than 1 millionth of a percent. Linearization of a multivariable function The equation for the linearization of a function at a point is: The general equation for the linearization of a multivariable function at a point is: where is the vector of variables, and is the linearization point of interest . Uses of linearization Linearization makes it possible to use tools for studying linear systems to analyze the behavior of a nonlinear function near a given point. The linearization of a function is the first order term of its Taylor expansion around the point of interest. For a system defined by the equation , the linearized system can be written as where is the point of interest and is the -Jacobian of evaluated at . Stability analysis In stability analysis of autonomous systems, one can use the eigenvalues of the Jacobian matrix evaluated at a hyperbolic equilibrium point to determine the nature of that equilibrium. This is the content of the linearization theorem. For time-varying systems, the linearization requires additional justification. Microeconomics In microeconomics, decision rules may be approximated under the state-space approach to linearization. Under this approach, the Euler equations of the utility maximization problem are linearized around the stationary steady state. A unique solution to the resulting system of dynamic equations then is found. Optimization In mathematical optimization, cost functions and non-linear components within can be linearized in order to apply a linear solving method such as the Simplex algorithm. The optimized result is reached much more efficiently and is deterministic as a global optimum. Multiphysics In multiphysics systems—systems involving multiple physical fields that interact with one another—linearization with respect to each of the physical fields may be performed. This linearization of the system with respect to each of the fields results in a linearized monolithic equation system that can be solved using monolithic iterative solution procedures such as the Newton–Raphson method. Examples of this include MRI scanner systems which results in a system of electromagnetic, mechanical and acoustic fields. See also Linear stability Tangent stiffness matrix Stability derivatives Linearization theorem Taylor approximation Functional equation (L-function) References External links Linearization tutorials Linearization for Model Analysis and Control Design Differential calculus Dynamical systems Approximations