(* ---------------------------------------------------------------------------- * $Id: SACEXT8.md,v 1.3 1993/05/11 10:51:40 kredel Exp $ * ---------------------------------------------------------------------------- * This file is part of MAS. * ---------------------------------------------------------------------------- * Copyright (c) 1989 - 1992 Universitaet Passau * ---------------------------------------------------------------------------- * $Log: SACEXT8.md,v $ * Revision 1.3 1993/05/11 10:51:40 kredel * Spelling errors corr. * * Revision 1.2 1992/02/12 17:34:55 pesch * Moved CONST definition to the right place * * Revision 1.1 1992/01/22 15:15:37 kredel * Initial revision * * ---------------------------------------------------------------------------- *) DEFINITION MODULE SACEXT8; (* SAC Extensions 8 Definition Module. *) (* Import lists and declarations. *) FROM MASSTOR IMPORT LIST;CONSTrcsid = "$Id: SACEXT8.md,v 1.3 1993/05/11 10:51:40 kredel Exp $";CONSTcopyright = "Copyright (c) 1989 - 1992 Universitaet Passau";PROCEDURE AFCOMP(MB,I,AL,BL: LIST): LIST; (*Algebraic number field comparison. MB is the integral minimal polynomial of a real algebraic number alpha. I is an acceptable isolating interval for alpha. a and b are elements of Q( alpha ). t eq SIGN(a-b).*)PROCEDURE AFFINT(M: LIST): LIST; (*Algebraic number field element from integer. M is an integer. A is M represented as an element of an algebraic number field.*)PROCEDURE AFFRN(R: LIST): LIST; (*Algebraic number field element from rational number. R is a rational number. A is R represented as an element of an algebraic number field.*)PROCEDURE AFPAFP(RL,M,AL,B: LIST): LIST; (*Algebraic number field polynomial algebraic number field element product. M is the rational minimal polynomial of an algebraic number alpha. a is an element of Q( alpha ). B is a polynomial over Q( alpha ) in r variables, r ge 1. C eq a cdot B.*)PROCEDURE AFPAFQ(RL,M,A,BL: LIST): LIST; (*Algebraic number field polynomial algebraic number field element quotient. M is the rational minimal polynomial of an algebraic number alpha. A is a polynomial over Q( alpha ) in r variables, r ge 1. b is an element of Q( alpha ). C eq A/b.*) (* -- depends indirectly on AFPRII :-( PROCEDURE AFPBRI(M,MB,I,L: LIST): LIST; (*Algebraic number field polynomial basis real root isolation. M is the rational minimal polynomial of a real algebraic number alpha. MB is the integral minimal polynomial of alpha. I is an acceptable isolating interval for alpha. L is a nonempty squarefree basis (A sub 1 ,... , A sub n ) of univariate polynomials over Q( alpha ). N is a list (i sub 1 ,b sub 1 ,... , i sub m ,b sub m ), m ge 0, where i sub 1 lt i sub 2 lt ... lt i sub m are strongly disjoint isolating intervals for all the real roots of a eq prod from (j eq 1) to n (a sub j). each i sub i has binary rational endpoints and is left open and right closed. b sub i is the unique a sub j which has a root in i sub i.*) *) (* -- depends on AFPRII :-( PROCEDURE AFPCLL(M,MB,I,A: LIST): LIST; (*Algebraic number field polynomial real root isolation, Collins-Loos algorithm, list output version. M is the rational minimal polynomial of a real algebraic number alpha. MB is the integral minimal polynomial of alpha. I is an acceptable isolating interval for alpha. A is a monic univariate polynomial of degree n ge 0 over Q( alpha ). If n eq 0 then L eq (). If n gt 0, then L eq (L sub 0 ,... , L sub n-1 ), where L sub i is a strong isolation list for the real roots of der sup i (A).*) *)PROCEDURE AFPDIF(RL,A,B: LIST): LIST; (*Algebraic number field polynomial difference. A and B are polynomials in r variables, r ge 0, over Q( alpha ), for some algebraic number alpha. C=A-B.*)PROCEDURE AFPDMV(RL,M,A: LIST): LIST; (*Algebraic number field polynomial derivative, main variable. M is the rational minimal polynomial of an algebraic number alpha. A is a polynomial over Q( alpha ) in r variables, r ge 1. B is the derivative of a with respect to its main variable.*)PROCEDURE AFPEMV(RL,M,A,AL: LIST): LIST; (*Algebraic number field polynomial evaluation of main variable. M is the rational minimal polynomial of an algebraic number alpha. A is a polynomial over Q( alpha ) in r variables, r ge 1. a is an element of Q( alpha ). B( x sub 1 ,... , x sub r-1 ) eq A(x sub 1 ,... , x sub r-1 ,a).*)PROCEDURE AFPEV(RL,M,A,IL,AL: LIST): LIST; (*Algebraic number field polynomial evaluation. M is the rational minimal polynomial of an algebraic number alpha. A is a polynomial in r ge 1 variables over Q( alpha ). i satisfies 1 le i le r, and a is an element of Q( alpha ). B(x sub 1 ,... , x sub i-1 , x sub i+1 ,... , x sub r) eq A(x sub 1 ,... , x sub i-1 , a , x sub i+1 ,... , x sub r ).*)PROCEDURE AFPFIP(RL,A: LIST): LIST; (*Algebraic number field polynomial from integral polynomial. A is an integral polynomial in r variables, r ge 1. B is a represented as a polynomial over an algebraic number field.*)PROCEDURE AFPFRP(RL,A: LIST): LIST; (*Algebraic number field polynomial from rational polynomial. A is a rational polynomial in r variables, r ge 1. B is a represented as a polynomial over an algebraic number field.*)PROCEDURE AFPINT(RL,M,A,BL: LIST): LIST; (*Algebraic number field polynomial integration. M is the rational minimal polynomial of an algebraic number alpha. A is a nonzero polynomial over Q( alpha ) in r variables, r ge 1. b is a polynomial over Q( alpha ) in r-1 variables. B eq B(x sub 1 ,... , x sub r ) is the integral of a with respect to its main variable, such that B(x sub 1 ,... , x sub r-1 ,0) eq b.*)PROCEDURE AFPME(RL,M,A,BL: LIST): LIST; (*Algebraic number field, polynomial multiple evaluation. M is the rational minimal polynomial of an algebraic number alpha. A eq A(x sub 1 ,... , x sub r ) is a polynomial in r ge 1 variables over Q( alpha ). b eq (b sub 1 ,... , b sub k ) is a list of k elements of Q( alpha ) for some k, 1 le k le r. B eq A(b sub 1 ,... , b sub k ,x sub k+1 ,... , x sub r ), an element of Q( alpha )(x sub k+1 ,... , x sub r ).*)PROCEDURE AFPMON(RL,M,A: LIST): LIST; (*Algebraic number field polynomial monic. A is a polynomial in r variables, r ge 1, over Q( alpha ) for some algebraic number alpha. M is the rational minimal polynomial of alpha. If A is nonzero then AP is the monic polynomial over Q( alpha ) similar to A. If A eq 0 then AP eq 0.*)PROCEDURE AFPMPR(M,MB,I,B,J,L: LIST;VARJS,JL: LIST); (*Algebraic number field polynomial minimal polynomial of a real root. M is the rational minimal polynomial of a real algebraic number alpha. MB is the integral minimal polynomial of alpha. I is an acceptable isolating interval for alpha. J is an interval with binary rational number endpoints which is either left-open and right-closed, or a one-point interval. B is a univariate polynomial over Q( alpha ) having a unique root beta of odd multiplicity in j. L is a nonempty list of positive irreducible univariate integral polynomials exactly one of which has beta as a root. j is the index in L of the unique element n of L having beta as a root, and js is a subinterval of j with binary rational endpoints which is an isolating interval for beta as a root of n. js is either left-open and right-closed or a one-point interval.*)PROCEDURE AFPNEG(RL,A: LIST): LIST; (*Algebraic number field polynomial negative. A is a polynomial in r variables, r ge 0, over Q( alpha ) for some algebraic number alpha. B=-A.*)PROCEDURE AFPNIP(MB,A: LIST): LIST; (*Algebraic number field polynomial normalize to integral polynomial. MB is the integral minimal polynomial of an algebraic number alpha. A is a univariate polynomial over Q( alpha ) of positive degree. l is a list (l sub 1 ,... , l sub n ), n ge 1, of the positive irreducible factors of positive degree of a univariate integral polynomial which has among its roots the roots of A.*)PROCEDURE AFPPR(RL,M,A,B: LIST): LIST; (*Algebraic number field polynomial product. A and B are polynomials in r variables, r ge 0, over Q( alpha ) for some algebraic number alpha. M is the rational minimal polynomial of alpha. C=A*B.*)PROCEDURE AFPQR(RL,M,A,B: LIST;VARQ,R: LIST); (*Algebraic number field polynomial quotient and remainder. A and B, B ne 0, are polynomials in r variables, r ge 1, over Q( alpha ), for some algebraic number alpha. M is the rational minimal polynomial of alpha. Q and R are the unique algebraic number field polynomials such that either B divides A, Q eq B/A, and r eq 0 or else B does not divide A and A eq BQ+R with degree(R) minimal.*) (* -- depends on AFPRII :-( PROCEDURE AFPRCL(M,MB,I,A: LIST): LIST; (*Algebraic number field polynomial real root isolation, Collins-Loos algorithm. M is the rational minimal polynomial of a real algebraic number alpha. MB is the integral minimal polynomial of alpha. I is an acceptable isolating interval for alpha. A is a monic univariate polynomial of degree n ge 0 over Q( alpha ). L is a strong isolation list for the real roots of a.*) *) (* -- zuviele goto's :-( PROCEDURE AFPRII(M,MB,J,A,AP,DL,LP: LIST): LIST; (*Algebraic number field polynomial real root isolation induction. M is the rational minimal polynomial of a real algebraic number alpha. MB is the integral minimal polynomial of alpha. J is an acceptable isolating interval for alpha. A is a positive univariate polynomial over Q( alpha ) of positive degree. AP is the derivative of A. d is a binary rational real root bound for A. LP is a strong isolation list for AP. L is a strong isolation list for A.*) *)PROCEDURE AFPRLS(M,MB,I,A1,A2,L1,L2: LIST;VARLS1,LS2: LIST); (*Algebraic number field polynomial real root list separation. M is the rational minimal polynomial of a real algebraic number alpha. MB is the integral minimal polynomial of alpha. I is an acceptable isolating interval for alpha. A1 and A2 are univariate polynomials over Q( alpha ) with no common roots and real roots of only odd multiplicity. L1 and L2 are strong isolation lists for the real roots of A1 and A2 respectively. Let L1 eq (i sub 1,1 ,m sub 1,1 ,... , i sub (1,r sub 1),m sub (1,r sub 1)), L2 eq (i sub 2,1 ,m sub 2,1 ,... , i sub (2,r sub 2),m sub (2,r sub 2)). Then i sub 1,1 lt i sub 1,2 lt ... lt i sub (1,r sub 1) and i sub 2,1 lt i sub 2,2 lt ... lt i sub (2,r sub 2) . l sub 1 sup * eq (i sub 1,1 sup * ,m sub 1,1 ,... , i sub (1,r sub 1) sup * ,m sub (1,r sub 1)) and l sub 2 sup * eq (i sub 2,1 sup * ,m sub 2,1 ,... , i sub (2,r sub 2) sup * ,m sub (2,r sub 2)), where i sub i,j sup * is a binary rational subinterval of i sub i,j containing the root of a sub i in i sub i,j. each i sub 1,j sup * is strongly disjoint from each i sub 2,j sup *.*)PROCEDURE AFPRRI(M,MB,I,A,B,J,SL1,TL1: LIST): LIST; (*Algebraic number field polynomial relative real root isolation. M is the rational minimal polynomial of a real algebraic number alpha. MB is the integral minimal polynomial of alpha. I is an acceptable isolating interval for alpha. A and B are univariate polynomials over Q( alpha ). J is a left open, right closed interval (a sub 1 ,a sub 2 ) where al sub 1 and al sub 2 are binary rational numbers with al sub 1 lt al sub 2. A and B have unique roots, alpha and beta respectively, in J, each of odd multiplicity and with alpha ne beta. sl sub 1 eq sign(a(al sub 1 +)) and tl sub 1 eq sign(b(al sub 1 +)). js eq (al sub 1 sup * ,al sub 2 sup * ) is al left-open, right-closed subinterval of j with al sub 1 sup * and al sub 2 sup * binary rational numbers and al sub 1 sup * lt al sub 2 sup *, such that js contains alpha but not beta.*)PROCEDURE AFPRRS(M,MB,I,A1,A2,I1,I2: LIST;VARIS1,IS2,SL: LIST); (*Algebraic number field polynomial real root separation. M is the rational minimal polynomial of a real algebraic number alpha. MB is the integral minimal polynomial of alpha. I is an acceptable isolating interval for alpha. A1 and A2 are univariate integral polynomials of positive degrees over Q( alpha ). I1 and I2 are intervals with binary rational number endpoints, each of which is either left-open and right-closed, or a one-point interval. I1 contains a unique root alpha sub 1 of A1 of odd multiplicity, and I2 contains a unique root alpha sub 2 ne alpha sub 1 of A2 of odd multiplicity. I sub 1 sup * and I sub 2 sup * are binary rational subintervals of I1 and I2 containing alpha sub 1 and alpha sub 2 respectively, with I sub 1 sup * and I sub 2 sup * strongly disjoint. If I1 is left-open and right-closed then so is I sub 1 sup *, and similarly for I2 and I sub 2 sup *. s eq -1 if I sub 1 sup * lt I sub 2 sup *, and s eq 1 if I sub 1 sup * gt I sub 2 sup *.*)PROCEDURE AFPSUM(RL,A,B: LIST): LIST; (*Algebraic number field polynomial sum. A and B are polynomials over Q( alpha ) in r variables, r ge 1, for some algebraic number alpha. C=A+B.*)PROCEDURE AFSUPB(M,A: LIST): LIST; (*Algebraic number field squarefree univariate polynomial squarefree basis. M is the rational minimal polynomial of an algebraic number alpha. A eq (a sub 1 ,... , a sub n ), n ge 0, is a list of monic squarefree univariate polynomials over Q( alpha ), each of which is of positive degree. B is a coarsest squarefree basis for A.*)PROCEDURE AFUPBA(M,A,B: LIST): LIST; (*Algebraic number field univariate polynomial squarefree basis augmentation. M is the rational minimal polynomial of an algebraic number alpha. A is a monic squarefree univariate polynomial over Q( alpha ), of positive degree. B eq (b sub 1 ,... , b sub s ), s ge 0, is a squarefree basis of univariate polynomials over Q( alpha ). BS is a coarsest squarefree basis for (a,b sub 1 ,... , b sub s ).*)PROCEDURE AFUPCB(M,A: LIST): LIST; (*Algebraic number field univariate polynomial coarsest squarefree basis. M is the rational minimal polynomial of an algebraic number alpha. A eq (a sub 1 ,... , a sub n ), n ge 0, is a list of monic univariate polynomials over Q( alpha ), each of which is of positive degree. B is a coarsest squarefree basis for A.*)PROCEDURE AFUPGC(M,A,B: LIST;VARC,AB,BB: LIST); (*Algebraic number field univariate polynomial greatest common divisor and cofactors. A and B are univariate polynomials over Q( alpha ) for some algebraic number alpha. M is the rational minimal polynomial of alpha. C eq gcd(A,B), a monic polynomial. If C ne 0, then AB eq A/C and BB eq B/C, otherwise AB eq 0 and BB eq 0. *)PROCEDURE AFUPGS(M,A: LIST): LIST; (*Algebraic number field polynomial greatest squarefree divisor. M is the rational minimal polynomial of an algebraic number alpha. A is a univariate polynomial over Q( alpha ). If A eq 0 then B eq 0. Otherwise B is the monic associate of the greatest squarefree divisor of A.*)PROCEDURE AFUPRB(MB,I,A: LIST): LIST; (*Algebraic number field univariate polynomial root bound. MB is the integral minimal polynomial of a real algebraic number alpha. I is an acceptable isolating interval for alpha. A is a monic univariate polynomial over Q( alpha ) of positive degree. B is a binary rational number which is a root bound for A. If A(x) eq x sup n + sum from (i eq 0) to n-1 (a sub i x sup i), then B is the smallest power of 2 such that 2 cdot (abs(a sub n-k )) sup 1/k le B for 1 le k le n. If a sub n-k eq 0 for 1 le k le n then B eq 1.*)PROCEDURE AFUPRL(M,A: LIST): LIST; (*Algebraic number field polynomial, root of a linear polynomial. A is an element of Q( alpha )(x) of degree one, for some algebraic number alpha. M is the rational minimal polynomial of alpha. a is the unique element of Q( alpha ) such that A(a) eq 0.*)PROCEDURE AFUPSF(M,A: LIST): LIST; (*Algebraic number field univariate polynomial squarefree factorization. M is the rational minimal polynomial of an algebraic number alpha. A is a monic univariate polynomial over Q( alpha ) of positive degree. L is the list ((e sub 1 ,a sub 1 ) ,... , (e sub k ,a sub k )), where A eq prod from (i eq 1) to k (a sub i sup (e sub i)) is the squarefree factorization of A, with 1 le e sub 1 lt e sub 2 le ... lt e sub k and each a sub i a monic squarefree polynomial of positive degree.*)PROCEDURE AFUPSR(M,MB,I,A,CL: LIST): LIST; (*Algebraic number field univariate polynomial, sign at a rational point. M is the rational minimal polynomial of a real algebraic number alpha. MB is the integral minimal polynomial of alpha. I is an acceptable isolating interval for alpha. A is a univariate polynomial over Q( alpha ). c is a rational number. s eq sign(A(c)).*)PROCEDURE ANDWR(M,I,NL: LIST); (*Algebraic number decimal write. M is the integral minimal polynomial of a real algebraic number alpha. I is an acceptable isolating interval for alpha. n is a nonnegative integer. alpha is approximated by a rational number r with inaccuracy of approximation at most 10 sup -(n+1). Then r is approximated by a decimal fraction d with n decimal digits following the decimal point and d is written in the output stream. The inaccuracy of the approximation of d to r is at most (1/2) 10 sup -n. If abs(d) gt abs(r) then the last digit is followed by $=-. If abs(d) lt abs(r) then by $=+.*)PROCEDURE ANFAF(M,I,AL: LIST;VARN,J: LIST); (*Algebraic number from algebraic number field element. M is the integral minimal polynomial of a real algebraic number alpha. I is an acceptable isolating interval for alpha. a is an element of Q( alpha ). N is the integral minimal polynomial of a, and J is an acceptable isolating interval for a.*)PROCEDURE ANIIPE(MB,I,NB,J,TL,L: LIST;VARS,KL,K: LIST); (*Algebraic number isolating interval for a primitive element. MB is the integral minimal polynomial of a real algebraic number alpha. I is a binary rational isolating interval for alpha which is either left-open and right-closed or a one-point interval. NB is the integral minimal polynomial of a real algebraic number beta. J is a binary rational isolating interval for beta which is either left-open and right-closed or a one-point interval. t is an integer such that Q( alpha +t beta ) eq Q( alpha , beta ). If degree(MB) eq 1 and degree(NB) eq 1, then L is a list containing a primitive positive integral polynomial p of degree 1, s eq p, k eq 1, and k is a binary rational isolating interval for the real root of p which is either left-open and right-closed or a one-point interval. If degree(mb) eq 1, degree(NB) gt 1, then l eq (NB), S eq NB, k eq 1, and k eq j. If degree(MB) gt 1, degree(NB) eq 1, then L eq (MB), S eq MB, k eq 1, and k eq i. If degree(MB) gt 1, degree(NB) gt 1, then L is a nonempty list of positive irreducible univariate integral polynomials exactly one of which has alpha +t beta as a root. S is the element of L having alpha +t beta as a root, k is the index of S in L, and k is a left-open, right-closed binary rational isolating interval for alpha +t beta as a root of S.*)PROCEDURE ANPEDE(MB,NB: LIST;VARTL,S,T: LIST); (*Algebraic number primitive element for a double extension. MB eq MB(x) is the integral minimal polynomial of a real algebraic number alpha. NB eq NB(x) is the integral minimal polynomial of a real algebraic number beta. t is an integer such that Q( alpha +t beta ) eq Q( alpha , beta ). If degree(MB) eq 1 and degree(NB) eq 1, then S eq (x), a list of length of length 1 containing the polynomial x. If degree(MB) eq 1 and degree(NB) gt 1, then S eq (NB). If degree(MB) gt 1 and degree(NB) eq 1, then S eq (MB). If degree(MB) gt 1 and degree(NB1) gt 1, then S is a list of the integral minimal polynomials of all algebraic numbers of the form alpha sub i + t beta sub j, where alpha sub i is some conjugate of alpha and beta sub j is some conjugate of beta. Where n ge 1 is the length of S, t is a list (m sub 1 sup * ,n sub 1 sup * ,... , m sub n sup * , n sub n sup * ). For 1 le k le n, where gamma sub k is a root of s sub k, m sub k sup * is the representation of alpha as an element of Q( gamma sub k ) and n sub k sup * is the representation of beta as an element of Q( gamma sub k ).*)PROCEDURE ANREPE(M,MB,A,B: LIST): LIST; (*Algebraic number represent element of a primitive extension. M is the rational minimal polynomial of an algebraic number gamma. MB is the integral minimal polynomial of gamma. A and B are elements of Q( gamma ) (y) which can be and are represented as bivariate integral polynomials, i.e. as elements of Z(x,y). A eq AP(x-ty) for a minimal polynomial AP of an algebraic number alpha, and B eq B(y) is the minimal polynomial of an algebraic number beta. gamma is a primitive element for alpha and beta. B is a univariate rational polynomial which is the representation for beta as an element of Q( gamma ).*)PROCEDURE APDWR(M,I,BL,NL: LIST); (*Algebraic point, decimal write. M,I, and b constitute the representation of an algebraic point in r-dimensional euclidean space for some r ge 1. n is a nonnegative integer. For each coordinate b sub i of b, b sub i is represented by a rational number r sub i with inaccuracy of approximation at most 10 sup -(n+1). Then r sub i is approximated by a decimal fraction d sub i with n decimal digits following the decimal point, and d sub i is written in the output stream. The inaccuracy of the approximation of d sub i to r sub i is at most (1/2) 10 sup -n.*)PROCEDURE IPAFME(RL,M,A,BL: LIST): LIST; (*Integral polynomial, algebraic number field multiple evaluation. A is an integral polynomial in r variables, r ge 1. M is the rational minimal polynomial of an algebraic number alpha. b eq (b sub 1 ,... , b sub k ) is a list of k elements of Q( alpha ), for some k, 1 le k le r. b eq a(b sub 1 ,... , b sub k , x sub k+1 ,... , x sub r ), an element of Q( alpha )(x sub k+1 ,... , x sub r ).*)PROCEDURE IUPMRN(R: LIST): LIST; (*Integral univariate polynomial minimal polynomial of a rational number. R is a rational number. M is the integral minimal polynomial of R.*)PROCEDURE RPAFME(RL,M,A,BL: LIST): LIST; (*Rational polynomial, algebraic number field multiple evaluation. A is a rational polynomial in r variables, r ge 1. M is the rational minimal polynomial of an algebraic number alpha. b eq (b sub 1 ,... , b sub k ) is a list of k elements of Q( alpha ), for some k, 1 le k le r. B eq A(b sub 1 ,... , b sub k , x sub k+1 ,... , x sub r ), an element of Q( alpha )(x sub k+1 ,... , x sub r ).*)ENDSACEXT8. (* -EOF- *)