Determining Sparse Jacobian Matrices Using Two-Sided Compression: An Algorithm and Lower Bounds

Conference paper


We study the determination of large and sparse derivative matrices using row and column compression. This sparse matrix determination problem has rich combinatorial structure which must be exploited to effectively solve any reasonably sized problem. We present a new algorithm for computing a two-sided compression of a sparse matrix. We give new lower bounds on the number of matrix-vector products needed to determine the matrix. The effectiveness of our algorithm is demonstrated by numerical testing on a set of practical test instances drawn from the literature.


Lower Bound Sparse Jacobian Matrix-vector Product (MVPs) Test Instances Dense Submatrix 
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.



This research is supported in part by Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant (Individual).


  1. 1.
    Coleman, T.F., Moré, J.J.: Estimation of sparse Jacobian matrices and graph coloring problems. SIAM J. Numer. Anal. 20 (1), 187–209 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Coleman, T.F., Verma, A.: The efficient computation of sparse Jacobian matrices using automatic differentiation. SIAM J. Sci. Comput. 19 (4), 1210–1233 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Curtis, A.R., Powell, M.J.D., Reid, J.K.: On the estimation of sparse Jacobian matrices. IMA J. Appl. Math. 13 (1), 117–119 (1974)CrossRefzbMATHGoogle Scholar
  4. 4.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)zbMATHGoogle Scholar
  5. 5.
    Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)zbMATHGoogle Scholar
  6. 6.
    Griewank, A., Toint, Ph.L.: On the unconstrained optimization of partially separable objective functions. In: Powell, M.J.D. (ed.), Nonlinear Optimization, pp. 301–312. Academic, London (1982)Google Scholar
  7. 7.
    Griewank, A., Walther, A.: Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia (2008)CrossRefzbMATHGoogle Scholar
  8. 8.
    Hossain, A.K.M.S.: On the computation of sparse Jacobian matrices and newton steps. Ph.D. Dissertation, Department of Informatics, University of Bergen (1998)Google Scholar
  9. 9.
    Hossain, A.K.M.S., Steihaug, T.: Computing a sparse Jacobian matrix by rows and columns. Optim. Methods Softw. 10, 33–48 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hossain, S., Steihaug, T.: Graph models and their efficient implementation for sparse Jacobian matrix determination. Discret. Appl. Math. 161 (2), 1747–1754 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Juedes, D., Jones, J.: Coloring Jacobians revisited: a new algorithm for star and acyclic bicoloring. Optim. Methods Softw. 27, 295–309 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Leighton, F.T.: A graph coloring algorithm for large scheduling problems. J. Res. Natl. Bur. Stand. 84, 489–505 (1979)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of LethbridgeLethbridgeCanada

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