(* ---------------------------------------------------------------------------- * $Id: SACLDIO.md,v 1.2 1992/02/12 17:33:12 pesch Exp $ * ---------------------------------------------------------------------------- * This file is part of MAS. * ---------------------------------------------------------------------------- * Copyright (c) 1989 - 1992 Universitaet Passau * ---------------------------------------------------------------------------- * $Log: SACLDIO.md,v $ * Revision 1.2 1992/02/12 17:33:12 pesch * Moved CONST definition to the right place * * Revision 1.1 1992/01/22 15:12:35 kredel * Initial revision * * ---------------------------------------------------------------------------- *) DEFINITION MODULE SACLDIO; (* SAC Linear Diophantine Equation System Definition Module. *) FROM MASSTOR IMPORT LIST;CONSTrcsid = "$Id: SACLDIO.md,v 1.2 1992/02/12 17:33:12 pesch Exp $";CONSTcopyright = "Copyright (c) 1989 - 1992 Universitaet Passau";PROCEDURE LDSMKB(A,BL: LIST;VARXLS,N: LIST); (*Linear diophantine system solution, modified Kannan and Bachem algorithm. A is an m by n integral matrix. A is represented column-wise. b is an integral m-vector. If the diophantine system A*x=b is consistent, then xs is a particular solution and N is a list of basis vectors of the solution module of A*x=0. Otherwise, xs and N are null lists. A and b are modified.*)PROCEDURE LDSSBR(A,BL: LIST;VARXLS,N: LIST); (*Linear diophantine system solution, based on Rosser ideas. A is an m by n integral matrix. A is represented column-wise. b is an integral m-vector. If the diophantine system A*x=b is consistent, then xs is a particular solution and N is a list of basis vectors of the solution module of A*x=0. Otherwise, xs and N are null lists. A and b are modified.*)PROCEDURE MAIPDE(RL,M: LIST): LIST; (*Matrix of integral polynomials determinant, exact division algorithm. M is a square matrix of integral polynomials in r variables, r ge 0, represented as a list. D is the determinant of M.*)PROCEDURE MAIPDM(RL,M: LIST): LIST; (*Matrix of integral polynomials determinant, modular algorithm. M is a square matrix of integral polynomials in r variables, r non-negative. D is the determinant of M.*)PROCEDURE MAIPHM(RL,ML,A: LIST): LIST; (*Matrix of integral polynomials homomorphism. A is a matrix of integral polynomials in r variables, r non-negative. m is a positive beta-integer. B is the matrix B(i,j) of polynomials in r variables over Z sub m such that B(i,j)=H(m)(A(i,j)).*)PROCEDURE MAIPP(RL,A,B: LIST): LIST; (*Matrix of integral polynomials product. A and B are matrices of integral polynomials in r variables, r ge 0, for which the matrix product A*B is defined. C=A*B.*)PROCEDURE MIAIM(A: LIST): LIST; (*Matrix of integers, adjoin identity matrix, A is an m by n matrix of integers. A is represented column-wise. B is the matrix obtained by adjoining an n by n identity matrix to the bottom of A. A is modified.*)PROCEDURE MICINS(A,V: LIST): LIST; (*Matrix of integers column insertion. A is an m by n integral matrix represented by the list (a(1),a(2), ...,a(n)), where a(i) is the list (a(1,i), ...,a(m,i)) representing column i of A and a(1,1) ge a(1,2) ge ... ge a(1,m). V=(v(1), ...,v(m)) is an integral vector with v(1) lt a(1,1). Let i be the largest integer such that a(1,i) ge v(1). Then B is the matrix represented by the list (a(1), ..., a(i),V,a(i+1), ...,a(n)). A is modified.*)PROCEDURE MICS(A: LIST): LIST; (*Matrix of integers column sort. A is an integral matrix with non- negative elements in first row. A is represented column-wise. B is an integral matrix obtained by sorting columns of A such that elements of the first row are in descending order. A is modified.*)PROCEDURE MINNCT(A: LIST): LIST; (*Matrix of integers, non-negative column transformation. A=(a(i,j)) is an m by n integral matrix. A is represented column-wise. B=(b(i,j)) is the m by n integral matrix with b(i,j)=a(i,j) if a(1,j) ge 0 and b(i,j)=-a(i,j) if a(1,j) lt 0. A is modified.*)PROCEDURE MMDDET(PL,M: LIST): LIST; (*Matrix of modular digits determinant. p is a prime beta-integer. M is a square matrix over GF(p), represented as a list. d is the determinant of M.*)PROCEDURE MMDNSB(PL,M: LIST): LIST; (*Matrix of modular digits null space basis. p is a prime beta- integer. M is an m by n matrix over Z sub p. B is a list (b(1), ..., b(r)) representing a basis for the null space of M, consisting of all x such that M*x=0. r is the dimension of the null space. B=() if the null space of M is 0. Each b(i) is a list representing an m-vector. M is modified. Alternatively, if M represents a matrix by columns, then B is a basis for the null space consisting of all x such that x*M=0.*)PROCEDURE MMPDMA(RL,PL,M: LIST): LIST; (*Matrix of modular polynomials determinant, modular algorithm. M is a square matrix of modular polynomials in r variables over Z sub p, r non-negative, p a prime beta-integer. D is the determinant of M.*)PROCEDURE MMPEV(RL,ML,A,KL,AL: LIST): LIST; (*Matrix of modular polynomials evaluation. A is a matrix of polynomials in r variables over Z sub m, m a positive beta-integer. 1 le k le r and a is an element of Z sub m. B is the matrix of polynomials b(i,j) where b(i,j)(x(1), ...,x(k-1),x(k+1), ...,x(r))= a(i,j)(x(1), ...,x(k-1),a,x(k+1), ...,x(r)).*)PROCEDURE VIAZ(A,NL: LIST): LIST; (*Vector of integers, adjoin zeros. A is the vector (a(1), ...,a(m)). n is a non-negative beta-integer. B is the vector (a(1), ...,a(m), 0, ...,0) of m+n components. A is modified.*)PROCEDURE VIDIF(A,B: LIST): LIST; (*Vector of integers difference. A and B are vectors in Z sup n. C=A-B.*)PROCEDURE VIERED(U,V,IL: LIST): LIST; (*Vector of integers, element reduction. U=(u(1), ...,u(n)) and V=(v(1), ...,v(n)) are integral n-vectors. 1 le i le n. v(i) ne 0. W=U-q*V, where q=INTEGER(u(i)/v(i)).*)PROCEDURE VILCOM(AL,BL,A,B: LIST): LIST; (*Vector of integers linear combination. a and b are integers. A and B are integral vectors in Z sup n. C=a*A+b*B.*)PROCEDURE VINEG(A: LIST): LIST; (*Vector of integers negation. A is an integral vector. B=-A.*)PROCEDURE VISPR(AL,A: LIST): LIST; (*Vector of integers scalar product. a is an integer. A is an integral vector. C=a*A.*)PROCEDURE VISUM(A,B: LIST): LIST; (*Vector of integers sum. A and B are vectors in Z sup n. C=A+B.*)PROCEDURE VIUT(U,V,IL: LIST;VARUP,VP: LIST); (*Vector of integers, unimodular transformation. U=(u(1), ...,u(n)) and V=(v(1), ...,v(n)) are vectors in Z sup n with u(i) ne 0. (UP,VP)=(U,V)*K where K is a unimodular matrix, depending on u(i) and v(i), whose elements are obtained from IDEGCD.*)ENDSACLDIO. (* -EOF- *)