Newton-Raphson method in multidimensions

As in one dimension, the Newton-Raphson method is based on local linear 

Though (2.33) is the well form of the Newton-Raphson correction formula, it is more efficient to solve the matrix equation for d k by decomposition and backward substitution. 

Two subroutines should be supplied by the user of the module. The subroutine starting at line 900 computes the left hand sides of the equations f (x) = 0, and stores them in array F. The subroutine starting at line 800 evaluates the elements of the Jacobian matrix and puts them into the array A. The subroutine starting at line 900 should return the error flag value ER 0 if the current estimate stored in array X is unfeasible. The matrix equation is solved by calling the modules M14 and M15, so that do not forget to merge these 

Example 2.3.2 Equilibrium of chemical reactions by Newton-Raphson method 

The problem is the one stated in the previous example. The equations are obtained rearranging (2.29). Since the Jacobian is always calculated after function evaluation, the subroutine starting at line 800 makes use of the computed mole numbers. We show the main program and the iterations, whereas the final results are the same as in the previous example and hence omitted from the output. 

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