# Difference between revisions of "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 and its generalization to function spaces, the theory of differential inequalities (cf. Differential inequality) and the method of dynamic programming. The simplest example illustrating a method of quasi-linearization is the use of the Newton–Raphson method for finding the root $r$ of a scalar monotone-decreasing strictly-convex function $f$. In case the original non-linear function $f$ is approximated at each stage of the iteration process by a linear function $\phi$, the root of $\phi$ is found, which serves as the next approximation, so that

$$x _ {n+} 1 = x _ {n} - \frac{f ( x _ {n} ) }{f ^ { \prime } ( x _ {n} ) } ,\ \ n = 0 , 1 ,\dots .$$

Under fairly general conditions the sequence so constructed has the property of monotonicity $( x _ {0} < x _ {1} < \dots < r )$ and quadratic convergence:

$$| x _ {n+} 1 - r | \leq k | x _ {n} - r | ^ {2} .$$

The application of quasi-linearization to solving the Riccati equation

$$v ^ \prime + v ^ {2} + p ( t) v + q ( t) = 0 ,\ \ v ( 0) = c$$

(it is assumed that a solution exists on an interval $[ 0 , t _ {0} ]$), is as follows. The original equation is replaced by the equivalent equation

$$v ^ \prime = \min _ { u } [ u ^ {2} - 2 u v - p ( t) v - q ( t) ] ,$$

where the minimum is taken over all functions $u$ defined on $[ 0 , t _ {0} ]$. This equation has a number of properties inherent to linear equations, and in order to solve it one uses the linear differential equation

$$w ^ \prime = u ^ {2} - 2 u w - p ( t) w - q ( t) ,\ \ w ( 0) = c ,$$

where $u$ is a fixed function. By appealing to the property $v ( t) \leq w ( t)$( equality holding when $u ( t) = v ( t)$), one can construct a system of successive approximations

$$v _ {1} \geq v _ {2} \geq \dots$$

satisfying the linear equations

$$v _ {n+} 1 ^ \prime = v _ {n} ^ {2} - 2 v _ {n} v _ {n+} 1 - p ( t) v _ {n+} 1 - q ( t) ,\ \ v _ {n+} 1 ( 0) = c .$$

The same recurrence relation can be obtained by applying the Newton–Kantorovich method (cf. 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

$$u ^ {\prime\prime} = f ( u ^ \prime , u , t ) ,\ \ t _ {1} \leq t \leq t _ {2} ,$$

$$g _ {1} ( u ( t _ {1} ) , u ^ \prime ( t _ {1} ) ) = 0 ,\ \ g _ {2} ( u ( t _ {2} ) , u ^ \prime ( t _ {2} ) ) = 0,$$

leads to a sequence of functions $\{ u _ {n} \}$ satisfying the linear equations

$$u _ {n+} 1 ^ {\prime\prime} = f ( u _ {n} ^ \prime , u _ {n} , t ) + f _ {u ^ \prime } ( u _ {n} ^ \prime , u _ {n} , t ) ( u _ {n+} 1 ^ \prime - u _ {n} ^ \prime ) +$$

$$+ f _ {n} ( u _ {n} ^ \prime , u _ {n} , t ) ( u _ {n+} 1 - u _ {n} ) ,$$

with the linearized boundary conditions

$$g _ {i} ( u _ {n} ( t _ {i} ) ,\ u _ {n} ^ \prime ( t _ {i} ) ) + g _ {iu} ( u _ {n} ( t _ {i} ) , u _ {n} ^ \prime ( t _ {i} ) ) ( u _ {n+} 1 ( t _ {i} ) - u _ {n} ( t _ {i} ) ) +$$

$$+ g _ {i u ^ \prime } ( u _ {n} ( t _ {i} ) , u _ {n} ^ \prime ( t _ {i} ) ) ( u _ {n+} 1 ^ \prime ( t _ {i} ) - u _ {n} ^ \prime ( t _ {i} ) ) = 0 .$$

The existence, uniqueness and quadratic convergence of the sequence follows from the corresponding convexity of the functions $f , g _ {1} , g _ {2}$ over a sufficiently small interval $[ t _ {1} , t _ {2} ]$.

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

 [1] R.E. Bellman, R.E. Kalaba, "Quasilinearization and nonlinear boundary-value problems" , Elsevier (1965) (Translated from Russian)