The “Seven Dwarfs” of Symbolic Computation

  • Erich L. Kaltofen
Part of the Texts & Monographs in Symbolic Computation book series (TEXTSMONOGR)


We present the Seven Dwarfs of Symbolic Computation, which are sequential and parallel algorithmic methods that today carry a great majority of all exact and hybrid symbolic compute cycles. We will elaborate on each dwarf and compare with Colella’s seven and the Berkeley team’s thirteen dwarfs of scientific computing.

SymDwf 1. Exact linear algebra, integer lattices


SymDwf 2. Exact polynomial and differential algebra, Gröbner bases


SymDwf 3. Inverse symbolic problems, e.g., interpolation and parameterization


SymDwf 4. Tarski’s algebraic theory of real geometry


SymDwf 5. Hybrid symbolic-numeric computation


SymDwf 6. Computation of closed form solutions


SymDwf 7. Rewrite rule systems and computational group theory



Symbolic Computation Exact Linear Algebra High Productivity Computing Systems (HPCS) Cylindrical Algebraic Decomposition Algorithm Sparse Signal Processing 
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.



I thank Bruno Salvy for his thoughtful comments. This material is based on work supported in part by the National Science Foundation under Grants CCF-0830347, CCF-0514585 and DMS-0532140.


  1. 1.
    Ben-Or, M., Tiwari, P.: A deterministic algorithm for sparse multivariate polynomial interpolation. In Proceeding of the Twentieth Annual ACM Symposium on Theory of Computing, pp. 301–309, ACM Press, New York (1988)Google Scholar
  2. 2.
    Bostan, A., Kauers, M.: The complete generating function for Gessel walks is algebraic. In Proceedings of the AMS, (2010); with an Appendix by Mark van Hoeij. To appear.
  3. 3.
    Boyle, A., Caviness, B.F. (ed.): Future Directions for Research in Symbolic Computation. SIAM, Philadelphia (1989); Report of a Workshop on Symbolic and Algebraic Computation April 29–30, 1988 Washington DC. Anthony C. Hearn Workshop Chairperson.
  4. 4.
    Brown, C.W.: Fast simplification of Tarski formulas. In ISSAC’09 Proceedings of the 2009 International Symposium on Symbolic and Algebraic Computation, pp. 63–70, New York, NY, USA (2009)Google Scholar
  5. 5.
    Buchberger, B.: Symbolic computation (an editorial). J. Symbolic Comput. 1(1), 1–6 (1985)CrossRefGoogle Scholar
  6. 6.
    Caviness, B.F., Johnson, J.R. (ed.): Quantifier Elimination and Cylindrical Algebraic Decomposition. Springer, Berlin (1998)zbMATHGoogle Scholar
  7. 7.
    Colella, P.: Defining software requirements for scientific computing. Slide of 2004 presentation included in David Patterson’s 2005 talk, (2004);
  8. 8.
    Coppersmith, D.: Solving homogeneous linear equations over GF(2) via block Wiedemann algorithm. Math. Comput. 62(205), 333–350 (1994)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Dumas, J.-G., Giorgi, P., Pernet, C.: Dense linear algebra over finite fields: the FFLAS and FFPACK packages. ACM Trans. Math. Software 35(3), 1–42 (2008)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Dumas, J.-G., Saunders, B.D., Villard, G.: On efficient sparse integer matrix Smith normal form computation. J. Symbolic Comput. 32(1/2), 71–99 (2001); Special issue on Computer Algebra and Mechanized Reasoning: Selected St. Andrews’ ISSAC/Calculemus Contributions. Guest editors: T. Recio and M. KerberGoogle Scholar
  11. 11.
    Eberly, W.: Yet another block Lanczos algorithm: How to simplify the computation and reduce reliance on preconditioners in the small field case. In [Watt 2010], page to appear, July 2010 International Symposium on Symbolic and Algebraic Computation, ACM, New York, NY, USA (2010)Google Scholar
  12. 12.
    Ferguson, H.R.P., Forcade, R.W.: Multidimensional Euclidean algorithms. J. Reine Angew. Math. 334, 171–181 (1982)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Gao, X.-S., der Hoeven, J.V., Yuan, C.M., Zhang, G.-L.: Characteristic set method for differential-difference polynomial systems. J. Symb. Comput. 44(9), 1137–1163 (2009)CrossRefzbMATHGoogle Scholar
  14. 14.
    Garg, S., Schost, É.: Interpolation of polynomials given by straight-line programs. Theor. Comput. Sci. 410(27–29), 2659–2662 (2009). ISSN 0304-3975.
  15. 15.
    Giesbrecht, M.: Fast computation of the Smith form of a sparse integer matrix. Comput. Complex. 10, 41–69 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Giesbrecht, M., Labahn, G., Shin Lee, W.: Symbolic-numeric sparse interpolation of multivariate polynomials. J. Symbolic Comput. 44, 943–959 (2009)Google Scholar
  17. 17.
    Giesbrecht, M, Roche, D.S.: Interpolation of shifted-lacunary polynomials. Comput. Complex. 19(3), 333–354 (2010)Google Scholar
  18. 18.
    Grabmeier, J., Kaltofen, E., Weispfenning, V. (ed.): Computer Algebra Handbook. Springer, Heidelberg, Germany (2003). ISBN 3-540-65466-6. 637 + xx pages + CD-ROM.zbMATHGoogle Scholar
  19. 19.
    Griewank, A.: Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation. SIAM Publications, Philadephia (2008)CrossRefGoogle Scholar
  20. 20.
    Håstad, J., Just, B., Lagarias, J.C., Schnorr, C.P.: Polynomial time algorithms for finding integer relations among real numbers. SIAM J. Comput. 18(5), 859–881 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hong, H., Safey El Din, M.: Variant real quantifier elimination: Algorithm and application. In ISSAC’09, ACM, pp. 183–190 (2009)Google Scholar
  22. 22.
    Hutton, S.E., Kaltofen, E.L., Zhi, L.: Computing the radius of positive semidefiniteness of a multivariate real polynomial via a dual of Seidenberg’s method. In [Watt 2010], page to appear, July 2010 International Symposium on Symbolic and Algebraic Computation, pp. 227–234, New York, NY, USA (2010);
  23. 23.
    Jeffrey, D. (ed.): ISSAC 2008. ACM Press. ISBN 978-1-59593-904-3Google Scholar
  24. 24.
    Kai, H., Sekigawa, H. (ed.): SNC’09 Proceeding 2009 International Workshop on Symbolic-Numeric Computation, pp. 28–31, ACM Press, New York, NY, USA (2009). ISBN 978-1-60558-664-9Google Scholar
  25. 25.
    Kaltofen, E.: Computer algebra algorithms. In: Traub, J.F. (eds.) Annual Review in Computer Science, vol. 2, pp. 91–118. Annual Reviews Inc., Palo Alto, California (1987);
  26. 26.
    Kaltofen, E.: Analysis of Coppersmith’s block Wiedemann algorithm for the parallel solution of sparse linear systems. Math. Comput. 64(210), 777–806 (1995);
  27. 27.
    Kaltofen, E., Lee, W.-S.: Early termination in sparse interpolation algorithms. J. Symbolic Comput. 36(3–4), 365–400 (2003); Special issue Internat. Symp. Symbolic Algebraic Comput. (ISSAC 2002). Guest editors: M. Giusti & L. M. Pardo. Google Scholar
  28. 28.
    Kaltofen, E., Li, B., Yang, Z., Zhi, L.: Exact certification of global optimality of approximate factorizations via rationalizing sums-of-squares with floating point scalars. In ISSAC’08, pp. 155–163 ACM Press, New York, NY, USA (2008);
  29. 29.
    Kaltofen, E., Saunders, B.D.: On Wiedemann’s method of solving sparse linear systems. In: Mattson, H.F., Mora, T., Rao, T.R.N. (eds.) Proceeding AAECC-9, Lect. Notes Comput. Sci., vol. 539, pp. 29–38, Springer, Heidelberg, Germany (1991);
  30. 30.
    Kaltofen, E., Yang, Z., Zhi, L.: On probabilistic analysis of randomization in hybrid symbolic-numeric algorithms. In ISSAC ’07 Proceedings of the 2007 international symposium on Symbolic and algebraic computation, pp. 11–17 ACM Press, New York, NY, USA (2007);
  31. 31.
    Kaltofen, E., Yang, Z., Zhi, L.: A proof of the Monotone Column Permanent (MCP) Conjecture for dimension 4 via sums-of-squares of rational functions. In SNC’09, pp. 65–69 (2009a);
  32. 32.
    Kaltofen, E., Yuhasz, G.: On the matrix Berlekamp-Massey algorithm, (2006); Manuscript, 29 pages. SubmittedGoogle Scholar
  33. 33.
    Kaltofen, E.L., Li, B., Yang, Z., Zhi, L.: Exact certification in global polynomial optimization via sums-of-squares of rational functions with rational coefficients, January (2009b); Accepted for publication in J. Symbolic Comput.
  34. 34.
    Kaltofen, E.L., Nehring, M.: Supersparse black box rational function interpolation. In: Leykin, A. (ed.) Proc. 2011 Internat. Symp. Symbolic Algebraic Comput, ISSAC 2011, pp. 177–185. Association for Computing Machinery, New York, (2011). ISBN 978-1-4503-0675-1Google Scholar
  35. 35.
    Klep, I., Schweighofer, M.: Sums of Hermitian squares and the BMV conjecture. J. Stat. Phys. 133, 739–760 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Kunkle, D., Cooperman, G.: Harnessing parallel disks to solve Rubik’s cube. J. Symbolic Comput. 44(7), 872–890 (2009); Google Scholar
  37. 37.
    Lenstra, A.K., Lenstra, Jr., H. W., Lovász, L.: Factoring polynomials with rational coefficients. Math. Ann. 261, 515–534 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Marshall, M.: Positive Polynomials and Sums of Squares. Amer. Math. Soc. 146, 187 (2008)Google Scholar
  39. 39.
    May, J.P. (ed.): ISSAC 2009 Proceeding 2009 International Symposium Symbolic Algebraic Computation, ACM, (2009). ISBN 978-1-60558-609-0Google Scholar
  40. 40.
    Novocin, A., Stehlé, D., Villard, G.: An LLL-reduction algorithm with quasi-linear time complexity: extended abstract. In: Proc. 43rd Annual ACM Symp. Theory Comput., pp. 403–412. ACM, New York (2011)Google Scholar
  41. 41.
    Peyrl, H., Parrilo, P.A.: Computing sum of squares decompositions with rational coefficients. Theor. Comput. Sci. 409, 269–281 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Püschel, M., Moura, J.M.F., Johnson, J., Padua, D., Veloso, M., Singer, B., Xiong, J., Franchetti, F., Gacic, A., Voronenko, Y., Chen, K., Johnson, R.W., Rizzolo, N.: SPIRAL: Code generation for DSP transforms. Proc. IEEE 93(2), 232–275 (2005); special issue on “Program Generation, Optimization, and Adaptation”, Google Scholar
  43. 43.
    Rosenkranz, M., Regensburger, G.: Solving and factoring boundary problems for linear ordinary differential equations in differential algebras. J. Symbolic Comput. 43(8), 515–544 (2008). ISSN 0747-7171Google Scholar
  44. 44.
    Safey El Din, M.: Computing the global optimum of a multivariate polynomial over the reals. In ISSAC’08 Proceedings of the twenty-first international symposium on Symbolic and algebraic computation, ACM Press, New York, NY (2008)Google Scholar
  45. 45.
    Sendra, J.R., Winkler, F., Pérez-Díaz, S.: Rational Algebraic Curves A Computer Algebra Approach, Algorithms and Computation in Mathematics. vol. 22, Springer, Heidelberg, Germany (2007). ISBN ISSN 1431-1550, ISBN 978-3-540-73724-7Google Scholar
  46. 46.
    van der Put, M., Singer, M.F.: Galois Theory of Linear Differential Equations, Grundlehren der mathematischen Wissenschaften. vol. 328 Springer, Heidelberg, Berlin (2003);
  47. 47.
    Verschelde, J., Watt, S.M. (ed.): SNC’07 Proceeding 2007 International Workshop on Symbolic-Numeric Computation, ACM Press, New York, NY, USA (2007). ISBN 978-1-59593-744-5Google Scholar
  48. 48.
    Villard, G.: Further analysis of Coppersmith’s block Wiedemann algorithm for the solution of sparse linear systems. In: Küchlin, W. (eds) ISSAC 97 Proceeding 1997 International Symposium Symbolic Algebraic Computation, pp. 32–39, ACM Press, New York, NY, USA (1997). ISBN 0-89791-875-4Google Scholar
  49. 49.
    Wang, D., Zhi, L. (ed.): Symbolic-Numeric Computation. Trends in Mathematics. Birkhäuser Verlag, Basel, Switzerland (2007). ISBN 978-3-7643-7983-4zbMATHGoogle Scholar
  50. 50.
    Watt, S.M. (ed.): Proceeding 2010 International Symposium Symbolic Algebraic Computation ISSAC 2010, Association for Computing Machinery (2010). ISBN 978-1-4503-0150-3Google Scholar
  51. 51.
    Wiedemann, D.: Solving sparse linear equations over finite fields. IEEE Trans. Inf. Theory 32(1), 54–62 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Yuan, Q., van Hoeij, M.: Finding all Bessel type solutions for linear differential equations with rational function coefficients. In [Watt 2010], page to appear, July 2010, pp.37–44 (2010)Google Scholar
  53. 53.
    Zippel, R.: Interpolating polynomials from their values. J. Symbolic Comput. 9(3), 375–403 (1990)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer-Verlag/Wien 2012

Authors and Affiliations

  • Erich L. Kaltofen
    • 1
  1. 1.Department of MathematicsNorth Carolina State UniversityRaleighUSA

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