View source for Quasi-linearization

A collection of methods for the numerical solution of non-linear problems by reducing them to a sequence of linear problems. Lying at the basis of the apparatus of quasi-linearization is the [[Newton method|Newton method]] and its generalization to function spaces, the theory of differential inequalities (cf. [[Differential inequality|Differential inequality]]) and the method of [[Dynamic programming|dynamic programming]]. The simplest example illustrating a method of quasi-linearization is the use of the Newton–Raphson method for finding the root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076590/q0765901.png" /> of a scalar monotone-decreasing strictly-convex function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076590/q0765902.png" />. In case the original non-linear function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076590/q0765903.png" /> is approximated at each stage of the iteration process by a linear function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076590/q0765904.png" />, the root of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076590/q0765905.png" /> is found, which serves as the next approximation, so that

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Under fairly general conditions the sequence so constructed has the property of monotonicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076590/q0765907.png" /> and quadratic convergence:

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The application of quasi-linearization to solving the [[Riccati equation|Riccati equation]]

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(it is assumed that a solution exists on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076590/q07659010.png" />), is as follows. The original equation is replaced by the equivalent equation

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where the minimum is taken over all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076590/q07659012.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076590/q07659013.png" />. This equation has a number of properties inherent to linear equations, and in order to solve it one uses the linear differential equation

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076590/q07659015.png" /> is a fixed function. By appealing to the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076590/q07659016.png" /> (equality holding when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076590/q07659017.png" />), one can construct a system of successive approximations

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satisfying the linear equations

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The same recurrence relation can be obtained by applying the Newton–Kantorovich method (cf. [[Kantorovich process|Kantorovich process]]) to the original non-linear equation.

The employment of a quasi-linearization scheme in the solution of boundary value problems for the non-linear second-order differential equation

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leads to a sequence of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076590/q07659022.png" /> satisfying the linear equations

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with the linearized boundary conditions

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The existence, uniqueness and quadratic convergence of the sequence follows from the corresponding convexity of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076590/q07659027.png" /> over a sufficiently small interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076590/q07659028.png" />.

The method of quasi-linearization finds application in the solution of two-point and multi-point boundary value problems for linear and non-linear ordinary differential equations, boundary value problems for elliptic and parabolic partial differential equations, variational problems, differential-difference and functional-differential equations, etc. As with every iteration scheme, the method of quasi-linearization is suitable for computer implementation and has various modifications enabling one to accelerate the convergence for narrower classes of problems. There exists a variety of examples of its use as a heuristic method for solving a number of problems in physics, technology and economy.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.E. Bellman,   R.E. Kalaba,   "Quasilinearization and nonlinear boundary-value problems" , Elsevier  (1965)  (Translated from Russian)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Bellman,   G. Adomian,   "Partial differential equations" , Reidel  (1985)  pp. Chapt. IV</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Bellman,   R. Vasudevan,   "Wave propagation. An invariant imbedding approach" , Reidel  (1986)</TD></TR></table>