Chkder Hybrd Hybrd1 Hybrd1_r Hybrd_r Hybrj Hybrj1 Hybrj1_r Hybrj_r Sos Sos_r

Hybrd() Sub Hybrd ( F As  LongPtr, N As  Long, X() As  Double, Fvec() As  Double, XTol As  Double, Diag() As  Double, Mode As  Long, Info As  Long, Optional Maxfev As  Long = 0, Optional Ml As  Long = -1, Optional Mu As  Long = -1, Optional Epsfcn As  Double = 0, Optional Factor As  Double = 0, Optional Nprint As  Long = 0, Optional Nfev As  Long  ) Solution of a system of nonlinear equations by Powell hybrid method (Jacobian not required) Purpose This routine finds a zero of a system of n nonlinear functions in n variables fi(x1, x2, ..., xn) = 0 (i = 1 to n) by a modification of the Powell hybrid method. The user must provide a subroutine which calculates the functions. Since the Jacobian is calculated by a forward difference approximation within the routine, the user is not required to provide the Jacobian. Parameters [in] F User supplied subroutine which calculates the functions fi(x) defined as follows. Sub F(N As Long, X() As Double, Fvec() As Double, IFlag As Long) If IFlag = 1 or IFlag = 2: Calculate the function values fi(x) at X() and return in Fvec() (i = 0 to n-1). Other variables should not be changed. If IFlag = 0: If Nprint is set to positive value, F is called every Nprint iterations with IFlag = 0 for outputting intermediate result X() and Fvec(). Any variables should not be changed. End Sub The value of Iflag should not be changed unless the user wants to terminate the execution. In this case, set Iflag to a negative integer. [in] N Number of functions and variables. (N > 0) [in,out] X() Array X(LX - 1) (LX >= N) [in] An initial estimate of the solution vector. [out] The obtained solution vector. [out] Fvec() Array Fvec(LFvec - 1) (LFvec >= N) The function values evaluated at the solution vector X(). [in] XTol Target relative tolerance. Termination occurs when the relative error between two consecutive iterations is at most XTol. (XTol >= 0) [in,out] Diag() Array Diag(LDiag - 1) (LDiag >= N) [in] If Mode = 2, Diag() must contain positive entries that serve as multiplicative scale factors for the variables. (Diag(i) > 0) [out] If Mode = 1, Diag() is set by the subroutine. [in] Mode = 1: The variables will be automatically scaled by the subroutine. = 2: The scaling is specified by the input Diag(). (For other values, Mode = 1 is assumed.) [out] Info = 0: Successful exit. (Relative error between two consecutive iterates is at most XTol) = -2: The argument N had an illegal value. (N <= 0) = -3: The argument X() is invalid. = -4: The argument Fvec() is invalid. = -5: The argument XTol had an illegal value. (XTol < 0) = -6: The argument Diag() is invalid. = 1: Number of calls to F with IFlag = 1 or 2 has reached Maxfev. = 2: XTol is too small. No further improvement in the approximate solution X is possible. = 3: Iteration is not making good progress, as measured by the improvement from the last five Jacobian evaluations. = 4: Iteration is not making good progress, as measured by the improvement from the last ten iterations. = 5: User imposed termination (returned from F with IFlag < 0). [in] Maxfev (Optional) Termination occurs when the number of calls to F with IFlag = 1 or 2 has reached this value. (default = 200*(N + 1)) (if Maxfev <= 0, the default value will be used) [in] Ml (Optional) Number of subdiagonals within the band of the Jacobian matrix (Ml >= 0). If the Jacobian is not banded, set Ml to at least N - 1. (default = N - 1) (if Ml < 0, the default value will be used) [in] Mu (Optional) Number of superdiagonals within the band of the Jacobian matrix (Mu >= 0). If the Jacobian is not banded, set Mu to at least N - 1. (default = N - 1) (if Mu < 0, the default value will be used) [in] Epsfcn (Optional) Used in determining a suitable step length for the forward-difference approximation. This approximation assumes that the relative errors in the functions are of the order of Epsfcn. If Epsfcn is less than the machine precision, it is assumed that the relative errors in the functions are of the order of the machine precision. (default = 0) [in] Factor (Optional) Used in determining the initial step bound. This bound is set to the product of Factor and the Euclidean norm of Diag*X if nonzero, or else to factor itself. In most cases factor should lie in the interval (0.1, 100). 100 is a generally recommended value. (default = 100) (if Factor <= 0, the default value will be used) [in] Nprint (Optional) Enables controlled printing of iterates. (default = 0) > 0: F is called with Iflag = 0 at the beginning of the first iteration and every nprint iterations thereafter and at the end of last iteration for printing intermediate result. <= 0: No special calls of F for printing intermediate results are made. [out] Nfev (Optional) Number of function evaluations (number of calls to F with IFlag = 1 or 2). Reference netlib/minpack Example Program Solve the following system of nonlinear equations. x1^2 - x2 - 1 = 0 (x1 - 2)^2 + (x2 - 0.5)^2 - 1 = 0 The initial approximation (x1, x2) = (0, 0) is used. Sub FHybrd(N As Long, X() As Double, Fvec() As Double, IFlag As Long) If IFlag = 1 Or IFlag = 2 Then Fvec(0) = X(0) ^ 2 - X(1) - 1 Fvec(1) = (X(0) - 2) ^ 2 + (X(1) - 0.5) ^ 2 - 1 End If End Sub Sub Ex_Hybrd() Const N = 2 Dim X(N - 1) As Double, Fvec(N - 1) As Double, XTol As Double Dim Diag(N - 1) As Double, Mode As Long, Info As Long X(0) = 0: X(1) = 0 XTol = 0.0000000001 '1.0e-10 Mode = 1 Call Hybrd(AddressOf FHybrd, N, X(), Fvec(), XTol, Diag(), Mode, Info) Debug.Print "X1, X2 =", X(0), X(1) Debug.Print "Info =", Info End Sub Example Results X1, X2 = 1.06734608580669 0.139227666886862 Info = 0