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

Dopn86() Sub Dopn86 ( N As  Long, F2 As  LongPtr, T As  Double, Y() As  Double, Yp() As  Double, Tout As  Double, Tend As  Double, RTol() As  Double, ATol() As  Double, Mode As  Long, Info As  Long, Optional Neval As  Long, Optional Nstep As  Long, Optional Naccept As  Long, Optional Nreject As  Long, Optional MaxIter As  Long = 0, Optional ErrCntl As  Long = 0, Optional Cnt As  Long = 0, Optional Hinit As  Double = 0, Optional Hmax As  Double = 0, Optional Fac1 As  Double = 0, Optional Fac2 As  Double = 0, Optional Safe As  Double = 0  ) Initial value problem of ordinary differential equations (8(6)-th order Runge-Kutta-Nystrom method) (for second order differential equations) Purpose This program integrates a system of second order ordinary differential equations of the form d2y/dt2 = f2(t, y), y = y0, y' = y'0 at t = t0 where y and y' are the n vectors and the system consists of n equations. t0, y0 and y'0 are the given initial values of t, y and y', respectively. This program is based on the Runge-Kutta-Nystroem method of order 8(6). See for details of coefficients (RKN8(6)9FM) and interpolant for dense output in the references (1) and (2), respectively. Parameters [in] N Number of differential equations. (N >= 1) [in] F2 The user supplied subroutine, which calculates the derivatives of the differential equations, defined as follows. Sub F2(N As Long, T As Double, Y() As Double, Ypp() As Double) Compute d2y/dt2 and store in Ypp(). End Sub where N is the number of equations, and d2y/dt2 (= f2(t, y)) is the computed second derivative at given T and Y(). [in,out] T Independent variable t. This program integrates from the initial value of t to Tend. Depending on the setting of Mode, this program may return at Tout or every successful step to provide an intermediate result. [in] An initial value of t (t0). [out] T = Tend if the integration has been completed. Or T = Tout or T of the latest step depending on Mode in the case of intermediate return. [in,out] Y() Array Y(LY - 1) (LY >= N) Dependent variable y. [in] Initial value of y (y0) at initial t (t0). [out] The value of y (computed numerical solution) at t = T. [in,out] Yp() Array Yp(LYp - 1) (LYp >= N) y', derivatives of y. [in] Initial values of derivtive y' (y'0) at initial t (t0). [out] The value of y' (computed numerical solution) at t = T. Note - This program does not return the computed value of y' (interpolated value in Yp()) in Mode = 3. If the value of y' as well as y is necessary, use Mode = 2. [in] Tout Set Tout to the point (t) at which an intermediate result is desired in mode = 2 or 3. Tout is not referenced in Mode = 0 or 1. In Mode 2 or 3, y at Tout is computed and returned with Info = 2 or 3 respectively. To continue the integration to get result at new Tout, it is possible to call this program again with new Tout and without changing other variables including Info. T < Tout <= Tend is required. However, if the integration is in backward direction, Tend <= Tout < T is required. If Tend is reached before Tout, the integration is terminated immediately and T = Tend and Info = 0 is returned. [in] Tend The point at which an integration is completed. If Tend is reached, the integration is terminated and T = Tend and Info = 0 is returned. The integration is possible for both forward (Tend > T) and backward (Tend < T) direction. [in] Rtol() Array Rtol(LRtol - 1) (LRtol >= 1) (all elements of Rtol() >= 0) The relative error tolerance(s) to specify how accurately compute the solution. This parameter may be a scalar or an array according to the values of LRtol and LAtol (Scalar if LRtol < N or LAtol < N, array if LRtol >= N and LAtol >= N). Rtol is used together with Atol in a local error test at each step. The integration is performed using step sizes which are automatically selected so as to achieve the following criteria. Scalar case (i = 0 to N - 1): (local error) <= max(Rtol(0)*Abs(Y(i)) + Atol(0), Rtol(0)*Abs(Yp(i)) + Atol(0)) (if ErrCntl = 0) (local error) <= Rtol(0)*Abs(Y(i)) + Atol(0) (if ErrCntl = 1) Both Rtol(0) and Atol(0) must not be 0 at the same time. Array case (i = 0 to N - 1): (local error) <= max(Rtol(i)*Abs(Y(i)) + Atol(i), Rtol(i)*Abs(Yp(i)) + Atol(i)) (if ErrCntl = 0) (local error) <= Rtol(i)*Abs(Y(i)) + Atol(i) (if ErrCntl = 1) Both Rtol(i) and Atol(i) must not be 0 at the same time. [in] Atol() Array Atol(LAtol - 1) (LAtol >= 1) (all elements of Atol() >= 0) The absolute error tolerance(s) to specify how accurately compute the solution. This parameter may be a scalar or an array according to the values of LRtol and LAtol (Scalar if LRtol < N or LAtol < N, array if LRtol >= N and LAtol >= N). Atol is used together with Rtol in a local error test at each step (Refer to Rtol above). [in] Mode Mode of operation. This program integrates from t0 to Tend. Four modes of operation are provided depending on the timing of returning imtermediate results. When Tend is reached, the integration is terminated and Info = 0 is returned in any of four modes. = 0: Not return until Tend. Tout is not referenced. = 1: Returns at every successful step (Info = 1 is returned). Tout is not referenced. The end point of the last step is returned in T. In the final step, the step size is adjusted to fit into Tend. = 2: Returns at Tout during the ingegration to provide an intermediate result (T = Tout and Info = 2 are returned). To continue the integration to get result at new Tout, it is possible to call this program again with new Tout. In the final step, the step size is adjusted to fit into Tout. = 3: Returns at Tout during the ingegration to provide an intermediate result (T = Tout and Info = 3 are returned). To continue the integration to get result at new Tout, it is possible to call this program again with new Tout. Different from Mode = 2, the value of Y() at Tout is computed by interpolation, and the step size is not adjusted even in the last step before Tout. The steps through integration are same with those of Mode = 1. Note - This program does not return the computed value of y' (interpolated value in Yp()) in Mode = 3. If the value of y' as well as y is necessary, use Mode = 2. [in,out] Info [in] Control code. = 0: Set Info = 0 on the initial call to start new problem. All variables will be initialized and the computation will begin. = 1, 2, 3: When returned with Info = 1, 2 or 3, it is possible to call this program again with new Tout and without changing Info to continue the integration. [out] Return code. = 0: Successful exit. Integration to Tend completed. < 0: The (-Info)-th argument is invalid. = 1: Returned to provide intermediate result in Mode = 1. It is possible to call this program again to forward to the next step. = 2: Returned at T = Tout to provide intermediate result in Mode = 2. It is possible to call this program again with new Tout. = 3: Returned at T = Tout to provide intermediate result in Mode = 3. It is possible to call this program again with new Tout. = 11: (Error) Maximum number of steps exceeded. = 12: (Error) Step size becomes too small. [out] Neval (Optional) Number of function evaluations. [out] Nstep (Optional) Number of all computed steps. [out] Naccept (Optional) Number of accepted steps. [out] Nreject (Optional) Number of rejected steps. [in] Maxiter (Optional) Maximum number of allowed steps. (default = 10000) [in] ErrCntl (Optional) Specifies how determine the acceptance of the step. (default = 0) = 0: Determine the acceptance of the step using the estimated errors of both y and y'. = 1: Determine the acceptance of the step using the estimated error of y only. [in] Cnt (Optional) Specifies when counters (Neval, Nstep, Naccept and Nreject) are reset to zero. (default = 0) = 0: Reset if Info = 0. = 1: Do not reset even if Info = 0. [in] Hinit (Optional) Initial step size. (default = estimated by the program) [in] Hmax (Optional) Maximaum step size. (default = Abs(Tend - T)) [in] Fac1,Fac2 (Optional) Parameters for step size selection. (default: Fac1 = 0.2, Fac2 = 10) The new step size is chosen subject to the restriction Fac1 <= Hnew/Hold <= Fac2. [in] Safe (Optional) The safety factor in step size prediction. (0.0001 < Safe < 1) (default = 0.9) References (1) J. R. Dormand, M. E. A. El-Mikkawy, P. J. Prince, "High Order Embedded Runge-Kutta-Nystrom Formulae", IMA J. of Numerical Analysis, 7, 423-430 (1987). (2) Ch. Tsitouras, G. Papageorgiou "Interpolating Runge-Kutta-Nystrom Methods of High Order", Intern. J. Computer Math., Vol. 41, 209-217 (1993). Example Program Solve the following initial value problem of ordinary differential equations. y1'' = -y1/R y2'' = -y2/R where R = (y1^2 + y2^2)^(3/2)/(π/4)^2 (y1 = 0.75, y2 = 0, y1' = 0, y2' = (π/4)√(1.25/0.75) at t = 0) Sub F2(N As Long, T As Double, Y() As Double, Yp2() As Double) Dim R As Double R = (Y(0) ^ 2 + Y(1) ^ 2) ^ (3 / 2) / Atn(1) ^ 2 Yp2(0) = -Y(0) / R Yp2(1) = -Y(1) / R End Sub Sub Ex_Dopn86() Const N = 2 Dim T As Double, Y(N - 1) As Double, Yp(N - 1) As Double, Tend As Double, Tout As Double Dim RTol(0) As Double, ATol(0) As Double, Mode As Long, Neval As Long, Info As Long RTol(0) = 0.0000000001 '1.0e-10 ATol(0) = RTol(0) Mode = 2 T = 0: Y(0) = 0.75: Y(1) = 0: Yp(0) = 0: Yp(1) = Atn(1) * Sqr(1.25 / 0.75) Tend = 12 Info = 0 Do Tout = T + 1 Call Dopn86(N, AddressOf F2, T, Y(), Yp(), Tout, Tend, RTol(), ATol(), Mode, Info, Neval) Debug.Print T, Y(0), Y(1) Loop While Tout < Tend Debug.Print Neval, Info End Sub Note - If Mode is chenged to 3 instead of 2 in the same program, the dense outout (interpolation) is used. Example Results 1 0.294417538897896 0.812178519819116 2 -0.490299790384155 0.939874998440987 3 -1.05403151439005 0.57570608221171 4 -1.24999999949284 4.43173611507476E-09 5 -1.05403151546368 -0.57570607453233 6 -0.490299789381966 -0.939874992716228 7 0.294417544255 -0.812178511126571 8 0.74999999972237 1.56983329918214E-08 9 0.294417533018338 0.812178529211862 10 -0.490299791335412 0.939875006183219 11 -1.05403151373083 0.575706094313534 12 -1.25000000207949 1.96437796056619E-08 713 0