Orthogonal Factorization and Linear Least Squares Problems

  • Efstratios Gallopoulos
  • Bernard Philippe
  • Ahmed H. Sameh
Part of the Scientific Computation book series (SCIENTCOMP)


Orthogonal factorization (or QR factorization) of a dense matrix is an essential tool in several matrix algorithms.


  1. 1.
    Halko, N., Martinsson, P., Tropp, J.: Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions. SIAM Rev. 53(2), 217–288 (2011). doi: 10.1137/090771806 MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning. Springer Series in Statistics. Springer, New York (2001)CrossRefMATHGoogle Scholar
  3. 3.
    Kontoghiorghes, E.: Handbook of Parallel Computing and Statistics. Chapman & Hall/CRC, New York (2005)CrossRefGoogle Scholar
  4. 4.
    Golub, G., Van Loan, C.: Matrix Computations, 4th edn. Johns Hopkins (2013)Google Scholar
  5. 5.
    Björck, Å.: Numerical Methods for Least Squares Problems. SIAM, Philadelphia (1996)CrossRefMATHGoogle Scholar
  6. 6.
    Sameh, A., Kuck, D.: On stable parallel linear system solvers. J. Assoc. Comput. Mach. 25(1), 81–91 (1978)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Modi, J., Clarke, M.: An alternative givens ordering. Numerische Mathematik 43, 83–90 (1984)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cosnard, M., Muller, J.M., Robert, Y.: Parallel QR decomposition of a rectangular matrix. Numerische Mathematik 48, 239–249 (1986)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cosnard, M., Daoudi, E.: Optimal algorithms for parallel Givens factorization on a coarse-grained PRAM. J. ACM 41(2), 399–421 (1994). doi: 10.1145/174652.174660 MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cosnard, M., Robert, Y.: Complexity of parallel QR factorization. J. ACM 33(4), 712–723 (1986)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Gentleman, W.M.: Least squares computations by Givens transformations without square roots. IMA J. Appl. Math. 12(3), 329–336 (1973)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Hammarling, S.: A note on modifications to the givens plane rotation. J. Inst. Math. Appl. 13, 215–218 (1974)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kontoghiorghes, E.: Parallel Algorithms for Linear Models: Numerical Methods and Estimation Problems. Advances in Computational Economics. Springer, New York (2000).
  14. 14.
    Lawson, C., Hanson, R., Kincaid, D., Krogh, F.: Basic linear algebra subprogams for Fortran usage. ACM Trans. Math. Softw. 5(3), 308–323 (1979)CrossRefMATHGoogle Scholar
  15. 15.
    Dongarra, J., Croz, J.D., Hammarling, S., Hanson, R.: An extended set of FORTRAN basic linear algebra subprograms. ACM Trans. Math. Softw. 14(1), 1–17 (1988)CrossRefMATHGoogle Scholar
  16. 16.
    Gallivan, K.A., Plemmons, R.J., Sameh, A.H.: Parallel algorithms for dense linear algebra computations. SIAM Rev. 32(1), 54–135 (1990). doi: 10.1137/1032002 MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Sameh, A.: Numerical parallel algorithms—a survey. In: Kuck, D., Lawrie, D., Sameh, A. (eds.) High Speed Computer and Algorithm Optimization, pp. 207–228. Academic Press, San Diego (1977)Google Scholar
  18. 18.
    Bischof, C., van Loan, C.: The WY representation for products of Householder matrices. SIAM J. Sci. Stat. Comput. 8(1), 2–13 (1987). doi: 10.1137/0908009 CrossRefGoogle Scholar
  19. 19.
    Schreiber, R., Parlett, B.: Block reflectors: theory and computation. SIAM J. Numer. Anal. 25(1), 189–205 (1988). doi: 10.1137/0725014 MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A., Sorensen, D.: LAPACK Users’ Guide, 3rd edn. Society for Industrial and Applied Mathematics, Philadelphia (1999)CrossRefGoogle Scholar
  21. 21.
    Dongarra, J., Du Croz, J., Hammarling, S., Duff, I.: A set of level-3 basic linear algebra subprograms. ACM Trans. Math. Softw. 16(1), 1–17 (1990)CrossRefMATHGoogle Scholar
  22. 22.
    Blackford, L., Choi, J., Cleary, A., D’Azevedo, E., Demmel, J., Dhillon, I., Dongarra, J., Hammarling, S., Henry, G., Petitet, A., Stanley, K., Walker, D., Whaley, R.: ScaLAPACK User’s Guide. SIAM, Philadelphia (1997).
  23. 23.
    Björck, Å.: Solving linear least squares problems by Gram-Schmidt orthogonalization. BIT 7, 1–21 (1967)CrossRefGoogle Scholar
  24. 24.
    Jalby, W., Philippe, B.: Stability analysis and improvement of the block Gram-Schmidt algorithm. SIAM J. Stat. Comput. 12(5), 1058–1073 (1991)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Sameh, A.: Solving the linear least-squares problem on a linear array of processors. In: Snyder, L., Gannon, D., Jamieson, L.H., Siegel, H.J. (eds.) Algorithmically Specialized Parallel Computers, pp. 191–200. Academic Press, San Diego (1985)Google Scholar
  26. 26.
    Sidje, R.B.: Alternatives for parallel Krylov subspace basis computation. Numer. Linear Algebra Appl. 4, 305–331 (1997)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Demmel, J., Grigori, L., Hoemmen, M., Langou, J.: Communication-optimal parallel and sequential QR and LU factorizations. SIAM J. Sci. Comput. 34(1), 206–239 (2012). doi: 10.1137/080731992 MathSciNetCrossRefGoogle Scholar
  28. 28.
    Chang, X.W., Paige, C.: An algorithm for combined code and carrier phase based GPS positioning. BIT Numer. Math. 43(5), 915–927 (2003)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Chang, X.W., Guo, Y.: Huber’s M-estimation in relative GPS positioning: computational aspects. J. Geodesy 79(6–7), 351–362 (2005)CrossRefMATHGoogle Scholar
  30. 30.
    Bomford, G.: Geodesy, 3rd edn. Clarendon Press, England (1971)Google Scholar
  31. 31.
    Golub, G., Plemmons, R.: Large scale geodetic least squares adjustment by dissection and orthogonal decomposition. Numer. Linear Algebra Appl. 35, 3–27 (1980)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Wolf, H.: The Helmert block method—its origin and development. In: Proceedings of the Second Symposium on Redefinition of North American Geodetic Networks, pp. 319–325 (1978)Google Scholar
  33. 33.
    Businger, P., Golub, G.H.: Linear least squares solutions by Householder transformations. Numer. Math. 7, 269–276 (1965)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Quintana-Ortí, G., Quintana-Ortí, E.: Parallel algorithms for computing rank-revealing QR factorizations. In: Cooperman, G., Michler, G., Vinck, H. (eds.) Workshop on High Performance Computing and Gigabit Local Area Networks. Lecture Notes in Control and Information Sciences, pp. 122–137. Springer, Berlin (1997). doi: 10.1007/3540761691_9 CrossRefGoogle Scholar
  35. 35.
    Quintana-Ortí, G., Sun, X., Bischof, C.: A BLAS-3 version of the QR factorization with column pivoting. SIAM J. Sci. Comput. 19, 1486–1494 (1998)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Bischof, C.: A parallel QR factorization algorithm using local pivoting. In: Proceedings of 1988 ACM/IEEE Conference on Supercomputing, Supercomputing’88, pp. 400–499. IEEE Computer Society Press, Los Alamitos (1988)Google Scholar
  37. 37.
    Bischof, C.H.: Incremental condition estimation. SIAM J. Matrix Anal. Appl. 11, 312–322 (1990)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Efstratios Gallopoulos
    • 1
  • Bernard Philippe
    • 2
  • Ahmed H. Sameh
    • 3
  1. 1.Computer Engineering and Informatics DepartmentUniversity of PatrasPatrasGreece
  2. 2.Campus de BeaulieuINRIA/IRISARennes CedexFrance
  3. 3.Department of Computer SciencePurdue UniversityWest LafayetteUSA

Personalised recommendations