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For the algorithms which require a Jacobian matrix of derivatives of
the fit functions, there are times when an analytic Jacobian may be
unavailable or too expensive to compute. Therefore GSL supports
approximating the Jacobian numerically using finite differences of the fit
functions. This is typically done by setting the relevant function pointers
of the `gsl_multifit_function_fdf`

data type to NULL, however the
following functions allow the user to access the approximate Jacobian
directly if needed.

- Function:
*int***gsl_multifit_fdfsolver_dif_df***(const gsl_vector **`x`, gsl_multifit_function_fdf *`fdf`, const gsl_vector *`f`, gsl_matrix *`J`) This function takes as input the current position

`x`with the function values computed at the current position`f`, along with`fdf`which specifies the fit function and parameters and approximates the`n`-by-`p`Jacobian`J`using forward finite differences:*J_ij = d f_i(x,params) / d x_j = (f_i(x^*,params) - f_i(x,params)) / d x_j*. where*x^**has the*j*th element perturbed by*\Delta x_j*and*\Delta x_j = \epsilon |x_j|*, where*\epsilon*is the square root of the machine precision`GSL_DBL_EPSILON`

.

- Function:
*int***gsl_multifit_fdfsolver_dif_fdf***(const gsl_vector **`x`, gsl_multifit_function_fdf *`fdf`, gsl_vector *`f`, gsl_matrix *`J`) This function computes the vector of function values

`f`and the approximate Jacobian`J`at the position vector`x`using the system described in`fdf`. See`gsl_multifit_fdfsolver_dif_df`

for a description of how the Jacobian is computed.