References related to numerically singular matrices sorted chronologically, most recent first. (sorted alphabetically)

T. A. Davis. Multifrontal multithreaded rank-revealing sparse qr factorization. submitted to ACM TOMS, 2008. [ .pdf ]

T. A. Davis. Algorithm 8xx: Suitesparseqr, a multifrontal multithreaded sparse qr factorization package. submitted to ACM TOMS, 2008. [ .pdf ]

Craig Gotsman and Sivan Toledo. On the computation of null spaces of sparse rectangular matrices. SIAM Journal on Matrix Analysis and Applications, 30(2):445-463, 2008. [ DOI | .pdf ]

James Demmel and Plamen Koev. Accurate SVDs of polynomial Vandermonde matrices involving orthonormal polynomials. Linear Algebra Appl., 417(2-3):382-396, 2006. [ .pdf ]

Froilán M. Dopico and Plamen Koev. Accurate symmetric rank revealing and eigendecompositions of symmetric structured matrices. SIAM J. Matrix Anal. Appl., 28(4):1126-1156 (electronic), 2006. [ .pdf ]

Leslie Foster and Rajesh Kommu. Algorithm 853: an efficient algorithm for solving rank-deficient least squares problems. ACM Transactions on Mathematical Software, 32(1):157-165, 2006. [ .pdf ]

Michael O'Sullivan and Michael Saunders. Lusol: A basis package for constrained optimization. In IFORS Triennial Conference on OR/MS, Honolulu, Hawaii, July 2005. [ .pdf ]

Ricardo D. Fierro and Per Christian Hansen. UTV expansion pack: Special-purpose rank-revealing algorithms. Numer. Algorithms, 40(1):47-66, 2005. [ .ps ]

D. A. Huckaby and T. F. Chan. Stewart's pivoted QLP decomposition for low-rank matrices. Numer. Linear Algebra Appl., 12(2-3):153-159, 2005. [ .ps.gz ]

Plamen Koev. Accurate eigenvalues and SVDs of totally nonnegative matrices. SIAM J. Matrix Anal. Appl., 27(1):1-23 (electronic), 2005. [ .pdf ]

T. Y. Li and Zhonggang Zeng. A rank-revealing method with updating, downdating and applications. SIAM J. Matrix Anal. Appl., 26(4):918-946, 2005. [ .pdf ]

James Demmel and Plamen Koev. Accurate SVDs of weakly diagonally dominant M-matrices. Numer. Math., 98(1):99-104, 2004. [ .pdf ]

M. Jacobsen. Modular Regularization Algorithms. PhD thesis, Informatics and Mathematical Modelling, Technical University of Denmark, DTU, Richard Petersens Plads, Building 321, DK-2800 Kgs. Lyngby, 2004. Supervised by Prof. Per Christian Hansen. [ http ]

J. M. Peña. LDU decompositions with L and U well conditioned. Electron. Trans. Numer. Anal., 18:198-208 (electronic), 2004. [ .pdf ]

Peter Benner, Enrique S. Quintana-Ortí, and Gregorio Quintana-Ortí. State-space truncation methods for parallel model reduction of large-scale systems. Parallel Comput., 29(11-12):1701-1722, 2003. Parallel and distributed scientific and engineering computing. [ .pdf ]

Leslie V. Foster. Solving rank-deficient and ill-posed problems using UTV and QR factorizations. SIAM J. Matrix Anal. Appl., 25(2):582-600 (electronic), 2003. [ .pdf ]

L. Miranian and M. Gu. Strong rank revealing LU factorizations. Linear Algebra Appl., 367:1-16, 2003. [ .pdf ]

Michael O'Sullivan and Michael Saunders. Sparse rank-revealing lu factorization. In Householder Symposium XV on Numerical Linear Algebra, Peebles, Scotland, June 2002. [ .pdf ]

I. Enting. Inverse Problems in atmospheric constituent transport. Cambridge University Press, Cambridge, 2002.

Ricardo D. Fierro and Per Christian Hansen. Truncated VSV solutions to symmetric rank-deficient problems. BIT, 42(3):531-540, 2002.

Nicholas J. Higham. Accuracy and stability of numerical algorithms. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, second edition, 2002.

Curtis R. Vogel. Computational methods for inverse problems, volume 23 of Frontiers in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. With a foreword by H. T. Banks.

Per Christian Hansen and Plamen Y. Yalamov. Computing symmetric rank-revealing decompositions via triangular factorization. SIAM J. Matrix Anal. Appl., 23(2):443-458 (electronic), 2001. [ .pdf ]

F. Natterer. The mathematics of computerized tomography, volume 32 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001. Reprint of the 1986 original.

C.-T. Pan. On the existence and computation of rank-revealing LU factorizations. Linear Algebra Appl., 316(1-3):199-222, 2000. Conference Celebrating the 60th Birthday of Robert J. Plemmons (Winston-Salem, NC, 1999).

E. Anderson, Z. Bai, S. Bischof, L. S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, S. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen. LAPACK users' guide, third edition. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. [ .html ]

James Demmel, Ming Gu, Stanley Eisenstat, Ivan Slapničar, Krešimir Veselić, and Zlatko Drmač. Computing the singular value decomposition with high relative accuracy. Linear Algebra Appl., 299(1-3):21-80, 1999. [ .pdf ]

Ricardo D. Fierro, Per Christian Hansen, and Peter Søren Kirk Hansen. UTV tools: Matlab templates for rank-revealing UTV decompositions. Numer. Algorithms, 20(2-3):165-194, 1999. [ .pdf ]

Ren-Cang Li. Relative perturbation theory. II. Eigenspace and singular subspace variations. SIAM J. Matrix Anal. Appl., 20(2):471-492 (electronic), 1999. [ .ps ]

Ching-Tsuan Pan and Ping Tak Peter Tang. Bounds on singular values revealed by QR factorizations. BIT, 39(4):740-756, 1999.

G. W. Stewart. The QLP approximation to the singular value decomposition. SIAM J. Sci. Comput., 20(4):1336-1348 (electronic), 1999.

Christian H. Bischof and Gregorio Quintana-Ortí. Computing rank-revealing QR factorizations of dense matrices. ACM Trans. Math. Software, 24(2):226-253, 1998. [ .ps.Z ]

Christian H. Bischof and Gregorio Quintana-Ortí. Algorithm 782: codes for rank-revealing QR factorizations of dense matrices. ACM Trans. Math. Software, 24(2):254-257, 1998.

Arnold Neumaier. Solving ill-conditioned and singular linear systems: a tutorial on regularization. SIAM Rev., 40(3):636-666 (electronic), 1998. [ .ps.gz ]

Gregorio Quintana-Ortí, Xiaobai Sun, and Christian H. Bischof. A BLAS-3 version of the QR factorization with column pivoting. SIAM J. Sci. Comput., 19(5):1486-1494 (electronic), 1998.

Ricardo D. Fierro and Per Christian Hansen. Low-rank revealing UTV decompositions. Numer. Algorithms, 15(1):37-55, 1997.

Ricardo D. Fierro, Leentje Vanhamme, and Sabine Van Huffel. Total least squares algorithms based on rank-revealing complete orthogonal decompositions. In Recent advances in total least squares techniques and errors-in-variables modeling (Leuven, 1996), pages 99-116. SIAM, Philadelphia, PA, 1997.

Daniel J. Pierce and John G. Lewis. Sparse multifrontal rank revealing qr factorization. SIAM Journal on Matrix Analysis and Applications, 18(1):159-180, 1997. [ DOI | http ]

Lloyd N. Trefethen and David Bau, III. Numerical linear algebra. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997.

Åke Björck. Numerical methods for least squares problems. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996.

Heinz W. Engl, Martin Hanke, and Andreas Neubauer. Regularization of inverse problems, volume 375 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht, 1996.

Gene H. Golub and Charles F. Van Loan. Matrix computations. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore, MD, third edition, 1996.

Ming Gu and Stanley C. Eisenstat. Efficient algorithms for computing a strong rank-revealing QR factorization. SIAM J. Sci. Comput., 17(4):848-869, 1996.

Stanley C. Eisenstat and Ilse C. F. Ipsen. Relative perturbation techniques for singular value problems. SIAM J. Numer. Anal., 32(6):1972-1988, 1995.

Ricardo D. Fierro and James R. Bunch. Bounding the subspaces from rank revealing two-sided orthogonal decompositions. SIAM J. Matrix Anal. Appl., 16(3):743-759, 1995.

Ricardo D. Fierro and Per Christian Hansen. Accuracy of TSVD solutions computed from rank-revealing decompositions. Numer. Math., 70(4):453-471, 1995.

Per Christian Hansen. Test matrices for regularization methods. SIAM J. Sci. Comput., 16(2):506-512, 1995.

Charles L. Lawson and Richard J. Hanson. Solving least squares problems, volume 15 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. Revised reprint of the 1974 original.

A. Thorpe and L. Scharf. Data adaptive rank shaping methods for solving least squares problems. IEEE Trans. Signal Process., 43:pp. 1591-1601, 1995.

Alfred S. Carasso. Overcoming Hölder continuity in ill-posed continuation problems. SIAM J. Numer. Anal., 31(6):1535-1557, 1994.

Tony F. Chan and Per Christian Hansen. Low-rank revealing QR factorizations. Numer. Linear Algebra Appl., 1(1):33-44, 1994.

Shivkumar Chandrasekaran and Ilse C. F. Ipsen. On rank-revealing factorisations. SIAM J. Matrix Anal. Appl., 15(2):592-622, 1994.

Per Christian Hansen. Regularization tools: a Matlab package for analysis and solution of discrete ill-posed problems. Numer. Algorithms, 6(1-2):1-35, 1994.

G. W. Stewart. UTV decompositions. In Numerical analysis 1993 (Dundee, 1993), volume 303 of Pitman Res. Notes Math. Ser., pages 225-236. Longman Sci. Tech., Harlow, 1994.

James W. Demmel and William Gragg. On computing accurate singular values and eigenvalues of matrices with acyclic graphs. Linear Algebra Appl., 185:203-217, 1993.

R. Mathias and G. W. Stewart. A block QR algorithm and the singular value decomposition. Linear Algebra Appl., 182:91-100, 1993.

V. A. Morozov. Regularization methods for ill-posed problems. CRC Press, Boca Raton, FL, 1993. Translated from the 1987 Russian original.

G. W. Stewart. Updating a rank-revealing ULV decomposition. SIAM J. Matrix Anal. Appl., 14(2):494-499, 1993.

Jesse L. Barlow and Udaya B. Vemulapati. Rank detection methods for sparse matrices. SIAM J. Matrix Anal. Appl., 13(4):1279-1297, 1992. [ http ]

Tony F. Chan and Per Christian Hansen. Some applications of the rank revealing QR factorization. SIAM J. Sci. Statist. Comput., 13(3):727-741, 1992.

Y. P. Hong and C.-T. Pan. Rank-revealing QR factorizations and the singular value decomposition. Math. Comp., 58(197):213-232, 1992.

G. W. Stewart. An updating algorithm for subspace tracking. IEEE Trans. on SP, 40:1535-1541, 1992. [ .pdf ]

Christian H. Bischof and Per Christian Hansen. Structure-preserving and rank-revealing QR-factorizations. SIAM J. Sci. Statist. Comput., 12(6):1332-1350, 1991. [ http ]

Esmond Ng. A scheme for handling rank-deficiency in the solution of sparse linear least squares problems. SIAM Journal on Scientific and Statistical Computing, 12(5):1173-1183, 1991. [ DOI | http ]

Christian H. Bischof, John G. Lewis, and Daniel J. Pierce. Incremental condition estimation for sparse matrices. SIAM J. Matrix Anal. Appl., 11(4):644-659, 1990.

Tony F. Chan and Per Christian Hansen. Computing truncated singular value decomposition least squares solutions by rank revealing QR-factorizations. SIAM J. Sci. Statist. Comput., 11(3):519-530, 1990.

Leslie V. Foster. The probability of large diagonal elements in the QR factorization. SIAM J. Sci. Statist. Comput., 11(3):531-544, 1990. [ http ]

G. W. Stewart and Ji Guang Sun. Matrix perturbation theory. Computer Science and Scientific Computing. Academic Press Inc., Boston, MA, 1990.

W. Menke. Geophysical data analysis, discrete inverse theory. Academic Press, San Diego, 1989.

M. Bertero, C. De Mol, and E. Pike. Applied inverse problems in optics. In Inverse and Ill-Posed Problems, pages 291-313. Academic Press, London, 1987.

Tony F. Chan. Rank revealing QR factorizations. Linear Algebra Appl., 88/89:67-82, 1987.

Peter Hall and D. M. Titterington. Common structure of techniques for choosing smoothing parameters in regression problems. J. Roy. Statist. Soc. Ser. B, 49(2):184-198, 1987.

Per Christian Hansen. The truncated SVD as a method for regularization. BIT, 27(4):534-553, 1987.

Nicholas J. Higham. A survey of condition number estimation for triangular matrices. SIAM Rev., 29(4):575-596, 1987.

G. Nolet. Seismic Tomography. Kluwer, Dordrecht, the Netherlands, 1987.

Ian J. D. Craig and John C. Brown. Inverse problems in astronomy. Adam Hilger Ltd., Bristol, 1986. A guide to inversion strategies for remotely sensed data.

Leslie V. Foster. Rank and null space calculations using matrix decomposition without column interchanges. Linear Algebra Appl., 74:47-71, 1986. [ .pdf ]

Tony F. Chan. On the existence and computation of LU-factorizations with small pivots. Math. Comp., 42(166):535-547, 1984.

G. W. Stewart. Rank degeneracy. SIAM J. Sci. Statist. Comput., 5(2):403-413, 1984.

J. ChristesenDalsgarrd, P.C. Hansen, and M. Thompson. Gsvd analysis of heliseismic inversions. Month. Not. R. Astr. Soc., 264:pp. 541-565, 1983.

N. J. Higham. Upper bounds for the condition number of a triangular matrix. Numerical Analysis Report No. 86, University of Manchester,England, 1983.

Herbert B. Keller. The bordering algorithm and path following near singular points of higher nullity. SIAM J. Sci. Statist. Comput., 4(4):573-582, 1983.

Michael T. Heath. Some extensions of an algorithm for sparse linear least squares problems. SIAM J. Sci. Statist. Comput., 3(2):223-237, 1982. [ http ]

Christopher C. Paige and Michael A. Saunders. LSQR: an algorithm for sparse linear equations and sparse least squares. ACM Trans. Math. Software, 8(1):43-71, 1982.

G. W. Stewart. The efficient generation of random orthogonal matrices with an application to condition estimators. SIAM J. Numer. Anal., 17(3):403-409 (loose microfiche suppl.), 1980.

J. J. Dongarra, J. R. Bunch, C. B. Moler, and G. W. Stewart. Linpack User's Guide. SIAM, Philadelphia, 1979.

H. Andrews and B. Hunt. Digital Image Restoration. Prentice-Hall, Englewood Cliffs, NJ, 1977.

Andrey N. Tikhonov and Vasiliy Y. Arsenin. Solutions of ill-posed problems. V. H. Winston & Sons, Washington, D.C.: John Wiley & Sons, New York, 1977. Translated from the Russian, Preface by translation editor Fritz John, Scripta Series in Mathematics.

Gene Golub, Virginia Klema, and G. W. Stewart. Rank degeneracy and least squares problems. Technical Report STAN-CS-76-559, Stanford, 1976. [ .pdf ]

W. B. Gragg and G. W. Stewart. A stable variant of the secant method for solving nonlinear equations. SIAM J. Numer. Anal., 13(6):889-903, 1976.

Peter Businger and Gene H. Golub. Handbook series linear algebra. Linear least squares solutions by Householder transformations. Numer. Math., 7:269-276, 1965.

Gene H. Golub. Numerical methods for solving linear least squares problems. Apl. Mat., 10:213-216, 1965.


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Maintained by Leslie Foster, last updated 18-Apr-2009.