a linear least-squares problem, which can be solved explicitly, yielding the normal equations in the algorithm. The normal equations are n simultaneous linear equations in the unknown increments . They may be solved in one step, using Cholesky decomposition, or, better, the QR factorization of . For large systems, an iterative method, such as the conjugate gradient method, may be more efficient. If there is a linear dependence between columns of Jr, the iterations will fail, as becomes singular. When is complex the conjugate form should be used: . Example In this example, the Gauss–Newton algorithm will be used to fit a model to some data by minimizing the sum of squares of errors between the data and model's predictions. In a biology experiment studying the relation between substrate concentration and reaction rate in an enzyme-mediated reaction, the data in the following table were obtained. It is desired to find a curve (model function) of the form that fits best the data in the least-squares sense, with the parameters and to be determined. Denote by and the values of and rate respectively, with . Let and . We will find and such that the sum of squares of the residuals is minimized. The Jacobian of the vector of residuals with respect to the unknowns is a matrix with the -th row having the entries Starting with the initial estimates of and , after five iterations of the Gauss–Newton algorithm, the optimal values and are obtained. The sum of squares of residuals decreased from the initial value of 1.445 to 0.00784 after the fifth iteration. The plot in the figure on the right shows the curve determined by the model for the optimal parameters with the observed data. Convergence properties The Gauss-Newton iteration is guaranteed to converge toward a local minimum point under 4 conditions: The functions are twice continuously differentiable in an open convex set , the Jacobian is of full column rank, the initial iterate is near , and the local minimum value is small. The convergence is quadratic if . It can be shown that the increment Δ is a descent direction for , and, if the algorithm converges, then the limit is a stationary point of . For large minimum value , however, convergence is not guaranteed, not even local convergence as in Newton's method, or convergence under the usual Wolfe conditions. The rate of convergence of