Chkder Hybrd Hybrd1 Hybrd1_r Hybrd_r Hybrj Hybrj1 Hybrj1_r Hybrj_r Sos Sos_r

Hybrj_r() Sub Hybrj_r ( N As  Long, X() As  Double, Fvec() As  Double, XTol As  Double, Diag() As  Double, Mode As  Long, Info As  Long, XX() As  Double, YY() As  Double, YYpd() As  Double, IRev As  Long, Optional Maxfev As  Long = 0, Optional Factor As  Double = 0, Optional Nprint As  Long = 0, Optional Nfev As  Long, Optional Njev As  Long  ) Solution of a system of nonlinear equations by Powell hybrid method (reverse communication version) 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 the calculated function values and the Jacobian according to IRev. Parameters [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] IRev = 0: The obtained solution vector.   IRev = 2: The abscissa where the Jacobian shoule be evaluated.   IRev = 3: Recent approximation of the solution vector. [out] Fvec() Array Fvec(LFvec - 1) (LFvec >= N) IRev = 0: The function values evaluated at the solution vector X(). IRev = 3: The function values evaluated at the recent approximation of the solution vector. [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) = -1: The argument N had an illegal value. (N <= 0) = -2: The argument X() is invalid. = -3: The argument Fvec() is invalid. = -4: The argument XTol had an illegal value. (XTol < 0) = -5: The argument Diag() is invalid. = -8: The argument XX() is invalid. = -9: The argument YY() is invalid. = -10: The argument YYpd() is invalid. = 1: Number of function evaluations (number of returns with IRev=1) 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. [out] XX() Array XX(LXX - 1) (LXX >= N) When returned with IRev = 1, XX() contains the abscissa where the function value should be evaluated and given in the next call. [in] YY() Array YY(LYY - 1) (LYY >= N) When returned with IRev=1, the function value fi(XX()) (i = 1 to N) should be given in YY() in the next call. [in,out] YYpd() Array YYpd(LYYpd1 - 1, LYYpd2 - 1) (LYYpd1 >= N, LYYpd2 >= N) [in] IRev = 2: Jacobian(dfi/dxj) at X() should be given in YYpd() in the next call. [out] IRev = 0: The orthogonal matrix Q produced by the QR factorization of the final approximate Jacobian. [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 process and call this routine again. = 0: Computation finished. See return code in Info. = 1: User should set the function values at XX() in YY(). Do not alter any variables other than YY(). = 2: User should set the Jacobian at X() in YYpd(). Do not alter any variables other than YYpd(). = 3: Display the intermediate result (X(), Fvec(), etc.) (in the case of Nprint > 0). Do not alter any variables. [in] Maxfev (Optional) Termination occurs when the number of function evaluations (number of returns with IRev = 1) has reached this value. (default = 100*(N + 1)) (if Maxfev <= 0, the default value will be used) [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: To be returned with IRev = 3 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 returns with IRev = 3 for printing intermediate results are made. [out] Nfev (Optional) Number of function evaluations (number of returns with IRev = 1). [out] Njev (Optional) Number of Jacobian evaluations (number of returns with IRev = 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 FHybrj(N As Long, X() As Double, Fvec() As Double, Fjac() As Double, IFlag As Long) If IFlag = 1 Then Fvec(0) = X(0) ^ 2 - X(1) - 1 Fvec(1) = (X(0) - 2) ^ 2 + (X(1) - 0.5) ^ 2 - 1 ElseIf IFlag = 2 Then Fjac(0, 0) = 2 * X(0) Fjac(1, 0) = 2 * X(0) - 4 Fjac(0, 1) = -1 Fjac(1, 1) = 2 * X(1) - 1 End If End Sub Sub Ex_Hybrj_r() 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 Dim XX(N - 1) As Double, YY(N - 1) As Double, YYpd(N - 1, N - 1) As Double Dim IRev As Long, Info As Long X(0) = 0: X(1) = 0 XTol = 0.0000000001 '1.0e-10 Mode = 1 IRev = 0 Do Call Hybrj_r(N, X(), Fvec(), XTol, Diag(), Mode, Info, XX(), YY(), YYpd(), IRev) If IRev = 1 Or IRev = 2 Then Call FHybrj(N, XX(), YY(), YYpd(), IRev) End If Loop While IRev <> 0 Debug.Print "X1, X2 =", X(0), X(1) Debug.Print "Info =", Info End Sub Example Results X1, X2 = 1.06734608580669 0.139227666886862 Info = 0