Contd5 Contd5_r Contd8 Contd8_r Contx1 Contx1_r Contx2 Contx2_r Derkf Derkf_r DerkfInt Dop853 Dop853_r Dopri5 Dopri5_r Doprin Doprin_r Dverk Dverk_r DverkInt Odex Odex2 Odex2_r Odex_r Retard Retard_r Ylag Ylag_r

Ylag() Function Ylag ( I As  Long, T As  Double, Phi As  LongPtr, RCont As  Double, ICont As  Long  ) Initial value problem of delay differential equations (5(4)-th order Dorman-Prince method) (Interpolation for back-values of solution) NOTE - THIS PROGRAM IS DEPRECATED AND WILL BE REMOVED IN THE NEXT VERSION. Purpose This routine is the support routine which computes the back-values of the solution by interpolation for solving an initial value problem of the delay differential equations and dense output by Retard. This routine can be used in the subroutine F which defines the differential equations and in the subroutine Solout for the dense output. Returns Double Interpolated solution Y(I) at T. Parameters [in] I Element number of solution to be interpolated. (0 <= I <= N - 1) [in] T T at which the interpolated solution is computed. T must be in the interval of the available back steps. Otherwise. 0 will be returned and Retard will be terminated with Info = 14. The number of back steps available through this routine depends on the allocated working memory, and it can be specified by the parameter Mxst of Retard. [in] Phi Used defined function which is called when an initial value is required. It must be defined as follows. Function Phi(I As Long, T As Double) As Double Phi = the function value Y(I) of component I at T (I = 0 to N - 1) where t0 - τ <= T <= t0. End Function where t0 is the initial value of t, and τ is the value of lag. That is, the function value at up to τ before t0 must be defined by Phi. For the components which do not need retarded argument in the definition of differential equations, Phi will not be called, i.e. the value needs not be defined for such element numbers in Phi. [in] RCont Control information for dense output. Enter parameter value of F or Solout without change. [in] ICont Integer control information for dense output. Enter parameter value of F or Solout without change. Reference 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 Define the following delay differential equation. y'(t) = -y(t-1), where t0 = 0, y(t) = 1 (-1 <= t <= 0) Sub F(N As Long, T As Double, Y() As Double, Yp() As Double, RCont As Double, ICont As Long) Yp(0) = -Ylag(0, T-1, AddressOf Phi, RCont, ICont) End Sub Function Phi(I As Long, X As Double) As Double If I = 0 And X >= -1 And X <= 0 Then Phi = 1 End Function