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() Sub Seulex ( N As  Long, F As  LongPtr, Ifcn As  Long, 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 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) 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. 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] 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] 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) = -5: The argument Y() is invalid. = -7: The argument RTol() had an illegal value. (RTol(i) < 0, RTol(i) = 0 and ATol(i) = 0) = -8: The argument ATol() had an illegal value. (ATol(i) < 0) = -9: the argument Info had an illegal value (Info <> 0 and Info <> 1) = -15: The argument Mljac or Mlmas had an illegal value. (Mlmas > Mljac) = -16: The argument Mujac or Mumas had an illegal value. (Mumas > Mujac) = -22: The argument M1 had an illegal value. (M1 < 0) = -23: The argument M1 or M2 had an illegal value. (M2 < 0 or M1 + M2 > N) = 1: Successful exit. = 2: Interrupted by Solout (normal return). = 11: Maximum number of steps exceeded. [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, RCont As Double, ICont As Long, 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 RCont and ICont. 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) = Contex(i, T2, RCont, ICont) 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-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() Const N = 2 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, I 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 Call Seulex(N, AddressOf F2, Ifcn, T, Y(), Tout, RTol(), ATol(), Info) If Info <> 1 Then Debug.Print "Error in Seulex: 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_2() Const N = 2 Dim Ifcn As Long, T As Double, Y(N - 1) As Double, Tend 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) Ifcn = 1 T = 0: Tend = 10: Y(0) = 1: Y(1) = 2 Info = 0 Call Seulex(N, AddressOf F2, Ifcn, T, Y(), Tend, RTol(), ATol(), Info, AddressOf SoloutEx) If Info <> 1 Then Debug.Print "Error in Seulex: Info =", Info End Sub Sub SoloutEx(Nr As Long, Told As Double, T As Double, Y() As Double, N As Long, RCont As Double, ICont As Long, Irtrn As Long) Dim Y0 As Double, Y1 As Double Static Tout As Double If Nr = 1 Then Tout = 1 While T >= Tout Y0 = Contex(0, Tout, RCont, ICont) Y1 = Contex(1, Tout, RCont, ICont) Debug.Print Tout, Y0, Y1 Tout = Tout + 1 Wend 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