| 1 | // $ID$ |
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| 2 | // ==================================================================== |
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| 3 | // library fast.lib |
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| 4 | // |
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| 5 | // ==================================================================== |
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| 6 | |
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| 7 | proc Tsimplify |
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| 8 | USAGE: simplify(id,n); id ideal, n integer |
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| 9 | RETURNS: list of two ideals the first containing the simplified |
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| 10 | elements, the second the variables which already have |
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| 11 | been eliminated |
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| 12 | ASSUME: basering has ordering dp, the elimination of the variables |
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| 13 | n+1,... is not supported |
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| 14 | EXAMPLE: example simplify; shows an example |
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| 15 | |
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| 16 | { |
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| 17 | def bsr=basering; |
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| 18 | int @n, @m, @k; |
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| 19 | option (redSB); |
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| 20 | ideal @i = interred(#[1]); |
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| 21 | ideal @j, @im, @va; |
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| 22 | for (@n=1;@n<=size(@i);@n=@n+1) |
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| 23 | { |
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| 24 | if (deg(@i[@n])==1) |
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| 25 | { |
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| 26 | @k=0; |
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| 27 | for (@m=#[2]+1;@m<=nvars(basering);@m=@m+1) |
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| 28 | { |
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| 29 | if (lead(@i[@n])/var(@m)!=0) |
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| 30 | { |
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| 31 | @k=1; |
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| 32 | break; |
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| 33 | } |
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| 34 | } |
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| 35 | if (@k==0) |
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| 36 | { |
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| 37 | @va=@va+lead(@i[@n]); |
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| 38 | @i[@n]=0; |
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| 39 | } |
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| 40 | } |
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| 41 | } |
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| 42 | @i=@i+@j; |
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| 43 | map @phi; |
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| 44 | for (@m=1;@m<=#[2];@m=@m+1) |
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| 45 | { |
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| 46 | for (@n=size(@i);@n>=1;@n=@n-1) |
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| 47 | { |
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| 48 | if (deg(@i[@n]/var(@m))==0) |
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| 49 | { |
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| 50 | @im = maxideal(1); |
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| 51 | @im[@m]=(-1/coeffs(@i[@n],var(@m))[2,1])*(@i[@n]-coeffs(@i[@n],var(@m))[2,1]*var(@m)); |
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| 52 | @phi = basering,@im; |
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| 53 | @i=@phi(@i); |
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| 54 | @i=@i+@j; |
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| 55 | @va=@va+var(@m); |
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| 56 | break;// hier muesste evt. sorgfaeltiger ausgewaehlt werden |
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| 57 | } |
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| 58 | } |
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| 59 | } |
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| 60 | @i=interred(@i); |
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| 61 | option(noredSB); |
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| 62 | return(@i,@va); |
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| 63 | } |
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| 64 | example |
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| 65 | { |
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| 66 | ring s=181,(x,y,z,t,u,v,w,a,b,c,d,f,e),dp; |
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| 67 | ideal j= |
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| 68 | w2+f2-1, |
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| 69 | x2+t2+a2-1, |
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| 70 | y2+u2+b2-1, |
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| 71 | z2+v2+c2-1, |
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| 72 | d2+e2-1, |
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| 73 | f4+2u, |
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| 74 | wa+tf, |
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| 75 | xy+tu+ab, |
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| 76 | yz+uv+bc, |
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| 77 | cd+ze, |
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| 78 | x+y+z+e+1, |
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| 79 | t+u+v+f-1, |
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| 80 | w+a+b+c+d; |
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| 81 | list l=Tsimplify(j,10); |
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| 82 | l; |
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| 83 | } |
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| 84 | |
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| 85 | |
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| 86 | proc simplifyInBadRings |
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| 87 | USAGE: simplify(id,n); id ideal, n product of the variables |
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| 88 | to be eliminated later |
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| 89 | RETURNS: list of two ideals the first containing the simplified |
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| 90 | elements, the second the variables which already have |
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| 91 | been eliminated |
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| 92 | EXAMPLE: example simplify; shows an example |
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| 93 | |
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| 94 | { |
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| 95 | int @i,@j; |
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| 96 | string @es=",("; |
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| 97 | string @ns="),dp;"; |
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| 98 | for (@i=1;@i<=nvars(basering);@i=@i+1) |
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| 99 | { |
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| 100 | if (deg(#[2]/var(@i))>=0) |
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| 101 | { |
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| 102 | @es=@es+varstr(@i)+","; |
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| 103 | @j=@j+1; |
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| 104 | } |
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| 105 | else |
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| 106 | { |
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| 107 | @ns=","+varstr(@i)+@ns; |
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| 108 | } |
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| 109 | } |
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| 110 | string @s=@es[1,size(@es)-1]+@ns; |
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| 111 | @ns=nameof(basering); |
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| 112 | ideal i=#[1]; |
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| 113 | execute "ring @gnir ="+charstr(basering)+@s; |
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| 114 | ideal @laedi=imap(`@ns`,i); |
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| 115 | list @l=Tsimplify(@laedi,@j); |
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| 116 | ideal @l1, @l2=@l[1], @l[2]; |
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| 117 | execute "setring "+@ns+";"; |
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| 118 | return(imap(@gnir,@l1),imap(@gnir,@l2)); |
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| 119 | } |
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| 120 | example |
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| 121 | { |
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| 122 | ring s=0,(e,f,x,y,z,t,u,v,w,a,b,c,d),dp; |
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| 123 | ideal j= |
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| 124 | w2+f2-1, |
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| 125 | x2+t2+a2-1, |
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| 126 | y2+u2+b2-1, |
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| 127 | z2+v2+c2-1, |
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| 128 | d2+e2-1, |
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| 129 | f4+2u, |
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| 130 | wa+tf, |
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| 131 | xy+tu+ab, |
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| 132 | yz+uv+bc, |
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| 133 | cd+ze, |
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| 134 | x+y+z+e+1, |
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| 135 | t+u+v+f-1, |
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| 136 | w+a+b+c+d; |
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| 137 | list l=simplifyInBadRings(j,xyztuvwabc); |
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| 138 | l; |
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| 139 | } |
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| 140 | |
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| 141 | |
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| 142 | proc fastElim |
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| 143 | USAGE: fastelim(id,v); id ideal, v Product of variables to |
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| 144 | be eliminated. |
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| 145 | RETURNS: list of two ideals the first containing the simplified |
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| 146 | elements, the second the variables which already have |
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| 147 | been eliminated |
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| 148 | EXAMPLE: example fastElim; shows an example. |
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| 149 | { |
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| 150 | string @oldgnir = nameof(basering); // Namen des alten Baserings merken |
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| 151 | string @varnames; |
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| 152 | int @i; |
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| 153 | poly @testpoly=1; // Variablen, die schon eliminiert sind (nach Simplify) |
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| 154 | poly @elimvarspoly=#[2]; // Variablen, die noch eliminiert werden sollen |
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| 155 | |
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| 156 | // das Ideal vereinfachen. Beachte: l[1] erzeugt nicht mehr id! |
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| 157 | list @l=simplifyInBadRings(#[1],#[2]); |
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| 158 | ideal @a, @b = @l[1], @l[2]; |
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| 159 | |
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| 160 | // ------------------------------------------------------------ |
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| 161 | // in einen Ring abbilden, der die in (2) angegebenen Variablen |
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| 162 | // nicht mehr enthaelt. |
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| 163 | // ------------------------------------------------------------ |
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| 164 | |
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| 165 | |
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| 166 | // Aus dem von simplifyInBadRings zurueckgegebenen Ideal(2), das die |
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| 167 | // Variablen anthealt, die eliminiert sind, ein Polynom bauen: |
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| 168 | for (@i=1; @i<=size(@b); @i=@i+1) |
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| 169 | { |
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| 170 | @testpoly=@testpoly*@b[@i]; |
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| 171 | @elimvarspoly = subst(@elimvarspoly, @b[@i], 1); |
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| 172 | } |
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| 173 | |
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| 174 | //Wenn keine Variablen mehr uebrig sind, sind wir fertig: |
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| 175 | if (@elimvarspoly==1) |
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| 176 | { |
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| 177 | // Falls das 4. Argument 1 ist, eine minimale Basis berechnen |
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| 178 | if (#[4]==1) |
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| 179 | { |
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| 180 | mres(@a,1,@a0); |
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| 181 | @a=@a0(1); |
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| 182 | } |
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| 183 | return(@a); |
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| 184 | } |
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| 185 | |
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| 186 | // Anhand dieses Polynoms ueberpruefen, welche Variablen unser |
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| 187 | // neuer Ring braucht: @varnames |
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| 188 | for (@i=1; @i<=nvars(basering); @i=@i+1) |
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| 189 | { |
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| 190 | if (@testpoly/var(@i)==0) |
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| 191 | { |
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| 192 | @varnames=@varnames + nameof(var(@i)) + ","; |
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| 193 | } |
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| 194 | } |
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| 195 | @varnames=@varnames[1, size(@varnames)-1]; |
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| 196 | |
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| 197 | // Den neuen Ring erzeugen und das Ideal a darin abbilden: |
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| 198 | execute("ring @r1=" + charstr(basering) + ",(" + @varnames + ")," + "dp;" ); |
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| 199 | ideal @a=imap(`@oldgnir`, @a); |
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| 200 | list #=imap(`@oldgnir`,#); |
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| 201 | |
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| 202 | // Nun die beste Reihenfolge fuer die Variablen berechnen: |
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| 203 | string @orderedvars=string(maxideal(1)); |
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| 204 | if (#[3]==1) |
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| 205 | { |
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| 206 | poly @elimvarspoly=imap(`@oldgnir`,@elimvarspoly); |
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| 207 | intvec @v= valvars(@a,#[5], @elimvarspoly,#[6]); |
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| 208 | @v; |
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| 209 | @orderedvars=string(sort1(maxideal(1),@v)); |
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| 210 | } |
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| 211 | |
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| 212 | // In einen Ring wechseln, der diese Reihenfolge beruecksichtig: |
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| 213 | execute("ring @r2=" + charstr(basering) + ",(" + @orderedvars + ",@homo),dp;" ); |
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| 214 | ideal @a= imap(@r1, @a); |
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| 215 | kill @r1; |
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| 216 | |
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| 217 | //das ideal homogenisieren |
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| 218 | @a=homog(@a,@homo); |
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| 219 | |
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| 220 | // Die Variablen holen, die wirklich noch zu eliminieren sind, und |
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| 221 | // danach eliminate aufrufen. |
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| 222 | poly @elimvarspoly= imap(`@oldgnir`, @elimvarspoly); |
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| 223 | ideal @b=eliminate(@a, @elimvarspoly); |
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| 224 | |
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| 225 | // Falls das 4. Argument 1 ist, eine minimale Basis berechnen |
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| 226 | if (#[4]==1) |
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| 227 | { |
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| 228 | mres(@b,1,@b0); |
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| 229 | @b=@b0(1); |
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| 230 | } |
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| 231 | |
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| 232 | // dehomogenisieren (@homo=1) und zurueck in den alten Ring |
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| 233 | @b=subst(@b,@homo,1); |
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| 234 | execute "setring "+@oldgnir+";"; |
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| 235 | return(imap(@r2,@b)); |
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| 236 | } |
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| 237 | example |
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| 238 | { |
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| 239 | ring s=31991,(e,f,x,y,z,t,u,v,w,a,b,c,d),dp; |
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| 240 | ideal j= |
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| 241 | w2+f2-1, |
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| 242 | x2+t2+a2-1, |
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| 243 | y2+u2+b2-1, |
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| 244 | z2+v2+c2-1, |
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| 245 | d2+e2-1, |
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| 246 | f4+2u, |
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| 247 | wa+tf, |
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| 248 | xy+tu+ab, |
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| 249 | yz+uv+bc, |
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| 250 | cd+ze, |
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| 251 | x+y+z+e+1, |
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| 252 | t+u+v+f-1, |
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| 253 | w+a+b+c+d; |
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| 254 | fastElim(j,xyztuvwabcd,1,0,0,0); |
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| 255 | } |
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| 256 | |
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| 257 | |
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| 258 | proc sort1 |
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| 259 | { |
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| 260 | intvec @i=#[2]; |
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| 261 | int @ii=size(@i); |
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| 262 | int @j; |
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| 263 | string @s=string(typeof(#[1]))+" @m;"; |
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| 264 | execute(@s); |
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| 265 | for (@j=1;@j<=@ii;@j=@j+1) |
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| 266 | { |
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| 267 | @m[@j] = #[1][@i[@j]]; |
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| 268 | } |
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| 269 | return(@m); |
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| 270 | } |
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| 271 | |
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| 272 | |
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| 273 | proc valvars |
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| 274 | |
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| 275 | USAGE: valvars(id, [n1, [m, [n2]]]); id (poly|ideal|vector|module); |
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| 276 | m monom; n1,n2 int |
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| 277 | RETURNS: an intvec describing the permutation such that the permuted |
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| 278 | ringvariables are ordered with respect to inreasing complexity |
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| 279 | [resp. decreasing complexity if n1 is present and non-zero]. |
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| 280 | If m is present, the variables occuring in m are sorted in one |
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| 281 | block before all variables not occuring in m. n2 controls the |
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| 282 | order of this block as n1 does for the other one. |
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| 283 | A variable x is more complex than y (with respect to id) if, in the |
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| 284 | input, either the maximal occuring power of x is bigger than that |
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| 285 | of y or both are equal and the coefficient of x to this power |
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| 286 | contains more monomials than that of y. |
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| 287 | EXAMPLE: example valvars; shows an example |
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| 288 | |
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| 289 | { |
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| 290 | int @order1, @order2, // order of vars not to elim resp. to elim |
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| 291 | @i1, // runs through all variables |
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| 292 | @i2, // # of vars to elim already found |
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| 293 | @N, @M, // N = total # of vars, M = # of vars to elim |
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| 294 | @i, @j, @k, @size, @degree; |
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| 295 | intvec @varvec, @degvec, @sizevec; |
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| 296 | poly @m; |
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| 297 | ideal @id; |
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| 298 | matrix @C; |
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| 299 | |
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| 300 | @id = matrix( #[1] ); @m = 0; |
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| 301 | @N = nvars( basering ); @M = 0; |
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| 302 | @order1 = 1; @order2 = 1; |
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| 303 | if ( size(#) >= 4 ) |
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| 304 | { @order2 = not( #[4] ) * 2 - 1; } |
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| 305 | if ( size(#) >= 3 ) |
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| 306 | { @m = #[3]; @M = deg( #[3] ); } |
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| 307 | if ( size(#) >= 2 ) |
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| 308 | { @order1 = not( #[2] ) * 2 - 1; } |
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| 309 | |
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| 310 | @i2 = 0; |
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| 311 | for ( @i1 = 1; @i1 <= @N; @i1++ ) |
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| 312 | { |
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| 313 | // get valuation of current variable. |
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| 314 | @C = coeffs( @id, var( @i1 )); |
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| 315 | @degree = nrows( @C ); |
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| 316 | @size = 0; |
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| 317 | for ( @j = ncols( @C ); @j >= 1; @j-- ) |
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| 318 | { @size = @size + size( @C[ @degree, @j ] ); } |
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| 319 | |
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| 320 | // mind order and calculate limits for search |
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| 321 | if ( @M and size( @m / var( @i1 ))) |
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| 322 | { |
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| 323 | @degree = @degree * @order2; @size = @size * @order2; |
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| 324 | @i2++; |
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| 325 | @i = @i2; @j = 1; |
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| 326 | } |
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| 327 | else |
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| 328 | { |
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| 329 | @degree = @degree * @order1; @size = @size * @order1; |
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| 330 | @i = @i1 - @i2 + @M; @j = @M+1; |
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| 331 | } |
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| 332 | |
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| 333 | // look where we have to insert the variable |
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| 334 | // "start = " + string( @j ) + ", end = " + string( @i ); |
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| 335 | @degvec[ @i ] = @degree + 1; @sizevec[ @i ] = @size + 1; |
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| 336 | while ( @degvec[ @j ] < @degree ) |
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| 337 | { @j++; } |
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| 338 | while ( @degvec[ @j ] == @degree and @sizevec[ @j ] < @size ) |
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| 339 | { @j++; } |
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| 340 | |
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| 341 | // now shift the variables behind the current one back |
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| 342 | // "inserting at " + string( @j ); |
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| 343 | for ( @k = @i-1; @k >= @j; @k-- ) |
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| 344 | { |
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| 345 | @varvec[ @k+1 ] = @varvec[ @k ]; |
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| 346 | @degvec[ @k+1 ] = @degvec[ @k ]; |
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| 347 | @sizevec[ @k+1 ] = @sizevec[ @k ]; |
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| 348 | } |
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| 349 | @varvec[ @j ] = @i1; // print( @varvec ); |
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| 350 | @degvec[ @j ] = @degree; // print( @degvec ); |
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| 351 | @sizevec[ @j ] = @size; // print( @sizevec ); |
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| 352 | } |
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| 353 | |
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| 354 | return( @varvec ); |
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| 355 | } |
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| 356 | example |
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| 357 | { |
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| 358 | echo = 1; ring @r=0,(a,b,c,x,y,z),lp; |
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| 359 | poly @f=a3+b4+xyz2+xyz+yz+1; |
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| 360 | valvars( @f ); |
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| 361 | valvars( @f, 1 ); |
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| 362 | valvars( @f, 0, abc ); |
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| 363 | valvars( @f, 0, xyz ); |
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| 364 | valvars( @f, 0, abc, 1 ); |
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| 365 | valvars( @f, 1, abc, 1 ); |
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| 366 | ideal @i=@f,z5,y5,-y5; |
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| 367 | valvars( @i ); |
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| 368 | valvars( @i, 0, abcxyz ); |
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| 369 | kill @r; echo = 0; |
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| 370 | } |
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