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2014-06-08T19:37:20Z

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2014-02-10T17:07:01Z

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Harmonics (electrical power)

2014-02-01T13:20:32Z

14.98.48.173: /* Current Harmonic */

In [[algebra]], '''like terms''' are [[Term (mathematics)|term]]s that have the same [[Variable (mathematics)|variable]]s and [[Exponentiation|power]]s. The [[coefficient]]s do not need to match.<ref>{{cite web |url=http://www.math.com/school/subject2/lessons/S2U2L4DP.html |title=Like terms in Depth |accessdate=2008-09-07 |work=Math Online |publisher=Math Online}}</ref>

'''Unlike terms''' are two or more [[Term (mathematics)|terms]] that are not like terms, i.e. they do not have the same variables or powers. The order of the variables does not matter unless there is a power. For example, 8''xyz''2 and −5''xyz''2 are like terms because they have the same variables and power while 3''abc'' and 3''ghi'' are unlike terms because they have different variables. Since the coefficient doesn't affect likeness, all [[constant term]]s are like terms.

==Generalization==
In this discussion, a "term" will refer to a string of numbers being multiplied or divided (remember that division is simply multiplication by a reciprocal) together. Terms are within the same expression and are combined by either addition or subtraction. For example, take the expression:

<math>ax+bx\,\!</math>

There are two terms in this expression. Notice that the two terms have a common factor, that is, both terms have an <math>x\,\!</math>. This means that we can factor out that common factor variable, resulting in

<math>(a+b)x\,\!</math>

If the expression in parentheses may be calculated, that is, if the variables in the expression in the parentheses are known numbers, then it is simpler to write the calculation <math>a+b\,\!</math>. and juxtapose that new number with the remaining unknown number. Terms combined in an expression with a common, unknown factor (or multiple unknown factors) are called like terms.

==Examples==

===General Example===
To provide an example for above, let <math>a\,\!</math> and <math>b\,\!</math> have arbitrary values, so that their sum may be calculated. For ease of calculation, let <math>a=5\,\!</math> and <math>b=3\,\!</math>. The original expression becomes

<math>5x+3x\,\!</math>

which may be factored into

<math>(5+3)x\,\!</math>

or, equally,

<math>8x\,\!</math>.

This demonstrates that

<math>5x+3x=8x\,\!</math>

The known values assigned to the unlike part of two or more terms are called coefficients. As this example shows, when like terms exist in an expression, they may be combined by adding or subtracting (whatever the expression indicates) the coefficients, and maintaining the common factor of both terms. Such combination is called combining like terms, and it is an important tool used for solving equations.

===Simplifying An Expression===
Take the expression, which is to be simplified:

<math>3(4x^2y-6y)+7x^2y-3y^2+2(8y-4y^2-4x^2y)\,\!</math>

The first step to grouping like terms in this expression is to get rid of the parentheses. Do this by distributing (multiplying) each number in front of a set of parentheses to each term in that set of parentheses:

<math>12x^2y-18y+7x^2y-3y^2+16y-8y^2-8x^2y\,\!</math>

The like terms in this expression are the terms that can be grouped together by having exactly the same set of unknown factors. Here, the sets of unknown factors are <math>x^2y,\,\!</math> <math>y^2,\,\!</math> and <math>y.\,\!</math>. By the rule in the first example, all terms with the same set of unknown factors, that is, all like terms, may be combined by adding or subtracting their coefficients, while maintaining the unknown factors. Thus, the expression becomes

<math>11x^2y-2y-11y^2\,\!</math>

The expression is considered simplified when all like terms have been combined, and all terms present are unlike. In this case, all terms now have different unknown factors, and are thus unlike, and so the expression is completely simplified.

==Footnotes==
{{reflist}}

[[Category:Elementary mathematics]]

[[es:Monomio#Monomios semejantes]]

Effective nuclear charge

2014-01-21T16:18:02Z

14.98.190.197:

In [[numerical analysis]], the '''shooting method''' is a method for solving a [[boundary value problem]] by reducing it to the solution of an [[initial value problem]]. The following exposition may be clarified by this [[#Example|illustration of the shooting method]].

For a boundary value problem of a second-order [[ordinary differential equation]], the method is stated as follows.
Let

:<math> y''(t) = f(t, y(t), y'(t)), \quad y(t_0) = y_0, \quad y(t_1) = y_1 </math>

be the boundary value problem.
Let ''y''(''t''; ''a'') denote the solution of the initial value problem

:<math> y''(t) = f(t, y(t), y'(t)), \quad y(t_0) = y_0, \quad y'(t_0) = a </math>

Define the function ''F''(''a'') as the difference between ''y''(''t''1; ''a'') and the specified boundary value ''y''1.

:<math> F(a) = y(t_1; a) - y_1 \,</math>

If the boundary value problem has a solution, then ''F'' has a [[root of a function|root]],
and that root is just the value of ''y''<nowiki>'</nowiki>(''t''0) which yields a solution ''y''(''t'') of the boundary value problem.

The usual methods for finding roots may be employed here,
such as the [[bisection method]] or [[Newton's method]].

== Linear shooting method ==
The boundary value problem is linear if ''f'' has the form
:<math> f(t, y(t), y'(t))=p(t)y'(t)+q(t)y(t)+r(t). \, </math>
In this case, the solution to the boundary value problem is usually given by:
:<math>y(t) = y_{(1)}(t)+\frac{y_1-y_{(1)}(t_1)}{y_{(2)}(t_1)}y_{(2)}(t)</math>
where <math>y_{(1)}(t)</math> is the solution to the initial value problem:
:<math>y_{(1)}''(t) = p(t)y_{(1)}'(t)+q(t)y_{(1)}(t)+r(t),\quad y_{(1)}(t_0) = y_0, \quad y_{(1)}'(t_0) = 0, </math>
and <math>y_{(2)}(t)</math> is the solution to the initial value problem:
:<math>y_{(2)}''(t) = p(t)y_{(2)}'(t)+q(t)y_{(2)}(t),\quad y_{(2)}(t_0) = 0, \quad y_{(2)}'(t_0) = 1. </math>
See [http://math.fullerton.edu/mathews/n2003/shootingmethod/ShootingProof.pdf the proof] for the precise condition under which this result holds.

== Example ==

A [[boundary value problem]] is given as follows by Stoer and Burlisch (Section 7.3.1).

:<math> w''(t) = \frac{3}{2} w^2, \quad w(0) = 4, \quad w(1) = 1 </math>

The [[initial value problem]]

:<math> w''(t) = \frac{3}{2} w^2, \quad w(0) = 4, \quad w'(0) = s</math>

was solved for ''s'' = −1, −2, −3, ..., −100, and ''F''(''s'') = ''w''(1;''s'') − 1 plotted in the first figure.
Inspecting the plot of ''F'',
we see that there are roots near −8 and −36.
Some trajectories of ''w''(''t'';''s'') are shown in the second figure.

Solutions of the initial value problem were computed by using the LSODE algorithm, as implemented in the mathematics package [[GNU Octave]].

Stoer and Bulirsch state that there are two solutions,
which can be found by algebraic methods.
These correspond to the initial conditions ''w''′(0) = −8 and ''w''′(0) = −35.9 (approximately).

[[Image:Shooting_method_error.png|thumb|center|400px|The function ''F''(''s'') = ''w''(1;''s'') − 1.]]
[[Image:Shooting_method_trajectories.png|thumb|center|400px|Trajectories ''w''(''t'';''s'') for ''s'' = ''w''<nowiki>'</nowiki>(0) equal to −7, −8, −10, −36, and −40 (red, green, blue, cyan, and magenta, respectively). The point (1,1) is marked with a red diamond.]]
 

== See also ==
* [[Direct multiple shooting method]]
* [[Computation of radiowave attenuation in the atmosphere]]

== References ==

* Josef Stoer and Roland Bulirsch. ''Introduction to Numerical Analysis''. New York: Springer-Verlag, 1980. ''(See Section 7.3.)''
*{{Cite book | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | publication-place=New York | isbn=978-0-521-88068-8 | chapter=Section 18.1. The Shooting Method | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=959}}

== External links ==
* [http://www.netlib.org/odepack/opks-sum Brief Description of ODEPACK] ''(at [[Netlib]]; contains LSODE)''
* [http://numericalmethods.eng.usf.edu/topics/shooting_method.html Shooting method of solving boundary value problems – Notes, PPT, Maple, Mathcad, Matlab, Mathematica] at ''Holistic Numerical Methods Institute'' [http://numericalmethods.eng.usf.edu]
* [http://math.fullerton.edu/mathews/n2003/ShootingMod.html Shooting Method for Boundary Value Problems]

[[Category:Numerical differential equations]]

Fixed-point arithmetic

2012-08-26T05:55:22Z

14.98.22.103: /* Representation */

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2012-08-22T15:00:09Z

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2012-08-20T04:58:08Z

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