Matrix multiplication via arithmetic progressions
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
Probabilistic computation of integer polynomial GCDs
Journal of Algorithms
Subresultants and Reduced Polynomial Remainder Sequences
Journal of the ACM (JACM)
On Euclid's Algorithm and the Computation of Polynomial Greatest Common Divisors
Journal of the ACM (JACM)
On Euclid's Algorithm and the Theory of Subresultants
Journal of the ACM (JACM)
The Exact Solution of Systems of Linear Equations with Polynomial Coefficients
Journal of the ACM (JACM)
The Design and Analysis of Computer Algorithms
The Design and Analysis of Computer Algorithms
STOC '73 Proceedings of the fifth annual ACM symposium on Theory of computing
Computation of distances for regular and context-free probabilistic languages
Theoretical Computer Science
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
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Two significant developments can be distinguished in the theory of algebraic algorithm design. One is that of fast algorithms in terms of counting the arithmetic operations, such as the asymptotically fast matrix multiplication procedures and related linear algebra algorithms or the asymptotically fast polynomial multiplication procedures and related polynomial manipulation algorithms. The other is to observe the actual bit complexity when such algorithms are performed for concrete fields, in particular the rational numbers. It was discovered in the mid-1960s that for rational polynomial GCD operations the classical algorithm can lead to exponential coefficient size growth. The beautiful theory of subresultants by Collins [7] and Brown & Traub [5] explains, however, that the size growth is not inherent but a consequence of over-simplified rational arithmetic. Similar phenomena were observed for the Gaussian elimination procedure [2], [9].