solving nonlinear algebraic equations can be written in our abstract form by introducing and then cos xd yx taylor expansions for multi-variable functions we follow the ideas of newton' method for one equation in one variableapproximate the nonlinear by linear function and find the root of that function when variables are involvedwe need to approximate vector function xby some linear function fq cwhere is an matrix and is some vector of length the technique for approximating by linear function is to use the first two terms in taylor series expansion given the value of and its partial derivatives with respect to at some point we can approximate the value at some point by the two first term in taylor series expansion around rf the next terms in the expansions are omitted here and of size jjx jj which are assumed to be small compared with the two terms above the expression rf is the matrix of all the partial derivatives of component ij in rf is @fi @xj for examplein our system ( )-( we can use sympy to compute the jacobianfrom sympy import symbols(' ' ** *cos(pi* * exp(- **(- diff( -pi* *sin(pi* * cos(pi* diff( - diff( **(- diff( exp(- we can then write rf @ @ @ @ @ @ @ @ cos sin the matrix rf is called the jacobian of and often denoted by