = fk" }{TEXT -1 8 ", where " }{TEXT 291 16 "f1(=x1), f 2, ..." }{TEXT -1 59 " is a certain complete enumeration of all the mo nomials in " }{TEXT 292 12 "MY_VARIABLES" }{TEXT -1 89 ". We order the monomials first by their total degree then by their order of occurenc e in " }{TEXT 293 21 "(x1+...+x.global_n)^k" }{TEXT -1 26 ". We call t his number the " }{TEXT 297 11 "enumeration" }{TEXT -1 74 " of the mon omial. Incidently, the number of linearly independent terms in " } {TEXT 294 13 "(x1+...+xn)^k" }{TEXT -1 4 " is " }{TEXT 295 20 "binomia l(k+n-1,n-1)." }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "enum_of_monomial(1);\nenum_of_monomial(x1);\nenum_of_monomial(x 2);\nenum_of_monomial(x1*x2);\nenum_of_monomial(x2^2);\nenum_of_monomi al(x1^2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "The procedure " } {TEXT 296 16 "monomial_of_enum" }{TEXT -1 66 " is the inverse of the p rocedure enum_of_monomial described above." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 125 "monomial_of_enum(1);\nmonomial_of_enum(2);\nmonomi al_of_enum(3);\nmonomial_of_enum(4);\nmonomial_of_enum(5);\nmonomial_o f_enum(6);" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "coeff_part" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 " \+ To find the list of coefficients of a polynomial p in " }{TEXT 299 12 "MY_VARIABLES" }{TEXT -1 6 ", type" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "p:=x1+2*x2-x1*x2+7*x1^2*x2;\ncoeff_part(p);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 " poly2vector, vector2poly" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 " T he procedure " }{TEXT 300 12 " poly2vector" }{TEXT -1 26 " converts a \+ polynomial in " }{TEXT 301 12 "MY_VARIABLES" }{TEXT -1 196 " into a ve ctor associated to the monomial terms via the enumeration. The dimensi on of the vector (actually, represented as a list) depends on the inp utted polynomial. The correspondence satisfies" }{TEXT 302 15 " fk |-- --> ek " }{TEXT -1 6 "where " }{TEXT 303 28 "ek = [0, ..., 0, 1, 0, . ..] " }{TEXT -1 73 "is the vector with a 1 in the kth position, zeroes elsewhere, and where " }{TEXT 304 11 "f1, f2, ..." }{TEXT -1 61 " is the complete listing of the monomials mentioned earlier. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "vp:=poly2vector(p);\nvq:=poly2vecto r(q);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 " The procedure " } {TEXT 305 11 "vector2poly" }{TEXT -1 23 " is the inverse to the " } {TEXT 306 11 "poly2vector" }{TEXT -1 11 " procedure." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "vector2poly(vp);p;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "poly_basis" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 22 " The procedure" }{TEXT 307 12 " poly_basis " }{TEXT -1 30 "takes a list of poynomials in " } {TEXT 308 12 "MY_VARIABLES" }{TEXT -1 67 " and returns a maximal subli st of linearly independent polynomials." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "poly_basis([p,q,r]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 5 "is_in" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 " " }{TEXT 309 5 "is_in" }{TEXT -1 279 " is a proc edure which returns 1 if the 1st argument belongs to the vector space \+ spanned by the vectors (represented as lists) in the 2nd argument. It \+ returns 0 otherwise. It is implicitly assumed the 2nd argument is a li st of basis vectors (All vectors are represented as lists.)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "is_in([1,1,0],[[1,0,0],[0,1, 0]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "with(linalg):\nv1 :=vector([1,2,0]);\nv2:=vector([1,0,0]);\nv3:=vector([0,1,0]);\nw0:=[0 ,0,1];\nfor i from 1 to 3 do w.i:=convert(v.i,list);od;\nis_in(w0,[w2, w3]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "is_in_poly" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 " " } {TEXT 311 10 "is_in_poly" }{TEXT -1 49 " is the analogous procedure us ing polynomials in " }{TEXT 310 12 "MY_VARIABLES" }{TEXT -1 21 " in pl ace of vectors." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "p:=x1+2 *x2-x1*x2+7*x1^2*x2:\nq:=12*x1^2+x1*x2+7*x1*x2^2:\nr:=p+q:\ns:=x2-x1*x 2+7*x1^2*x2:\nis_in_poly(p,[q,s]);\nis_in_poly(p,[q,r]);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "solve_pol y" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 " The procedure " }{TEXT 312 10 "solve_poly" }{TEXT -1 43 " takes two arguments: the 1st a poly nomial " }{TEXT 315 1 "f" }{TEXT -1 35 " and the 2nd a list of polynom ials " }{TEXT 313 15 "p1, p2, ..., pr" }{TEXT -1 47 ". It tries to fin d a linear combination of the " }{TEXT 314 4 "pi's" }{TEXT -1 14 " whi ch equals " }{TEXT 316 1 "f" }{TEXT -1 81 ". Returns 0 if no linear co mbination exists and a list of coefficients otherwise." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "solve_poly(p,[q,s]);\nsolve_poly(p, [q,r]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "diff_inner" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 " \+ The procedure" }{TEXT 317 11 " diff_inner" }{TEXT -1 65 " computes th e \"differential inner product\" of two polynomials in " }{TEXT 318 12 "MY_VARIABLES" }{TEXT -1 49 ": \+ " }{TEXT 319 43 " diff_inner(f,g)=f(D1,...,Dn)(g(x1,...,xn) ," }{TEXT -1 8 " where " }{TEXT 320 2 "Di" }{TEXT -1 40 " denotes the derivative with respect to " }{TEXT 321 2 "xi" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "f:=x1^3*x2^2;\ndiff_inner(f, f);\ng:=x1^3*x2^2-x1^5*x2;\ndiff_inner(g,g);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "h:=monomial_part(-x1^5*x2);\ndiff_inner(f,g);\nd iff_inner(h,f);\ndiff_inner(f,f);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }}{MARK "0 15 0" 70 }{VIEWOPTS 1 1 0 1 1 1803 }