Linearization

Linearizations of a function are lines—usually lines that can be used for purposes of calculation. Linearization is an effective method for approximating the output of a function $y\; =\; f(x)$ at any $x\; =\; a$ based on the value and slope of the function at $x\; =\; b$, given that $f(x)$ is differentiable on $[a,\; b]$ (or $[b,\; a]$) and that $a$ is close to $b$. In short, linearization approximates the output of a function near $x\; =\; a$.

For example, $\backslash sqrt\{4\}\; =\; 2$. However, what would be a good approximation of $\backslash sqrt\{4.001\}\; =\; \backslash sqrt\{4\; +\; .001\}$?

For any given function $y\; =\; f(x)$, $f(x)$ can be approximated if it is near a known differentiable point. The most basic requisite is that $L\_a(a)\; =\; f(a)$, where $L\_a(x)$ is the linearization of $f(x)$ at $x\; =\; a$. The point-slope form of an equation forms an equation of a line, given a point $(H,\; K)$ and slope $M$. The general form of this equation is: $y\; -\; K\; =\; M(x\; -\; H)$.

Using the point $(a,\; f(a))$, $L\_a(x)$ becomes $y\; =\; f(a)\; +\; M(x\; -\; a)$. Because differentiable functions are locally linear, the best slope to substitute in would be the slope of the line tangent to $f(x)$ at $x\; =\; a$.

While the concept of local linearity applies the most to points arbitrarily close to $x\; =\; a$, those relatively close work relatively well for linear approximations. The slope $M$ should be, most accurately, the slope of the tangent line at $x\; =\; a$.

Image:Tangent-calculus.svg

Visually, the accompanying diagram shows the tangent line of $f(x)$ at $x$. At $f(x+h)$, where $h$ is any small positive or negative value, $f(x+h)$ is very nearly the value of the tangent line at the point $(x+h,\; L(x+h))$.

The final equation for the linearization of a function at $x\; =\; a$ is:

$$y\; =\; (f(a)\; +\; f\text{'}(a)(x\; -\; a))$$

For $x\; =\; a$, $f(a)\; =\; f(x)$. The derivative of $f(x)$ is $f\text{'}(x)$, and the slope of $f(x)$ at $a$ is $f\text{'}(a)$.

To find $\backslash sqrt\{4.001\}$, we can use the fact that $\backslash sqrt\{4\}\; =\; 2$. The linearization of $f(x)\; =\; \backslash sqrt\{x\}$ at $x\; =\; a$ is $y\; =\; \backslash sqrt\{a\}\; +\; \backslash frac\{1\}\{2\; \backslash sqrt\{a\}\}(x\; -\; a)$, because the function $f\text{'}(x)\; =\; \backslash frac\{1\}\{2\; \backslash sqrt\{x\}\}$ defines the slope of the function $f(x)\; =\; \backslash sqrt\{x\}$ at $x$. Substituting in $a\; =\; 4$, the linearization at 4 is $y\; =\; 2\; +\; \backslash frac\{x-4\}\{4\}$. In this case $x\; =\; 4.001$, so $\backslash sqrt\{4.001\}$ is approximately $2\; +\; \backslash frac\{4.001-4\}\{4\}\; =\; 2.00025$. The true value is close to 2.00024998, so the linearization approximation has a relative error of less than 1 millionth of a percent.

{{See also|Taylor series#In several variables}}

The equation for the linearization of a function $f(x,y)$ at a point $p(a,b)$ is:

:$f(x,y)\; \backslash approx\; f(a,b)\; +\; \backslash left.\; \{\backslash frac\{\{\backslash partial\; f(x,y)\}\}\{\{\backslash partial\; x\}\}\}\; \backslash right|\_\{a,b\}\; (x\; -\; a)\; +\; \backslash left.\; \{\backslash frac\{\{\backslash partial\; f(x,y)\}\}\{\{\backslash partial\; y\}\}\}\; \backslash right|\_\{a,b\}\; (y\; -\; b)$

The general equation for the linearization of a multivariable function $f(\backslash mathbf\{x\})$ at a point $\backslash mathbf\{p\}$ is:

:$f(\{\backslash mathbf\{x\}\})\; \backslash approx\; f(\{\backslash mathbf\{p\}\})\; +\; \backslash left.\; \{\backslash nabla\; f\}\; \backslash right|\_\{\backslash mathbf\{p\}\}\; \backslash cdot\; (\{\backslash mathbf\{x\}\}\; -\; \{\backslash mathbf\{p\}\})$

where $\backslash mathbf\{x\}$ is the vector of variables, $\{\backslash nabla\; f\}$ is the gradient, and $\backslash mathbf\{p\}$ is the linearization point of interest

.[http://www.ece.jhu.edu/~pi/Courses/454/linear.pdf Linearization. The Johns Hopkins University. Department of Electrical and Computer Engineering] {{webarchive|url=https://web.archive.org/web/20100607120539/http://www.ece.jhu.edu/~pi/Courses/454/linear.pdf |date=2010-06-07 }}

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

:$\backslash frac\{d\backslash mathbf\{x\}\}\{dt\}\; =\; \backslash mathbf\{F\}(\backslash mathbf\{x\},t)$,

the linearized system can be written as

:$\backslash frac\{d\backslash mathbf\{x\}\}\{dt\}\; \backslash approx\; \backslash mathbf\{F\}(\backslash mathbf\{x\_0\},t)\; +\; D\backslash mathbf\{F\}(\backslash mathbf\{x\_0\},t)\; \backslash cdot\; (\backslash mathbf\{x\}\; -\; \backslash mathbf\{x\_0\})$

where $\backslash mathbf\{x\_0\}$ is the point of interest and $D\backslash mathbf\{F\}(\backslash mathbf\{x\_0\},t)$ is the $\backslash mathbf\{x\}$-Jacobian of $\backslash mathbf\{F\}(\backslash mathbf\{x\},t)$ evaluated at $\backslash mathbf\{x\_0\}$.

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.{{cite journal |first=G. A. |last=Leonov |first2=N. V. |last2=Kuznetsov |title=Time-Varying Linearization and the Perron effects |journal=International Journal of Bifurcation and Chaos |volume=17 |issue=4 |year=2007 |pages=1079–1107 |doi=10.1142/S0218127407017732 |bibcode=2007IJBC...17.1079L }}

In microeconomics, decision rules may be approximated under the state-space approach to linearization.Moffatt, Mike. (2008) About.com [http://economics.about.com/od/economicsglossary/g/statespace.htm State-Space Approach] {{Webarchive|url=https://web.archive.org/web/20160304055023/http://economics.about.com/od/economicsglossary/g/statespace.htm |date=2016-03-04 }} Economics Glossary; Terms Beginning with S. Accessed June 19, 2008. 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.

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.

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.{{cite journal |first=S. |last=Bagwell |first2=P. D. |last2=Ledger |first3=A. J. |last3=Gil |first4=M. |last4=Mallett |first5=M. |last5=Kruip |year=2017 |title=A linearised hp–finite element framework for acousto-magneto-mechanical coupling in axisymmetric MRI scanners |journal=International Journal for Numerical Methods in Engineering |volume=112 |issue=10 |pages=1323–1352 |doi=10.1002/nme.5559 |bibcode=2017IJNME.112.1323B |doi-access=free }}

• Linear stability
• Tangent stiffness matrix
• Stability derivatives
• Linearization theorem
• Taylor approximation
• Functional equation (L-function)
• Quasilinearization

{{Reflist}}

• [http://www.mathworks.com/discovery/linearization.html Linearization for Model Analysis and Control Design]

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Category:Differential calculus

Category:Dynamical systems

Category:Approximations