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Cited by 144Matrix Theory and AlgorithmsAdvanced Combinatorial MathematicsMathematical and Computational Methods

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Algorithm 358: singular value decomposition of a complex matrix [F1, 4, 5]

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Businger, Peter A., Golub, Gene H.·1969·Communications of the ACM
DOI 10.1145/363235.363249
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Abstract

n, x, t, S); integer n, x, l; integer array S; comment Ising generates n-sequences ($1, "", S,) of zeros and ones where x = ~i~ S~ and t = ~,-~1 I S~+I -S~ I are given.The main idea is to interleave compositions of x and n --x objects and resort to a lexicographic generation of compositions.We call these sequences Ising configurations since we believe they first appeared in the study of the so-called Ising problem (See Hill [1], Ising [2]).The number R(n, x, t) of distinct configurations with fixed n, x, t is well known [1, 2]: Now define a block of l's (or zeros) in the sequence as a set of a maximum number of consecutive l's (or zeros) eventually consisting of a single element.For given n, x, t, the number p of blocks of l's may easily be deduced from t, as well as the number q of blocks of zeros.In fact, a block of l's including either $1 or S, yields one variation and each one of the others yields two variations; hence we get p = q ~ m + 1 when t = 2m + 1 (t odd requires $1 # S,) and either p = m + 1, q = m (S~ = S, = 1), or p = m, q = m+l ($1 = S, = 0) when t--2m.Clearly, there is a 1-1 correspondence between the compositions of x with p parts and the distributions of the x l's into p blocks.And for each distribution of l's, distinct distributions of the n -x zeros into g blocks correspond to distinct configurations.The main body of the algorithm is compose, which generates compositions of an integer x with k parts and stores them in the array L. The role of sort and bisort is to form the final sequence ($1 , ... , S,) from the structure of one-blocks Li and zeroblocks M~.The Ising problem was brought to my attention by Dr. B. Dejon during an informal visit to the IBM Research Laboratory in Zurich.Thanks are also due to Prof. Paul ErdSs for pointing out to me reference [1] and to Prof. A. A. Zykov for correspondence.The procedure was tested on the NCR 4130 of the Labora-tSrio de C£1culo Autom~tico, Universidade do Porto.Thanks are also due to the Director and his Staff.

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  • The Eckart-Young theorem guarantees that truncating a matrix's SVD to retain the top r singular values and their singular vectors yields the optimal rank-r approximation, minimizing the approximation error in the Frobenius norm.

    “For any matrix A ∈ R m × n , its optimal low-rank approximation can be obtained by truncating its SVD (Businger and Golub, 1969) to retain the top r singular values along with their corresponding singular vectors.”

    From Knowledge Decoupling via Orthogonal Projection for Lifelong Editing of Large Language Models (2025) · cites this paper as Businger and Golub, 1969

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Metadata

Authors
Businger, P. A., & Golub, G. H.
Affiliations
Bell Telephone Labs, Inc., Murray Hill, NJ
Venue
Communications of the ACM
Publisher
Association for Computing Machinery (ACM)
Published
1969-10
Volume
12
Issue
10
Pages
564-565
ISSN
  • 0001-0782 (ISSN-L)
  • 1557-7317
DOI
10.1145/363235.363249
OpenAlex
W1970916490
Cited by
143
References
3
Citations by year
2026: 1 · 2025: 3 · 2024: 1 · 2023: 2 · 2022: 4 · 2021: 2 · 2020: 17 · 2019: 20 · 2018: 3 · 2017: 3 · 2016: 3 · 2015: 6 · 2013: 1 · 2012: 1
FWCI
1.90
Type
journal article
Language
en
Open access
Licence
www.acm.org/publications/policies/copyright_policy#Background (vor)
Locations
Communications of the ACM (publishedVersion)
Topics
  • Physical Sciences › Computer Science › Computational Theory and Mathematics › Matrix Theory and Algorithms
  • Physical Sciences › Mathematics › Discrete Mathematics and Combinatorics › Advanced Combinatorial Mathematics
  • Physical Sciences › Mathematics › Modeling and Simulation › Mathematical and Computational Methods
Keywords
binary sequences, Ising problem, singular value decomposition, complex matrices, numerical linear algebra, FORTRAN algorithm, combinatorial generation, matrix factorization, eigenvalue computation, matrix eigenvalue computation, matrix conditioning, matrix rank

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