// \$Id: random.lib,v 1.5 1998-05-05 11:55:36 krueger Exp \$ //system("random",787422842); //(GMG/BM, last modified 22.06.96) /////////////////////////////////////////////////////////////////////////////// version="\$Id: random.lib,v 1.5 1998-05-05 11:55:36 krueger Exp \$"; info=" LIBRARY: random.lib PROCEDURES OF RANDOM MATRIX AND POLY OPERATIONS genericid(id[,p,b]); generic sparse linear combinations of generators of id randomid(id,[k,b]); random linear combinations of generators of id randommat(n,m[,id,b]); nxm matrix of random linear combinations of id sparseid(k,u[,o,p,b]); ideal of k random sparse poly's of degree d [u<=d<=o] sparsemat(n,m[,p,b]); nxm sparse integer matrix with random coefficients sparsepoly(u[,o,p,b]); random sparse polynomial with terms of degree in [u,o] sparsetriag(n,m[..]); nxm sparse lower-triag intmat with random coefficients (parameters in square brackets [] are optional) "; LIB "inout.lib"; LIB "general.lib"; //////////////////////////////////////////////////////////////////////////////// proc genericid (id, list #) "USAGE: genericid(id,[,p,b]); id ideal/module, k,p,b integers RETURN: system of generators of id which are generic, sparse, triagonal linear combinations of given generators with coefficients in [1,b] and sparsety p percent, bigger p being sparser (default: p=75, b=30000) NOTE: For performance reasons try small bound b in characteristic 0 EXAMPLE: example genericid; shows an example " { //----------------------------- set defaults ---------------------------------- if( size(#)>=2 ) { int p=#[1]; int b=#[2]; } if( size(#)==1 ) { int p=#[1]; int b=30000} if( size(#)==0 ) { int p=75; int b=30000; } //---------------- use sparsetriag for creation of genericid ------------------ def i = simplify(id,10); i = i*sparsetriag(ncols(i),ncols(i),p,b); return(i); } example { "EXAMPLE:"; echo = 2; ring r=0,(t,x,y,z),ds; ideal i= x3+y4,z4+yx,t+x+y+z; genericid(i,0,10); module m=[x,0,0,0],[0,y2,0,0],[0,0,z3,0],[0,0,0,t4]; print(genericid(m)); } //////////////////////////////////////////////////////////////////////////////// proc randomid (id, list #) "USAGE: randomid(id,[k,b]); id ideal/module, b,k integers RETURN: ideal/module having k generators which are random linear combinations of generators of id with coefficients in the interval [-b,b] (default: b=30000, k=size(id)) NOTE: For performance reasons try small bound b in characteristic 0 EXAMPLE: example randomid; shows an example " { //----------------------------- set defaults ---------------------------------- if( size(#)>=2 ) { int k=#[1]; int b=#[2]; } if( size(#)==1 ) { int k=#[1]; int b=30000; } if( size(#)==0 ) { int k=size(id); int b=30000; } //--------------------------- create randomid --------------------------------- def i = id; i = matrix(id)*random(b,ncols(id),k); return(i); } example { "EXAMPLE:"; echo = 2; ring r=0,(x,y,z),dp; randomid(maxideal(2),2,9); module m=[x,0,1],[0,y2,0],[y,0,z3]; show(randomid(m)); } //////////////////////////////////////////////////////////////////////////////// proc randommat (int n, int m, list #) "USAGE: randommat(n,m[,id,b]); n,m,b integers, id ideal RETURN: nxm matrix, entries are random linear combinations of elements of id and coefficients in [-b,b] [default: (id,b) = (maxideal(1),30000)] NOTE: For performance reasons try small bound b in char 0 EXAMPLE: example randommat; shows an example " { //----------------------------- set defaults ---------------------------------- if( size(#)>=2 ) { ideal id=#[1]; int b=#[2]; } if( size(#)==1 ) { ideal id=#[1]; int b=30000; } if( size(#)==0 ) { ideal id=maxideal(1); int b=30000; } //--------------------------- create randommat -------------------------------- id=simplify(id,2); int g=ncols(id); matrix rand[n][m]; matrix ra[1][m]; for (int k=1; k<=n; k=k+1) { ra = id*random(b,g,m); rand[k,1..m]=ra[1,1..m]; } return(rand); } example { "EXAMPLE:"; echo = 2; ring r=0,(x,y,z),dp; matrix A=randommat(3,3,maxideal(2),9); print(A); A=randommat(2,3); print(A); } /////////////////////////////////////////////////////////////////////////////// proc sparseid (int k, int u, list #) "USAGE: sparseid(k,u[,o,p,b]); k,u,o,p,b integers RETURN: ideal having k generators in each degree d, u<=d<=o, p percent of terms in degree d are 0, the remaining have random coefficients in the interval [1,b], (default: o=u=d, p=75, b=30000) EXAMPLE: example sparseid; shows an example " { //----------------------------- set defaults ---------------------------------- if( size(#)>=3 ) { int o=#[1]; int p=#[2]; int b=#[3]; } if( size(#)==2 ) { int o=#[1]; int p=#[2]; int b=30000; } if( size(#)==1 ) { int o=#[1]; int p=75; int b=30000; } if( size(#)==0 ) { int o=u; int p=75; int b=30000; } //------------------ use sparsemat for creation of sparseid ------------------- int ii; ideal i; intmat m; for ( ii=u; ii<=o; ii=ii+1) { m = sparsemat(size(maxideal(ii)),k,p,b); i = i+ideal(matrix(maxideal(ii))*m); } return(i); } example { "EXAMPLE:"; echo = 2; ring r = 0,(a,b,c,d),ds; sparseid(3,4);""; sparseid(2,2,5,90,9); } /////////////////////////////////////////////////////////////////////////////// proc sparsemat (int n, int m, list #) "USAGE: sparsemat(n,m[,p,b]); n,m,p,b integers RETURN: nxm integer matrix, p percent of the entries are 0, the remaining are random coefficients >=1 and <= b; [defaults: (p,b) = (75,1)] EXAMPLE: example sparsemat; shows an example " { int r,h,ii; int t = n*m; intmat v[1][t]; //----------------------------- set defaults ---------------------------------- if( size(#)>=2 ) { int p=#[1]; int b=#[2]; } if( size(#)==1 ) { int p=#[1]; int b=1; } if( size(#)==0 ) { int p=75; int b=1; } //------------------------- check trivial cases ------------------------------ if( p<0 ) { p = 0; } if(p>100) { p=100; } //--------------- this is faster for not very sparse matrices ---------------- if( p<40 ) { for( ii=1; ii<=t; ii=ii+1 ) { r=( random(1,100)>p ); v[1,ii]=r*random(1,b); h=h+r; } } int bb = t*(100-p); if( 100*h > bb ) { while( 100*h > bb ) { r=random(1,t); h=h-( v[1,r]>0 ); v[1,r]=0; } } else { //------------------- this is faster for sparse matrices --------------------- while ( 100*h < bb ) { r=random(1,t); h=h+(v[1,r]==0); v[1,r]=random(1,b); } } intmat M[n][m] = v[1,1..t]; return(M); } example { "EXAMPLE:"; echo = 2; sparsemat(5,5);""; sparsemat(5,5,95);""; sparsemat(5,5,5);""; sparsemat(5,5,50,100); } /////////////////////////////////////////////////////////////////////////////// proc sparsepoly (int u, list #) "USAGE: sparsepoly(u[,o,p,b]); u,o,p,b integers RETURN: poly having only terms in degree d, u<=d<=o, p percent of the terms in degree d are 0, the remaining have random coefficients in [1,b), (defaults: o=u=d, p=75, b=30000) EXAMPLE: example sparsepoly; shows an example " { //----------------------------- set defaults ---------------------------------- if( size(#)>=3 ) { int o=#[1]; int p=#[2]; int b=#[3]; } if( size(#)==2 ) { int o=#[1]; int p=#[2]; int b=30000; } if( size(#)==1 ) { int o=#[1]; int p=75; int b=30000; } if( size(#)==0 ) { int o=u; int p=75; int b=30000; } int ii; poly f; //----------------- use sparseid for creation of sparsepoly ------------------- for( ii=u; ii<=o; ii=ii+1 ) { f=f+sparseid(1,ii,ii,p,b)[1]; } return(f); } example { "EXAMPLE:"; echo = 2; ring r=0,(x,y,z),dp; sparsepoly(5);""; sparsepoly(3,5,90,9); } /////////////////////////////////////////////////////////////////////////////// proc sparsetriag (int n, int m, list #) "USAGE: sparsetriag(n,m[,p,b]); n,m,p,b integers RETURN: nxm lower triagonal integer matrix, diagonal entries equal to 1, about p percent of lower diagonal entries are 0, the remaining are random integers >=1 and <= b; [defaults: (p,b) = (75,1)] EXAMPLE: example sparsetriag; shows an example " { int ii,min,l,r; intmat M[n][m]; int t=(n*(n-1)) div 2; //----------------------------- set defaults ---------------------------------- if( size(#)>=2 ) { int p=#[1]; int b=#[2]; } if( size(#)==1 ) { int p=#[1]; int b=1; } if( size(#)==0 ) { int p=75; int b=1; } //---------------- use sparsemat for creation of sparsetriag ------------------ intmat v[1][t]=sparsemat(1,t,p,b); if( n<=m ) { min=n-1; M[n,n]=1; } else { min=m; } for( ii=1; ii<=min; ii=ii+1 ) { l=r+1; r=r+n-ii; M[ii..n,ii]=1,v[1,l..r]; } return(M); } example { "EXAMPLE:"; echo = 2; sparsetriag(5,7);""; sparsetriag(7,5,90);""; sparsetriag(5,5,0);""; sparsetriag(5,5,50,100); } ///////////////////////////////////////////////////////////////////////////////