%0 Journal Article %1 sidi2020vector %A Sidi, Avram %D 2020 %J Advances in Computational Mathematics %K 41a20-approximation-by-rational-functions 41a21-pade-approximation 62m10-time-series-auto-correlation-regression 65d15-algorithms-for-approximation-of-functions 65f20-overdetermined-systems-pseudoinverses 65f50-sparse-matrices 65h10-systems-of-nonlinear-algebraic-equations prony var %N 2 %P 30 %R 10.1007/s10444-020-09751-9 %T Vector versions of Prony's algorithm and vector-valued rational approximations %U https://link.springer.com/article/10.1007/s10444-020-09751-9#citeas %V 46 %X Given the scalar sequence \$\backslash\f\_\m\\backslash\^\\backslashinfty \\_\m=0\\$ that satisfiesfm=∑i=1kai$\zeta$im,m=0,1,łdots,\$\$ f\_\m\ = \backslashsum\backslashlimits\_ı=1\^\k\ \a\_ı\\\\backslashzeta\\_ı\^\m\,\backslashquad m=0,1,\backslashldots, \$\$ where \$a\_ı\, \backslashzeta \_ı\\backslashin \backslashmathbb \C\\$ and $\zeta$i are distinct, the algorithm of Prony concerns the determination of the ai and the $\zeta$i from a finite number of the fm. This algorithm is also related to Padé approximants from the infinite power series \$\\backslashsum \^\\backslashinfty \\_\j=0\f\_\j\z^\j\\$. In this work, we discuss ways of extending Prony's algorithm to sequences of vectors \$\\backslash\\backslashboldsymbol \f\\_\m\\backslash\\^\\backslashinfty \\_\m=0\\$ in \$\backslashmathbb \C\^\N\\$ that satisfyfm=∑i=1kai$\zeta$im,m=0,1,łdots,\$\$ \backslashboldsymbol\f\\_\m\ = \backslashsum\backslashlimits\_ı=1\^\k\ \backslashboldsymbol\a\\_ı\ \\backslashzeta\\_ı\^\m\, \backslashquad m=0,1,\backslashldots, \$\$ where \$\backslashboldsymbol \a\\_ı\\backslashin \backslashmathbb \C\^\N\\$ and \$\backslashzeta \_ı\\backslashin \backslashmathbb \C\\$. Two distinct problems arise depending on whether the vectors ai are linearly independent or not. We consider different approaches that enable us to determine the ai and $\zeta$i for these two problems, and develop suitable methods. We concentrate especially on extensions that take into account the possibility of the components of the ai being coupled. One of the applications we consider concerns the case in whichfm=∑i=1rai$\zeta$im,m=0,1,łdots,rlarge,\$\$ \backslashboldsymbol\f\\_\m\ = \backslashsum\backslashlimits\_ı=1\^\r\ \backslashboldsymbol\a\\_ı\ \\backslashzeta\\_ı\^\m\, \backslashquad m=0,1,\backslashldots,\backslashquad r \backslashtext\ large\, \$\$ and we would like to approximate/determine of a number of the pairs ($\zeta$i, ai) for which |$\zeta$i| are largest. We present the related theory and provide numerical examples that confirm this theory. This application can be extended to the more general case in whichfm=∑i=1rpi(m)$\zeta$im,m=0,1,łdots,\$\$ \backslashboldsymbol\f\\_\m\ = \backslashsum\backslashlimits\_ı=1\^\r\ \backslashboldsymbol\p\\_ı\ (m)\\backslashzeta\\_ı\^\m\, \backslashquad m=0,1,\backslashldots, \$\$ where \$\backslashboldsymbol \p\\_ı\(m)\backslashin \backslashmathbb \C\^\N\\$ are some (vector-valued) polynomials in m, and \$\backslashzeta \_ı\\backslashin \backslashmathbb \C\\$ are distinct. Finally, the methods suggested here can be extended to vector sequences in infinite dimensional spaces in a straightforward manner.

@article{sidi2020vector, abstract = {Given the scalar sequence {\$}{\backslash}{\{}f{\_}{\{}m{\}}{\backslash}{\}}^{\{}{\backslash}infty {\}}{\_}{\{}m=0{\}}{\$} that satisfiesfm=∑i=1kai$\zeta$im,m=0,1,{\ldots},{\$}{\$} f{\_}{\{}m{\}} = {\backslash}sum{\backslash}limits{\_}{\{}i=1{\}}^{\{}k{\}} {\{}a{\_}{\{}i{\}}{\}}{\{}{\backslash}zeta{\}}{\_}{\{}i{\}}^{\{}m{\}},{\backslash}quad m=0,1,{\backslash}ldots, {\$}{\$} where {\$}a{\_}{\{}i{\}}, {\backslash}zeta {\_}{\{}i{\}}{\backslash}in {\backslash}mathbb {\{}C{\}}{\$} and $\zeta$i are distinct, the algorithm of Prony concerns the determination of the ai and the $\zeta$i from a finite number of the fm. This algorithm is also related to Pad{\'e} approximants from the infinite power series {\$}{\{}{\backslash}sum {\}}^{\{}{\backslash}infty {\}}{\_}{\{}j=0{\}}f{\_}{\{}j{\}}z^{\{}j{\}}{\$}. In this work, we discuss ways of extending Prony's algorithm to sequences of vectors {\$}{\{}{\backslash}{\{}{\backslash}boldsymbol {\{}f{\}}{\_}{\{}m{\}}{\backslash}{\}}{\}}^{\{}{\backslash}infty {\}}{\_}{\{}m=0{\}}{\$} in {\$}{\backslash}mathbb {\{}C{\}}^{\{}N{\}}{\$} that satisfyfm=∑i=1kai$\zeta$im,m=0,1,{\ldots},{\$}{\$} {\backslash}boldsymbol{\{}f{\}}{\_}{\{}m{\}} = {\backslash}sum{\backslash}limits{\_}{\{}i=1{\}}^{\{}k{\}} {\backslash}boldsymbol{\{}a{\}}{\_}{\{}i{\}} {\{}{\backslash}zeta{\}}{\_}{\{}i{\}}^{\{}m{\}}, {\backslash}quad m=0,1,{\backslash}ldots, {\$}{\$} where {\$}{\backslash}boldsymbol {\{}a{\}}{\_}{\{}i{\}}{\backslash}in {\backslash}mathbb {\{}C{\}}^{\{}N{\}}{\$} and {\$}{\backslash}zeta {\_}{\{}i{\}}{\backslash}in {\backslash}mathbb {\{}C{\}}{\$}. Two distinct problems arise depending on whether the vectors ai are linearly independent or not. We consider different approaches that enable us to determine the ai and $\zeta$i for these two problems, and develop suitable methods. We concentrate especially on extensions that take into account the possibility of the components of the ai being coupled. One of the applications we consider concerns the case in whichfm=∑i=1rai$\zeta$im,m=0,1,{\ldots},rlarge,{\$}{\$} {\backslash}boldsymbol{\{}f{\}}{\_}{\{}m{\}} = {\backslash}sum{\backslash}limits{\_}{\{}i=1{\}}^{\{}r{\}} {\backslash}boldsymbol{\{}a{\}}{\_}{\{}i{\}} {\{}{\backslash}zeta{\}}{\_}{\{}i{\}}^{\{}m{\}}, {\backslash}quad m=0,1,{\backslash}ldots,{\backslash}quad r {\backslash}text{\{} large{\}}, {\$}{\$} and we would like to approximate/determine of a number of the pairs ($\zeta$i, ai) for which |$\zeta$i| are largest. We present the related theory and provide numerical examples that confirm this theory. This application can be extended to the more general case in whichfm=∑i=1rpi(m)$\zeta$im,m=0,1,{\ldots},{\$}{\$} {\backslash}boldsymbol{\{}f{\}}{\_}{\{}m{\}} = {\backslash}sum{\backslash}limits{\_}{\{}i=1{\}}^{\{}r{\}} {\backslash}boldsymbol{\{}p{\}}{\_}{\{}i{\}} (m){\{}{\backslash}zeta{\}}{\_}{\{}i{\}}^{\{}m{\}}, {\backslash}quad m=0,1,{\backslash}ldots, {\$}{\$} where {\$}{\backslash}boldsymbol {\{}p{\}}{\_}{\{}i{\}}(m){\backslash}in {\backslash}mathbb {\{}C{\}}^{\{}N{\}}{\$} are some (vector-valued) polynomials in m, and {\$}{\backslash}zeta {\_}{\{}i{\}}{\backslash}in {\backslash}mathbb {\{}C{\}}{\$} are distinct. Finally, the methods suggested here can be extended to vector sequences in infinite dimensional spaces in a straightforward manner.}, added-at = {2021-07-13T05:45:10.000+0200}, author = {Sidi, Avram}, biburl = {https://www.bibsonomy.org/bibtex/2b9a4046a1c8431180ee887fbe8ba7b11/gdmcbain}, day = 19, doi = {10.1007/s10444-020-09751-9}, interhash = {ffc0e63bfa9ffd6a27a8cbfebfc739f7}, intrahash = {b9a4046a1c8431180ee887fbe8ba7b11}, issn = {1572-9044}, journal = {Advances in Computational Mathematics}, keywords = {41a20-approximation-by-rational-functions 41a21-pade-approximation 62m10-time-series-auto-correlation-regression 65d15-algorithms-for-approximation-of-functions 65f20-overdetermined-systems-pseudoinverses 65f50-sparse-matrices 65h10-systems-of-nonlinear-algebraic-equations prony var}, month = mar, number = 2, pages = 30, timestamp = {2021-07-13T07:19:39.000+0200}, title = {Vector versions of Prony's algorithm and vector-valued rational approximations}, url = {https://link.springer.com/article/10.1007/s10444-020-09751-9#citeas}, volume = 46, year = 2020 }