Additive preconditioning and aggregation in matrix computations

Computers and Mathematics with Applications

Volumn 55, Issue 8, 2008, Pages 1870-1886

  Pan, Victor Y ^a,b   Ivolgin, Dmitriy ^b   Murphy, Brian ^a   Rosholt, Rhys Eric ^a   Taj Eddin, Islam ^b   Tang, Yuqing ^b   Yan, Xiaodong ^b  

^a CITY UNIVERSITY OF NEW YORK   (United States)

^b CITY UNIVERSITY OF NEW YORK   (United States)

Author keywords

Additive preconditioning; Aggregation; Matrix computations; MSAs

Indexed keywords

ALGORITHMS; COMPUTATION THEORY; EIGENVALUES AND EIGENFUNCTIONS; LINEAR SYSTEMS; PROBLEM SOLVING;

ADDITIVE PRECONDITIONING; MATRIX COMPUTATIONS; MATRIX STRUCTURES; SINGULAR LINEAR SYSTEMS;

SINGULAR VALUE DECOMPOSITION;

EID: 40649113345     PISSN: 08981221     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.camwa.2004.03.022     Document Type: Article

References (68)

1. Greenbaum A. Iterative Methods for Solving Linear Systems (1997), SIAM, Philadelphia
2. Benzi M. Preconditioning techniques for large linear systems: A survey. J. Comput. Phys. 182 (2002) 418-477
3. Chen K. Matrix Preconditioning Techniques and Applications (2005), Cambridge University Press, Cambridge, England
4. V.Y. Pan, D. Ivolgin, B. Murphy, R.E. Rosholt, Y. Tang, X. Yan, Additive preconditioning in matrix computations, Technical Report TR 2005009, CUNY Ph.D. Program in Computer Science, Graduate Center, City University of New York, July 2005
5. V.Y. Pan, D. Ivolgin, B. Murphy, R.E. Rosholt, I. Taj-Eddin, Y. Tang, X. Yan, Additive preconditioning and aggregation in matrix computations, Technical Report TR 2006006, CUNY Ph.D. Program in Computer Science, Graduate Center, City University of New York, May 2006
6. V.Y. Pan, D. Ivolgin, B. Murphy, R.E. Rosholt, Y. Tang, X. Yan, Additive preconditioning and aggregation in matrix computations, Technical Report TR 2007002, CUNY Ph.D. Program in Computer Science, Graduate Center, City University of New York, March 2007
7. V.Y. Pan, D. Ivolgin, B. Murphy, R.E. Rosholt, Y. Tang, X. Yan, Additive preconditioning for matrix computations, Technical Report TR 2007003, CUNY Ph.D. Program in Computer Science, Graduate Center, City University of New York, March 2007
8. V.Y. Pan, X. Yan, Additive preconditioning, eigenspaces, and inverse iteration, Technical Report TR 2007004, CUNY Ph.D. Program in Computer Science, Graduate Center, City University of New York, 2007
9. V.Y. Pan, Null aggregation and extensions, Technical Report TR 2007009, CUNY Ph.D. Program in Computer Science, Graduate Center, City University of New York, 2007
10. V.Y. Pan, B. Murphy, G. Qian, R.E. Rosholt, Numerical computation of determinants with additive preconditioning, Technical Report TR 2007011, CUNY Ph.D. Program in Computer Science, Graduate Center, City University of New York, 2007
11. Pan V.Y., Murphy B., Rosholt R.E., and Tabanjeh M. The Schur aggregation for solving linear systems of equations. In: Masa M.M., and Watt S. (Eds). Proceedings of the third International Workshop on Symbolic-Numeric Computation, SNC 2007. July 2007, London, Ontario, Canada (2007), ACM Press, New York 142-151
12. Pan V.Y., and Yan X. Null space and eigenspace computations with additive preprocessing. In: Masa M.M., and Watt S. (Eds). Proceedings of the International Workshop on Symbolic-Numeric Computation, SNC 2007. July 2007, London, Ontario, Canada (2007), ACM Press, New York 152-160
13. V.Y. Pan, B. Murphy, G. Qian, R.E. Rosholt, Error-free computations via floating-point operations, Technical Reports TR 2007010 and TR 2007013, CUNY Ph.D. Program in Computer Science, Graduate Center, City University of New York, 2007, Comput. Math. Appl. (in press).
14. Golub G.H., and Van Loan C.F. Matrix Computations. 3rd ed. (1996), The Johns Hopkins University Press, Baltimore, Maryland
15. Stewart G.W. Basic Decompositions. Matrix Algorithms vol. I (1998), SIAM, Philadelphia
16. Wang X. Affect of small rank modification on the condition number of a matrix. Comput. Math. Appl. 54 (2007) 819-825
17. Pan V.Y. Structured Matrices and Polynomials: Unified Superfast Algorithms (2001), Birkhäuser, Springer, Boston, New York
18. Dongarra J.J., Duff I.S., Sorensen D.C., and van Der Vorst H.A. Numerical Linear Algebra for High-Performance Computers (1998), SIAM, Philadelphia
19. Duff I.S., Erisman A.M., and Reid J.K. Direct Methods for Sparse Matrices (1986), Clarendon Press, Oxford, England
20. Lipton R.J., Rose D., and Tarjan R.E. Generalized nested dissection. SIAM J. Numer. Anal. 16 2 (1979) 346-358
21. Pan V.Y., and Reif J. Fast and efficient parallel solution of sparse linear systems. SIAM J. Comput. 22 6 (1993) 1227-1250
22. Vandebril R., Van Barel M., Golub G., and Mastronardi N. A bibliography on semiseparable matrices. Calcolo 42 3-4 (2005) 249-270
23. Lawson C.L., and Hanson R.J. Solving Least Squares Problems (1974), Prentice-Hall, Englewood Cliffs, New Jersey Reissued with a survey of recent developments by SIAM, Philadelphia, 1995
24. Björck Å. Numerical Methods for Least Squares Problems (1996), SIAM, Philadelphia
25. Stewart G.W. Eigensystems. Matrix Algorithms. first ed. vol II (1998), SIAM, Philadelphia 2001, second ed
26. Brönnimann H., Emiris I.Z., Pan V.Y., and Pion S. Sign determination in residue number systems. Theoret. Comput. Sci. 210 1 (1999) 173-197
27. Pan V.Y., and Yu Y. Certification of numerical computation of the sign of the determinant of a matrix. Algorithmica 30 (2001) 708-724
28. Shevchuk J. Adaptive precision floating-point arithmetic and fast robust geometric predicates. Discrete Comput. Geom. 18 (1997) 305-363. Available at: www.cs.cmu.edu/quake/robust.html
29. Demmel J., and Hida Y. Fast and accurate floating point summation with application to computational geometry. Numer. Algorithms 37 1-4 (2004) 101-112
30. Emiris I.Z., and Pan V.Y. Improved algorithms for computing determinants and resultants. J. Complexity 21 1 (2005) 43-71
31. Gautschi W., and Inglese G. Lower bounds for the condition number of vandermonde matrices. Numer. Math. 52 (1988) 241-250
32. Tyrtyshnikov E.E. How bad are hankel matrices?. Numer. Math. 67 2 (1994) 261-269
33. Pan V.Y. On computations with dense structured matrices. Math. Comput. 55 191 (1990) 179-190
34. Gohberg I., Kailath T., and Olshevsky V. Fast gaussian elimination with partial pivoting for matrices with displacement structure. Math. Comput. 64 (1995) 1557-1576
35. Heinig G. Inversion of generalized cauchy matrices and the other classes of structured matrices. Linear Algebra for Signal Processing, IMA Volume in Math. and Its Applications, vol. 69 (1995) 95-114
36. Higham N.J. Accuracy and Stability in Numerical Analysis. second ed. (2002), SIAM, Philadelphia
37. Demmel J., and Hida Y. Accurate and efficient floating point summation. SIAM J. Sci. Comput. 25 (2003) 1214-1248
38. Li X., Demmel J., Bailey D., Henry G., Hida Y., Iskandar J., Kahan W., Kang S., Kapur A., Martin M., Thompson B., Tung T., and Yoo D. Design, implementation and testing of extended and mixed precision BLAS. ACM Trans. Math. Software 28 (2002) 152-205. http://crd.lbl.gov/xiaoye/XBLAS/
39. Ogita T., Rump S.M., and Oishi S. Accurate sum and dot product. SIAM J. Sci. Comput. 26 6 (2005) 1955-1988
40. Kulish U., and Miranker W.L. Computer Arithmetic in Theory and Practice (1980), Academic Press, New York
41. Malcolm M. On accurate floating-point summation. Commun. ACM 4 (1971) 731-736
42. Priest D.M. Algorithms for arbitrary precision floating point arithmetic. In: Kornerup R., and Matula D.W. (Eds). Proc. 10th Symposium on Computer Arithmetic (1991), IEEE Press 132-145
43. T.J. Dekker, A Floating-point Technique for Extending the Available Precision
44. V.Y. Pan, Can we utilize the cancellation of the most significant digits? Tech. Report TR 92 061, The International Computer Science Institute, Berkeley, California, 1992
45. Emiris I.Z., Pan V.Y., and Yu Y. Modular arithmetic for linear algebra computations in the real field. J. Symbolic Comput. 21 (1998) 1-17
46. IA-32 Intel Architecture Software Developers Manual, Volume 1: Basic Architecture (2001), Intel Corporation, Mt. Prospect, Illinois Order Number 245470
47. Agner Fog, How to optimize for the Pentium family of microprocessors. www.agner.org, 1996-2004. Last updated 2004-04-16
48. Chu M.T., Funderlic R.E., and Golub G.H. A rank-one reduction formula and its applications to matrix factorizations. SIAM Rev. 37 4 (1995) 512-530
49. Hubert L., Meulman J., and Heiser W. Two purposes for matrix factorization: A historical appraisal. SIAM Rev. 42 1 (2000) 68-82
50. Golub G.H. Some modified matrix eigenvalue problems. SIAM Rev. 15 (1973) 318-334
51. V.Y. Pan, Univariate polynomial root-finding with lower precision and higher convergence rate, Technical Report TR 2002003, CUNY Ph.D. Program in Computer Science, Graduate Center, City University of New York, 2002
52. Bini D.A., Gemignani L., and Pan V.Y. Inverse power and Durand/Kerner iteration for univariate polynomial root-finding. Comput. Math. Appl. 47 2-3 (2004) 447-459 Also Technical Report TR 2002020, CUNY Ph.D. Program in Computer Science, Graduate Center, City University of New York, 2002
53. Bini D.A., Gemignani L., and Pan V.Y. Fast and stable QR eigenvalue algorithms for generalized companion matrices and secular equation. Numer. Math. 3 (2005) 373-408 Also Technical Report 1470, Department of Math., University of Pisa, Pisa, Italy, July 2003
54. Bini D.A., Gemignani L., and Pan V.Y. Improved initialization of the accelerated and robust QR-like polynomial root-finding. Electron. Trans. Numer. Anal. 17 (2004) 195-205
55. Pan V.Y., Ivolgin D., Murphy B., Rosholt R.E., Tang Y., Wang X., and Yan X. Root-finding with Eigen-solving. In: Wang D., and Zhi L. (Eds). Symbolic-Numerical Computation (2007), Birkhäuser, Basel, Boston 219-245
56. Miranker W.L., and Pan V.Y. Methods of aggregations. Linear Algebra Appl. 29 (1980) 231-257
57. Pan V.Y. How can we speed up matrix multiplication?. SIAM Rev. 26 3 (1984) 393-415
58. Coppersmith D., and Winograd S. Matrix multiplication via arithmetic progressions. J. Symbolic Comput. 9 3 (1990) 251-280
59. Laderman J., Pan V.Y., and Sha H.X. On practical algorithms for accelerated matrix multiplication. Linear Algebra Appl. 162-164 (1992) 557-588
60. Kaporin I. A practical algorithm for faster matrix multiplication. Numer. Linear Algebra Appl. 6 8 (1999) 687-700
61. Kaporin I. The aggregation and cancellation techniques as a practical tool for faster matrix multiplication. Theoret. Comput. Sci. 315 2-3 (2004) 469-510
62. Bondyfalat D., Mourrain B., and Pan V.Y. Computation of a specified root of a polynomial system of equations using eigenvectors. Linear Algebra Appl. 319 (2000) 193-209
63. Mourrain B., and Pan V.Y. Multivariate polynomials, duality and structured matrices. J. Complexity 16 1 (2000) 110-180
64. NAG Fortran Library Manual, Mark 13, vol. 1, 1988
65. Pan V.Y. Solving a polynomial equation: Some history and recent progress. SIAM Rev. 39 2 (1997) 187-220
66. Malek F., and Vaillancourt R. Polynomial zerofinding iterative matrix algorithms. Comput. Math. Appl. 29 1 (1995) 1-13
67. Malek F., and Vaillancourt R. A composite polynomial zerofinding matrix algorithm. Comput. Math. Appl. 30 2 (1995) 37-47
68. Fortune S. An iterated eigenvalue algorithm for approximating roots of univariate polynomials. J. Symbolic Comput. 33 5 (2002) 627-646 Proc. version in Proc. Intern. Symp. on Symbolic and Algebraic Computation, ISSAC'01, 121-128 ACM Press, New York, 2001