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Merouane Debbah  One of the best experts on this subject based on the ideXlab platform.

convolution operations arising from Vandermonde matrices
IEEE Transactions on Information Theory, 2011CoAuthors: Øyvind Ryan, Merouane DebbahAbstract:Different types of convolution operations involving large Vandermonde matrices are considered. The convolutions parallel those of large Gaussian matrices and additive and multiplicative free convolution, and include additive and multiplicative convolution of Vandermonde matrices and deterministic diagonal matrices, and cases where two independent Vandermonde matrices are involved. It is also shown that the convergence of any combination of Vandermonde matrices is almost sure. The convolutions are divided into two types: those which depend on the phase distribution of the Vandermonde matrices, and those which depend only on the spectra of the matrices. A general criterion is presented to find which type applies for any given convolution. A simulation is presented, verifying the results. Implementations of the presented convolutions are provided and discussed. The implementation is based on the technique of FourierMotzkin elimination, and is quite general as it can be applied to virtually any combination of Vandermonde matrices. Connections with related matrices, such as Toeplitz and Hankel matrices, are also discussed.

Convolution operations arising from Vandermonde matrices
IEEE Transactions on Information Theory, 2011CoAuthors: Øyvind Ryan, Merouane DebbahAbstract:Different types of convolution operations involving large Vandermonde matrices are considered. The convolutions parallel those of large Gaussian matrices and additive and multiplicative free convolution. First additive and multiplicative convolution of Vandermonde matrices and deterministic diagonal matrices are considered. After this, several cases of additive and multiplicative convolution of two independent Vandermonde matrices are considered. It is also shown that the convergence of any combination of Vandermonde matrices is almost sure. We will divide the considered convolutions into two types: those which depend on the phase distribution of the Vandermonde matrices, and those which depend only on the spectra of the matrices. A general criterion is presented to find which type applies for any given convolution. A simulation is presented, verifying the results. Implementations of all considered convolutions are provided and discussed, together with the challenges in making these implementations efficient. The implementation is based on the technique of FourierMotzkin elimination, and is quite general as it can be applied to virtually any combination of Vandermonde matrices. Generalizations to related random matrices, such as Toeplitz and Hankel matrices, are also discussed.

Asymptotic behavior of random Vandermonde matrices with entries on the unit circle
IEEE Transactions on Information Theory, 2009CoAuthors: Øyvind Ryan, Merouane DebbahAbstract:Analytical methods for finding moments of random Vandermonde matrices with entries on the unit circle are developed. Vandermonde Matrices play an important role in signal processing and wireless applications such as direction of arrival estimation, precoding, and sparse sampling theory, just to name a few. Within this framework, we extend classical freeness results on random matrices with independent, identically distributed (i.i.d.) entries and show that Vandermonde structured matrices can be treated in the same vein with different tools. We focus on various types of matrices, such as Vandermonde matrices with and without uniform phase distributions, as well as generalized Vandermonde matrices. In each case, we provide explicit expressions of the moments of the associated Gram matrix, as well as more advanced models involving the Vandermonde matrix. Comparisons with classical i.i.d. random matrix theory are provided, and deconvolution results are discussed. We review some applications of the results to the fields of signal processing and wireless communications.

random Vandermonde matrices part i fundamental results
arXiv: Information Theory, 2008CoAuthors: Øyvind Ryan, Merouane DebbahAbstract:In this first part, analytical methods for finding moments of random Vandermonde matrices are developed. Vandermonde Matrices play an important role in signal processing and communication applications such as direction of arrival estimation, precoding or sparse sampling theory for example. Within this framework, we extend classical freeness results on random matrices with i.i.d entries and show that Vandermonde structured matrices can be treated in the same vein with different tools. We focus on various types of Vandermonde matrices, namely Vandermonde matrices with or without uniformly distributed phases, as well as generalized Vandermonde matrices (with nonuniform distribution of powers). In each case, we provide explicit expressions of the moments of the associated Gram matrix, as well as more advanced models involving the Vandermonde matrix. Comparisons with classical i.i.d. random matrix theory are provided and free deconvolution results are also discussed.
Øyvind Ryan  One of the best experts on this subject based on the ideXlab platform.

convolution operations arising from Vandermonde matrices
IEEE Transactions on Information Theory, 2011CoAuthors: Øyvind Ryan, Merouane DebbahAbstract:Different types of convolution operations involving large Vandermonde matrices are considered. The convolutions parallel those of large Gaussian matrices and additive and multiplicative free convolution, and include additive and multiplicative convolution of Vandermonde matrices and deterministic diagonal matrices, and cases where two independent Vandermonde matrices are involved. It is also shown that the convergence of any combination of Vandermonde matrices is almost sure. The convolutions are divided into two types: those which depend on the phase distribution of the Vandermonde matrices, and those which depend only on the spectra of the matrices. A general criterion is presented to find which type applies for any given convolution. A simulation is presented, verifying the results. Implementations of the presented convolutions are provided and discussed. The implementation is based on the technique of FourierMotzkin elimination, and is quite general as it can be applied to virtually any combination of Vandermonde matrices. Connections with related matrices, such as Toeplitz and Hankel matrices, are also discussed.

Convolution operations arising from Vandermonde matrices
IEEE Transactions on Information Theory, 2011CoAuthors: Øyvind Ryan, Merouane DebbahAbstract:Different types of convolution operations involving large Vandermonde matrices are considered. The convolutions parallel those of large Gaussian matrices and additive and multiplicative free convolution. First additive and multiplicative convolution of Vandermonde matrices and deterministic diagonal matrices are considered. After this, several cases of additive and multiplicative convolution of two independent Vandermonde matrices are considered. It is also shown that the convergence of any combination of Vandermonde matrices is almost sure. We will divide the considered convolutions into two types: those which depend on the phase distribution of the Vandermonde matrices, and those which depend only on the spectra of the matrices. A general criterion is presented to find which type applies for any given convolution. A simulation is presented, verifying the results. Implementations of all considered convolutions are provided and discussed, together with the challenges in making these implementations efficient. The implementation is based on the technique of FourierMotzkin elimination, and is quite general as it can be applied to virtually any combination of Vandermonde matrices. Generalizations to related random matrices, such as Toeplitz and Hankel matrices, are also discussed.

Asymptotic behavior of random Vandermonde matrices with entries on the unit circle
IEEE Transactions on Information Theory, 2009CoAuthors: Øyvind Ryan, Merouane DebbahAbstract:Analytical methods for finding moments of random Vandermonde matrices with entries on the unit circle are developed. Vandermonde Matrices play an important role in signal processing and wireless applications such as direction of arrival estimation, precoding, and sparse sampling theory, just to name a few. Within this framework, we extend classical freeness results on random matrices with independent, identically distributed (i.i.d.) entries and show that Vandermonde structured matrices can be treated in the same vein with different tools. We focus on various types of matrices, such as Vandermonde matrices with and without uniform phase distributions, as well as generalized Vandermonde matrices. In each case, we provide explicit expressions of the moments of the associated Gram matrix, as well as more advanced models involving the Vandermonde matrix. Comparisons with classical i.i.d. random matrix theory are provided, and deconvolution results are discussed. We review some applications of the results to the fields of signal processing and wireless communications.

random Vandermonde matrices part i fundamental results
arXiv: Information Theory, 2008CoAuthors: Øyvind Ryan, Merouane DebbahAbstract:In this first part, analytical methods for finding moments of random Vandermonde matrices are developed. Vandermonde Matrices play an important role in signal processing and communication applications such as direction of arrival estimation, precoding or sparse sampling theory for example. Within this framework, we extend classical freeness results on random matrices with i.i.d entries and show that Vandermonde structured matrices can be treated in the same vein with different tools. We focus on various types of Vandermonde matrices, namely Vandermonde matrices with or without uniformly distributed phases, as well as generalized Vandermonde matrices (with nonuniform distribution of powers). In each case, we provide explicit expressions of the moments of the associated Gram matrix, as well as more advanced models involving the Vandermonde matrix. Comparisons with classical i.i.d. random matrix theory are provided and free deconvolution results are also discussed.
Vadim Olshevsky  One of the best experts on this subject based on the ideXlab platform.

fast inversion of polynomial Vandermonde matrices for polynomial systems related to order one quasiseparable matrices
2013CoAuthors: T Bella, Yuli Eidelman, Vadim Olshevsky, E E TyrtyshnikovAbstract:While Gaussian elimination is well known to require O(n 3) operations to invert an arbitrary matrix, Vandermonde matrices may be inverted using O(n 2) operations by a method of Traub [24]. While this original version of the Traub algorithm was noticed to be unstable, it was shown in [12] that with a minor modification, the Traub algorithm can typically yield a very high accuracy. This approach has been extended from classical Vandermonde matrices to polynomialVandermonde matrices involving real orthogonal polynomials [3], [10], and Szegő polynomials [19]. In this paper we present an algorithm for inversion of a class of polynomialVandermonde matrices with special structure related to order one quasiseparable matrices, generalizing monomials, real orthogonal polynomials, and Szegő polynomials. We derive a fast O(n 2) inversion algorithm applicable in this general setting, and explore its reduction in the previous special cases. Some very preliminary numerical experiments are presented, demonstrating that, as observed by our colleagues in previous work, good forward accuracy is possible in some circumstances, which is consistent with previous work of this type.

a traub like algorithm for hessenberg quasiseparable Vandermonde matrices of arbitrary order
Operator Theory: Advances and Applications, 2010CoAuthors: T Bella, Yuli Eidelman, Israel Gohberg, Vadim Olshevsky, Pavel Zhlobich, E E TyrtyshnikovAbstract:Although Gaussian elimination uses O(n 3) operations to invert an arbitrary matrix, matrices with a special Vandermonde structure can be inverted in only O(n 2) operations by the fast Traub algorithm. The original version of Traub algorithm was numerically unstable although only a minor modification of it yields a high accuracy in practice. The Traub algorithm has been extended from Vandermonde matrices involving monomials to polynomialVandermonde matrices involving real orthogonal polynomials, and the Szego polynomials.

a traub like algorithm for hessenbergquasiseparable Vandermonde matrices of arbitrary order
Operator Theory: Advances and Applications, 2010CoAuthors: T Bella, Yuli Eidelman, Israel Gohberg, Vadim Olshevsky, Pavel Zhlobich, E E TyrtyshnikovAbstract:Although Gaussian elimination uses O(n 3) operations to invert an arbitrary matrix, matrices with a special Vandermonde structure can be inverted in only O(n 2) operations by the fast Traub algorithm. The original version of Traub algorithm was numerically unstable although only a minor modification of it yields a high accuracy in practice. The Traub algorithm has been extended from Vandermonde matrices involving monomials to polynomialVandermonde matrices involving real orthogonal polynomials, and the Szego polynomials.

a bjorck pereyra type algorithm for szego Vandermonde matrices based on properties of unitary hessenberg matrices
Linear Algebra and its Applications, 2007CoAuthors: T Bella, Yuli Eidelman, Israel Gohberg, I Koltracht, Vadim OlshevskyAbstract:Abstract In this paper we carry over the BjorckPereyra algorithm for solving Vandermonde linear systems to what we suggest to call SzegoVandermonde systems V Φ ( x ), i.e., polynomialVandermonde systems where the corresponding polynomial system Φ is the Szego polynomials . The properties of the corresponding unitary Hessenberg matrix allow us to derive a fast O( n 2 ) computational procedure. We present numerical experiments that indicate that for illconditioned matrices the new algorithm yields better forward accuracy than Gaussian elimination.

displacement structure approach to polynomial Vandermonde and related matrices
Linear Algebra and its Applications, 1997CoAuthors: T Kailath, Vadim OlshevskyAbstract:Abstract We introduce a new class of what we call polynomial Vandermondelike matrices. This class generalizes the polynomial Vandermonde matrices studied earlier by various authors, who derived explicit inversion formulas and fast algorithms for inversion and for solving the associated linear systems. A displacementstructure approach allows us to carry over all these results to the wider class of polynomial Vandermondelike matrices.
T Bella  One of the best experts on this subject based on the ideXlab platform.

fast inversion of polynomial Vandermonde matrices for polynomial systems related to order one quasiseparable matrices
2013CoAuthors: T Bella, Yuli Eidelman, Vadim Olshevsky, E E TyrtyshnikovAbstract:While Gaussian elimination is well known to require O(n 3) operations to invert an arbitrary matrix, Vandermonde matrices may be inverted using O(n 2) operations by a method of Traub [24]. While this original version of the Traub algorithm was noticed to be unstable, it was shown in [12] that with a minor modification, the Traub algorithm can typically yield a very high accuracy. This approach has been extended from classical Vandermonde matrices to polynomialVandermonde matrices involving real orthogonal polynomials [3], [10], and Szegő polynomials [19]. In this paper we present an algorithm for inversion of a class of polynomialVandermonde matrices with special structure related to order one quasiseparable matrices, generalizing monomials, real orthogonal polynomials, and Szegő polynomials. We derive a fast O(n 2) inversion algorithm applicable in this general setting, and explore its reduction in the previous special cases. Some very preliminary numerical experiments are presented, demonstrating that, as observed by our colleagues in previous work, good forward accuracy is possible in some circumstances, which is consistent with previous work of this type.

a traub like algorithm for hessenbergquasiseparable Vandermonde matrices of arbitrary order
Operator Theory: Advances and Applications, 2010CoAuthors: T Bella, Yuli Eidelman, Israel Gohberg, Vadim Olshevsky, Pavel Zhlobich, E E TyrtyshnikovAbstract:Although Gaussian elimination uses O(n 3) operations to invert an arbitrary matrix, matrices with a special Vandermonde structure can be inverted in only O(n 2) operations by the fast Traub algorithm. The original version of Traub algorithm was numerically unstable although only a minor modification of it yields a high accuracy in practice. The Traub algorithm has been extended from Vandermonde matrices involving monomials to polynomialVandermonde matrices involving real orthogonal polynomials, and the Szego polynomials.

a traub like algorithm for hessenberg quasiseparable Vandermonde matrices of arbitrary order
Operator Theory: Advances and Applications, 2010CoAuthors: T Bella, Yuli Eidelman, Israel Gohberg, Vadim Olshevsky, Pavel Zhlobich, E E TyrtyshnikovAbstract:Although Gaussian elimination uses O(n 3) operations to invert an arbitrary matrix, matrices with a special Vandermonde structure can be inverted in only O(n 2) operations by the fast Traub algorithm. The original version of Traub algorithm was numerically unstable although only a minor modification of it yields a high accuracy in practice. The Traub algorithm has been extended from Vandermonde matrices involving monomials to polynomialVandermonde matrices involving real orthogonal polynomials, and the Szego polynomials.

a bjorck pereyra type algorithm for szego Vandermonde matrices based on properties of unitary hessenberg matrices
Linear Algebra and its Applications, 2007CoAuthors: T Bella, Yuli Eidelman, Israel Gohberg, I Koltracht, Vadim OlshevskyAbstract:Abstract In this paper we carry over the BjorckPereyra algorithm for solving Vandermonde linear systems to what we suggest to call SzegoVandermonde systems V Φ ( x ), i.e., polynomialVandermonde systems where the corresponding polynomial system Φ is the Szego polynomials . The properties of the corresponding unitary Hessenberg matrix allow us to derive a fast O( n 2 ) computational procedure. We present numerical experiments that indicate that for illconditioned matrices the new algorithm yields better forward accuracy than Gaussian elimination.
E E Tyrtyshnikov  One of the best experts on this subject based on the ideXlab platform.

fast inversion of polynomial Vandermonde matrices for polynomial systems related to order one quasiseparable matrices
2013CoAuthors: T Bella, Yuli Eidelman, Vadim Olshevsky, E E TyrtyshnikovAbstract:While Gaussian elimination is well known to require O(n 3) operations to invert an arbitrary matrix, Vandermonde matrices may be inverted using O(n 2) operations by a method of Traub [24]. While this original version of the Traub algorithm was noticed to be unstable, it was shown in [12] that with a minor modification, the Traub algorithm can typically yield a very high accuracy. This approach has been extended from classical Vandermonde matrices to polynomialVandermonde matrices involving real orthogonal polynomials [3], [10], and Szegő polynomials [19]. In this paper we present an algorithm for inversion of a class of polynomialVandermonde matrices with special structure related to order one quasiseparable matrices, generalizing monomials, real orthogonal polynomials, and Szegő polynomials. We derive a fast O(n 2) inversion algorithm applicable in this general setting, and explore its reduction in the previous special cases. Some very preliminary numerical experiments are presented, demonstrating that, as observed by our colleagues in previous work, good forward accuracy is possible in some circumstances, which is consistent with previous work of this type.

a traub like algorithm for hessenberg quasiseparable Vandermonde matrices of arbitrary order
Operator Theory: Advances and Applications, 2010CoAuthors: T Bella, Yuli Eidelman, Israel Gohberg, Vadim Olshevsky, Pavel Zhlobich, E E TyrtyshnikovAbstract:Although Gaussian elimination uses O(n 3) operations to invert an arbitrary matrix, matrices with a special Vandermonde structure can be inverted in only O(n 2) operations by the fast Traub algorithm. The original version of Traub algorithm was numerically unstable although only a minor modification of it yields a high accuracy in practice. The Traub algorithm has been extended from Vandermonde matrices involving monomials to polynomialVandermonde matrices involving real orthogonal polynomials, and the Szego polynomials.

a traub like algorithm for hessenbergquasiseparable Vandermonde matrices of arbitrary order
Operator Theory: Advances and Applications, 2010CoAuthors: T Bella, Yuli Eidelman, Israel Gohberg, Vadim Olshevsky, Pavel Zhlobich, E E TyrtyshnikovAbstract:Although Gaussian elimination uses O(n 3) operations to invert an arbitrary matrix, matrices with a special Vandermonde structure can be inverted in only O(n 2) operations by the fast Traub algorithm. The original version of Traub algorithm was numerically unstable although only a minor modification of it yields a high accuracy in practice. The Traub algorithm has been extended from Vandermonde matrices involving monomials to polynomialVandermonde matrices involving real orthogonal polynomials, and the Szego polynomials.