Deabm Deabm_r Derkfa Derkfa_r Dop853a Dop853a_r Dopn1210 Dopn1210_r Dopn43 Dopn43_r Dopn64 Dopn64_r Dopn86 Dopn86_r Dopri5a Dopri5a_r Dverka Dverka_r Odex2a Odex2a_r Odexa Odexa_r Retarda Retarda_r Ylaga

Ylaga() Sub Ylaga ( I As  Long, N As  Long, T As  Double, Y() As  Double, Phi As  LongPtr, Cont() As  Double, ICont() As  Long, Optional Info As  Long  ) Initial value problem of delay differential equations (5(4)-th order Dorman-Prince method) (Computation of back-values of solution) Purpose This program is the support program which computes the back-values of the solution by interpolation for solving an initial value problem of the delay differential equations by Retard. A system of first order delay ordinary differential equations of the form dy/dt = f(t, y(t), y(t - τ), y(t - τ2), ...), y = y0 at t = t0 where f depends on the values at previous points y(t - τ), y(t - τ2), .... These values can be computed by using this program. This code is the rewritten version of YLAG used for the 5(4)-th order Dormand-Prince code for delay equations, RETARD (Reference (1)). Parameters [in] I Element number of y to be interpolated. (1 <= I <= N or I = 0) 1 <= I <= N: Only the I-th element of y is computed. I = 0: All elements of y (Y(0) 〜 Y(N - 1)) are computed. [in] N Number of differential equations. (N >= 1) [in] T Independent variable t. To compute back-value y(t - τ), for example, T = t - τ is given to this argument. [out] Y() Array Y(LY - 1) (LY >= 1 (1 <= I <= N), LY >= N (I = 0)) The values of dependent variable y at T (interpolation of numerical solution). 1 <= I <= N: The computed I-th element of y is returned in Y(0). I = 0: All computed elements are returned in Y(0) 〜 Y(N - 1). [in] Phi The user supplied subroutine, which returns the values of y at t < t0, defined as follows. Sub Phi(I As Long, N As Long, T As Double, Y() As Double, Info As Long) y() = values of y at T (where t0 - τ < T < t0). Info = 0 End Sub If I = 0, all elements must be stored in Y(0) 〜 Y(N - 1). Otherwise, I-th element must be stored in Y(0). Only the values of I, which are used in Ylaga by the user, may be supported. Ylaga will terminate if Phi returns Info <> 0. [in] Cont() Array Cont(LCont - 1) Control information area. The contents of this array must not be altered. The same array which is used when calling Retarda or Retarda_r. [in] ICont() Array Icont(LICont - 1) (LICont >= 50) Integer control information area. The contents of this array must not be altered. The same array which is used when calling Retarda or Retarda_r. [out] Info (Optional) Return code. = 0: Successful exit. < 0: The (-Info)-th argument is invalid. = 11: (Error) Invalid control information. = 12: (Error) t is too small. Mxst of Retarda may be too small. = 13: (Error) t is too large. = Others: Phi returnd this value (<> 0) as Info. Reference (1) E. Hairer, S.P. Norsett and G. Wanner, "Solving Ordinary Differential Equations. Nonstiff Problems. 2nd edition", Springer Series in Computational Mathematics, Springer-Verlag (1993) Example Program See example program of Retarda.