View source for Undetermined coefficients, method of

The determination of an unknown function in the form of an exact or approximate linear combination (finite or infinite) of known functions. This linear combination is taken with unknown coefficients, which are determined in one way or another from the conditions of the problem in question. As a rule, one obtains a system of algebraic equations for them.

A classic example of the method of undetermined coefficients is its use in the expansion of a regular rational function in a complex or real domain in elementary fractions. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u0951601.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u0951602.png" /> be algebraic polynomials with complex coefficients, where the degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u0951603.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u0951604.png" /> is less than the degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u0951605.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u0951606.png" /> and the coefficient of the highest term of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u0951607.png" /> is 1, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u0951608.png" /> be a root of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u0951609.png" /> of multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516012.png" />, so that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516013.png" /></td> </tr></table>

The regular rational function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516014.png" /> can be represented uniquely in the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516016.png" /> are yet unknown complex numbers (altogether <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516017.png" />). To find them, both parts of the equality are brought to a common denominator. After this, by disregarding inessential terms and reduction of similar terms on the right-hand side one obtains an equality on each side of which there stand polynomials of degree at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516018.png" />; on the left-hand side with known coefficients, on the right-hand side in the form of linear combinations of the unknown numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516019.png" />. By equating coefficients at equal powers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516020.png" />, one obtains a system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516021.png" /> linear equations in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516022.png" /> which, owing to the existence and uniqueness of the expansion (1), has a unique solution. Occasionally it is convenient to use a somewhat different device for finding the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516023.png" />. For example, suppose that all roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516024.png" /> are simple, so that (1) takes the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516025.png" /></td> </tr></table>

After bringing the two sides to a common denominator and reduction of similar terms, one obtains the equality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516026.png" /></td> </tr></table>

When one sets in it in succession <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516028.png" />, one readily obtains

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516029.png" /></td> </tr></table>

In the general case it is useful to combine these two devices for finding the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516030.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516032.png" /> be polynomials with real coefficients,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516033.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516034.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516035.png" /> are the real roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516036.png" /> of multiplicities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516037.png" />, and the quadratic trinomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516038.png" /> with real coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516040.png" /> is the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516041.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516042.png" /> is a complex root of multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516044.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516045.png" /> and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516046.png" /></td> </tr></table>

Then for the regular rational function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516047.png" /> there is one and only one expansion of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516048.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516049.png" /></td> </tr></table>

where the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516056.png" />, are real numbers. The method for finding them is the same as in the complex case described above: the equality (2) is brought to a common denominator, inessential terms are disregarded, and after collecting similar terms the coefficients at equal powers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516057.png" /> on both sides are equated. As a result one obtains a system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516058.png" /> equations in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516059.png" /> unknowns <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516060.png" />, which has a unique solution.

The expansion of regular rational functions into elementary ones is applied, for example, to find their [[Laurent series|Laurent series]] (in particular, their [[Taylor series|Taylor series]]), and to integrate them. The method of undetermined coefficients is also used to integrate rational functions by means of the [[Ostrogradski method|Ostrogradski method]], and to integrate functions of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516061.png" />. In this case the integral becomes

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516062.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516063.png" /></td> </tr></table>

where the degree of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516064.png" /> is one less than that of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516065.png" />. To find the coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516066.png" /> and the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516067.png" /> one differentiates (3). Then one brings both sides to a common denominator, disregards inessential terms, collects similar terms, and equates coefficients at equal powers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095160/u09516068.png" />. As a result one obtains again a system of linear equations with a unique solution. Similar methods of integration can be applied in certain other cases.

The method of undetermined coefficients is applied in finding solutions of (ordinary and partial) differential equations in the form of power series. For this purpose, in a neighbourhood of the point in question a power series with undetermined coefficients is substituted in the given equation. Sometimes one obtains as a result relations between the coefficients of the series from which by means of given initial or boundary conditions one succeeds in finding these coefficients and, consequently, a solution of the equation in the form of a series. For example, when one solves the [[Hypergeometric equation|hypergeometric equation]] in this manner, one can obtain an expansion in a series of hypergeometric functions (cf. [[Hypergeometric function|Hypergeometric function]]).

The method of undetermined coefficients is also applied in other ways when solving differential equations, for example, the [[Galerkin method|Galerkin method]], the [[Ritz method|Ritz method]] and the [[Trefftz method|Trefftz method]]; it is also used in numerical methods: in Krylov's method for obtaining the coefficients of the secular equation, and in the approximate solution of integral equations.



====Comments====
The phrase  "method of undetermined coefficients" , or Lagrange's method of undetermined coefficients, is also used to refer to the method of [[Lagrange multipliers|Lagrange multipliers]] when investigating conditional extrema, cf. e.g. [[#References|[a3]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W.I. [V.I. Smirnov] Smirnow,   "Lehrgang der höhren Mathematik" , '''1''' , Deutsch. Verlag Wissenschaft.  (1963)  pp. 181–377; 397ff  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Schröder (ed.) , ''Mathematik für die Praxis'' , '''II''' , Deutsch. Verlag Wissenschaft.  (1966)  pp. 49ff</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Rektorys (ed.) , ''Applicable mathematics'' , Iliffe  (1969)  pp. 480–481</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  D.M. Young,   R.T. Gregory,   "A survey of numerical mathematics" , '''1''' , Dover, reprint  (1988)  pp. 259–261</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M.D. Greenberg,   "Foundations of applied mathematics" , Prentice-Hall  (1978)  pp. 419ff</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  P. Henrici,   "Applied and computational complex analysis" , '''1''' , Wiley  (1974)  pp. 553; 562</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  A.I. Markushevich,   "Theory of functions of a complex variable" , '''1''' , Chelsea  (1965)  pp. 352ff  (Translated from Russian)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  D.G. Zill,   M.R. Cullen,   "Advanced engineering mathematics" , PWS-Kent  (1992)  pp. 139–157; 658ff</TD></TR></table>