메뉴 건너뛰기




Volumn 432, Issue 12, 2010, Pages 3272-3318

Randomized Preprocessing of Homogeneous Linear Systems of Equations

Author keywords

Conditioning; Linear systems of equations; Randomized preprocessing

Indexed keywords

CONDITION NUMBERS; CPU TIME; DISPLACEMENT RANKS; FREE SOLUTIONS; HOMOGENEOUS LINEAR SYSTEMS; LINEAR SYSTEMS OF EQUATIONS; MATRIX; MATRIX INVERSIONS; MATRIX PRODUCTS; MOORE-PENROSE; NEWTON'S ITERATION; NONSINGULAR LINEAR SYSTEMS; NUMERICAL STABILITIES; ORTHOGONALIZATION; POLYNOMIAL SYSTEMS; ROOTFINDING; SECULAR EQUATIONS; SHERMAN-MORRISON-WOODBURY FORMULAS; SOLUTION TIME; STRUCTURED MATRIXES; TOEPLITZ;

EID: 77950369121     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.laa.2010.01.023     Document Type: Article
Times cited : (20)

References (91)
  • 1
    • 0002671024 scopus 로고
    • Stability analysis of the method of seminormal equations
    • Björck Å. Stability analysis of the method of seminormal equations. Linear Algebra Appl. 88/89 (1987) 31-48
    • (1987) Linear Algebra Appl. , vol.88-89 , pp. 31-48
    • Björck, Å.1
  • 2
    • 0000585494 scopus 로고
    • Numerics of Gram-Schmidt orthogonalization
    • Björck Å. Numerics of Gram-Schmidt orthogonalization. Linear Algebra Appl. 197/198 (1994) 297-316
    • (1994) Linear Algebra Appl. , vol.197-198 , pp. 297-316
    • Björck, Å.1
  • 4
    • 0000081585 scopus 로고
    • Asymptotically fast solution of Toeplitz and related systems of linear equations
    • Bitmead R.R., and Anderson B.D.O. Asymptotically fast solution of Toeplitz and related systems of linear equations. Linear Algebra Appl. 34 (1980) 103-116
    • (1980) Linear Algebra Appl. , vol.34 , pp. 103-116
    • Bitmead, R.R.1    Anderson, B.D.O.2
  • 6
    • 0034396488 scopus 로고    scopus 로고
    • Design, analysis, and implementation of a multiprecision polynomial rootfinder
    • Bini D.A., and Fiorentino G. Design, analysis, and implementation of a multiprecision polynomial rootfinder. Numer. Algorithms 23 (2000) 127-173
    • (2000) Numer. Algorithms , vol.23 , pp. 127-173
    • Bini, D.A.1    Fiorentino, G.2
  • 7
    • 1242307407 scopus 로고    scopus 로고
    • Inverse power and Durand/Kerner iteration for univariate polynomial root-finding
    • Also Technical Report TR 2002 020, CUNY Ph.D. Program in Computer Science, Graduate Center, City University of New York, 2002.
    • Bini D.A., Gemignani L., and Pan V.Y. Inverse power and Durand/Kerner iteration for univariate polynomial root-finding. Comput. Math. (with Appl.) 47 2/3 (2004) 447-459. http://www.cs.gc.cuny.edu/tr/techreport.php?id=352 Also Technical Report TR 2002 020, CUNY Ph.D. Program in Computer Science, Graduate Center, City University of New York, 2002.
    • (2004) Comput. Math. (with Appl.) , vol.47 , Issue.2-3 , pp. 447-459
    • Bini, D.A.1    Gemignani, L.2    Pan, V.Y.3
  • 8
    • 84947068118 scopus 로고    scopus 로고
    • Fast and stable QR eigenvalue algorithms for generalized companion matrices and secular equation
    • (Also Technical Report 1470, Department of Math., University of Pisa, Pisa, Italy, July 2003)
    • 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)
    • (2005) Numer. Math. , vol.3 , pp. 373-408
    • Bini, D.A.1    Gemignani, L.2    Pan, V.Y.3
  • 10
  • 11
    • 0016025765 scopus 로고
    • Generalized inverses of certain Toeplitz matrices
    • Cline R.E., Plemmons R.J., and Worm G. Generalized inverses of certain Toeplitz matrices. Linear Algebra Appl. 8 (1974) 25-33
    • (1974) Linear Algebra Appl. , vol.8 , pp. 25-33
    • Cline, R.E.1    Plemmons, R.J.2    Worm, G.3
  • 12
    • 85023205150 scopus 로고
    • Matrix multiplication via arithmetic progressions
    • Coppersmith D., and Winograd S. Matrix multiplication via arithmetic progressions. J. Symbolic Comput. 9 3 (1990) 251-280
    • (1990) J. Symbolic Comput. , vol.9 , Issue.3 , pp. 251-280
    • Coppersmith, D.1    Winograd, S.2
  • 13
    • 0001290746 scopus 로고
    • Estimating extremal eigenvalues and condition numbers of matrices
    • Dixon J.D. Estimating extremal eigenvalues and condition numbers of matrices. SIAM J. Numer. Anal. 20 4 (1983) 812-814
    • (1983) SIAM J. Numer. Anal. , vol.20 , Issue.4 , pp. 812-814
    • Dixon, J.D.1
  • 14
    • 84966237760 scopus 로고
    • The probability that a numerical analysis problem is difficult
    • Demmel J. The probability that a numerical analysis problem is difficult. Math. Comp. 50 (1988) 449-480
    • (1988) Math. Comp. , vol.50 , pp. 449-480
    • Demmel, J.1
  • 17
    • 4043118275 scopus 로고    scopus 로고
    • Accurate and efficient floating point summation
    • Demmel J., and Hida Y. Accurate and efficient floating point summation. SIAM J. Sci. Comput. 25 (2003) 1214-1248
    • (2003) SIAM J. Sci. Comput. , vol.25 , pp. 1214-1248
    • Demmel, J.1    Hida, Y.2
  • 18
    • 0000006297 scopus 로고
    • A probabilistic remark on algebraic program testing
    • Demillo R.A., and Lipton R.J. A probabilistic remark on algebraic program testing. Inform. Process. Lett. 7 4 (1978) 193-195
    • (1978) Inform. Process. Lett. , vol.7 , Issue.4 , pp. 193-195
    • Demillo, R.A.1    Lipton, R.J.2
  • 19
    • 0012070851 scopus 로고    scopus 로고
    • Local operator theory, random matrices, and Banach spaces
    • Johnson W.B., and Lindenstrauss J. (Eds), North Holland, Amsterdam
    • Davidson K.R., and Szarek S.J. Local operator theory, random matrices, and Banach spaces. In: Johnson W.B., and Lindenstrauss J. (Eds). Handbook on the Geometry of Banach Spaces (2001), North Holland, Amsterdam 317-368
    • (2001) Handbook on the Geometry of Banach Spaces , pp. 317-368
    • Davidson, K.R.1    Szarek, S.J.2
  • 20
    • 0000324960 scopus 로고
    • Eigenvalues and condition numbers of random matrices
    • (Ph.D. Thesis (106 pp.), Math Dept., MIT, 1989)
    • Edelman A. Eigenvalues and condition numbers of random matrices. SIAM J. Matrix Anal. Appl. 9 4 (1988) 543-560 (Ph.D. Thesis (106 pp.), Math Dept., MIT, 1989)
    • (1988) SIAM J. Matrix Anal. Appl. , vol.9 , Issue.4 , pp. 543-560
    • Edelman, A.1
  • 22
    • 0036226901 scopus 로고    scopus 로고
    • Symbolic and numerical methods for exploiting structure in constructing resultant matrices
    • Emiris I.Z., and Pan V.Y. Symbolic and numerical methods for exploiting structure in constructing resultant matrices. J. Symbolic Comput. 33 (2002) 393-413
    • (2002) J. Symbolic Comput. , vol.33 , pp. 393-413
    • Emiris, I.Z.1    Pan, V.Y.2
  • 23
    • 0001070999 scopus 로고
    • Some modified matrix eigenvalue problems
    • Golub G.H. Some modified matrix eigenvalue problems. SIAM Rev. 15 (1973) 318-334
    • (1973) SIAM Rev. , vol.15 , pp. 318-334
    • Golub, G.H.1
  • 24
    • 0032366597 scopus 로고    scopus 로고
    • Stable and efficient algorithms for structured systems of linear equations
    • Gu M. Stable and efficient algorithms for structured systems of linear equations. SIAM J. Matrix Anal. Appl. 19 (1998) 279-306
    • (1998) SIAM J. Matrix Anal. Appl. , vol.19 , pp. 279-306
    • Gu, M.1
  • 25
    • 0000835513 scopus 로고
    • Lower bounds for the condition number of Vandermonde matrices
    • Gautschi W., and Inglese G. Lower bounds for the condition number of Vandermonde matrices. Numer. Math. 52 (1988) 241-250
    • (1988) Numer. Math. , vol.52 , pp. 241-250
    • Gautschi, W.1    Inglese, G.2
  • 26
    • 84968505008 scopus 로고
    • Fast Gaussian elimination with partial pivoting for matrices with displacement structure
    • Gohberg I., Kailath T., and Olshevsky V. Fast Gaussian elimination with partial pivoting for matrices with displacement structure. Math. Comp. 64 (1995) 1557-1576
    • (1995) Math. Comp. , vol.64 , pp. 1557-1576
    • Gohberg, I.1    Kailath, T.2    Olshevsky, V.3
  • 28
    • 0002638254 scopus 로고
    • Inversion of generalized Cauchy matrices and other classes of structured matrices
    • Springer, New York pp. 63-81
    • Heinig G. Inversion of generalized Cauchy matrices and other classes of structured matrices. Linear Algebra for Signal Processing, IMA Vol. Math. Appl. vol. 69 (1995), Springer, New York pp. 63-81
    • (1995) Linear Algebra for Signal Processing, IMA Vol. Math. Appl. , vol.69
    • Heinig, G.1
  • 30
    • 2142700139 scopus 로고    scopus 로고
    • The aggregation and cancellation techniques as a practical tool for faster matrix multiplication
    • Kaporin I. The aggregation and cancellation techniques as a practical tool for faster matrix multiplication. Theoret. Comput. Sci. 315 2-3 (2004) 469-510
    • (2004) Theoret. Comput. Sci. , vol.315 , Issue.2-3 , pp. 469-510
    • Kaporin, I.1
  • 33
    • 0003476369 scopus 로고
    • Prentice-Hall, Englewood Cliffs, New Jersey Reissued with a survey of recent developments by SIAM, Philadelphia, 1995
    • 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
    • (1974) Solving Least Squares Problems
    • Lawson, C.L.1    Hanson, R.J.2
  • 34
    • 38249013295 scopus 로고
    • On practical algorithms for accelerated matrix multiplication
    • Laderman J., Pan V.Y., and Sha H.X. On practical algorithms for accelerated matrix multiplication. Linear Algebra Appl. 162-164 (1992) 557-588
    • (1992) Linear Algebra Appl. , vol.162-164 , pp. 557-588
    • Laderman, J.1    Pan, V.Y.2    Sha, H.X.3
  • 36
    • 0019146429 scopus 로고
    • Doubling algorithms for Toeplitz and related equations
    • IEEE Press, Piscataway, New Jersey
    • Morf M. Doubling algorithms for Toeplitz and related equations. Proceedings of IEEE International Conference on ASSP (1980), IEEE Press, Piscataway, New Jersey 954-959
    • (1980) Proceedings of IEEE International Conference on ASSP , pp. 954-959
    • Morf, M.1
  • 37
    • 0031540780 scopus 로고    scopus 로고
    • A unifying convergence analysis of second-order methods for secular equations
    • Melman A. A unifying convergence analysis of second-order methods for secular equations. Math. Comp. 66 (1997) 333-344
    • (1997) Math. Comp. , vol.66 , pp. 333-344
    • Melman, A.1
  • 39
    • 0000692349 scopus 로고    scopus 로고
    • Multivariate polynomials, duality and structured matrices
    • Proceedings Version in STOC'98
    • Mourrain B., and Pan V.Y. Multivariate polynomials, duality and structured matrices. J. Complexity 16 1 (2000) 110-180 Proceedings Version in STOC'98
    • (2000) J. Complexity , vol.16 , Issue.1 , pp. 110-180
    • Mourrain, B.1    Pan, V.Y.2
  • 40
    • 0012058635 scopus 로고
    • Error analysis of complex arithmetic
    • Computational Aspect of Complex Analysis. Reidel D. (Ed), Dordrecht, Holland
    • Olver F.W.J. Error analysis of complex arithmetic. In: Reidel D. (Ed). Computational Aspect of Complex Analysis. NATO Advanced Study Institute Series C vol. 102 (1983), Dordrecht, Holland 279-292
    • (1983) NATO Advanced Study Institute Series C , vol.102 , pp. 279-292
    • Olver, F.W.J.1
  • 41
  • 43
    • 67349229944 scopus 로고    scopus 로고
    • Superfast inversion of two-level Toeplitz matrices using Newton iteration and tensor-displacement structure
    • Olshevsky V., Oseledets I.V., and Tyrtyshnikov E.E. Superfast inversion of two-level Toeplitz matrices using Newton iteration and tensor-displacement structure. Oper. Theory Adv. Appl. 179 (2008) 229-240
    • (2008) Oper. Theory Adv. Appl. , vol.179 , pp. 229-240
    • Olshevsky, V.1    Oseledets, I.V.2    Tyrtyshnikov, E.E.3
  • 44
    • 27644573654 scopus 로고    scopus 로고
    • Approximate inversion of matrices in the process of solving a hypersingular integral equation
    • Translated from JVM i MF 45 (2) (2005) 315-326
    • Oceledetz I.V., and Tyrtyshnikov E.E. Approximate inversion of matrices in the process of solving a hypersingular integral equation. Comput. Math. Math. Phys. 45 2 (2005) 302-313 Translated from JVM i MF 45 (2) (2005) 315-326
    • (2005) Comput. Math. Math. Phys. , vol.45 , Issue.2 , pp. 302-313
    • Oceledetz, I.V.1    Tyrtyshnikov, E.E.2
  • 45
    • 85140868432 scopus 로고    scopus 로고
    • I. Oseledets, E.E. Tyrtyshnikov, Breaking the curse of dimensionality, or how to use SVD in many dimensions, SIAM J. Sci. Comput., submitted by publication. Also ICM HKBU Research Report 09-03, February 2009. .
    • I. Oseledets, E.E. Tyrtyshnikov, Breaking the curse of dimensionality, or how to use SVD in many dimensions, SIAM J. Sci. Comput., submitted by publication. Also ICM HKBU Research Report 09-03, February 2009. .
  • 46
    • 0038954992 scopus 로고
    • On schemes for the evaluation of products and inverses of matrices
    • (in Russian)
    • Pan V.Y. On schemes for the evaluation of products and inverses of matrices. Uspekhi Mat. Nauk 27 5 (167) (1972) 249-250 (in Russian)
    • (1972) Uspekhi Mat. Nauk , vol.27 , Issue.5 167 , pp. 249-250
    • Pan, V.Y.1
  • 47
    • 0021452919 scopus 로고
    • How can we speed up matrix multiplication?
    • Pan V.Y. How can we speed up matrix multiplication?. SIAM Rev. 26 3 (1984) 393-415
    • (1984) SIAM Rev. , vol.26 , Issue.3 , pp. 393-415
    • Pan, V.Y.1
  • 48
    • 84966229708 scopus 로고
    • On computations with dense structured matrices
    • Pan V.Y. On computations with dense structured matrices. Math. Comput. 55 191 (1990) 179-190
    • (1990) Math. Comput. , vol.55 , Issue.191 , pp. 179-190
    • Pan, V.Y.1
  • 49
    • 38249015327 scopus 로고
    • Parallel solution of Toeplitz-like linear systems
    • Pan V.Y. Parallel solution of Toeplitz-like linear systems. J. Complexity 8 (1992) 1-21
    • (1992) J. Complexity , vol.8 , pp. 1-21
    • Pan, V.Y.1
  • 50
    • 0027595846 scopus 로고
    • Concurrent iterative algorithm for Toeplitz-like linear systems
    • Pan V.Y. Concurrent iterative algorithm for Toeplitz-like linear systems. IEEE Trans. Parallel Distributed Syst. 4 5 (1993) 592-600
    • (1993) IEEE Trans. Parallel Distributed Syst. , vol.4 , Issue.5 , pp. 592-600
    • Pan, V.Y.1
  • 51
    • 21144462386 scopus 로고
    • Decreasing the displacement rank of a matrix
    • Pan V.Y. Decreasing the displacement rank of a matrix. SIAM J. Matrix Anal. Appl. 14 1 (1993) 118-121
    • (1993) SIAM J. Matrix Anal. Appl. , vol.14 , Issue.1 , pp. 118-121
    • Pan, V.Y.1
  • 53
    • 77950368381 scopus 로고    scopus 로고
    • Newton's iteration for matrix inversion, advances and extensions
    • Olshevsky V., and Tyrtyshnikov E. (Eds), World Scientific Publishing ISBN: (978-981-283-601-4)-(981-283-601-2), Dedicated to the Memory of Gene Golub
    • Pan V.Y. Newton's iteration for matrix inversion, advances and extensions. In: Olshevsky V., and Tyrtyshnikov E. (Eds). Matrix Methods: Theory, Algorithms and Applications (2008), World Scientific Publishing 364-381 ISBN: (978-981-283-601-4)-(981-283-601-2), Dedicated to the Memory of Gene Golub
    • (2008) Matrix Methods: Theory, Algorithms and Applications , pp. 364-381
    • Pan, V.Y.1
  • 54
    • 77950371177 scopus 로고    scopus 로고
    • A note on the O (n)-storage implementation of the GKO algorithm, subjects: Numerical analysis (math.NA
    • Available from:
    • F. Poloni, A note on the O (n)-storage implementation of the GKO algorithm, subjects: numerical analysis (math.NA), 2009. Available from: .
    • (2009)
    • Poloni, F.1
  • 56
    • 55749102088 scopus 로고    scopus 로고
    • Schur aggregation for linear systems and determinants
    • Bini D.A., Pan V.Y., and Verschelde J. (Eds) Special Issue on Symbolic-Numerical Algorithms
    • Pan V.Y., Grady D., Murphy B., Qian G., Rosholt R.E., and Ruslanov A. Schur aggregation for linear systems and determinants. In: Bini D.A., Pan V.Y., and Verschelde J. (Eds). Theoret. Comput. Sci. 409 2 (2008) 255-268 Special Issue on Symbolic-Numerical Algorithms
    • (2008) Theoret. Comput. Sci. , vol.409 , Issue.2 , pp. 255-268
    • Pan, V.Y.1    Grady, D.2    Murphy, B.3    Qian, G.4    Rosholt, R.E.5    Ruslanov, A.6
  • 59
    • 44649102752 scopus 로고    scopus 로고
    • Additive preconditioning for matrix computations
    • Proc. of the Third International Computer Science Symposium in Russia CSR
    • V.Y. Pan, D. Ivolgin, B. Murphy, R.E. Rosholt, Y. Tang, X. Yan, Additive preconditioning for matrix computations, in: Proc. of the Third International Computer Science Symposium in Russia (CSR 2008), Lecture Notes in Computer Science (LNCS), vol. 5010, 2008, pp. 372-383.
    • (2008) Lecture Notes in Computer Science (LNCS , vol.5010 , pp. 372-383
    • Pan, V.Y.1    Ivolgin, D.2    Murphy, B.3    Rosholt, R.E.4    Tang, Y.5    Yan, X.6
  • 60
    • 77950371242 scopus 로고    scopus 로고
    • Error-free computations via floating-point operations
    • Also Tech. Report (with computer codes) TR 2007010, Ph.D. Program in Computer Science, Graduate Center, the City University of New York, 2007.
    • Pan V.Y., Murphy B., Qian G., and Rosholt R.E. Error-free computations via floating-point operations. Comput. Math. (with Appl.) 57 (2009) 560-564. http://www.cs.gc.cuny.edu/tr/techreport.php?id=352 Also Tech. Report (with computer codes) TR 2007010, Ph.D. Program in Computer Science, Graduate Center, the City University of New York, 2007.
    • (2009) Comput. Math. (with Appl.) , vol.57 , pp. 560-564
    • Pan, V.Y.1    Murphy, B.2    Qian, G.3    Rosholt, R.E.4
  • 61
    • 77950369231 scopus 로고    scopus 로고
    • V.Y. Pan, G. Qian, On solving linear system with randomized augmentation, Tech. Report TR 2009009, Ph.D. Program in Computer Science, Graduate Center, the City University of New York, 2009. .
    • V.Y. Pan, G. Qian, On solving linear system with randomized augmentation, Tech. Report TR 2009009, Ph.D. Program in Computer Science, Graduate Center, the City University of New York, 2009. .
  • 62
    • 77950370930 scopus 로고    scopus 로고
    • V.Y. Pan, G. Qian, A. Zheng, Randomized preprocessing versus pivoting, Tech. Report TR 2009010, Ph.D. Program in Computer Science, Graduate Center, the City University of New York, 2009. .
    • V.Y. Pan, G. Qian, A. Zheng, Randomized preprocessing versus pivoting, Tech. Report TR 2009010, Ph.D. Program in Computer Science, Graduate Center, the City University of New York, 2009. .
  • 63
    • 77950371630 scopus 로고    scopus 로고
    • Matrix computations and root-finding with randomized preprocessing
    • preprint
    • V.Y. Pan, G. Qian, A. Zheng, Z. Chen, Matrix computations and root-finding with randomized preprocessing, preprint, 2009.
    • (2009)
    • Pan, V.Y.1    Qian, G.2    Zheng, A.3    Chen, Z.4
  • 64
    • 0027799021 scopus 로고
    • Fast and efficient parallel solution of sparse linear systems
    • Pan V.Y., and Reif J. Fast and efficient parallel solution of sparse linear systems. SIAM J. Comput. 22 6 (1993) 1227-1250
    • (1993) SIAM J. Comput. , vol.22 , Issue.6 , pp. 1227-1250
    • Pan, V.Y.1    Reif, J.2
  • 65
    • 27644515683 scopus 로고    scopus 로고
    • Newton's iteration for the inversion of structured matrices
    • Bini D., Tyrtyshnikov E., and Yalamov P. (Eds), Nova Science Publishers, USA
    • Pan V.Y., and Rami Y. Newton's iteration for the inversion of structured matrices. In: Bini D., Tyrtyshnikov E., and Yalamov P. (Eds). Structured Matrices: Recent Developments in Theory and Computation (2001), Nova Science Publishers, USA 79-90
    • (2001) Structured Matrices: Recent Developments in Theory and Computation , pp. 79-90
    • Pan, V.Y.1    Rami, Y.2
  • 66
    • 84924422763 scopus 로고    scopus 로고
    • Structured matrices and Newton's iteration: unified approach
    • Pan V.Y., Rami Y., and Wang X. Structured matrices and Newton's iteration: unified approach. Linear Algebra Appl. 343-344 (2002) 233-265
    • (2002) Linear Algebra Appl. , vol.343-344 , pp. 233-265
    • Pan, V.Y.1    Rami, Y.2    Wang, X.3
  • 67
    • 0012063917 scopus 로고
    • An improved Newton iteration for the generalized inverse of a matrix, with applications
    • Pan V.Y., and Schreiber R. An improved Newton iteration for the generalized inverse of a matrix, with applications. SIAM J. Sci. Stat. Comput. 12 5 (1991) 1109-1131
    • (1991) SIAM J. Sci. Stat. Comput. , vol.12 , Issue.5 , pp. 1109-1131
    • Pan, V.Y.1    Schreiber, R.2
  • 68
    • 2042437068 scopus 로고    scopus 로고
    • Iterative inversion of structured matrices
    • Special Issue on Algebraic and Numerical Computing
    • Pan V.Y., Van Barel M., Wang X., and Codevico G. Iterative inversion of structured matrices. Theoret. Comput. Sci. 315 2-3 (2004) 581-592 Special Issue on Algebraic and Numerical Computing
    • (2004) Theoret. Comput. Sci. , vol.315 , Issue.2-3 , pp. 581-592
    • Pan, V.Y.1    Van Barel, M.2    Wang, X.3    Codevico, G.4
  • 69
    • 49249100712 scopus 로고    scopus 로고
    • Degeneration of integer matrices modulo an integer
    • Pan V.Y., and Wang X. Degeneration of integer matrices modulo an integer. Linear Algebra Appl. 429 (2008) 2113-2130
    • (2008) Linear Algebra Appl. , vol.429 , pp. 2113-2130
    • Pan, V.Y.1    Wang, X.2
  • 70
    • 54149114308 scopus 로고    scopus 로고
    • Additive preconditioning, eigenspaces, and the inverse iteration
    • Pan V.Y., and Yan X. Additive preconditioning, eigenspaces, and the inverse iteration. Linear Algebra Appl. 430 (2009) 186-203
    • (2009) Linear Algebra Appl. , vol.430 , pp. 186-203
    • Pan, V.Y.1    Yan, X.2
  • 71
    • 71649110841 scopus 로고    scopus 로고
    • Root-finding with eigen-solving and preprocessing
    • preprint
    • V.Y. Pan, A. Zheng, Root-finding with eigen-solving and preprocessing, preprint.
    • Pan, V.Y.1    Zheng, A.2
  • 72
    • 34547224513 scopus 로고    scopus 로고
    • Fast solution of Toeplitz- and Cauchy-like least squares problems
    • Rodriguez G. Fast solution of Toeplitz- and Cauchy-like least squares problems. SIAM J. Matrix Anal. Appl. 28 3 (2006) 724-748
    • (2006) SIAM J. Matrix Anal. Appl. , vol.28 , Issue.3 , pp. 724-748
    • Rodriguez, G.1
  • 73
    • 55049129860 scopus 로고    scopus 로고
    • Accurate floating-point summation, Part I: faithful rounding
    • Rump S.M., Ogita T., and Oishi S. Accurate floating-point summation, Part I: faithful rounding. SIAM J. Sci. Comput. 31 1 (2008) 169-224
    • (2008) SIAM J. Sci. Comput. , vol.31 , Issue.1 , pp. 169-224
    • Rump, S.M.1    Ogita, T.2    Oishi, S.3
  • 74
    • 67649610606 scopus 로고    scopus 로고
    • Accurate floating-point summation, Part II: sign, K-fold faithful and rounding to nearest
    • Rump S.M., Ogita T., and Oishi S. Accurate floating-point summation, Part II: sign, K-fold faithful and rounding to nearest. SIAM J. Sci. Comput. 31 2 (2008) 1269-1302
    • (2008) SIAM J. Sci. Comput. , vol.31 , Issue.2 , pp. 1269-1302
    • Rump, S.M.1    Ogita, T.2    Oishi, S.3
  • 75
    • 84976779342 scopus 로고
    • Fast probabilistic algorithms for verification of polynomial identities
    • Schwartz J.T. Fast probabilistic algorithms for verification of polynomial identities. J. ACM 27 4 (1980) 701-717
    • (1980) J. ACM , vol.27 , Issue.4 , pp. 701-717
    • Schwartz, J.T.1
  • 76
    • 33748875998 scopus 로고    scopus 로고
    • Matrix Algorithms
    • SIAM, Philadelphia
    • Stewart G.W. Matrix Algorithms. Basic Decompositions vol. I (1998), SIAM, Philadelphia
    • (1998) Basic Decompositions , vol.I
    • Stewart, G.W.1
  • 77
    • 33746243736 scopus 로고    scopus 로고
    • Matrix Algorithms
    • SIAM, Philadelphia
    • Stewart G.W. Matrix Algorithms. Eigensystems vol. II (1998), SIAM, Philadelphia
    • (1998) Eigensystems , vol.II
    • Stewart, G.W.1
  • 78
    • 34247360062 scopus 로고    scopus 로고
    • Smoothed analysis of the condition numbers and growth factors of matrices
    • Sankar A., Spielman D., and Teng S.-H. Smoothed analysis of the condition numbers and growth factors of matrices. SIAM J. Matrix Anal. 28 2 (2006) 446-476
    • (2006) SIAM J. Matrix Anal. , vol.28 , Issue.2 , pp. 446-476
    • Sankar, A.1    Spielman, D.2    Teng, S.-H.3
  • 80
    • 21344486577 scopus 로고
    • How bad are Hankel matrices?
    • Tyrtyshnikov E.E. How bad are Hankel matrices?. Numer. Math. 67 2 (1994) 261-269
    • (1994) Numer. Math. , vol.67 , Issue.2 , pp. 261-269
    • Tyrtyshnikov, E.E.1
  • 81
    • 77950370994 scopus 로고    scopus 로고
    • ∼marc/software/index.html>.
    • ∼marc/software/index.html>.
  • 82
    • 0036054763 scopus 로고    scopus 로고
    • A stabilized superfast solver for nonsymmetric Toeplitz systems
    • Van Barel M., Heinig G., and Kravanja P. A stabilized superfast solver for nonsymmetric Toeplitz systems. SIAM J. Matrix Anal. Appl. 23 2 (2001) 494-510
    • (2001) SIAM J. Matrix Anal. Appl. , vol.23 , Issue.2 , pp. 494-510
    • Van Barel, M.1    Heinig, G.2    Kravanja, P.3
  • 83
    • 0039260594 scopus 로고    scopus 로고
    • A stabilized superfast solver for indefinite Hankel systems
    • Van Barel M., and Kravanja P. A stabilized superfast solver for indefinite Hankel systems. Linear Algebra Appl. 284 1-3 (1998) 335-355
    • (1998) Linear Algebra Appl. , vol.284 , Issue.1-3 , pp. 335-355
    • Van Barel, M.1    Kravanja, P.2
  • 85
  • 89
    • 1242344845 scopus 로고    scopus 로고
    • Smoothed analysis of κ (a)
    • Wschebor M. Smoothed analysis of κ (a). J. Complexity 20 (2004) 97-107
    • (2004) J. Complexity , vol.20 , pp. 97-107
    • Wschebor, M.1
  • 90
    • 34547147441 scopus 로고    scopus 로고
    • Affect of small rank modification on the condition number of a matrix
    • Wang X. Affect of small rank modification on the condition number of a matrix. Comput. Math. (with Appl.) 54 (2007) 819-825
    • (2007) Comput. Math. (with Appl.) , vol.54 , pp. 819-825
    • Wang, X.1
  • 91
    • 84925463455 scopus 로고
    • Probabilistic algorithms for sparse polynomials
    • Proceedings of EUROSAM'79, Springer, Berlin
    • Zippel R.E. Probabilistic algorithms for sparse polynomials. Proceedings of EUROSAM'79. Lecture Notes in Computer Science vol. 72 (1979), Springer, Berlin 216-226
    • (1979) Lecture Notes in Computer Science , vol.72 , pp. 216-226
    • Zippel, R.E.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.