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

Radau5() Sub Radau5 ( N As  Long, F As  LongPtr, T As  Double, Y() As  Double, Tout As  Double, RTol() As  Double, ATol() As  Double, Info As  Long, Optional Solout As  LongPtr = NullPtr, Optional Jac As  LongPtr = NullPtr, Optional Mljac As  Long = -1, Optional Mujac As  Long, Optional Mas As  LongPtr = NullPtr, 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 MaxNit As  Long, Optional StartN As  Long, Optional Nind1 As  Long, Optional Nind2 As  Long, Optional Nind3 As  Long, Optional Pred As  Long, Optional Hess As  Long, Optional Safe As  Double, Optional Thet As  Double, Optional FNewt As  Double, Optional Quot1 As  Double, Optional Quot2 As  Double, Optional Facl As  Double, Optional Facr As  Double, Optional Cnt As  Long  ) Initial value problem of ordinary differential equations (5-th order implicit Runge-Kutta method (Radau IIA)) 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). Radau5 is the implicit Runge-Kutta code based on the 5-th order Radau IIA method. It is provided with the step control algorithm and the continuous output feature. See for details in the reference below. Parameters [in] N Number of differential equations. (N >= 1) [in] F The user supplied subroutine, which calculates the derivatives of the differential equations, defined as follows. Sub F(N As Long, T As Double, Y() As Double, Yp() As Double) Yp(i) = computed derivative at given T and Y() (i = 0 to N-1) End Sub where N is the number of equations, and Yp() is the computed derivatives at given T and Y(), i.e. Yp(i) = dyi/dt = fi(T, Y(0), ..., Y(N-1)) (i = 0 to N-1). The other variables than Yp() should not be altered. In many cases, M is an identity matrix. However, it can be defined in subroutine Mas if necessary. [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] 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 Info had an illegal value (Info <> 0 and Info <> 1) = -14: The argument Mljac or Mlmas had an illegal value. (Mlmas > Mljac) = -15: The argument Mujac or Mumas had an illegal value. (Mumas > Mujac) = -21: The argument M1 had an illegal value. (M1 < 0) = -22: The argument M1 or M2 had an illegal value. (M2 < 0 or M1 + M2 > N) = -28: The argument Nind1, Nind2 or Nind3 had an illegal value. (Nind1 + Nind2 + Nind3 <> N) = -35: The argument FNewt had an illegal value. (FNewt too small) = 1: Successful exit. = 2: Interrupted by Solout (normal return). = 11: Maximum number of steps exceeded. = 12: Step size becomes too small. = 13: Matrix is repeatedly singular. [in] Solout (Optional) The user supplied subroutine to print out the intermediate solutions, which is called after every successful step, defined as follows. (default = NullPtr) Sub Solout(Nr As Long, Told As Double, T As Double, Y() As Double, N As Long, Cont As Double, Irtrn As Long) Output the Y() values at Nr-th step T. Told is the previous value of T. N is the order of equations. The value of Irtrn will be 0, 1 or 2 in the first, intermediate or last call of Solout, respectively. Irtrn also serves to interrupt the integration. If Irtrn is set to the negative value in Solout, the integration will be interrupted and exit with Info = 2. Dense output is supported by the control information Cont. The solution Y(i) (0 <= i <= N-1) at the arbitrary point T2 in the interval [Told, T] can be computed by the function call Y(i) = Contr5(i, T2, Cont) End Sub If Solout is not provided (if Solout = NullPtr), the intermediate solutions will not be printed out. [in] Jac (Optional) The user supplied subroutine, which computes the Jacobian dfi(t, y)/dyj, defined as follows. (default = NullPtr) Sub Jac(N As Long, T As Double, Y() As Double, Ypd() As Double) Ypd(i,j) = the calculated value of dfi/dyj at T and Y() (for i = 0 to N-1, j = 0 to N-1). End Sub Ypd() is two dimensional full matrix if Mljac = N. It is in band matrix form if 0 <= Mljac < N. The other variables than Ypd() should not be altered. If Jac is not provided (if Jac = NullPtr), the Jacobian will be computed by finite differences. [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] Mas (Optional) The user supplied subroutine, which provides the mass matrix M, defined as follows (default = NullPtr). Sub Mas(N As Long, Am() As Double) Am(i,j) = the value of mass matrix Mij (i = 0 to N-1, j = 0 to N-1). End Sub Am() is two dimensional full matrix if Mlmas = N. It is in band matrix form if 0 <= Mlmas < N. The other variables than Am() should not be altered. If Mas is not provided (if Mas = NullPtr), the mass matrix M is supposed to be the identity matrix. [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 in Jac. 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 Mas. [in] Hinit (Optional) Initial step size. (default = 1.0e-6) H = 1/||f'||, usually 1.0e-3 or 1.0e-5 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] MaxNit (Optional) Maximum number of Newton iterations for the solution of the implicit system in each step. (default = 7) (if MaxNit <= 0, the default value will be used) [in] StartN (Optional) Starting value for Newton's method. (default = 0) = 0: Extrapolated collocation solution. <> 0: Zero starting value. The latter is recommended if Newton's method has difficulties with convergence. [in] Nind1,Nind2,Nind3 (Optional) Dimension of the index 1, 2 and 3 variables, respectively. (Nind1 > 0, Nind1 + Nind2 + Nind3 = N) (default: Nind1 = N, Nind2 = 0, Nind3 = 0) These parameters are important for differential algebraic equations (DAEs) of index > 1. The function subroutine should be written such that the index 1, 2 and 3 variables appear in this order. For ordinary defferential equations (ODEs), Nind1 equals to the number of equations of the system. [in] Pred (Optional) Switch for step size strategy. (default = 1) = 1: Model predictive controller (Gustafsson). = 2: Classical step size control. (For other values, the default value will be assumed) [in] Hess (Optional) Switch to transform the Jacobian matrix to Hessenberg form. (default = 0) = 0: Do not transform. (This option is not supported. For other values, 0 will be assumed.) [in] Safe (Optional) The safety factor in step size prediction. (default = 0.9) (If Safe <= 0.001 or Safe >= 1, the default value will be used) [in] Thet (Optional) Decides whether the Jacobian should be recomputed. (Thet < 1) (default = 0.001) Increase Thet (e.g. 0.1) when Jacobian evaluations are costly. For small systems, Thet should be smaller (e.g. 0.001). Negative Thet forces the code to compute the Jacobian after every accepted step. (If Thet = 0 or Thet >= 1, the default value will be used) [in] FNewt (Optional) Stopping criterion for Newton's method. (Usually FNewt < 1) (default = automatically computed from RTol(0)) Smaller values make the code slower, but safer. (If FNewt = 0, the default value will be used) [in] Quot1,Quot2 (Optional) If Quot1 < Hnew/Hold < Quot2, the step size is not chenged. (Quot1 <= 1, Quot2 >= 1) (default: Quot1 = 1, Quot2 = 1.2) Quot1 = 1 and Quot2 = 1.2 for small systems, Quot1 = 0.99 and Quot2 = 2 for large systems might be good. (If Quot1 = 0 or Quot1 > 1, Quot2 = 0 or Quot2 < 1, the default values will be used respectively) [in] Facl,Facr (Optional) Parameters for step size selection. (Facl <= 1, Facr >= 1) (default: Facl = 0.2, Facr = 8) The new step size is chosen subject to the restriction Facl < Hnew/Hold < Facr. (If Facl = 0 or Facl > 1, Facr = 0 or Facr < 1, the default values will be used respectively) [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_Radau5() Const N = 2 Dim 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, I As Long RTol(0) = 0.0000000001 '1.0e-10 ATol(0) = RTol(0) T = 0: Tend = 10: Y(0) = 1: Y(1) = 2 Info = 0 Do Tout = T + 1 Call Radau5(N, AddressOf F2, T, Y(), Tout, RTol(), ATol(), Info) If Info <> 1 Then Debug.Print "Error in Radau5: Info =", Info Exit Do End If Debug.Print T, Y(0), Y(1) Loop While Tout < Tend End Sub Example Results 1 0.367879441187316 0.90818174710239 2 0.135335283255029 -0.280811553607639 3 4.97870683911592E-02 -0.940205428513689 4 1.83156389200067E-02 -0.635327984370112 5 6.73794702805148E-03 0.290400132651525 6 2.47875220047626E-03 0.96264903967345 7 9.11881982753585E-04 0.754814137453754 8 3.35462642879546E-04 -0.145164572401227 9 1.23409800092344E-04 -0.911006854141312 10 4.53999251935239E-05 -0.839026134868127 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_Radau5_2() Const N = 2 Dim T As Double, Y(N - 1) As Double, Tend As Double Dim RTol(0) As Double, ATol(0) As Double, Info As Long RTol(0) = 0.0000000001 '1.0e-10 ATol(0) = RTol(0) T = 0: Tend = 10: Y(0) = 1: Y(1) = 2 Info = 0 Call Radau5(N, AddressOf F2, T, Y(), Tend, RTol(), ATol(), Info, AddressOf SoloutR5) If Info <> 1 Then Debug.Print "Error in Radau5: Info =", Info End Sub Sub SoloutR5(Nr As Long, Told As Double, T As Double, Y() As Double, N As Long, Cont As Double, Irtrn As Long) Dim Y0 As Double, Y1 As Double Static Tout As Double If Nr = 1 Then Tout = 1 While T >= Tout Y0 = Contr5(0, Tout, Cont) Y1 = Contr5(1, Tout, Cont) Debug.Print Tout, Y0, Y1 Tout = Tout + 1 Wend End Sub Example Results 1 0.36787944514433 0.90818175052153 2 0.135335281095574 -0.280811550970382 3 4.97870683819513E-02 -0.940205439380229 4 1.83156414146801E-02 -0.635327955429466 5 6.73794660800666E-03 0.29040008345163 6 2.47875304716379E-03 0.962649009570966 7 9.11881374205137E-04 0.754814090253208 8 3.35462420859941E-04 -0.145164626236664 9 1.23409926575974E-04 -0.911006958450603 10 4.53999332681047E-05 -0.839026159221514