References related to numerically singular matrices sorted alphabetically by author. (sorted by date)

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 ]

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

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

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 ]

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.

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 ]

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.

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.

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 ]

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

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

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

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

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

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.

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.

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.

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

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.

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

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

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 ]

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

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.

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

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 ]

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

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.

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

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.

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

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

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

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 ]

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.

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 ]

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

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

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 ]

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

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

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.

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 ]

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.

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.

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.

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.

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

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 ]

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 ]

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

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

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

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

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 ]

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 ]

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.

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

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.

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

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 ]

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

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

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

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

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.

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

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 ]

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

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

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.

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).

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

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

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 ]

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.

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.

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

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

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

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.

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

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

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

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.

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

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.


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