solving nonlinear algebraic equations has xd sin cos just move all terms to the left-hand side and then the formula to the left of the equality sign is xsowhen do we really need to solve algebraic equations beyond the simplest types we can treat with pen and paperthere are two major application areas one is when using implicit numerical methods for ordinary differential equations these give rise to one or system of algebraic equations the other major application type is optimizationi finding the maxima or minima of function these maxima and minima are normally found by solving the algebraic equation xd if xis the function to be optimized differential equations are very much used throughout science and engineeringand actually most engineering problems are optimization problems in the endbecause one wants design that maximizes performance and minimizes cost we first consider one algebraic equation in one variablewith our usual emphasis on how to program the algorithms systems of nonlinear algebraic equations with many variables arise from implicit methods for ordinary and partial differential equations as well as in multivariate optimization our attention will be restricted to newton' method for such systems of nonlinear algebraic equations terminology when solving algebraic equations xd we often say that the solution is root of the equation the solution process itself is thus often called root finding brute force methods the representation of mathematical function xon computer takes two forms one is python function returning the function value given the argumentwhile the other is collection of points xf /along the function curve the latter is the representation we use for plottingtogether with an assumption of linear variation between the points this representation is also very suited for equation solving and optimizationwe simply go through all points and see if the function crosses the axisor for optimizationtest for local maximum or minimum point because there is lot of work to examine huge number of pointsand also because the idea is extremely simplesuch approaches are often referred to as brute force methods howeverwe are not embarrassed of explaining the methods in detail and implementing them