Bibliography for Ill-Conditioned Linear Systems

unabridged

  1. An upper bound for the condition number of a matrix in spectral norm
    Piazza, G.; Politi, T.
    Journal of Computational and Applied Mathematics, 2002, vol. 143, no. 1, pp. 141-144, Ingenta
  2. A survey of preconditioners for ill-conditioned Toeplitz systems
    Chan, R. H.; Ng, M. K.; Yip, A. M.
    Contemporary Mathematics, 2001, vol. 281, pp. 175-192, Ingenta
  3. Least-Squares Polynomial Filters for Ill-Conditioned Linear Systems
    Erhel, J.; Guyomarc h, F.; Saad, Y.
    Rapport de Recherche- Institut National de Recherche en Informatique et en Automatique, 2001, no. 4175, pp. ALL, Ingenta
  4. Preconditioners for Ill-Conditioned Toeplitz Systems Constructed from Positive Kernels
    Potts, D.; Steidl, G.
    Siam Journal on Scientific Computing, 2001, vol. 22, no. 5, pp. 1741-1761, Ingenta
  5. Convergence of the multigrid method for ill-conditioned block Toeplitz systems
    Sun, H. W.; Jin, X. Q.; Chang, Q. S.
    Bit, 2001, vol. 41, no. 1, pp. 179-190, Ingenta
  6. Composite structure method for ill-conditioned linear equations
    Zou, J.; Qian, J.
    Journal- Tsinghua University, 2001, vol. 41, no. 4/5, pp. 231-234, Ingenta
  7. Inversion error, condition number, and approximate inverses of uncertain matrices
    El Ghaoui, L.
    Linear Algebra and Its Applications, 2001, vol. 343-344, pp. 171-193, Ingenta
  8. Condition number and determinant of a matrix. (Spanish)
    Lemagne Pérez, Jorge
    Investigación Oper. 21 (2000), no. 1, 27--31, MathSciNet.  
  9. Pin-Pointing Solution of Ill-Conditioned Square Systems of Linear Equations
    Volokh, K. Y.
    Applied Mathematics Letters, 2000, vol. 13, no. ER7, pp. 119-124, Ingenta
  10. Condition number and backward error for the generalized singular value decomposition.
    Sun, Ji-Guang
    SIAM J. Matrix Anal. Appl. 22 (2000), no. 2, 323--341 (electronic), MathSciNet.  
  11. The condition number of real Vandermonde, Krylov and positive definite Hankel matrices.
    Beckermann, Bernhard
    Numer. Math. 85 (2000), no. 4, 553--577, MathSciNet.  
  12. Structured backward error and condition number for linear systems of the type A* Ax = b.
    Fraysse, V.
    Bit, 2000, vol. 40, no. 1, pp. 74, Ingenta
  13. Ill-Conditioned Convex Processes and Conic Linear Systems.
    Lewis, Adrian
    Mathematics of operations research, 1999, vol. 24, no. 4, pp. 829, Ingenta
  14. Preconditioners for ill-conditioned Toeplitz matrices.
    Potts, D.; Steidl, G.
    BIT, 1999, vol. 39, no. 3, pp. 513, Ingenta
  15. Condition Number of the Matrix of Transition to the Normal Jordan Form.
    Grabovskii, O. I.
    Ukrainian mathematical journal, 1999, vol. 51, no. 1, pp. 131, Ingenta
  16. The condition number of the Schur complement in domain decomposition.
    Brenner, S.C.
    Numerische mathematik, 1999, vol. 83, no. 2, pp. 187, Ingenta
  17. Condition numbers of matrices and correlativity of column vectors. (Chinese)
    Xu, Ming Hua; Wang, Jia Song
    Nanjing Daxue Xuebao Shuxue Bannian Kan 16 (1999), no. 1, 82--87, MathSciNet.  
  18. Extrapolation techniques for ill-conditioned linear systems.
    Brezinski, C.; Redivo-Zaglia, M.; Seatzu, S.
    Numerische mathematik, 1998, vol. 81, no. 1, pp. 1, Ingenta
  19. Ill-Conditioned Covariance Matrices in the First-Order Two-Step Estimator.
    Garrison, J.L.; Axelrad, P.; Kasdin, N.J.
    Journal of guidance, control, and dynamics, 1998, vol. 21, no. 5, pp. 754, Ingenta
  20. Multigrid Method for Ill-Conditioned Symmetric Toeplitz Systems.
    Chan, Raymond H.; Chang, Qian-Shun; Sun, Hai-Wei
    SIAM journal on scientific computing, 1998, vol. 19, no. 2, pp. 516, Ingenta
  21. Reliable computation of the condition number of a tridiagonal matrix in O(n) time.
    Dhillon, Inderjit S.
    SIAM J. Matrix Anal. Appl. 19 (1998), no. 3, 776--796 (electronic), MathSciNet.  
  22. Condition numbers of random triangular matrices.
    Viswanath, D.; Trefethen, L. N.
    SIAM J. Matrix Anal. Appl. 19 (1998), no. 2, 564--581 (electronic), MathSciNet.  
  23. On Matrix Unitarily Invariant Norm Condition Number.
    Zheng, Dao-sheng
    J. Comput. Math. 16 (1998), no. 2, 121--128, MathSciNet.  
  24. A note on properties of condition numbers.
    Gonzaga, Clovis C.; Lara, Hugo J.
    Linear Algebra Appl. 261 (1997), 269--273, MathSciNet.  
  25. An upper bound for the spectral condition number of a diagonalizable matrix.
    Jiang, Erxiong; Lam, Peter C. B.
    Linear Algebra Appl. 262 (1997), 165--178, MathSciNet.  
  26. A Simple Estimate of the Condition Number of a Linear System   
    Heinrich W. Guggenheimer, Alan S. Edelman, Charles R. Johnson   
    College Math Journal: Volume 26, Number 1, (1995), Pages: 2-5, Ingenta
  27. On solving linear algebraic equations with an ill-conditioned matrix.
    Moszynski, K.
    Zastosowanie matematyki, 1995, vol. 22, no. 4, pp. 499, Ingenta
  28. Round-off, Batting Averages, and Ill-Conditioning  
    Edward Rozema  
    College Math Journal: Volume 25, Number 4, (1994), Pages: 314-317   
  29. Preconditioning strategies for asymptotically ill-conditioned block Toeplitz systems.
    Serra, Stefano
    BIT, 1994, no. 4, pp. 579, Ingenta
  30. Interval methods and condition numbers of linear algebraic systems. SCAN-93 (Vienna, 1993).
    Christiansen, Søren
    Interval Comput. Interval. Vychisl. 1994, no. 4, 5--27, MathSciNet.  
  31. Minimizing the Euclidean condition number.
    Braatz, Richard D.; Morari, Manfred
    SIAM J. Control Optim. 32 (1994), no. 6, 1763--1768, MathSciNet.  
  32. Perturbation Analysis of a Condition Number for Linear Systems.
    Luo, Zhi-Quan; Tseng, Paul
    SIAM J. Matrix Anal. Appl. 15 (1994), no. 2, 636--660, MathSciNet.  
  33. Some results on condition numbers and bounds of matrix norms. (Chinese)
    Zheng, Dao Sheng
    J. East China Norm. Univ. Natur. Sci. Ed. 1993, no. 2, 1--8, MathSciNet.  
  34. A posteriori bounds on the condition number in spectral norm for tridiagonal matrices. (Italian)
    Mastronardi, N.; Piazza, G.
    Calcolo 29 (1992), no. 1-2, 81--95 (1993), MathSciNet.  
  35. On the Distribution of a Scaled Condition Number  
    Alan Edelman  
    Mathematics of Computation, Vol. 58, No. 197. (Jan., 1992), pp. 185-190, Jstor.  
  36. On the approximate computation of extreme eigenvalues and the condition number of nonsingular matrices.
    Lei, Guang Yao
    Appl. Math. Mech. (English Ed.) 13 (1992), no. 2, 199--204; translated from
    Appl. Math. Mech. 13 (1992), no. 2, 181--186(Chinese), MathSciNet.  
  37. On the Distribution of a Scaled Condition Number.
    Edelaman, A.
    Mathematics of computation, 1992, vol. 58, no. 197, pp. 185, Ingenta
  38. A method for solving ill-conditioned systems of linear algebraic equations.
    Abramov, A.A.
    Computational mathematics and mathematical physics, 1991, vol. 31, no. 4, pp. 1, Ingenta
  39. Condition numbers of random matrices.
    Szarek, Stanislaw J.
    J. Complexity 7 (1991), no. 2, 131--149, MathSciNet.  
  40. Assessing the Effects of Multiple Rows on the Condition Number of a Matrix (in Theory and Methods)  
    Ali S. Hadi, Martin T. Wells  
    Journal of the American Statistical Association, Vol. 85, No. 411. (Sep., 1990), pp. 786-792, Jstor.  
  41. Estimators of the condition number of matrices. (Spanish)
    González, G.; Vargas, C.
    XXIInd National Congress of the Mexican Mathematical Society (Spanish) (Puebla, 1989), 47--64, Aportaciones Mat. Comun., 8, Soc. Mat. Mexicana, México, 1990, MathSciNet.  
  42. Interpretation and Improved Solution Approach for Ill-Conditioned Linear Equations.
    Ojalvo, I.U.; Ting, T.
    AIAA journal, 1990, vol. 28, no. 11, pp. 1976, Ingenta
  43. An improved Lanczos algorithm for solving ill-conditioned linear equations.
    Shao, P.-L.
    Computers & mathematics with applications, 1990, vol. 20, no. 12, pp. 25, Ingenta
  44. Assessing the effects of multiple rows on the condition number of a matrix.
    Hadi, Ali S.; Wells, Martin T.
    J. Amer. Statist. Assoc. 85 (1990), no. 411, 786--792, MathSciNet.  
  45. Applications of the condition number to numerical analysis in the identification of large-scale systems. (Chinese)
    Zhang, Hong Yi; Davison, E. J.
    Control Theory Appl. 7 (1990), no. 4, 93--97, MathSciNet.  
  46. Improved Condition Number for Spectral Methods  
    Wilhelm Heinrichs  
    Mathematics of Computation, Vol. 53, No. 187. (Jul., 1989), pp. 103-119, Jstor.  
  47. An easily computed upper bound for the condition number of a matrix.
    Popov, Wladimir
    Serdica 15 (1989), no. 3, 192--196, MathSciNet.  
  48. New condition numbers for matrices and linear systems.
    Rohn, J.
    Computing 41 (1989), no. 1-2, 167--169, MathSciNet.  
  49. Eigenvalues and condition numbers of random matrices.
    Edelman, Alan
    SIAM J. Matrix Anal. Appl. 9 (1988), no. 4, 543--560, MathSciNet.  
  50. A Survey of Condition Number Estimation for Triangular Matrices  
    Nicholas J. Higham  
    SIAM Review, Vol. 29, No. 4. (Dec., 1987), pp. 575-596, Jstor.  
  51. A Simple Estimate of the Condition Number of a Linear System  
    Heinrich W. Guggenheimer, Alan S. Edelman, Charles R. Johnson  
    College Math Journal: Volume 26, Number 1, (1995), Pages: 2-5.
  52. Ill-conditioning: A constant Surprise in Computational Mathematics   
    Bruce H. Edwards and Patricia L. Sharpe   
    College Math Journal: Volume 16, Number 2, (1985), Pages: 141-148.   
  53. Efficient algorithms for computing the condition number of a tridiagonal matrix.
    Higham, Nicholas J.
    SIAM J. Sci. Statist. Comput. 7 (1986), no. 1, 150--165, MathSciNet.  
  54. The Condition Number of Equivalence Transformations that Block Diagonalize Matrix Pencils  
    James Demmel  
    SIAM Journal on Numerical Analysis, Vol. 20, No. 3. (Jun., 1983), pp. 599-610, Jstor.  
  55. Tricks or Treats with the Hilbert Matrix
    Man-Duen Choi
    American Mathematical Monthly, Vol. 90, No. 5. (May, 1983), pp. 301-312, Jstor.  
  56. A fast method for estimating the condition number of a matrix.
    Natori, Makoto; Tsukamoto, Atsuko
    J. Inform. Process. 6 (1983), no. 3, 138--140, MathSciNet.  
  57. Estimating extremal eigenvalues and condition numbers of matrices.
    Dixon, John D.
    SIAM J. Numer. Anal. 20 (1983), no. 4, 812--814, MathSciNet.  
  58. A remark on Hilbert's matrix.    
    Grünbaum, F. Alberto     
    Linear Algebra Appl. 43 (1982), 119--124, MathSciNet.  
  59. Estimating matrix condition numbers.
    O'Leary, Dianne Prost
    SIAM J. Sci. Statist. Comput. 1 (1980), no. 2, 205--209, MathSciNet.  
  60. An Estimate for the Condition Number of a Matrix  
    A. K. Cline, C. B. Moler, G. W. Stewart, J. H. Wilkinson  
    SIAM Journal on Numerical Analysis, Vol. 16, No. 2. (Apr., 1979), pp. 368-375, Jstor.  
  61. Some numerical problems in connection with the Hilbert matrix. (Swedish)    
    Fröberg, Carl-Erik    
    Nordisk Mat. Tidskr. 25/26 (1978), no. 3-4, 188--190, 210, MathSciNet.  
  62. On the Condition Number of Local Bases for Piecewise Cubic Polynomials  
    J. M. Varah  
    Mathematics of Computation, Vol. 31, No. 137. (Jan., 1977), pp. 37-44, Jstor.  
  63. Ill-Conditioned Eigensystems and the Computation of the Jordan Canonical Form  
    G. H. Golub, J. H. Wilkinson  
    SIAM Review, Vol. 18, No. 4. (Oct., 1976), pp. 578-619, Jstor.  
  64. Some condition-number bounds for the Gaussian elimination process.
    Broyden, C. G.
    J. Inst. Math. Appl. 12 (1973), 273--286, MathSciNet.  
  65. On the Numerical Solution of Ill-Conditioned Linear Systems with Applications to Ill-Posed Problems  
    J. M. Varah  
    SIAM Journal on Numerical Analysis, Vol. 10, No. 2. (Apr., 1973), pp. 257-267, Jstor.  
  66. An Algorithm for the Ill-Conditioned Generalized Eigenvalue Problem  
    G. Fix, R. Heiberger  
    SIAM Journal on Numerical Analysis, Vol. 9, No. 1. (Mar., 1972), pp. 78-88, Jstor.  
  67. Some Elementary Properties of Ill Conditioned Matrices and Linear Equations
    N. S. Mendelsohn
    American Mathematical Monthly, Vol. 63, No. 5. (May, 1956), pp. 285-295, Jstor.  
  68. Computational problems concerning the Hilbert matrix.    
    Todd, John    
    J. Res. Nat. Bur. Standards Sect. B 65B 1961 19--22, MathSciNet.  
  69. Effect of propagated error on inverse of Hilbert matrix.    
    Atta, Susie E.     
    J. Assoc. Comput. Mach. 4 (1957), 36--40, MathSciNet.  
  70. On the Hilbert Matrix  
    Tosio Kato  
    Proceedings of the American Mathematical Society, Vol. 8, No. 1. (Feb., 1957), pp. 73-81, Jstor.  
  71. Some Elementary Properties of Ill Conditioned Matrices and Linear Equations  
    N. S. Mendelsohn  
    American Mathematical Monthly, Vol. 63, No. 5. (May, 1956), pp. 285-295, Jstor.  
  72. Solving Systems of Linear Equations With a Positive Definite, Symmetric, but Possibly Ill-Conditioned Matrix  
    James D. Riley  
    Mathematical Tables and Other Aids to Computation, Vol. 9, No. 51. (Jul., 1955), pp. 96-101, Jstor.  

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2003