%0 Report %1 noauthororeditor %A Sidi, Avram %C Haifa 32000, Israel %D 2018 %K 41a20-approximation-by-rational-functions 62m10-time-series-auto-correlation-regression 65f20-overdetermined-systems-pseudoinverses 65f50-sparse-matrices 65h10-systems-of-nonlinear-algebraic-equations prony %N CS-2018-04 %T Vector versions of Prony's algorithm and vector-valued rational approximations %U https://www.cs.technion.ac.il/users/wwwb/cgi-bin/tr-get.cgi/2018/CS/CS-2018-04.pdf %X Given the scalar sequencefm1m=0that satisfiesfm=k∑i=1aimi; m= 0;1; : : : ;whereai; i∈Candiare distinct, the algorithm of Prony concerns the determi-nation of theaiand theifrom a finite number of thefm. This algorithm is alsorelated to Pad ́e approximants from the infinite power series∑1j=0fjzj.In this work, we discuss ways of extending Prony’s algorithm to sequences ofvectorsfm1m=0inCNthat satisfyfm=k∑i=1aimi; m= 0;1; : : : ;whereai∈CNandi∈C. Two distinct problems arise depending on whetherthe vectorsaiare linearly independent or not. We consider different approachesthat enable us to determine theaiandifor these two problems, and developsuitable methods. We concentrate especially on extensions that take into accountthe possibility of the components of theaibeing coupled. One of the applicationsconcern the determination of a number of the pairs (i;ai) for which|i|are largest.These applications can be applied to the more general case in whichfm=k∑i=1pi(m)mi; m= 0;1; : : : ;wherepi(m)∈CNare some (vector-valued) polynomials inm, andi∈Caredistinct. Finally, the methods suggested here can be extended to vector sequencesin infinite dimensional spaces in a straightforward manner.

@techreport{noauthororeditor, abstract = {Given the scalar sequence{fm}1m=0that satisfiesfm=k∑i=1aimi; m= 0;1; : : : ;whereai; i∈Candiare distinct, the algorithm of Prony concerns the determi-nation of theaiand theifrom a finite number of thefm. This algorithm is alsorelated to Pad ́e approximants from the infinite power series∑1j=0fjzj.In this work, we discuss ways of extending Prony’s algorithm to sequences ofvectors{fm}1m=0inCNthat satisfyfm=k∑i=1aimi; m= 0;1; : : : ;whereai∈CNandi∈C. Two distinct problems arise depending on whetherthe vectorsaiare linearly independent or not. We consider different approachesthat enable us to determine theaiandifor these two problems, and developsuitable methods. We concentrate especially on extensions that take into accountthe possibility of the components of theaibeing coupled. One of the applicationsconcern the determination of a number of the pairs (i;ai) for which|i|are largest.These applications can be applied to the more general case in whichfm=k∑i=1pi(m)mi; m= 0;1; : : : ;wherepi(m)∈CNare some (vector-valued) polynomials inm, andi∈Caredistinct. Finally, the methods suggested here can be extended to vector sequencesin infinite dimensional spaces in a straightforward manner.}, added-at = {2021-07-13T07:27:13.000+0200}, address = {Haifa 32000, Israel}, author = {Sidi, Avram}, biburl = {https://www.bibsonomy.org/bibtex/25d8de1bffc88affe252e2905096494f8/gdmcbain}, institution = {Computer Science Department, Technion - Israel Institute of Technology}, interhash = {357f1219784c287327b0c528903a3233}, intrahash = {5d8de1bffc88affe252e2905096494f8}, keywords = {41a20-approximation-by-rational-functions 62m10-time-series-auto-correlation-regression 65f20-overdetermined-systems-pseudoinverses 65f50-sparse-matrices 65h10-systems-of-nonlinear-algebraic-equations prony}, number = {CS-2018-04}, timestamp = {2021-07-13T07:27:13.000+0200}, title = {Vector versions of Prony's algorithm and vector-valued rational approximations}, url = {https://www.cs.technion.ac.il/users/wwwb/cgi-bin/tr-get.cgi/2018/CS/CS-2018-04.pdf}, year = 2018 }