A Survey of Componentwise Perturbation Theory

Perturbation bounds in numerical linear algebra are traditionally derived and expressed using norms. Norm bounds cannot reflect the scaling or sparsity of a problem and its perturbation, and so can be unduly weak. If the problem data and its perturbation are measured componentwise, much smaller and more revealing bounds can be obtained. A survey is given of componentwise perturbation theory in numerical linear algebra, covering linear systems, the matrix inverse, matrix factorizations, the least squares problem, and the eigenvalue and singular value problems. Most of the results described have been published in the last five years.

[1]  C. T. Fike,et al.  Norms and exclusion theorems , 1960 .

[2]  J. H. Wilkinson.  Error analysis of floating-point computation , 1960 .

[3]  W. Prager,et al.  Compatibility of approximate solution of linear equations with given error bounds for coefficients and right-hand sides , 1964 .

[4]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .

[5]  J. H. Wilkinson,et al.  Admissible Solutions of Linear Systems with Not Sharply Defined Coefficients , 1965 .

[6]  W. Kahan.  Accurate eigenvalues of a symmetric tri-diagonal matrix , 1966 .

[7]  J. H. Wilkinson,et al.  Note on the iterative refinement of least squares solution , 1966 .

[8]  J. L. Rigal,et al.  On the Compatibility of a Given Solution With the Data of a Linear System , 1967, JACM.

[9]  P. Businger.  Matrices which can be optimally scaled , 1968 .

[10]  A. Sluis.  Condition numbers and equilibration of matrices , 1969 .

[11]  W. Kahan,et al.  The Rotation of Eigenvectors by a Perturbation. III , 1970 .

[12]  G. Stewart.  Error and Perturbation Bounds for Subspaces Associated with Certain Eigenvalue Problems , 1973 .

[13]  P. Wedin.  Perturbation theory for pseudo-inverses , 1973 .

[14]  Z. Kovarik.  Compatibility of approximate solutions of inaccurate linear equations , 1976 .

[15]  Donald B. Rubin,et al.  The Acceptability of Regression Solutions: Another Look at Computational Accuracy , 1976 .

[16]  G. Stewart.  Perturbation Bounds for the $QR$ Factorization of a Matrix , 1977 .

[17]  Robert D. Skeel,et al.  Scaling for Numerical Stability in Gaussian Elimination , 1979, JACM.

[18]  R. Skeel.  Iterative refinement implies numerical stability for Gaussian elimination , 1980 .

[19]  B. Parlett.  The Symmetric Eigenvalue Problem , 1981 .

[20]  A. J. Geurts,et al.  A contribution to the theory of condition , 1982 .

[21]  Optimally scaled matrices, necessary and sufficient conditions , 1982 .

[22]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[23]  J. Barlow,et al.  Computing accurate eigensystems of scaled diagonally dominant matrices: LAPACK working note No. 7 , 1988 .

[24]  Nicholas J. Higham,et al.  FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation , 1988, TOMS.

[25]  Nicholas J. Higham,et al.  The accuracy of solutions to triangular systems , 1989 .

[26]  J. Demmel,et al.  The strong stability of algorithms for solving symmetric linear systems , 1989 .

[27]  J. Demmel,et al.  The bidiagonal singular value decomposition and Hamiltonian mechanics: LAPACK working note No. 11 , 1989 .

[28]  I. Duff,et al.  On the augmented system approach to sparse least-squares problems , 1989 .

[29]  J. Demmel,et al.  On Floating Point Errors in Cholesky , 1989 .

[30]  A. S. Deif.  Realistic a Priori and a Posteriori Error Bounds for Computed Eigenvalues , 1990 .

[31]  G. Stewart,et al.  Matrix Perturbation Theory , 1990 .

[32]  G. W. Stewart,et al.  Stochastic Perturbation Theory , 1990, SIAM Rev..

[33]  James Demmel,et al.  Accurate Singular Values of Bidiagonal Matrices , 1990, SIAM J. Sci. Comput..

[34]  A. Barrlund.  Perturbation bounds for theLDLH andLU decompositions , 1991 .

[35]  Ji-guang Sun.  Perturbation bounds for the Cholesky andQR factorizations , 1991 .

[36]  N. Higham.  Iterative refinement enhances the stability ofQR factorization methods for solving linear equations , 1991 .

[37]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[38]  Å. Björck.  Component-wise perturbation analysis and error bounds for linear least squares solutions , 1991 .

[39]  W. Gragg,et al.  On computing accurate singular values and eigenvalues of matrices with acyclic graphs , 1992 .

[40]  Sven G. Bartels,et al.  The structured sensitivity of Vandermonde-like systems , 1992 .

[41]  Nicholas J. Higham,et al.  Backward Error and Condition of Structured Linear Systems , 1992, SIAM J. Matrix Anal. Appl..

[42]  James Demmel,et al.  Jacobi's Method is More Accurate than QR , 1989, SIAM J. Matrix Anal. Appl..

[43]  N. Higham,et al.  Stability of methods for matrix inversion , 1992 .

[44]  James Demmel,et al.  The Componentwise Distance to the Nearest Singular Matrix , 1992, SIAM J. Matrix Anal. Appl..

[45]  Svatopluk Poljak,et al.  Checking robust nonsingularity is NP-hard , 1993, Math. Control. Signals Syst..

[46]  H. Zha.  A componentwise perturbation analysis of the QR decomposition , 1993 .

[47]  J. Demmel,et al.  Improved Error Bounds for Underdetermined System Solvers , 1993, SIAM J. Matrix Anal. Appl..

[48]  Israel Koltracht,et al.  Mixed componentwise and structured condition numbers , 1993 .

[49]  G. Stewart.  On the perturbation of LU, Cholesky, and QR factorizations , 1993 .

[50]  Ivan Slapničar,et al.  Floating-point perturbations of Hermitian matrices , 1993 .

[51]  N. Higham,et al.  COMPONENTWISE ERROR ANALYSIS FOR STATIONARY ITERATIVE METHODS , 1993 .

[52]  Israel Koltracht,et al.  On accurate computations of the Perron root , 1993 .

[53]  C. O'Cinneide.  Entrywise perturbation theory and error analysis for Markov chains , 1993 .

[54]  Ilse C. F. Ipsen,et al.  On Rank-Revealing Factorisations , 1994, SIAM J. Matrix Anal. Appl..

[55]  B. Parlett,et al.  Accurate singular values and differential qd algorithms , 1994 .

[56]  Ilse C. F. Ipsen,et al.  Uniform Stability of Markov Chains , 1994, SIAM J. Matrix Anal. Appl..

[57]  B. Parlett,et al.  Accurate singular values and differential qd algorithms , 1994 .

[58]  Ilse C. F. Ipsen,et al.  Relative perturbation techniques for singular value problems , 1995 .

[59]  Ji-Guang Sun,et al.  Optimal backward perturbation bounds for the linear least squares problem , 1995, Numer. Linear Algebra Appl..

[60]  Desmond J. Higham,et al.  Condition numbers and their condition numbers , 1995 .