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**1 - 6**of**6**### Fast and accurate con-eigenvalue algorithm for optimal rational approximations

- SIAM J. Matrix Anal. Appl
, 2012

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### Accurate computations with Said-Ball-Vandermonde matrices

, 2008

"... A generalization of the Vandermonde matrices which arise when the power basis is replaced by the Said-Ball basis is considered. When the nodes are inside the interval (0,1), then those matrices are strictly totally positive. An algorithm for computing the bidiagonal decomposition of those Said-Ball- ..."

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A generalization of the Vandermonde matrices which arise when the power basis is replaced by the Said-Ball basis is considered. When the nodes are inside the interval (0,1), then those matrices are strictly totally positive. An algorithm for computing the bidiagonal decomposition of those Said-Ball-Vandermonde matrices is presented, which allows to use known algorithms for totally positive matrices represented by their bidiagonal decomposition. The algorithm is shown to be fast and to guarantee high relative accuracy. Some numerical experiments which illustrate the good behaviour of the algorithm are included.

### ACCURATE COMPUTATIONS WITH TOTALLY POSITIVE BERNSTEIN-VANDERMONDE MATRICES

, 2013

"... The accurate solution of some of the main problems in numerical linear algebra (linear system solving, eigenvalue computation, singular value computation and the least squares problem) for a totally positive Bernstein–Vandermonde matrix is considered. Bernstein–Vandermonde matrices are a generalizat ..."

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The accurate solution of some of the main problems in numerical linear algebra (linear system solving, eigenvalue computation, singular value computation and the least squares problem) for a totally positive Bernstein–Vandermonde matrix is considered. Bernstein–Vandermonde matrices are a generalization of Vandermonde matrices arising when considering for the space of the algebraic polynomials of degree less than or equal to n the Bernstein basis instead of the monomial basis. The approach in this paper is based on the computation of the bidiagonal factorization of a totally positive Bernstein–Vandermonde matrix or of its inverse. The explicit expressions obtained for the determinants involved in the process make the algorithm both fast and accurate. The error analysis of this algorithm for computing this bidiagonal factorization and the perturbation theory for the bidiagonal factorization of totally positive Bernstein–Vandermonde matrices are also carried out. Several applications of the computation with this type of matrices are also pointed out.

### Extending Summation Precision for Network Reduction Operations

"... Abstract—Double precision summation is at the core of numerous important algorithms such as Newton-Krylov methods and other operations involving inner products, but the effectiveness of summation is limited by the accumulation of rounding errors, which are an increasing problem with the scaling of m ..."

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Abstract—Double precision summation is at the core of numerous important algorithms such as Newton-Krylov methods and other operations involving inner products, but the effectiveness of summation is limited by the accumulation of rounding errors, which are an increasing problem with the scaling of modern HPC systems and data sets. To reduce the impact of precision loss, researchers have proposed increasedand arbitrary-precision libraries that provide reproducible error or even bounded error accumulation for large sums, but do not guarantee an exact result. Such libraries can also increase computation time significantly. We propose big integer (BigInt) expansions of double precision variables that enable arbitrarily large summations without error and provide exact and reproducible results. This is feasible with performance comparable to that of double-precision floating point summation, by the inclusion of simple and inexpensive logic into modern NICs to accelerate performance on large-scale systems. I.

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"... Accurate computations with totally positive Bernstein-Vandermonde matrices ..."

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Accurate computations with totally positive Bernstein-Vandermonde matrices