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TOM code


TOM a software for the numerical solution of Boundary Value Problems



TOM: Solve two-point boundary value problems for ODEs using the Top Order Method of order 2 and 6. Nonlinear problems are solved using quasilinearization. Mesh selection is based on the conditioning of the discrete linear problems.

The Matlab solver TOM .

Numerical Experiments .

References .

The Matlab solver TOM


The code TOM, release 2004 based on the paper [2,3,4,5,6], consists of four files:

tom.m contains the functions that implement the integration procedure;

tominit.m Form the initial guess for TOM.

tomget.m Get TOM OPTIONS parameters.

tomset.m Set TOM OPTIONS parameters.

If you retrieve the software, please send a message to so that we may keep you updated on any changes. Also any bug reports are appreciated.

Numerical Experiments

The code has been tested on many difficult stiff test problems. For example, those contained in the Jeff Cash's home page.

Read also the slides of SCICADE 03 for more information about the mesh selection strategy used in TOM.

A modification of  TOM with a continuation algorithm in the initial state has been used in  p2pOC: a pde2path add-on library for solving spatially distributed optimal control problems.


  1. L. Brugnano and D. Trigiante, Solving Differential Problems by Multistep Initial and Boundary Value Methods, Gordon & Breach.
  2. L. Brugnano and D. Trigiante, A New Mesh Selection Strategy for ODEs,Appl. Numer. Math. (1997), 24, 1-21.
  3. F. Mazzia, I. Sgura, Numerical Approximation of Nonlinear BVPs by means of BVMs}, Appl. Numer. Math.,{\bf 42}(2002), 337–352.
  4. L. Aceto, F. Mazzia and D. Trigiante, The performances of the code TOM on the Holt problem, Modeling, simulation, and optimization of integrated circuits (Oberwolfach, 2001), Internat. Ser. Numer. Math., 146, Birkhauser, Basel, 2003, 349–360.
  5. F. Mazzia, D. Trigiante, A Hybrid Mesh Selection Strategy Based on Conditioning for Boundary Value ODE Problems, Numerical Algorithms, 36 (2004), no.2, 169–187.
  6. J. Cash, F. Mazzia, N. Sumarti, D. Trigiante, The role of conditioning in mesh selection algorithms for first order systems of linear two-point boundary value problems, Journal of Computation and Applied Mathematics Volume: 185 Issue: 2 Pages: 212-224,JAN 15 2006
  7. Mazzia, F.; Sestini, A.; Trigiante, D., BS linear multistep methods on non-uniform meshes, JNAIAM J. Numer. Anal. Indust. Appl. Math Volume: 1 Issue: 1 Pages: 131-144,2006
  8. Mazzia, Francesca; Sestini, Alessandra; Trigiante, Donato, B-spline linear multistep methods and their continuous extensions, Siam Journal on Numerical Analysis Volume: 44 Issue: 5 Pages: 1954-1973,2006
  9. Capper, S.; Cash, J.; Mazzia, F., On the development of effective algorithms for the numerical solution of singularly perturbed two-point boundary value problems,International Journal of Computing Science and Mathematics Volume: 1 Issue: 1 Pages: 42-57, 2007
  10. Mazzia, F.; Trigiante, D.,Efficient strategies for solving nonlinear problems in BVPs codes, Nonlinear Studies Volume: 17 Issue: 4 Pages: 309-326,2010
  11. Mazzia, Francesca; Sestini, Alessandra; Trigiante, Donato,The continuous extension of the B-spline linear multistep methods for BVPs on non-uniform meshes, Applied Numerical Mathematics Volume: 59 Issue: 3-4 Pages: 723-738, MAR-APR 2009


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