  By using the Hadamard matrix product concept, this paper introduces two
generalized matrix formulation forms of numerical analogue of nonlinear
differential operators. The SJT matrix-vector product approach is found to be a
simple, efficient and accurate technique in the calculation of the Jacobian
matrix of the nonlinear discretization by finite difference, finite volume,
collocation, dual reciprocity BEM or radial functions based numerical methods.
We also present and prove simple underlying relationship (theorem (3.1))
between general nonlinear analogue polynomials and their corresponding Jacobian
matrices, which forms the basis of this paper. By means of theorem 3.1,
stability analysis of numerical solutions of nonlinear initial value problems
can be easily handled based on the well-known results for linear problems.
Theorem 3.1 also leads naturally to the straightforward extension of various
linear iterative algorithms such as the SOR, Gauss-Seidel and Jacobi methods to
nonlinear algebraic equations. Since an exact alternative of the quasi-Newton
equation is established via theorem 3.1, we derive a modified BFGS quasi-Newton
method. A simple formula is also given to examine the deviation between the
approximate and exact Jacobian matrices. Furthermore, in order to avoid the
evaluation of the Jacobian matrix and its inverse, the pseudo-Jacobian matrix
is introduced with a general applicability of any nonlinear systems of
equations. It should be pointed out that a large class of real-world nonlinear
problems can be modeled or numerically discretized polynomial-only algebraic
system of equations. The results presented here are in general applicable for
all these problems. This paper can be considered as a starting point in the
research of nonlinear computation and analysis from an innovative viewpoint.
