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

derkfa_r() void derkfa_r ( int  n, double *  t, double  y[], double  tout, double  tend, double *  rtol, double *  atol, int  itol, int  mode, double  work[], int  lwork, int  iwork[], int  liwork, int *  info, double *  tt, double  yy[], double  yyp[], int *  irev  ) Initial value problem of a system of first order ordinary differential equations (5(4)-th order Runge-Kutta-Fehlberg method) (Reverse communication version) Purpose This program integrates a system of first order ordinary differential equations of the form dy/dt = f(t, y), y = y0 at t = t0 where y is the n vector and the system consists of n equations. t0 and y0 are the given initial values of t and y, respectively. This program is the rewrite of the 5(4)-order Runge-Kutta-Fehrberg program DDERKF (Reference (1)) by adding a dense output function using the interpolation formula of Enright et al. (Reference (2)). This program is the reverse communication version of derkfa. Parameters [in] n Number of differential equations. (n >= 1) [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. t = tout or t of the latest step depending on mode in the case of intermediate return. [in,out] y[] Array y[ly] (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. [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 Scalar if itol = 0, or array rtol[lrtol] if itol = 1 (lrtol >= n) (rtol or rtol[i] >= 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 itol. 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 equations. (local error) <= rtol*abs(y[i]) + atol (if itol = 0) or (local error) <= rtol[i]*abs(y[i]) + atol[i] (if itol = 1) for each component of y[] (i = 0 to n-1). Both rtol and atol, or, rtol[i] and atol[i] (i = 0 to n-1) should not be 0 at the same time. [in] atol Scalar if itol = 0, or array atol[latol] if itol = 1 (latol >= n) (atol or all components 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 itol. atol is used together with rtol in a local error test at each step (Refer to rtol above). [in] itol Specifies whether the parameters rtol and atol are scalars or arrays. = 0: rtol and atol are scalars (or arrays with one element). = 1: rtol and atol are arrays. [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. [in,out] work[] Array work[lwork] Work array. work[0] to work[19] serve as parameters for the program. If the input parameter is set to 0, the default parameter value defined for each parameter will be used. [in] work[0]: Initial step size. (default = estimated by the program) [in] lwork The size of array work[]. (abs(lwork) >= 9*n + 40) If lwork < 0, the abs(lwork) is used, and all parameters work[0] to work[19] are cleared to 0. [in,out] iwork[] Array iwork[liwork] Integer work array. iwork[0] to iwork[19] serve as parameters for the program. If the input parameter is set to 0, the default parameter value defined for each parameter will be used. [in] iwork[1]: Maximum number of allowed steps. (default = 10000) iwork[12]: Specifies when counters iwork[13] to iwork[17] are reset to zero. (default = 0) = 0: Reset if info = 0. = 1: Do not reset even if info = 0. [out] Statistical information (counters). iwork[13]: Number of function evaluations. iwork[15]: Number of all computed steps. iwork[16]: Number of accepted steps. iwork[17]: Number of rejected steps. [in] liwork The size of array iwork[]. (abs(liwork) >= 40) If liwork < 0, the abs(liwork) is used, and all parameters iwork[0] to iwork[19] are cleared to 0. [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. = 13: (Error) Problem is probably stiff. = 14: (Error) Error test failed. Tolerances may be too stringent. [out] tt irev = 1〜10: Refer to yyp[]. [out] yy[] Array yy[lyy] (lyy >= n) irev = 1〜10: Refer to yyp[]. [in] yyp[] Array yyp[lyyp] (lyyp >= n) irev = 1〜10: Computed derivatives dy/dt = f(t, y) at given t = tt and y = yy[] should be set in yyp[] in the next call. [in,out] irev Control variable for reverse communication interface. [in] irev should be initialized to zero in the first call. In succeeding calls, irev should not be altered. [out] If irev is not zero, complete the following tasks and call this program again without changing irev. = 0: Computation finished. Check return code in info. = 1 to 10: Set the computed derivatives at t = tt and y = yy[] in yyp[]. Do not alter any variables other than yyp[]. Reference (1) netlib/ode (2) W. H. Enright, et al.: "Interpolants for Runge-Kutta Formulas", ACM Transactions on Mathematocal Software, Vol.12, No.3. (1986)