dfzero hybrd1 rpzero2

hybrd1() def hybrd1 ( f  , n  , x  , fvec  , xtol  = 1.0e-10  ) Solution of a system of nonlinear equations by Powell hybrid method (Jacobian not required) (simple driver) Purpose hybrd1 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. hybrd1 is the simple driver for standard use, which calls the hybrd with default parameters. Returns info (int) = 0: Successful exit (relative error between two consecutive iterates is at most xtol) = -1: The argument f is invalid. = -2: The argument n had an illegal value (n < 1) = -3: The argument x is invalid. = -4: The argument fvec is invalid. = -5: The argument xtol had an illegal value (xtol < 0) = 1: Number of calls to fcn with iflag = 1 or 2 has reached the limit (200*(n + 1)) = 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 Parameters [in] f User supplied subroutine which calculates the functions fi(x) defined as follows. _CODE def f(n, x, fvec, iflag): if iflag == 1 or iflag == 2: fvec[0] = function value f1(x[0], x[1], ..., x[n-1]) fvec[1] = function value f2(x[0], x[1], ..., x[n-1]) ... fvec[n-1] = function value fn(x[0], x[1], ..., x[n-1]) _ENDCODE When iflag = 1 or 2, calculate the function values fi(x) and set to fvec. [in] n Number of functions and variables. (n > 0) [in,out] x Numpy ndarray (1-dimensional, float, length n) [in] An initial estimate of the solution vector. [out] The obtained solution vector. [out] fvec Numpy ndarray (1-dimensional, float, length n) The function values evaluated at the solution vector x. [in] xtol (Optional) Target relative tolerance. Termination occurs when the relative error between two consecutive iterations is at most xtol. (xtol >= 0) (default = 1.0e-10) 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. def f(n, x, fvec, iflag): if iflag == 1 or iflag == 2: fvec[0] = x[0]**2 - x[1] - 1 fvec[1] = (x[0] - 2)**2 + (x[1] - 0.5)**2 - 1 def TestHybrd1(): n = 2 x = np.array([0.0, 0.0]) fvec = np.empty(n) info = hybrd1(f, n, x, fvec) print(x) print(fvec) print(info) Example Results >>> TestHybrd1() [1.06734609 0.13922767] [-1.11022302e-16 0.00000000e+00] 0