Numerical Methods The Newton Raphson Technique

The MESH equations form a large system of interrelated, nonlinear, algebraic equations. The mathematical method used to solve all or part of these equations as a group is the Newton-Raphson method. An understanding of the numerical method is needed to understand the performance of all column methods. Detailed discussion of the Newton-Raphson method and its variations can be found in Holland's (1981) text.

The Newton-Raphson is an approximation technique. It assumes in the derivatives that the MESH equations are linear over short distances and the slopes will point towards the answers. The MESH equations can be far from linear and the predictions can take the next trial well off the curves, and move away from the solution. In some rigorous methods based on Newton-Raphson, a poor set of starting values can cause the calculation never to approach a solution. Also, the calculation can oscillate, with values swinging to either side of the solution. The independent variables calculated in a trial need to move the column to a solution. The software should include means to prevent or detect these problems and improve stability, e.g. by damping or limiting the change to the next set of variables. A Newton-Raph-son method will normally take even steps toward the solution.

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