Matrix methods in mathematical programming

  • Gene Golub
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 193)


Linear Constraint Triangular Matrix Orthogonal Matrix Gaussian Elimination Permutation Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Beale, E.M.L., "Numerical Methods", in Nonlinear Programming, J. Abadie (ed.). John Wiley, New York, 1967; pp. 133–205.Google Scholar
  2. [2]
    Björck, Å., "Iterative Refinement of Linear Least Squares Solutions II", BIT 8 (1968), pp. 8–30.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    _____ and G. H. Golub, "Iterative Refinement of Linear Least Squares Solutions by Householder Transformations", BIT 7 (1967), pp. 322–37.CrossRefGoogle Scholar
  4. [4]
    _____ and V. Pereyra, "Solution of Vandermonde Systems of Equations", Publicaion 70-02, Universidad Central de Venezuela, Caracas, Venezuela, 1970.Google Scholar
  5. [5]
    Cottle, R. W. and G. B. Dantzig, "Complementary Pivot Theory of Mathematical Programming", Mathematics of the Decision Sciences, Part 1, G. B. Dantzig and A. F. Veinott (eds.), American Mathematical Society (1968), pp. 115–136.Google Scholar
  6. [6]
    Dantzig, G. B., R. P. Harvey, R. D. McKnight, and S. S. Smith, "Sparse Matrix Techniques in Two Mathematical Programming Codes", Proceedings of the Symposium on Sparse Matrices and Their Applications, T. J. Watson Research Publications RA1, no. 11707, 1969.Google Scholar
  7. [7]
    Fletcher, R., "A Technique for Orthogonalization", J. Inst. Maths. Applics. 5 (1969), pp. 162–66.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Forsythe, G. E., and G. H. Golub, "On the Stationary Values of a Second-Degree Polynomial on the Unit Sphere", J. SIAM, 13 (1965), pp. 1050–68.MathSciNetzbMATHGoogle Scholar
  9. [9]
    _____ and C. B. Moler, Computer Solution of Linear Algebraic Systems, Prentice-Hall, Englewood Cliffs, New Jersey, 1967.zbMATHGoogle Scholar
  10. [10]
    Francis, J., "The QR Transformation. A Unitary Analogue to the LR Transformation," Comput. J. 4 (1961–62), pp. 265–71.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Golub, G. H., and C. Reinsch, "Singular Value Decomposition and Least Squares Solutions", Numer. Math., 14(1970), pp. 403–20.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    _____ and R. Underwood, "Stationary Values of the Ratio of Quadratic Forms Subject to Linear Constraints", Technical Report No. CS 142, Computer Science Department, Stanford University, 1969.Google Scholar
  13. [13]
    Hanson, R. J., "Computing Quadratic Programming Problems: Linear Inequality and Equality Constraints", Technical Memorandum No. 240, Jet Propulsion Laboratory, Pasadena, California, 1970.Google Scholar
  14. [14]
    _____ and C. L. Lawson, "Extensions and Applications of the Householder Algorithm for Solving Linear Least Squares Problems", Math. Comp., 23 (1969), pp. 787–812.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Householder, A.S., "Unitary Triangularization of a Nonsymmetric Matrix", J. Assoc. Comp. Mach., 5 (1968), pp. 339–42.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Lanozos, C., Linear Differential Operators. Van Nostrand, London, 1961. Chapter 3.Google Scholar
  17. [17]
    Leringe, O., and P. Wedin, "A Comparison Between Different Methods to Compute a Vector x Which Minimizes ‖Ax − b‖2 When Gx = h", Technical Report, Department of Computer Sciences, Lund University, Sweden.Google Scholar
  18. [18]
    Levenberg, K., "A Method for the solution of Certain Non-Linear Problems in Least Squares", Quart. Appl. Math., 2 (1944), pp. 164–68.MathSciNetzbMATHGoogle Scholar
  19. [19]
    Marquardt, D. W., "An Algorithm for Least-Squares Estimation of Non-Linear Parameters", J. SIAM, 11 (1963), pp. 431–41.MathSciNetzbMATHGoogle Scholar
  20. [20]
    Meyer, R. R., "Theoretical and Computational Aspects of Nonlinear Regression", P-1819, Shell Development Company, Emeryville, California.Google Scholar
  21. [21]
    Penrose, R., "A Generalized Inverse for Matrices", Proceedings of the Cambridge Philosophical Society, 51 (1955), pp. 406–13.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Peters, G., and J. H. Wilkinson, "Eigenvalues of Ax = λB x with Band Symmetric A and B", Comput. J., 12 (1969), pp. 398–404.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    Powell, M.J.D., "Rank One Methods for Unconstrained Optimization", T. P. 372, Atomic Energy Research Establishment, Harwell, England, (1969).Google Scholar
  24. [24]
    Rosen, J. B., "Gradient Projection Method for Non-linear Programming. Part I. Linear Constraints", J. SIAM, 8 (1960), pp. 181–217.zbMATHGoogle Scholar
  25. [25]
    Shanno, D. C. "Parameter Selection for Modified Newton Methods for Function Minimization", J. SIAM, Numer. Anal., Ser. B, 7 (1970).Google Scholar
  26. [26]
    Stoer, J., "On the Numerical Solution of Constrained Least Squares Problems", (private communication), 1970.Google Scholar
  27. [27]
    Tewarson, R. P., "The Gaussian Elimination and Sparse Systems", Proceedings of the Symposium on Sparse Matrices and Their Applications, T. J. Watson Research Publication RA1, no. 11707, 1969.Google Scholar
  28. [28]
    Wilkinson, J. H., "Error Analysis of Direct Methods of Matrix Inversion", J. Assoc. Comp. Mach., 8 (1961), pp. 281–330.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    _____ "Error Analysis of Transformations Based on the Use of Matrices of the Form I — 2wwH", in Error in Digital Computation, Vol. ii, L. B. Rall (ed.), John Wiley and Sons, Inc., New York, 1965, pp. 77–101.Google Scholar
  30. [30]
    _____ The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965.zbMATHGoogle Scholar
  31. [31]
    Zoutendijk, G., Methods of Feasible Directions, Elsevier Publishing Company, Amsterdam (1960), pp. 80–90.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1971

Authors and Affiliations

  • Gene Golub
    • 1
  1. 1.Stanford UniversityUSA

Personalised recommendations