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Structured matrices and polynomials: unified superfast algorithmsAugust 2001
- Author:
- Victor Y. Pan
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Abstract
Structure matrices serve as a natural bridge between the areas of algebraic computations with polynomials and numerical matrix computations, allowing cross-fertilization of both fields. This book covers most fundamental numerical and algebraic computations with Toeplitz, Hankel, Vandermonde, Cauchy, and other popular structured matrices. Throughout the computations, the matrices are represented by their compressed images, called displacements, enabling both a unified treatment of various matrix structures and dramatic saving of computer time and memory. The resulting superfast algorithms allow further dramatic parallel acceleration using FFT and fast sine and cosine transforms.
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Ding C, Wang S, Liu N, Xu K, Wang Y and Liang Y REQ-YOLO Proceedings of the 2019 ACM/SIGDA International Symposium on Field-Programmable Gate Arrays, (33-42)- Thomas A, Gu A, Dao T, Rudra A and Ré C Learning compressed transforms with low displacement rank Proceedings of the 32nd International Conference on Neural Information Processing Systems, (9066-9078)
- Deng C, Liao S, Xie Y, Parhi K, Qian X and Yuan B PermDNN Proceedings of the 51st Annual IEEE/ACM International Symposium on Microarchitecture, (189-202)
- Huang R (2018). A Periodic qd-Type Reduction for Computing Eigenvalues of Structured Matrix Products to High Relative Accuracy, Journal of Scientific Computing, 75:3, (1229-1261), Online publication date: 1-Jun-2018.
- Ding C, Ren A, Yuan G, Ma X, Li J, Liu N, Yuan B and Wang Y Structured Weight Matrices-Based Hardware Accelerators in Deep Neural Networks Proceedings of the 2018 on Great Lakes Symposium on VLSI, (353-358)
- Charif F, Benchabane A and Bennia A (2018). A Simplified Architecture of the Zhang Neural Network for Toeplitz Linear Systems Solving, Neural Processing Letters, 47:2, (391-401), Online publication date: 1-Apr-2018.
- Wang S, Li Z, Ding C, Yuan B, Qiu Q, Wang Y and Liang Y C-LSTM Proceedings of the 2018 ACM/SIGDA International Symposium on Field-Programmable Gate Arrays, (11-20)
- Ye K and Lim L (2018). Fast Structured Matrix Computations, Foundations of Computational Mathematics, 18:1, (45-95), Online publication date: 1-Feb-2018.
- De Sa C, Gu A, Puttagunta R, Ré C and Rudra A A two-pronged progress in structured dense matrix vector multiplication Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, (1060-1079)
- Liao S, Li Z, Lin X, Qiu Q, Wang Y and Yuan B Energy-efficient, high-performance, highly-compressed deep neural network design using block-circulant matrices Proceedings of the 36th International Conference on Computer-Aided Design, (458-465)
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- Zhao L, Liao S, Wang Y, Li Z, Tang J and Yuan B Theoretical properties for neural networks with weight matrices of low displacement rank Proceedings of the 34th International Conference on Machine Learning - Volume 70, (4082-4090)
- Hyun S, Lebreton R and Schost É Algorithms for Structured Linear Systems Solving and Their Implementation Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation, (205-212)
- Freitas M, Martins W, Lima Filho E and Silva Júnior W (2017). New Designs for Reduced-Redundancy Transceivers, Circuits, Systems, and Signal Processing, 36:5, (2075-2101), Online publication date: 1-May-2017.
- Blondel M, Fujino A, Ueda N and Ishihata M Higher-order factorization machines Proceedings of the 30th International Conference on Neural Information Processing Systems, (3359-3367)
- Bender M, Faugère J, Perret L and Tsigaridas E A Superfast Randomized Algorithm to Decompose Binary Forms Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation, (79-86)
- Eberly W Selecting Algorithms for Black Box Matrices Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation, (207-214)
- Pan V and Tsigaridas E (2016). Nearly optimal refinement of real roots of a univariate polynomial, Journal of Symbolic Computation, 74:C, (181-204), Online publication date: 1-May-2016.
- Pan V and Zhao L Randomized Circulant and Gaussian Pre-processing Proceedings of the 17th International Workshop on Computer Algebra in Scientific Computing - Volume 9301, (361-375)
- Pan V and Zhao L Polynomial Real Root Isolation by Means of Root Radii Approximation Proceedings of the 17th International Workshop on Computer Algebra in Scientific Computing - Volume 9301, (349-360)
- Saunders B Matrices with Two Nonzero Entries per Row Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation, (323-330)
- Henzinger M, Krinninger S, Nanongkai D and Saranurak T Unifying and Strengthening Hardness for Dynamic Problems via the Online Matrix-Vector Multiplication Conjecture Proceedings of the forty-seventh annual ACM symposium on Theory of Computing, (21-30)
- Chowdhury M, Jeannerod C, Neiger V, Schost E and Villard G (2015). Faster Algorithms for Multivariate Interpolation With Multiplicities and Simultaneous Polynomial Approximations, IEEE Transactions on Information Theory, 61:5, (2370-2387), Online publication date: 1-May-2015.
- Pan V and Tsigaridas E Nearly optimal computations with structured matrices Proceedings of the 2014 Symposium on Symbolic-Numeric Computation, (21-30)
- Pan V Polynomial Evaluation and Interpolation and Transformations of Matrix Structures Proceedings of the 15th International Workshop on Computer Algebra in Scientific Computing - Volume 8136, (273-287)
- Pan V, Qian G and Zheng A Real and complex polynomial root-finding by means of eigen-solving Proceedings of the 14th international conference on Computer Algebra in Scientific Computing, (271-282)
- Pan V Univariate polynomial root-finding by arming with constraints Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation, (112-121)
- Pan V and Zheng A (2011). Root-finding by expansion with independent constraints, Computers & Mathematics with Applications, 62:8, (3164-3182), Online publication date: 1-Oct-2011.
- Murphy B Acceleration of the inversion of triangular Toeplitz matrices and polynomial division Proceedings of the 13th international conference on Computer algebra in scientific computing, (321-332)
- Hart W and Novocin A Practical divide-and-conquer algorithms for polynomial arithmetic Proceedings of the 13th international conference on Computer algebra in scientific computing, (200-214)
- Pan V, Qian G and Zheng A Randomized preconditioning of the MBA algorithm Proceedings of the 36th international symposium on Symbolic and algebraic computation, (281-288)
- Kotsireas I, Mourrain B and Pan V (2011). Preface, Theoretical Computer Science, 412:16, (1443-1444), Online publication date: 1-Apr-2011.
- Jeannerod C and Mouilleron C Computing specified generators of structured matrix inverses Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, (281-288)
- Pan V and Zheng A Real and complex polynomial root-finding with eigen-solving and preprocessing Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, (219-226)
- Jeannerod C, Mouilleron C and Villard G (2010). Extending Cardinal's algorithm to a broader class of structured matrices, ACM Communications in Computer Algebra, 43:3/4, (84-85), Online publication date: 24-Jun-2010.
- Pan V, Qian G and Zheng A Advancing matrix computations with randomized preprocessing Proceedings of the 5th international conference on Computer Science: theory and Applications, (303-314)
- Paturi R and Pudlak P On the complexity of circuit satisfiability Proceedings of the forty-second ACM symposium on Theory of computing, (241-250)
- Martins W and Diniz P (2010). Block-based transceivers with minimum redundancy, IEEE Transactions on Signal Processing, 58:3, (1321-1333), Online publication date: 1-Mar-2010.
- Emiris I and Pan V Applications of FFT and structured matrices Algorithms and theory of computation handbook, (18-18)
- Emiris I, Pan V and Tsigaridas E Algebraic and numerical algorithms Algorithms and theory of computation handbook, (17-17)
- Stehlé D, Steinfeld R, Tanaka K and Xagawa K Efficient Public Key Encryption Based on Ideal Lattices Proceedings of the 15th International Conference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology, (617-635)
- Pan V, Murphy B and Rosholt R Nearly optimal symbolic-numerical algorithms for structured integer matrices and polynomials Proceedings of the 2009 conference on Symbolic numeric computation, (105-114)
- Pan V, Qian G and Zheng A Preconditioning, randomization, solving linear systems, eigen-solving, and root-finding Proceedings of the 2009 conference on Symbolic numeric computation, (5-6)
- Bella T, Eidelman Y, Gohberg I and Olshevsky V (2008). Computations with quasiseparable polynomials and matrices, Theoretical Computer Science, 409:2, (158-179), Online publication date: 10-Dec-2008.
- Pan V, Grady D, Murphy B, Qian G, Rosholt R and Ruslanov A (2008). Schur aggregation for linear systems and determinants, Theoretical Computer Science, 409:2, (255-268), Online publication date: 10-Dec-2008.
- Bini D, Pan V and Verschelde J (2008). Preface, Theoretical Computer Science, 409:2, (155-157), Online publication date: 10-Dec-2008.
- Pan V, Ivolgin D, Murphy B, Rosholt R, Tang Y and Yan X Additive preconditioning for matrix computations Proceedings of the 3rd international conference on Computer science: theory and applications, (372-383)
- Li B and Wang D An Algorithm for Transforming Regular Chain into Normal Chain Computer Mathematics, (236-245)
- Bostan A, Jeannerod C and Schost É Solving toeplitz- and vandermonde-like linear systems with large displacement rank Proceedings of the 2007 international symposium on Symbolic and algebraic computation, (33-40)
- Pan V, Murphy B, Rosholt R, Qian G and Tang Y Real root-finding Proceedings of the 2007 international workshop on Symbolic-numeric computation, (161-169)
- Pan V and Yan X Null space and eigenspace computations with additive preprocessing Proceedings of the 2007 international workshop on Symbolic-numeric computation, (152-160)
- Pan V, Murphy B, Rosholt R and Tabanjeh M The schur aggregation for solving linear systems of equations Proceedings of the 2007 international workshop on Symbolic-numeric computation, (142-151)
- Williams R Matrix-vector multiplication in sub-quadratic time Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms, (995-1001)
- Bostan A, Jeannerod C and Schost É (2006). Solving structured linear systems of large displacement rank, ACM Communications in Computer Algebra, 40:2, (42-44), Online publication date: 1-Jun-2006.
- Pan V (2005). The amended DSeSC power method for polynomial root-finding, Computers & Mathematics with Applications, 49:9-10, (1515-1524), Online publication date: 1-May-2005.
- Pan V, Van Barel M, Wang X and Codevico G (2004). Iterative inversion of structured matrices, Theoretical Computer Science, 315:2-3, (581-592), Online publication date: 6-May-2004.
- Grabmeier J, Kaltofen E and Weispfenning V Cited References Computer algebra handbook, (493-622)
- Pan V and Wang X Acceleration of Euclidean algorithm and extensions Proceedings of the 2002 international symposium on Symbolic and algebraic computation, (207-213)
- Pan V (2002). Univariate polynomials, Journal of Symbolic Computation, 33:5, (701-733), Online publication date: 1-May-2002.
- Pan V Univariate polynomials Proceedings of the 2001 international symposium on Symbolic and algebraic computation, (253-267)
Index Terms
- Structured matrices and polynomials: unified superfast algorithms
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