The DOI matches its Crossref record; 10 of the 10 references it cites were found in a registry or an index.
Solution of Vandermonde systems of equations
Open accessAbstract
We obtain in this paper a considerable improvement over a method developed earlier by Ballester and Pereyra for the solution of systems of linear equations with Vandermonde matrices of coefficients. This is achieved by observing that a part of the earlier algorithm is equivalent to Newton’s interpolation method. This allows also to produce a progressive algorithm which is significantly more efficient than previous available methods. Algol-60 programs and numerical results are included. Confluent Vandermonde systems are also briefly discussed.
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Citations
Cites
Iterative Methods for the Solution of Equations.
Young, D. M., & Traub, J. F. (1967) · American Mathematical Monthly · doi:10.2307/2316079 · Cited by 2,191
On inverses of Vandermonde and confluent Vandermonde matrices
Gautschi, W. (1962) · Numerische Mathematik · doi:10.1007/bf01386302 · Cited by 136
On inverses of Vandermonde and confluent Vandermonde matrices. II
Gautschi, W. (1963) · Numerische Mathematik · doi:10.1007/bf01385906 · Cited by 117
Associated Polynomials and Uniform Methods for the Solution of Linear Problems
Traub, J. F. (1966) · SIAM Review · doi:10.1137/1008061 · Cited by 76
Van der Monde systems and numerical differentiation
Lyness, J. N., & Moler, C. (1966) · Numerische Mathematik · doi:10.1007/bf02166671 · Cited by 41
Solving confluent Vandermonde systems of Hermite type
Galimberti, G., & Pereyra, V. L. (1971) · Numerische Mathematik · doi:10.1007/bf01398458 · Cited by 33
Self-citationOn the construction of discrete approximations to linear differential expressions
Ballester, C., & Pereyra, V. L. (1967) · Mathematics of Computation · doi:10.1090/s0025-5718-1967-0228167-8 · Cited by 27
Self-citationNumerical differentiation and the solution of multidimensional Vandermonde systems
Galimberti, G., & Pereyra, V. L. (1970) · Mathematics of Computation · doi:10.1090/s0025-5718-1970-0275668-2 · Cited by 16
Self-citationOn the Construction of Discrete Approximations to Linear Differential Expressions
Ballester, C., & Pereyra, V. L. (1967) · Mathematics of Computation · doi:10.2307/2003232 · Cited by 5
Self-citationNumerical Differentiation and the Solution of Multidimensional Vandermonde Systems
Galimberti, G., & Pereyra, V. L. (1970) · Mathematics of Computation · doi:10.2307/2004482 · Cited by 3
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The Accurate and Efficient Solution of a Totally Positive Generalized Vandermonde Linear System
Demmel, J., & Koev, P. (2005) · SIAM Journal on Matrix Analysis and Applications · doi:10.1137/s0895479804440335 · Cited by 154
Explicit factorization of the Vandermonde matrix
Oruç, H., & Phillips, G. M. (2000) · Linear Algebra and its Applications · doi:10.1016/s0024-3795(00)00124-5 · Cited by 57
Reichel, L., & Opfer, G. (1991) · Mathematics of Computation · doi:10.1090/s0025-5718-1991-1094957-9 · Cited by 55
Displacement structure approach to polynomial Vandermonde and related matrices
Kailath, T., & Olshevsky, V. (1997) · Linear Algebra and its Applications · doi:10.1016/s0024-3795(96)00315-1 · Cited by 50
Symmetric functions and the Vandermonde matrix
Oruç, H., & Akmaz, H. K. (2004) · Journal of Computational and Applied Mathematics · doi:10.1016/j.cam.2004.01.032 · Cited by 25
Fast approximate computations with Cauchy matrices and polynomials
Pan, V. (2017) · Mathematics of Computation · doi:10.1090/mcom/3204 · Cited by 21
“unded (e.g., double) precision of computing rules out using the cited fast algorithms, and the known double precision algorithms for Problems 1{4 require quadratic arithmetic time (cf. [BF00], [P64], [BP70], [BEGO08]). This pessimistic outcome, however, does not apply to the important special case where the knots s i…”
MethodBella, T., Eidelman, Y., Gohberg, I., Koltracht, I., & Olshevsky, V. (2007) · Linear Algebra and its Applications · doi:10.1016/j.laa.2006.08.032 · Cited by 17
A fast method to diagonalize a Hankel matrix
Boley, D. L., Luk, F. T., & Vandevoorde, D. (1998) · Linear Algebra and its Applications · doi:10.1016/s0024-3795(98)10101-5 · Cited by 17
A Fast DVM Algorithm for Wideband Time-Delay Multi-Beam Beamformers
Perera, S. M., Lingsch, L., Madanayake, A., Mandal, S., & Mastronardi, N. (2022) · IEEE Transactions on Signal Processing · doi:10.1109/tsp.2022.3231182 · Cited by 11
Thiran Filters for Wideband DSP-Based Multi-Beam True Time Delay RF Sensing Applications
Perera, S. M., Rathnasekara, G., & Madanayake, A. (2024) · Sensors · doi:10.3390/s24020576 · Cited by 5
Metadata
- Authors
- Björck, Ȧ., & Pereyra, V.
- Venue
- Mathematics of Computation
- Publisher
- American Mathematical Society (AMS)
- Published
- 1970
- Volume
- 24
- Issue
- 112
- Pages
- 893-903
- ISSN
- 0025-5718 (ISSN-L)
- 1088-6842
- OpenAlex
- W2020298878
- Cited by
- 398
- References
- 10
- Citations by year
- 2026: 7 · 2025: 6 · 2024: 6 · 2023: 14 · 2022: 7 · 2021: 20 · 2020: 15 · 2019: 15 · 2018: 16 · 2017: 10 · 2016: 20 · 2015: 6 · 2014: 17 · 2013: 13 · 2012: 8
- FWCI
- 5.86
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Mathematics of Computation (publishedVersion)
- Funders
- Sweden-America Foundation
- Topics
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Matrix Theory and Algorithms
- Physical Sciences › Mathematics › Applied Mathematics › Mathematical functions and polynomials
- Physical Sciences › Mathematics › Modeling and Simulation › Mathematical and Computational Methods
- Keywords
- Vandermonde systems, systems of linear equations, Vandermonde matrix, Newton interpolation, progressive algorithms, numerical linear algebra
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