newton' method fundamental idea of numerical methods for nonlinear equations is to construct series of linear equations (since we know how to solve linear equationsand hope that the solutions of these linear equations bring us closer and closer to the solution of the nonlinear equation the idea will be clearer when we present newton' method and the secant method deriving and implementing newton' method figure shows the xfunction in our model equation numerical methods for algebraic equations require us to guess at solution first herethis guess is called the fundamental idea of newton' method is to approximate the original function xby straight linei linear functionsince it is straightforward to solve linear equations there are infinitely many choices of how to approximate xby straight line newton' method applies the tangent of xat see the rightmost tangent in fig this linear tangent function crosses the axis at point we call this is (hopefullya better approximation to the solution of xd than the next fundamental idea is to repeat this process we find the tangent of at compute where it crosses the axisat point called and repeat the process again figure shows that the process brings us closer and closer to the left it remainshoweverto see if we hit or come sufficiently close to this solution how do we compute the tangent of function xat point the tangent functionhere called fq /is linear and has two propertiesfig illustrates the idea of newton' method with xd repeatedly solving for crossing of tangent lines with the axis