| hermann on Wed, 20 Dec 2023 02:14:29 +0100 |
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| Re: Question on "qfminim()" for quadratic form |
On 2023-12-18 20:08, hermann@stamm-wilbrandt.de wrote:
So question was, whether there is an efficient method to
compute either [711, -153, -7]~ from M with minimal norml2 of 528979.
Or compute vector [311585, -67063, -3085]~ from M with
maximal norml2 of 101592175419.
...
Script used, tqf.gp has to be in same directory:
$ cat S2.b.gp
readvec("tqf.gp");
n=eval(getenv("n"));
Q=get_tqf(n);
M=Q~*Q;
S=[x|x<-Vec(qfminim(M,n)[3]),qfeval(M,x)==n];
S2=vecsort(concat(S,-S),norml2);
{if(!eval(getenv("mindbg")),
foreach(S2,s,print(norml2(s)," ",s," ",Q*s)),
print(norml2(S2[1])," ",S2[1]," ",Q*S2[1]);
print("...");
print(norml2(S2[#S2])," ",S2[#S2]," ",Q*S2[#S2]))}
print("#S2=",#S2);
print("12*h(-4*n)=",12*qfbclassno(-4*n));
forqfvec(v,M,n,if(qfeval(M,v)==n,V=v;break()));
print(norml2(V)," ",V," ",Q*V);
$
With this little diff it is used again as it should: $ diff S2.b.gp S2.c.gp 3c3 < Q=get_tqf(n); ---
Q=qflllgram(get_tqf(n))^-1;
$And for that Q, with M=Q~*Q we know the minimal L2 norm vector, no search needed.
It is [0,0,1]~ by Dirichlet's construction. All L2 norms of the middle column vectors are much lower than before. So only question remaining is, whether one can efficiently determine the last vector with maximal L2 norm 3142707 of S2. $ n=101 gp -q < S2.c.gp 101=norml2([1, 6, 8]) all asserts OK 1 [0, 0, 1]~ [-1, -6, -8]~ 1 [0, 0, -1]~ [1, 6, 8]~ 651 [25, -5, -1]~ [6, 1, 8]~ 651 [-25, 5, 1]~ [-6, -1, -8]~ 6886 [81, -18, -1]~ [-8, 6, -1]~ 6886 [-81, 18, 1]~ [8, -6, 1]~ 23826 [151, -32, -1]~ [6, -8, -1]~ 23826 [-151, 32, 1]~ [-6, 8, 1]~ 23891 [151, -33, -1]~ [-8, 1, -6]~ 23891 [-151, 33, 1]~ [8, -1, 6]~ 40181 [196, -42, -1]~ [1, -8, -6]~ 40181 [-196, 42, 1]~ [-1, 8, 6]~ 43561 [204, -44, -3]~ [-1, 6, 8]~ 43561 [-204, 44, 3]~ [1, -6, -8]~ 43970 [205, -44, -3]~ [2, 4, 9]~ 43970 [-205, 44, 3]~ [-2, -4, -9]~ 48350 [215, -46, -3]~ [4, 2, 9]~ 48350 [-215, 46, 3]~ [-4, -2, -9]~ 49307 [217, -47, -3]~ [-4, 7, 6]~ 49307 [-217, 47, 3]~ [4, -7, -6]~ 57835 [235, -51, -3]~ [-6, 7, 4]~ 57835 [-235, 51, 3]~ [6, -7, -4]~ 59731 [239, -51, -3]~ [6, -1, 8]~ 59731 [-239, 51, 3]~ [-6, 1, -8]~ 74662 [267, -58, -3]~ [-8, 6, 1]~ 74662 [-267, 58, 3]~ [8, -6, -1]~ 77357 [272, -58, -3]~ [7, -4, 6]~ 77357 [-272, 58, 3]~ [-7, 4, -6]~ 94105 [300, -64, -3]~ [7, -6, 4]~ 94105 [-300, 64, 3]~ [-7, 6, -4]~ 96781 [304, -66, -3]~ [-9, 4, -2]~ 96781 [-304, 66, 3]~ [9, -4, 2]~ 115417 [332, -72, -3]~ [-9, 2, -4]~ 115417 [-332, 72, 3]~ [9, -2, 4]~ 118762 [337, -72, -3]~ [6, -8, 1]~ 118762 [-337, 72, 3]~ [-6, 8, -1]~ 139475 [365, -79, -3]~ [-8, -1, -6]~ 139475 [-365, 79, 3]~ [8, 1, 6]~ 142411 [369, -79, -3]~ [4, -9, -2]~ 142411 [-369, 79, 3]~ [-4, 9, 2]~ 156667 [387, -83, -3]~ [2, -9, -4]~ 156667 [-387, 83, 3]~ [-2, 9, 4]~ 158386 [389, -84, -3]~ [-6, -4, -7]~ 158386 [-389, 84, 3]~ [6, 4, 7]~ 166606 [399, -86, -3]~ [-4, -6, -7]~ 166606 [-399, 86, 3]~ [4, 6, 7]~ 167405 [400, -86, -3]~ [-1, -8, -6]~ 167405 [-400, 86, 3]~ [1, 8, 6]~ 182014 [417, -90, -5]~ [-4, 6, 7]~ 182014 [-417, 90, 5]~ [4, -6, -7]~ 206377 [444, -96, -5]~ [-7, 6, 4]~ 206377 [-444, 96, 5]~ [7, -6, -4]~ 228114 [467, -100, -5]~ [6, -4, 7]~ 228114 [-467, 100, 5]~ [-6, 4, -7]~ 270987 [509, -109, -5]~ [6, -7, 4]~ 270987 [-509, 109, 5]~ [-6, 7, -4]~ 356957 [584, -126, -5]~ [-7, -4, -6]~ 356957 [-584, 126, 5]~ [7, 4, 6]~ 375467 [599, -129, -5]~ [-4, -7, -6]~ 375467 [-599, 129, 5]~ [4, 7, 6]~ 393242 [613, -132, -7]~ [-2, 4, 9]~ 393242 [-613, 132, 7]~ [2, -4, -9]~ 432542 [643, -138, -7]~ [4, -2, 9]~ 432542 [-643, 138, 7]~ [-4, 2, -9]~ 478341 [676, -146, -7]~ [-9, 4, 2]~ 478341 [-676, 146, 7]~ [9, -4, -2]~ 574411 [741, -159, -7]~ [4, -9, 2]~ 574411 [-741, 159, 7]~ [-4, 9, -2]~ 604545 [760, -164, -7]~ [-9, -2, -4]~ 604545 [-760, 164, 7]~ [9, 2, 4]~ 661315 [795, -171, -7]~ [-2, -9, -4]~ 661315 [-795, 171, 7]~ [2, 9, 4]~ 708739 [823, -177, -9]~ [0, 1, 10]~ 708739 [-823, 177, 9]~ [0, -1, -10]~ 717349 [828, -178, -9]~ [1, 0, 10]~ 717349 [-828, 178, 9]~ [-1, 0, -10]~ 729706 [835, -180, -9]~ [-6, 4, 7]~ 729706 [-835, 180, 9]~ [6, -4, -7]~ 745541 [844, -182, -9]~ [-7, 4, 6]~ 745541 [-844, 182, 9]~ [7, -4, -6]~ 819406 [885, -190, -9]~ [4, -6, 7]~ 819406 [-885, 190, 9]~ [-4, 6, -7]~ 845531 [899, -193, -9]~ [4, -7, 6]~ 845531 [-899, 193, 9]~ [-4, 7, -6]~ 872459 [913, -197, -9]~ [-10, 1, 0]~ 872459 [-913, 197, 9]~ [10, -1, 0]~ 899410 [927, -200, -9]~ [-10, 0, -1]~ 899410 [-927, 200, 9]~ [10, 0, 1]~ 980369 [968, -208, -9]~ [1, -10, 0]~ 980369 [-968, 208, 9]~ [-1, 10, 0]~ 998710 [977, -210, -9]~ [0, -10, -1]~ 998710 [-977, 210, 9]~ [0, 10, 1]~ 1013281 [984, -212, -9]~ [-7, -6, -4]~ 1013281 [-984, 212, 9]~ [7, 6, 4]~ 1023571 [989, -213, -9]~ [-6, -7, -4]~ 1023571 [-989, 213, 9]~ [6, 7, 4]~ 1112366 [1031, -222, -11]~ [-4, 2, 9]~ 1112366 [-1031, 222, 11]~ [4, -2, -9]~ 1114429 [1032, -222, -11]~ [-1, 0, 10]~ 1114429 [-1032, 222, 11]~ [1, 0, -10]~ 1125219 [1037, -223, -11]~ [0, -1, 10]~ 1125219 [-1037, 223, 11]~ [0, 1, -10]~ 1177826 [1061, -228, -11]~ [2, -4, 9]~ 1177826 [-1061, 228, 11]~ [-2, 4, -9]~ 1211721 [1076, -232, -11]~ [-9, 2, 4]~ 1211721 [-1076, 232, 11]~ [9, -2, -4]~ 1296490 [1113, -240, -11]~ [-10, 0, 1]~ 1296490 [-1113, 240, 11]~ [10, 0, -1]~ 1329299 [1127, -243, -11]~ [-10, -1, 0]~ 1329299 [-1127, 243, 11]~ [10, 1, 0]~ 1338331 [1131, -243, -11]~ [2, -9, 4]~ 1338331 [-1131, 243, 11]~ [-2, 9, -4]~ 1408221 [1160, -250, -11]~ [-9, -4, -2]~ 1408221 [-1160, 250, 11]~ [9, 4, 2]~ 1415190 [1163, -250, -11]~ [0, -10, 1]~ 1415190 [-1163, 250, 11]~ [0, 10, -1]~ 1437209 [1172, -252, -11]~ [-1, -10, 0]~ 1437209 [-1172, 252, 11]~ [1, 10, 0]~ 1469371 [1185, -255, -11]~ [-4, -9, -2]~ 1469371 [-1185, 255, 11]~ [4, 9, 2]~ 1632531 [1249, -269, -13]~ [-6, 1, 8]~ 1632531 [-1249, 269, 13]~ [6, -1, -8]~ 1679987 [1267, -273, -13]~ [-8, 1, 6]~ 1679987 [-1267, 273, 13]~ [8, -1, -6]~ 1725001 [1284, -276, -13]~ [1, -6, 8]~ 1725001 [-1284, 276, 13]~ [-1, 6, -8]~ 1801037 [1312, -282, -13]~ [1, -8, 6]~ 1801037 [-1312, 282, 13]~ [-1, 8, -6]~ 1949830 [1365, -294, -13]~ [-8, -6, -1]~ 1949830 [-1365, 294, 13]~ [8, 6, 1]~ 1978410 [1375, -296, -13]~ [-6, -8, -1]~ 1978410 [-1375, 296, 13]~ [6, 8, 1]~ 2227502 [1459, -314, -15]~ [-4, -2, 9]~ 2227502 [-1459, 314, 15]~ [4, 2, -9]~ 2239819 [1463, -315, -15]~ [-6, -1, 8]~ 2239819 [-1463, 315, 15]~ [6, 1, -8]~ 2258042 [1469, -316, -15]~ [-2, -4, 9]~ 2258042 [-1469, 316, 15]~ [2, 4, -9]~ 2295347 [1481, -319, -15]~ [-8, -1, 6]~ 2295347 [-1481, 319, 15]~ [8, 1, -6]~ 2316769 [1488, -320, -15]~ [-1, -6, 8]~ 2316769 [-1488, 320, 15]~ [1, 6, -8]~ 2367217 [1504, -324, -15]~ [-9, -2, 4]~ 2367217 [-1504, 324, 15]~ [9, 2, -4]~ 2404757 [1516, -326, -15]~ [-1, -8, 6]~ 2404757 [-1516, 326, 15]~ [1, 8, -6]~ 2456149 [1532, -330, -15]~ [-9, -4, 2]~ 2456149 [-1532, 330, 15]~ [9, 4, -2]~ 2478307 [1539, -331, -15]~ [-2, -9, 4]~ 2478307 [-1539, 331, 15]~ [2, 9, -4]~ 2517382 [1551, -334, -15]~ [-8, -6, 1]~ 2517382 [-1551, 334, 15]~ [8, 6, -1]~ 2536699 [1557, -335, -15]~ [-4, -9, 2]~ 2536699 [-1557, 335, 15]~ [4, 9, -2]~ 2549842 [1561, -336, -15]~ [-6, -8, 1]~ 2549842 [-1561, 336, 15]~ [6, 8, -1]~ 2992266 [1691, -364, -17]~ [-6, -4, 7]~ 2992266 [-1691, 364, 17]~ [6, 4, -7]~ 3024245 [1700, -366, -17]~ [-7, -4, 6]~ 3024245 [-1700, 366, 17]~ [7, 4, -6]~ 3027646 [1701, -366, -17]~ [-4, -6, 7]~ 3027646 [-1701, 366, 17]~ [4, 6, -7]~ 3077675 [1715, -369, -17]~ [-4, -7, 6]~ 3077675 [-1715, 369, 17]~ [4, 7, -6]~ 3124657 [1728, -372, -17]~ [-7, -6, 4]~ 3124657 [-1728, 372, 17]~ [7, 6, -4]~ 3142707 [1733, -373, -17]~ [-6, -7, 4]~ 3142707 [-1733, 373, 17]~ [6, 7, -4]~ #S2=168 12*h(-4*n)=168 708739 [823, -177, -9]~ [0, 1, 10]~ $ Regards, Hermann.