Minimal Absorption Index for an n-Dimensional Simplex
Author
Alexey Ukhalov
Title
Minimal Absorption Index for an n-Dimensional Simplex
Description
Wolfram Language code for automated computation of a minimal absorption index for an n-dimensional simplex
Category
Academic Articles & Supplements
Keywords
simplex, absorption index, convex geometry
URL
http://www.notebookarchive.org/2020-02-b23ogar/
DOI
https://notebookarchive.org/2020-02-b23ogar
Date Added
2020-02-24
Date Last Modified
2020-02-24
File Size
34.91 kilobytes
Supplements
Rights
Redistribution rights reserved
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Minimal Absorption Index for an n-Dimensional Simplex
Definitions, basic facts and formulas can be found in the following papers
In English:
Nevskii, M.V. & Ukhalov, A.Y. On Minimal Absorption Index for an n-Dimensional Simplex.
Automatic Control and Computer Sciences. 2018. Volume 52, Issue 7, pp 680–687.
https://doi.org/10.3103/S0146411618070209
Nevskii M., and Ukhalov A. Perfect simplices in R^5.
Beitrage zur Algebra und Geometrie / Contributions to Algebra and Geometry. 2018. Vol 59, Issue 3, pp. 501-521.
In Russian:
Nevskii M.V., Ukhalov A.Y. On Minimal Absorption Index for an n-Dimensional Simplex.
Modeling and Analysis of Information Systems. 2018. Volume 25, No 1, pp. 140-150. (In Russ.)
https://doi.org/10.18255/1818-1015-2018-1-140-150
See also book (in Russian)
Nevskii, M.\,V. Geometric estimates in polynomial interpolation. P. G. Demidov Yaroslavl State University, Yaroslavl, 2012.
In English:
Nevskii, M.V. & Ukhalov, A.Y. On Minimal Absorption Index for an n-Dimensional Simplex.
Automatic Control and Computer Sciences. 2018. Volume 52, Issue 7, pp 680–687.
https://doi.org/10.3103/S0146411618070209
Nevskii M., and Ukhalov A. Perfect simplices in R^5.
Beitrage zur Algebra und Geometrie / Contributions to Algebra and Geometry. 2018. Vol 59, Issue 3, pp. 501-521.
In Russian:
Nevskii M.V., Ukhalov A.Y. On Minimal Absorption Index for an n-Dimensional Simplex.
Modeling and Analysis of Information Systems. 2018. Volume 25, No 1, pp. 140-150. (In Russ.)
https://doi.org/10.18255/1818-1015-2018-1-140-150
See also book (in Russian)
Nevskii, M.\,V. Geometric estimates in polynomial interpolation. P. G. Demidov Yaroslavl State University, Yaroslavl, 2012.
Print[TimeObject[Now],"Start"];ClearAll[n,A,S,L,i,k,x,y,z,T,te,tb,i,j,bound];n=4;S=Array[x,{n+1,n}];A=Map[Append[#,1]&,S];(*Print[A//MatrixForm];*)L=Inverse[A];(*Print[L//MatrixForm];*)L=Transpose[L];T={};For[i=0,i≤2^n-1,i++,(*Print["i=",i];*)y=IntegerDigits[i,2];k=Length[y];(*Print["k= ",k];*)If[k<n,z=ConstantArray[0,n-k];y=Flatten[Prepend[y,z]];(*Print["vertex= ",y];*)];AppendTo[y,1];(*Print["y=",y];*)te=-L.y;AppendTo[T,te];];T=Flatten[T];(*Print[T]*)ksi[S]:=1+(n+1)*Max[T];bound=True;For[i=1,i≤n+1,i++,For[j=1,j≤n,j++,tb=0≤A[[i]][[j]]≤1;bound=And[bound,tb]]];bound=And[bound];(*Print[bound];*)Vars=Flatten[S];Print[Vars];Print[TimeObject[Now],"MINIMIZATION IS RUNNING"];tmin=100;For[i=0,i≤100,i++,Print[TimeObject[Now]," i= ",i];newmin=NMinimize[{ksi[S],bound},Vars,Method"SimulatedAnnealing","RandomSeed"i];If[newmin[[1]]<tmin,Print[TimeObject[Now]," Attempt Number: ",i," New Min was found!!!! Min= "];Print[newmin];tmin=newmin[[1]];NotebookSave[]];If[Mod[i,10]0,Print[TimeObject[Now],"Attempt N ",i," - no news... "];Print["Current results: ",newmin];NotebookSave[]];]
15:03:13
{x[1,1],x[1,2],x[1,3],x[1,4],x[2,1],x[2,2],x[2,3],x[2,4],x[3,1],x[3,2],x[3,3],x[3,4],x[4,1],x[4,2],x[4,3],x[4,4],x[5,1],x[5,2],x[5,3],x[5,4]}
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15:03:51
{4.12296,{x[1,1]0.227316,x[1,2]0.,x[1,3]2.91924×,x[1,4]0.,x[2,1]1.,x[2,2]0.765197,x[2,3]1.78669×,x[2,4]1.,x[3,1]0.,x[3,2]1.,x[3,3]0.478579,x[3,4]0.496521,x[4,1]0.22793,x[4,2]0.000605567,x[4,3]1.,x[4,4]0.99997,x[5,1]0.998519,x[5,2]0.763728,x[5,3]0.996175,x[5,4]0.0120005}}
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Current results: {4.12296,{x[1,1]0.227316,x[1,2]0.,x[1,3]2.91924×,x[1,4]0.,x[2,1]1.,x[2,2]0.765197,x[2,3]1.78669×,x[2,4]1.,x[3,1]0.,x[3,2]1.,x[3,3]0.478579,x[3,4]0.496521,x[4,1]0.22793,x[4,2]0.000605567,x[4,3]1.,x[4,4]0.99997,x[5,1]0.998519,x[5,2]0.763728,x[5,3]0.996175,x[5,4]0.0120005}}
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15:09:29
{4.11426,{x[1,1]0.767565,x[1,2]0.48352,x[1,3]1.,x[1,4]1.,x[2,1]6.52136×,x[2,2]1.,x[2,3]0.232361,x[2,4]0.515562,x[3,1]1.,x[3,2]0.592361,x[3,3]0.,x[3,4]0.41253,x[4,1]0.,x[4,2]0.,x[4,3]0.232361,x[4,4]0.515607,x[5,1]0.767565,x[5,2]0.48352,x[5,3]1.,x[5,4]0.}}
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Cite this as: Alexey Ukhalov, "Minimal Absorption Index for an n-Dimensional Simplex" from the Notebook Archive (2020), https://notebookarchive.org/2020-02-b23ogar
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