Re: How to do quickest

• To: mathgroup at smc.vnet.net
• Subject: [mg116566] Re: How to do quickest
• From: DrMajorBob <btreat1 at austin.rr.com>
• Date: Sun, 20 Feb 2011 05:24:53 -0500 (EST)

```Here's a more readable code (IMHO). I like variable names that mean
something.

Timing[
poly = x^8 - x - 1;
order = Exponent[poly, x];
partitions = IntegerPartitions@order;
np = Length@partitions;
index[_] = 0;
Do[index[partitions[[j]]] = j, {j, np}];
aa = ConstantArray[0, np];
n = total = 0;
While[total < order!,
p = Prime[++n];
factors = FactorList[poly, Modulus -> p][[All, 1]];
ndx = index@Reverse@Sort@Rest@Exponent[factors, x];
Positive@ndx && (total++; aa[[ndx]]++)
];
aa]

{9.84093, {4996, 5781, 3361, 3449, 2653, 4055, 1360, 1249, 3360, 1321,
2470, 412, 1103, 1114, 1652, 1129, 105, 102, 416, 206, 25, 1}}

The following yields the tidbit of information that ndx == 0 EXACTLY ONCE:

Timing[
poly = x^8 - x - 1;
order = Exponent[poly, x];
partitions = IntegerPartitions@order;
np = Length@partitions;
Clear[counts, index];
counts[_] = 0;
index[_] = 0;
Do[index@partitions[[j]] = j, {j, np}];
n = total = 0;
While[total < order!,
p = Prime[++n];
factors = FactorList[poly, Modulus -> p][[All, 1]];
ndx = index@Reverse@Sort@Rest@Exponent[factors, x];
counts[ndx]++;
Positive@ndx && total++
];
Table[counts@i, {i, 0, np}]]

{10.2563, {1, 4996, 5781, 3361, 3449, 2653, 4055, 1360, 1249, 3360,
1321, 2470, 412, 1103, 1114, 1652, 1129, 105, 102, 416, 206, 25, 1}}

If we can COUNT on that being so for some reason, the code becomes

Timing[
poly = x^8 - x - 1;
order = Exponent[poly, x];
partitions = IntegerPartitions@order;
np = Length@partitions;
Clear[counts, index];
counts[_] = 0;
index[_] = 0;
Do[index@partitions[[j]] = j, {j, np}];
n = 0;
While[n < order!,
p = Prime[++n];
factors = FactorList[poly, Modulus -> p][[All, 1]];
ndx = index@Reverse@Sort@Rest@Exponent[factors, x];
counts[ndx]++;
ndx == 0 &&
Print@{n, p, factors, Reverse@Sort@Rest@Exponent[factors, x]};
];
Array[counts, np]]

{5,11,{1,9+x,8+5 x+3 x^2+10 x^3+x^4+4 x^5+x^6},{6,1}}

{9.95797, {4996, 5781, 3361, 3449, 2653, 4055, 1360, 1249, 3360, 1321,
2470, 412, 1103, 1114, 1651, 1129, 105, 102, 416, 206, 25, 1}}

where I have shown the exception.

The problem is a black box to me, but finding just ONE exception is not
what I would have expected.

Bobby

On Sat, 19 Feb 2011 12:02:22 -0600, Artur <grafix at csl.pl> wrote:

> Thank You for procedure! On my computer
> {12.312, {4996, 5781, 3361, 3449, 2653, 4055, 1360, 1249, 3360, 1321,
>    2470, 412, 1103, 1114, 1652, 1129, 105, 102, 416, 206, 25,1}}
>
> Bob try Your quickest steps combined with following which is still
> little quickest
>
> {10.313, {4996, 5781, 3361, 3449, 2653, 4055, 1360, 1249, 3360, 1321,
>    2470, 412, 1103, 1114, 1652, 1129, 105, 102, 416, 206, 25, 1}}
>
> (*Daniel Lichtblau modified by Artur Jasinski*)
>
> Timing[cc = {}; pol = x^8 - x - 1;
>   nn = Length[CoefficientList[pol, x]] - 1;
>   pp = IntegerPartitions[nn];
>   Do[htab[pp[[j]]] = j, {j, Length[pp]}];
>   aa = Table[0, {Length[pp]}];
>   n = 1; cn = 0;
>   While[cn < nn!, p = Prime[n];
>    n++;
>    kk = FactorList[pol, Modulus -> p];
>    ww = Rest[Exponent[kk[[All, 1]], x]];
>    ww = Reverse[Sort[ww]];
>    pos = htab[ww];
>    If[pos == 0, , cn++; aa[[pos]] = aa[[pos]] + 1]];
>   aa]
>
> {10.313, {4996, 5781, 3361, 3449, 2653, 4055, 1360, 1249, 3360, 1321,
>    2470, 412, 1103, 1114, 1652, 1129, 105, 102, 416, 206, 25, 1}}
>
> W dniu 2011-02-19 17:37, DrMajorBob pisze:
>> This is easier to read, if no faster.
>>
>> Timing[
>>  Clear[a, c];
>>  pol = x^8 - x - 1;
>>  nn = Length@CoefficientList[pol, x] - 1;
>>  If[
>>   IrreduciblePolynomialQ[pol],
>>   a[i_] = {};
>>   c[i_] := Length@Flatten[a@i];
>>   pp = IntegerPartitions@nn;
>>   b = FactorInteger[Discriminant[pol, x]][[All, 1]];
>>   n = 1;
>>   cn = 0;
>>   While[cn < nn!, p = Prime@n;
>>    If[! MemberQ[b, p],
>>     cn++;
>>     k = Reverse@Rest@FactorList[pol, Modulus -> p][[All, 1]];
>>     w = Length@CoefficientList[#, x] - 1 & /@ k;
>>     pos = Position[pp, w, 1, 1][[1, 1]];
>>     a[pos] = {a[pos], p}];
>>    n++]];
>>  Array[c, Length@pp]
>>  ]
>>
>> {10.8518, {4996, 5781, 3361, 3449, 2653, 4055, 1360, 1249, 3360, 1321,
>>    2470, 412, 1103, 1114, 1652, 1129, 105, 102, 416, 206, 25, 1}}
>>
>> Bobby
>>
>> On Sat, 19 Feb 2011 04:12:14 -0600, Sjoerd C. de Vries
>> <sjoerd.c.devries at gmail.com> wrote:
>>
>>> Without changing the basic operation of your algorithm I've changed a
>>> couple of details. The difference is not huge, but about 20% of speed
>>> gain is still nice.
>>>
>>> pol = x^8 - x - 1;
>>>  nn = Length[CoefficientList[pol, x]] - 1;
>>>  If[IrreduciblePolynomialQ[pol],
>>>   pp = IntegerPartitions[nn];
>>>   aa = Table[{}, {n, 1, Length[pp]}]; Print[aa];
>>>   ff = FactorInteger[Discriminant[pol, x]];
>>>   bb = Table[ff[[n, 1]], {n, 1, Length[ff]}];
>>>   n = 1;
>>>   cn = 0;
>>>   While[cn < nn!,
>>>    p = Prime[n];
>>>    If[MemberQ[bb, p],
>>>     (*True*),
>>>     cn++;
>>>     kk = FactorList[pol, Modulus -> p];
>>>     ww = Table[
>>>       Length[CoefficientList[kk[[m, 1]], x]] - 1,
>>>       {m, Length[kk], 2, -1}
>>>       ];
>>>     pos = Position[pp, ww, 1, 1][[1, 1]];
>>>     aa[[pos]] = {aa[[pos]], p};
>>>     ];
>>>    n++
>>>    ]
>>>   ]; aa = Map[Flatten, aa, {1}];
>>>  Table[Length[aa[[m]]], {m, 1, Length[aa]}]
>>>  ]
>>>
>>>
>>> Cheers -- Sjoerd
>>>
>>>
>>> On Feb 15, 12:33 pm, Artur <gra... at csl.pl> wrote:
>>>> Dear Mathematica Gurus,
>>>> How to do following procedure quickest?
>>>> (*start*)
>>>> pol = x^8 - x - 1; nn = Length[CoefficientList[pol, x]] - 1; If[
>>>>  IrreduciblePolynomialQ[pol], pp = IntegerPartitions[nn]; aa = {};
>>>>  Do[AppendTo[aa, {}], {n, 1, Length[pp]}]; Print[aa];
>>>>  ff = FactorInteger[Discriminant[pol, x]]; bb = {};
>>>>  Do[AppendTo[bb, ff[[n]][[1]]], {n, 1, Length[ff]}]; n = 1; cn = 0;
>>>>  While[cn < nn!, p = Prime[n];
>>>>   If[MemberQ[bb, p], , cn = cn + 1;
>>>>    kk = FactorList[pol, Modulus -> p]; ww = {};
>>>>    Do[cc = Length[CoefficientList[kk[[m]][[1]], x]];
>>>>     AppendTo[ww, cc - 1], {m, 2, Length[kk]}]; ww = Reverse[ww];
>>>>    pos = Position[pp, ww][[1]][[1]]; AppendTo[aa[[pos]], Prime[n]]]=
>>> ;
>>>>   n++]]; Table[Length[aa[[m]]], {m, 1, Length[aa]}]
>>>> (*end*)
>>>> Best wishes
>>>> Artur
>>>
>>>
>>
>>

--
DrMajorBob at yahoo.com

```

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