Contex Contex_r Contr5 Contr5_r Contra Contra_r Contro Contro_r Contrp Contrp_r Dassl Dassl_r Radau Radau5 Radau5_r Radau_r Radaup Radaup_r Rodas Rodas_r Seulex Seulex_r

Seulex_r() Sub Seulex_r ( N As  Long, Ifcn As  Long, T As  Double, Y() As  Double, Tout As  Double, RTol() As  Double, ATol() As  Double, RCont() As  Double, ICont() As  Long, Info As  Long, TT As  Double, YY() As  Double, YYp() As  Double, YYpd() As  Double, Irtrn As  Long, IRev As  Long, Optional Iout As  Long, Optional Ijac As  Long, Optional Mljac As  Long = -1, Optional Mujac As  Long, Optional Imas As  Long, Optional Mlmas As  Long = -1, Optional Mumas As  Long, Optional Neval As  Long, Optional Njac As  Long, Optional Nstep As  Long, Optional Naccept As  Long, Optional Nreject As  Long, Optional M1 As  Long, Optional M2 As  Long, Optional Hinit As  Double, Optional Hmax As  Double, Optional MaxIter As  Long, Optional Km As  Long, Optional Nsequ As  Long, Optional Lambda As  Long, Optional Hess As  Long, Optional Thet As  Double, Optional Fac1 As  Double, Optional Fac2 As  Double, Optional Fac3 As  Double, Optional Fac4 As  Double, Optional Safe1 As  Double, Optional Safe2 As  Double, Optional Cnt As  Long  ) Initial value problem of ordinary differential equations (extrapolation method based on the linearly implicit Euler method) (reverse communication version) NOTE - THIS PROGRAM IS DEPRECATED AND WILL BE REMOVED IN THE NEXT VERSION. Purpose This routine computes a numerical solution of a stiff (or differential algebraic) system of first order ordinary differential equations of the form M * dy/dt = f(t, y), y = y0 at t = t0 where t0 and y0 are the given initial values of t and y, respectively. y may be a vector if the above is a system of differential equations. M is the mass matrix. The system can be linearly implicit (M is not I (identity matrix)) or explicit (M = I). Seulex is the code of extrapolation method based on the linearly implicit Euler method. It is provided with the step control algorithm and the continuous output feature. See for details in the reference below. Seulex_r is the reverse communication version of Seulex. Parameters [in] N Number of differential equations. (N >= 1) [in] Ifcn Whether f() depends on t or not. = 0: Independent of t: dy/dt = f(y) (autonomous). = 1: May depend on t: dy/dt = f(y, t) (non-autonomous). [in,out] T This routine integrates from T to Tout. The initial point of the integration is to be given, and the last point of the final step will be returned. [in] Initial value of the independent variable T. [out] Last value of the independent variable T of the final step (normally equals to Tout). The solution was successfully advanced to this point. It is possible to continue the integration to new point by recalling this routine with the new Tout value with setting Info = 1. [in,out] Y() Array Y(LY - 1) (LY >= N) [in] Initial values of the dependent variables Y() at initial T. [out] Computed solution approximation at last T (normally equals to Tout). [in] Tout Set Tout to the point at which a solution is desired. Integration either forward in T (Tout > T) or backward in T (Tout < T) is permitted. The routine advances the solution from T to Tout using step sizes which are automatically selected so as to achieve the desired accuracy. [in] RTol() Array RTol(LRTol - 1) (LRTol >= 1) (all components of RTol() >= 0) The relative error tolerance(s) to tell the code how accurately you want the solution to be computed. This parameter may be a scalar (LRTol = 1) or a vector (LRTol = N). If LRTol = 2, ... or N-1, LRTol = 1 is assumed. If LRTol > N, LRTol = N is assumed. Even if LRTol = N, it is assumed to be 1 if LATol = 1. The tolerances are used by the code in a local error test at each step which requires roughly that   abs(local error of Y(i)) <= RTol(i)*abs(Y(i)) + ATol(i) for each component of Y() (i = 0 to LRTol-1). Setting RTol(i) = 0 results in a pure absolute error test on that component. RTol(i) and ATol(i) should not be zero at the same time (i = 0 to LRTol-1). [in] ATol() Array ATol(LATol - 1) (LATol >= 1) (all components of ATol() >= 0) The absolute error tolerance(s) to tell the code how accurately you want the solution to be computed. This parameter may be a scalar (LATol = 1) or a vector (LATol = N). If LATol = 2, ... or N-1, LATol = 1 is assumed. If LATol > N, LATol = N is assumed. Even if LATol = N, it is assumed to be 1 if LRTol = 1. The tolerances are used by the code in a local error test at each step which requires roughly that   abs(local error of Y(i)) <= RTol(i)*abs(Y(i)) + ATol(i) for each component of Y() (i = 0 to LATol-1). Setting ATol(i) = 0 results in a pure relative error test on that component. RTol(i) and ATol(i) should not be zero at the same time (i = 0 to LRTol-1). [in,out] RCont() Array RCont(LRCont - 1) (LRCont >= (Km + 2)*N) Control information for dense output. (Not referenced if Iout = 0) [in,out] ICont() Array ICont(LICont - 1) (LICont >= N) Control information for dense output. (Not referenced if Iout = 0) [in,out] Info [in] = 0: Initialize and start computation (Solve new problem). = 1: Continue computation with new Tout value (Resume computation of previous call). [out] = -1: The argument N had an illegal value. (N < 1) = -4: The argument Y() is invalid. = -6: The argument RTol() had an illegal value. (RTol(i) < 0, RTol(i) = 0 and ATol(i) = 0) = -7: The argument ATol() had an illegal value. (ATol(i) < 0) = -8: The argument RCont() is invalid. = -9: The argument ICont() is invalid. = -10: the argument Info had an illegal value (Info <> 0 and Info <> 1) = -12: The argument YY() is invalid. = -13: The argument YYp() is invalid. = -14: The argument YYpd() is invalid. = -16: the argument IRev had an illegal value. (IRev <> 0, 1, 2, 3, 4 nor 5) = -22: The argument Mljac or Mlmas had an illegal value. (Mlmas > Mljac) = -23: The argument Mujac or Mumas had an illegal value. (Mumas > Mujac) = -29: The argument M1 had an illegal value. (M1 < 0) = -30: The argument M1 or M2 had an illegal value. (M2 < 0 or M1 + M2 > N) = 1: Successful exit. = 2: Interrupted by Irtrn (normal return). = 11: Maximum number of steps exceeded. [out] TT IRev = 1, 2 or 3: The value of independent variable t where the derivatives or Jacobian should be evaluated and provided in YYp(). [out] YY() Array YY(LYY - 1) (LYY >= N) IRev = 1, 2 or 3: The value of dependent variables y where the derivatives or Jacobian should be evaluated and provided in YYp(). [in] YYp() Array YYp(LYYp - 1) (LYYp >= N) IRev = 1: The computed derivatives at TT and YY(), i.e. dyi/dt = fi(TT, YY()) (i = 0 to N-1) should be provided in the next call. IRev = 3: The computed derivatives of f with respect to t at TT and YY(), i.e. dfi(TT, YY())/dt (i = 0 to N-1) should be provided in the next call. [in] YYpd() Array YYpd(LYYpd1 - 1, LYYpd2 - 1) (LYYpd1 >= max(Ljac, Lmas), LYYpd2 >= N) IRev = 2: The Jacobian (dfi/dyj) at TT and YY() should be provided in the next call.   Note - If Mljac = N, a matrix is stored as N x N full matrix (Ljac = N). If Mljac < N, a matrix is stored in band matrix form (Ljac = Mljac + Mujac + 1). If Ijac = 0, Ljac is 0. IRev = 4: The mass matrix M should be provided in the next call.   Note - If Mlmas = N, a matrix is stored as N x N full matrix (Lmas = N). If Mlmas < N, a matrix is stored in band matrix form (Lmas = Mlmas + Mumas + 1). If Imas = 0, Lmas is 0. [in,out] Irtrn [in] IRev = 5: Do not alter unless user want to interrupt the integration. If Irtrn is set to the negative value, the integration will be interrupted and exit with Info = 2. [out] IRev = 5: Returns 0, 1 or 2 in the first, intermediate or last return with IRev = 5, respectively. [in,out] IRev Control variable for reverse communication. [in] Before first call, IRev should be initialized to zero. On succeeding calls, IRev should not be altered. [out] If IRev is not zero, complete the following tasks and call this routine again without changing IRev. = 0: Computation finished. See return code in Info. = 1: User should set the computed derivative values at TT and YY() in YYp(). Do not alter any variables other than YYp(). = 2: User should set the computed Jacobian (dfi/dxj) at TT and YY() in YYpd(). Do not alter other variables. = 3: User should set the computed partial derivatives of fi with respect to t (dfi/dt) at TT and YY() in YYp(). Do not alter other variables. = 4: User should set the mass matrix M in YYpd(). Do not alter other variables. = 5: User may output the intermediate result. Do not alter any variables. (See Iout) [in] Iout (Optional) Specifies if the intermediate result output is required. (default = 0) = 0: Output is not required. (Not return with IRev = 5) = 1: Returns after every successful step with IRev = 5 to output the intermediate results. This is same as calling Solout in the case of normal Rodas (non reverse communication version). The corresponding information is as follows. Nr = Naccept + 1, Told = previous T, T = current T, Y() = current Y(). RCont() and ICont() are used in the same way with Solout for the dense output. Y(i) = Contex_r(i, T2, RCont(), ICont()) (If other value is specified for Iout, Iout = 0 will be assumed) [in] Ijac (Optional) Switch for calculation of Jacobian. (default = 0) = 0: Jacobian is calculated by finite differences. (Not return with IRev = 2) = 1: Jacobian is calculated by user externally. (Return with IRev = 2 when calculation is required) (For other values, the default value will be used) [in] Mljac (Optional) The lower bandwidth of Jacobian. (0 <= Mljac <= N) (default = N) If Mljac = N, Jacobian is stored as N x N full matrix. If Mljac < N, Jacobian is stored in band matrix form. (If Mljac < 0 or Mljac > N, the default value will be used) [in] Mujac (Optional) The upper bandwidth of Jacobian. (0 <= Mujac <= N) (default = 0) If Mljac = N, Mujac is ignored. (If Mujac < 0 or Mujac > N, the default value will be used) [in] Imas (Optional) Calculation method of the mass matrix M (default = 0) = 0: M is supposed to be the identity matrix (never return with IRev = 4) = 1: M is calculated and given externally (return with IRev = 4 when calculation is required) [in] Mlmas (Optional) The lower bandwidth of mass matrix M. (0 <= Mlmas <= N) (default = N) If Mlmas = N, M is stored as N x N full matrix. If Mlmas < N, M is stored in band matrix form. (If Mlmas < 0 or Mlmas > N, the default value will be used) [in] Mumas (Optional) The upper bandwidth of mass matrix M. (0 <= Mumas <= N) (default = 0) If Mlmas = N, Mumas is ignored. (If Mumas < 0 or Mumas > N, the default value will be used) [out] Neval (Optional) Number of function evaluations. (Those for Jacobian evaluations are not included) [out] Njac (Optional) Number of Jacobian evaluations. (Those by finite differences are included) [out] Nstep (Optional) Number of computed steps. [out] Naccept (Optional) Number of accepted steps. [out] Nreject (Optional) Number of rejected steps. (Step rejections in the first step are not counted) [in] M1,M2 (Optional) If the first M1 equations has the following form   y'(i) = y(i + M2) for i = 1 to M1, with M1 a multiple of M2, and the remaining equations do not explicitly depend on y'(M1), ..., y'(N-1), efficient computation can be achieved by setting parameters M1 and M2 to nonzero values. (M1 > 0, M2 > 0, M1 + M2 <= N) (default M1 = M2 = 0) When parameters are set to nonzero, only the elements of non-trivial part of the Jacobian (rows M1+1 to N) heve to be stored in (N - M1) x N array. Also only the elements of right lower block of order N - M1 of the mass matrix M have to be stored in (N - M1) x (N - M1) array. [in] Hinit (Optional) Initial step size. (default = 1.0e-6) H = 1/||f'||, usually 1.0e-2 or 1.0e-3 is good for stiff equations with initial transient. (If Hinit = 0, 1.0e-6 will be used) [in] Hmax (Optional) Maximal step size. (default = Tout - T) (If Hmax = 0, the default value will be used) [in] MaxIter (Optional) Maximum number of allowed steps. (default = 100000) (If MaxIter <= 0, the default value will be used) [in] Km (Optional) Maximum number of columns in the extrapolation table. (Km >= 3) (default = 12) (If Km < 3, the default value will be used) [in] Nsequ (Optional) Switch for the step size sequence. (default = 2) = 1: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, ... = 2: 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, ... = 3: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... = 4: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ... (If other value is specified, the default value will be used) [in] Lambda (Optional) Parameter "lambda" of dense output. (Lambda = 0 or 1) (default = 0) (If Lambda <> 0 and Lambda <> 1, the default value will be used) [in] Hess (Optional) Switch to transform the Jacobian matrix to Hessenberg form. (default = 0) = 0: Do not transform. <> 0: Transform. This is particularly advantageous for large systems with full Jacobian. It does not work for banded Jacobian and not for implicit systems (Mas <> NullPtr). [in] Thet (Optional) Decides whether the Jacobian should be recomputed. (default = min(0.0001, RTol)) Increase Thet (e.g. 0.01) when Jacobian evaluations are costly. For small systems, Thet should be smaller. (If Thet = 0, the default value will be used) [in] Fac1,Fac2 (Optional) Parameters for step size selection. (default: Fac1 = 0.1, Fac2 = 4) The new step size for the j-th diagonal entry is chosen subject to the restriction:   Facmin/Fac2 <= Hnew(j)/Hold <= 1/Facmin where Facmin = Fac1^(1/(j - 1)). (If Fac1 = 0 or Fac2 = 0, the default values will be used) [in] Fac3,Fac4 (Optional) Parameters for the order selection. (default: Fac3 = 0.7, Fac4 = 0.9)   Step size is decreased if W(k-1) <= W(k)*Fac3.   Step size is increased if W(k) <= W(k-1)*Fac4. (If Fac3 = 0 or Fac4 = 0, the default values will be used) [in] Safe1,Safe2 (Optional) Safety factors for step control algorithm. (default: Safe1 = 0.6, Safe2 = 0.93)   Hnew(j) = H*Safe2*(Safe1*Tol/Err)^(1/(j - 1)) (If Safe1 = 0 or Safe2 = 0, the default values will be used) [in] Cnt (Optional) Specifies when Neval, Njac, Nstep, Naccept and Nreject are reset to zero. (default = 0) = 0: Reset whenever this routine is called. <> 0: Reset only if this routine is called with Info = 0. Reference E. Hairer, S.P. Norsett and G. Wanner, "Solving Ordinary Differential Equations II. Stiff and differential-algebraic Problems. 2nd edition", Springer Series in Computational Mathematics, Springer-Verlag (1996) Example Program (1) Solve the following initial value problem of ordinary differential equations (stiff problem). dy1/dt = -2*y1 + y2 - cos(t) dy2/dt = 1998*y1 - 1999*y2 + 1999*cos(t) - sin(t) (y1 = 1, y2 = 2 at t = 0) Sub F2(N As Long, T As Double, Y() As Double, Yp() As Double) Yp(0) = -2 * Y(0) + Y(1) - Cos(T) Yp(1) = 1998 * Y(0) - 1999 * Y(1) + 1999 * Cos(T) - Sin(T) End Sub Sub Ex_Seulex_r() Const N = 2, KM = 12 Dim Ifcn As Long, T As Double, Y(N - 1) As Double, Tend As Double, Tout As Double Dim RTol(0) As Double, ATol(0) As Double, Info As Long Dim RCont((KM + 2) * N) As Double, ICont(N + 1) As Long Dim TT As Double, YY(N - 1) As Double, YYp(N - 1) As Double, YYpd(0) As Double Dim Irtrn As Long, IRev As Long, Iout As Long RTol(0) = 0.0000000001 '1.0e-10 ATol(0) = RTol(0) Ifcn = 1 T = 0: Tend = 10: Y(0) = 1: Y(1) = 2 Info = 0 Do Tout = T + 1 IRev = 0 Do Call Seulex_r(N, Ifcn, T, Y(), Tout, RTol(), ATol(), RCont(), ICont(), Info, TT, YY(), YYp(), YYpd(), Irtrn, IRev, Iout) If IRev = 1 Then Call F2(N, TT, YY(), YYp()) Loop While IRev <> 0 If Info <> 1 Then Debug.Print "Error in Seulex_r: Info =", Info Exit Do End If Debug.Print T, Y(0), Y(1) Loop While Tout < Tend End Sub Example Results 1 0.367879441171429 0.908181747019678 2 0.135335283238996 -0.280811553295101 3 4.97870683828775E-02 -0.940205428219208 4 1.83156388962834E-02 -0.635327981969723 5 6.73794700708669E-03 0.290400132470383 6 2.47875218151303E-03 0.962649038831868 7 9.11881988566409E-04 0.754814136332881 8 3.35462640858368E-04 -0.145164571166839 9 1.23409810850164E-04 -0.911006852073842 10 4.53999366716977E-05 -0.839026129140713 Example Program (2) Solve the following initial value problem of ordinary differential equations (stiff problem) (using dense output). dy1/dt = -2*y1 + y2 - cos(t) dy2/dt = 1998*y1 - 1999*y2 + 1999*cos(t) - sin(t) (y1 = 1, y2 = 2 at t = 0) Sub F2(N As Long, T As Double, Y() As Double, Yp() As Double) Yp(0) = -2 * Y(0) + Y(1) - Cos(T) Yp(1) = 1998 * Y(0) - 1999 * Y(1) + 1999 * Cos(T) - Sin(T) End Sub Sub Ex_Seulex_r_2() Const N = 2, KM = 12 Dim Ifcn As Long, T As Double, Y(N - 1) As Double, Tend As Double, Tout As Double Dim RTol(0) As Double, ATol(0) As Double, Info As Long Dim RCont((KM + 2) * N) As Double, ICont(N + 1) As Long Dim TT As Double, YY(N - 1) As Double, YYp(N - 1) As Double, YYpd(0) As Double Dim Irtrn As Long, IRev As Long, Iout As Long Dim Y0 As Double, Y1 As Double RTol(0) = 0.0000000001 '1.0e-10 ATol(0) = RTol(0) Iout = 1 Ifcn = 1 T = 0: Tend = 10: Y(0) = 1: Y(1) = 2 Tout = 1 Info = 0 IRev = 0 Do Call Seulex_r(N, Ifcn, T, Y(), Tend, RTol(), ATol(), RCont(), ICont(), Info, TT, YY(), YYp(), YYpd(), Irtrn, IRev, Iout) If IRev = 1 Then Call F2(N, TT, YY(), YYp()) ElseIf IRev = 5 Then While T >= Tout Y0 = Contex_r(0, Tout, RCont(), ICont()) Y1 = Contex_r(1, Tout, RCont(), ICont()) Debug.Print Tout, Y0, Y1 Tout = Tout + 1 Wend End If Loop While IRev <> 0 If Info <> 1 Then Debug.Print "Error in Seulex_r: Info =", Info End Sub Example Results 1 0.367879441171669 0.90818174654094 2 0.135335283289468 -0.280811548008303 3 4.97870684964011E-02 -0.940205424371012 4 1.83156530207606E-02 -0.635328169713382 5 6.73794698286098E-03 0.290400132401757 6 2.47877855763725E-03 0.962655193543705 7 9.11881827350402E-04 0.754814133775546 8 3.35470947611422E-04 -0.145189025251434 9 1.23409759086634E-04 -0.911006868342305 10 4.53999113068494E-05 -0.839026129166472