Vandermonde

Vandermonde matrices often arise when reconstructing PDFs from their moments, and efficient numerical methods exist for solving (B.51) (Press et al. 1992). Although Vandermonde matrices are notoriously ill-conditioned for large Ne, this will usually not be a problem in applications of the DQMOM-IEM model since it is most attractive (relative to Monte-Carlo methods) when Ne is small. 

In Ref. [76], an elegant proof was instigated through the theory of Vandermonde determinants. Here, we will offer an alternative route, i.e., by directly computing (n > 1)... 

HSVD T Y N N N AT3 Linear prediction and Vandermonde decomposition Good... 

This determinant is nonsingular. This is due to the inequality Vk =/=0 that holds true because the roots Uj are never equal to m, for/ / j. The quotient Vjt+i/Vjfc is computed via the Vandermonde determinant with the following ... 

It was not until 1786, however, that scientists learned what it was in steel that made it a more useful metal than iron. Three researchers, Gaspard Monge (1746—1818), C. A. Vandermonde, and Claude Louis Berthollet (1748—1822), solved the puzzle. They found that a small amount of carbon mixed with iron produced a strong alloy. That alloy was steel. Today, the vast amount of iron used in so many applications is used in the form of steel, not pure iron. 

For each value of a, the final step is to solve the linear system generated by Eq. (3.74), which assumes again the Vandermonde form ... 

In this equation, the matrix V represents the so-called Vandermond s matrix [17] with the determinant... 

To get the correct results the eigenvalues with the index k have to be omitted. As discussed in eq. (2.54), IVJ is the Vandermond determinant given by... 

Taking all brackets, of which the eigenvalues are part, equal to one, the different values of are obtained from the Vandermond determinant V. ... 

If equal eigenvalues result in a Vandermond determinant, which equals zero, eq. (2.53) can no longer be solved with respect to p,. With... 

By using the support abscissas of the element, it is possible to write the following matrix (Vandermonde matrix) ... 

B. Toeplitz, Hankel, and Vandermonde Systems and Their Correlation to Polynomial Operations... 

Vandermonde systems have the coefficient matrix A of the form A = 1, , ] Solving system... 

Numerical stability is frequently a problem in computations with polynomials and thus in the Toeplitz, Hankel, and Vandermonde computations. 

Toeplitz, Hankel, and Vandermonde matrices are special cases of more general classes of structured matrices associated with operators of shift (displacement) and scaling such an association enables us to extend the algorithms for Toeplitz linear systems to the systems of the cited general classes (see author s book with D. Bini). 

Charles Auguste Vandermonde (Paris 1735-1 January 1796), director (1782) of the Conservatoire pour les arts-et-metiers, was also an eminent mathematician ( Vandermonde s theorem , etc.) and wrote on the theory of music. He published on the cold of 1776 with Bezout and Lavoisier, and on the manufacture of bayonets and sword blades at Klingental, department of the Lower Rhine. ... 

Ce fut le 24 juin 1783 que nous fimes cette experience, M. de Laplace et moi, en presence de MM. le Roi, de Vandermonde, de plusieurs autres academiciens, et de M. Blagden, aujourd hui secretaire de la Societe royale de Londres ce dernier nous apprit que M. Cavendish avait deja essaye, a Londres, de bruler de I air inflammable dans des vaisseaux fermes, et qu il avait obtenu une quantite d eau tres-sensible. 

We did not know that M. Monge was occupied with the same subject, and we were not informed till some days later by a letter he addressed to M. Vandermonde, which the latter read to the Academy . . . his experiment is much more conclusive than ours and leaves nothing to be desired the result he obtained was pure water, the weight of which was found to be very nearly equal to that of the two airs. ... 

These discoveries were made in a time of revolutions, one scientific and one political. The report from Vandermonde et al. was edited in 1786, the Bastille was stormed in 1789, the French king was executed in 1793 and Lavoisier guillotined in 1794. 

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