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Iterations of Newton's Method for Two Nonlinear Equations

Drag the locator to see how the iterations proceed when solving a set of two nonlinear equations using Newton's method. The red and blue curves represent the points in the plane where the two equations hold. The two points of intersection of the red and blue curves are the solutions of the nonlinear system. The small light-blue points are initial points that lead to convergence to the point in the first quadrant. From the white parts of the plot, the convergence is to the point in the third quadrant.

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The snapshots show that the solution process and the point of convergence can be very different even if the starting points are very near to each other.
The two equations we consider are and . We supply the Mathematica built-in function FindRoot with one starting point, the one shown graphically by the locator, and display it numerically in the upper-left corner. Newton's method is used to solve the nonlinear system.
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