Properties of basic Lagrange polynomials
Author
Alexey Ukhalov
Title
Properties of basic Lagrange polynomials
Description
Wolfram Language code for demonstration of properties of basic Lagrange polynomials
Category
Academic Articles & Supplements
Keywords
simplex, basic Lagrange polynomial
URL
http://www.notebookarchive.org/2020-02-b23tbbv/
DOI
https://notebookarchive.org/2020-02-b23tbbv
Date Added
2020-02-24
Date Last Modified
2020-02-24
File Size
22.95 kilobytes
Supplements
Rights
CC BY 4.0
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Properties of basic Lagrange polynomials
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.
x[1]={0,0};x[2]={3/2,1/2};x[3]={2/3,1};A={{x[1][[1]],x[1][[2]],1},{x[2][[1]],x[2][[2]],1},{x[3][[1]],x[3][[2]],1}};L=Inverse[A];(*L//MatrixForm*)p1[x1_,x2_]:=L[[1]][[1]]*x1+L[[2]][[1]]*x2+L[[3]][[1]];gr=Graphics[{Black,FaceForm[None],EdgeForm[Directive[Thickness[0.01]]],Simplex[{x[1],x[2],x[3]}],Thickness[0.01],PointSize[0.03],Point[{0,0}],Text[Style["",Large],{-0.1,-0.10}],Text[Style["\!\(\*SubscriptBox[\(λ\), \(1\)]\)(x)=1",Large],{1.0,-0.35}],Text[Style["\!\(\*SubscriptBox[\(λ\), \(1\)]\)(x)=0",Large],{2.2,0.3}]},AxesFalse,AxesStyleDirective[Thickness[0.01],20],AxesLabel{X1,X2},TicksNone];pl0=ContourPlot[p1[x1,x2]0,{x1,0.4,2.0},{x2,-0.5,2.0},ContourStyleBlack];pl1=ContourPlot[p1[x1,x2]1,{x1,-0.2,3.0},{x2,-0.5,2.0},ContourStyleBlack];Show[gr,pl0,pl1]
(1)
x
x[1]={0,0};x[2]={3/2,1/2};x[3]={2/3,1};A={{x[1][[1]],x[1][[2]],1},{x[2][[1]],x[2][[2]],1},{x[3][[1]],x[3][[2]],1}};L=Inverse[A];p1[x1_,x2_]:=L[[1]][[1]]*x1+L[[2]][[1]]*x2+L[[3]][[1]];Manipulate[gr=Graphics[{FaceForm[None],EdgeForm[Directive[Thickness[0.01]]],Simplex[{x[1],x[2],x[3]}],Thickness[0.01],PointSize[0.03],Point[{0,0}],Text[Style["\!\(\*SubscriptBox[\(v\), \(1\)]\)",Large],{-0.1,-0.10}],Text[Style["\!\(\*SubscriptBox[\(λ\), \(1\)]\)(x)=t",Large],{2.2,0.3}]},AxesFalse,AxesStyleDirective[Thickness[0.01],20],AxesLabel{X1,X2},TicksNone,PlotRange{{-1,3},{-1,2}}];pl0=ContourPlot[p1[x1,x2]t,{x1,-1,3.0},{x2,-2.0,2.0},ContourStyleRed,PlotRange{{-1,3},{-1,2}}];Show[gr,pl0],{t,0,1}]
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Cite this as: Alexey Ukhalov, "Properties of basic Lagrange polynomials" from the Notebook Archive (2020), https://notebookarchive.org/2020-02-b23tbbv
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