# The development of the boundary-value codes in the ordinary differential equations chapter of the NAG library

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## Abstract

We discuss the historical development of the ordinary differential equations chapter of the Numerical Algorithms Groups Library with special emphasis on boundary-value codes. Much of this development has been motivated by the need to solve practical problems. We give six examples of problems which have influenced us and we consider in some detail how these problems can be solved using the shooting and matching codes in the NAG library. We also briefly describe other boundary-value codes in the NAG library and discuss future plans. In an appendix we give a classified list of the current NAG library boundary-value codes.

## Keywords

Newton Iteration Nonlinear Eigenvalue Problem Chebyshev Series Match Code Modify Newton Method
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## References

- Albansiny, E.L. (1978). A Subroutine for Solving a System of Differential Equations in Chebyshev Series, pages 280–286 of these proceedings.Google Scholar
- Ascher, U., Christiansen, J. & Russell, R.D. (1978). COLSYS—A Collocation Code for Boundary-Value Problems, pages 164–185 of these proceedings.Google Scholar
- Bailey, P.B. (1978). An Eigenvalue-Eigenfunction Code for Sturm-Liouville Problems. Report SAND77-2044. Sandia Laboratories, Alburquerque, New Mexico.Google Scholar
- Bailey, P.B. & Shampine, L.F. (1978). Automatic Solution of Sturm-Liouville Eigenvalue Problems, pages 274–279 of these proceedings.Google Scholar
- Bramley, J.S. (1978). Calculation of Eigenvalues of Systems of ODE's Using the Riccati Transformation, pages 319–324 of these proceedings.Google Scholar
- Bus, J.C.P. & Dekker, T.J. (1975). Two Efficient Algorithms with Guaranteed Convergence for Finding a Zero of a Function. TOMS, 1, pp.330–345.CrossRefGoogle Scholar
- Davey, A. (1978). On the Removal of the Singularities from the Riccati method. J. Comp. Phys. To appear.Google Scholar
- Deuflhard, P. (1974). A Modified Newton Method for the Solution of Ill-Conditioned Systems of Nonlinear Equations with Application to Multiple Shooting. Num. Math. 22, pp.289–315.Google Scholar
- Gay, D. (1976). On Modifying Singular Values to Solve Possibly Singular Systems of Nonlinear Equations. Working Paper No.125. Computer Research Center for Economics and Management Science. NBER. Cambridge, Mass.Google Scholar
- Gladwell, I. (1978). On the Numerical Solution of a Differential Nonlinear Eigenvalue Problem on an Infinite Range. App. Math. and Comp. To appear.Google Scholar
- Hargrave, B. & Pryce, J.D. (1977). NPARAM: Report on a Program to Solve Multiparameter Sturm-Liouville Problem. Bristol U. Computer Science Dept.Google Scholar
- Hatton, L. (1973). On the Dynamics of Concentrated Atmospheric Vortices. Ph.D. Thesis. University of Manchester.Google Scholar
- Moss, D.M. (1973. Stars in Radiative Equilibrium Containing Multiple Magnetic Fields. Mon. Not. R. astr. Soc. 164, pp. 33–51.Google Scholar
- Numerical Algorithms Group Manual. Mark 7 (1978). NAG, 7 Banbury Rd., Oxford.Google Scholar
- Numerical Algorithms Group Manual. Mark 8 To appear. NAG, 7 Banbury Rd., Oxford.Google Scholar
- Pereyra, V. (1978). An Adaptive Finite Difference Fortran Program for First Order Nonlinear, Ordinary Boundary Problems, pages 67–88 of these proceedings.Google Scholar
- Picken, S.M. (1970). Algorithms for the Solution of Differential Equations in Chebyshev Series by the Selected Points Method. Report Math. 94. NPL, Teddington, Middlesex.Google Scholar
- Scott, M.R. & Watts, H.A. (1978). Superposition, Orthonormalization, Quasilinearization and Two-Point Boundary-Value Problems, pages 109–121 of these proceedings.Google Scholar
- Walker, R.S. (1972). Boundary Layers in Rotating Bodies. Ph.D. Thesis. University of Manchester.Google Scholar
- Walton, I.C. (1972). Problems in Laminar Boundary Layer Flow. Ph.D. Thesis. University of Manchester.Google Scholar
- Walsh, S.K. & Wilson, S.D.R. (1978). Boundary Layer Flow in Forced-Convection Film-Boiling on a Wedge. Submitted for Publication.Google Scholar
- Watson, E.J. (1976). Similarity Solutions in Fluid Dynamics, Paper given at a conference on "Partial Differential Equations in Industry, University of Manchester, 1976".Google Scholar
- Wilson, S.D.R. and Gladwell, I. (1978). The Stability of a Two-Dimensional Stagnation Flow to Three-Dimensional Disturbances. J. Fluid Mech. 84, pp. 517–527.Google Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 1979