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| In [[mathematics]], the '''harmonic mean''' (sometimes called the '''subcontrary mean''') is one of several kinds of [[average]]. Typically, it is appropriate for situations when the average of [[rate (mathematics)|rate]]s is desired.
| | A lagging computer is really annoying and is quite a headache. Almost each person who utilizes a computer faces this issue some time or the alternative. If a computer additionally suffers within the same issue, you will find it difficult to continue working because normal. In such a condition, the thought, "what must I do to create my PC run faster?" is repeated and infuriating. There's a solution, yet!<br><br>Google Chrome crashes on Windows 7 by the corrupted cache contents and issues with all the stored browsing data. Delete the browsing information plus well-defined the contents of the cache to solve this problem.<br><br>So what if you look for when you compare registry products. Many of the registry products accessible now, have extremely synonymous qualities. The leading ones which we should be searching for are these.<br><br>The way to fix this issue is to initially reinstall the program(s) causing the mistakes. There are a lot of different programs which employ this file, however one will have placed their own faulty version of the file onto your program. By reinstalling any programs which are causing the error, you'll not merely enable a PC to run the system properly, yet a brand-new file is placed onto the program - exiting a computer running as smoothly because possible again. If you try this, and find it does not function, then we should look to update the system & any software we have on your PC. This will probably update the Msvcr71.dll file, permitting the computer to read it correctly again.<br><br>Another common cause of PC slow down is a corrupt registry. The registry is a very important component of computers running on Windows platform. When this gets corrupted a PC will slowdown, or worse, not begin at all. Fixing the registry is easy with the employ of the system and [http://bestregistrycleanerfix.com/registry-reviver registry reviver].<br><br>S/w associated error handling - If the blue screen physical memory dump happens after the installation of s/w application or perhaps a driver it may be which there is system incompatibility. By booting into safe mode and removing the software you can immediately fix this error. You can also try out a "program restore" to revert to an earlier state.<br><br>The initially reason the computer will be slow is because it demands more RAM. You'll notice this problem right away, specifically in the event you have less than a gig of RAM. Most unique computers come with a least which much. While Microsoft says Windows XP will run on 128 MB, it and Vista want at least a gig to run smoothly and enable you to run multiple programs at when. Fortunately, the cost of RAM has dropped significantly, plus you can get a gig installed for $100 or less.<br><br>What I would recommend is to look on your for registry cleaners. You are able to do this with a Google search. If you find treatments, look for reviews plus reviews regarding the product. Next you are able to see how others like the product, and how well it works. |
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| The harmonic mean ''H'' of the positive [[real number]]s ''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub> > 0 is defined to be
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| :<math>H = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n}} = \frac{n}{\sum_{i=1}^n \frac{1}{x_i}} = \frac{n \cdot \prod_{j=1}^n x_j }{ \sum_{i=1}^n \frac{\prod_{j=1}^n x_j}{x_i}}.</math>
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| From the third formula in the above equation it is more apparent that the harmonic mean is related to the [[arithmetic mean]] and [[geometric mean]].
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| Equivalently, the harmonic mean is the [[Multiplicative inverse|reciprocal]] of the [[arithmetic mean]] of the reciprocals. As a simple example, the harmonic mean of 1, 2, and 4 is <math>\frac{3}{\frac{1}{1}+\frac{1}{2}+\frac{1}{4}} = \frac{1}{\frac{1}{3}(\frac{1}{1}+\frac{1}{2}+\frac{1}{4})} = \frac{12}{7}\,.</math>
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| ==Relationship with other means==
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| [[Image:MathematicalMeans.svg|thumb|right|A geometric construction of the three [[Pythagorean means]] of two numbers, ''a'' and ''b''. Harmonic mean is denoted by ''H'' in purple color. The ''Q'' denotes a fourth mean, the [[quadratic mean]]. ]]
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| The harmonic mean is one of the three [[Pythagorean means]]. For all positive data sets ''containing at least one pair of nonequal values'', the harmonic mean is always the least of the three means, while the [[arithmetic mean]] is always the greatest of the three and the [[geometric mean]] is always in between. (If all values in a nonempty dataset are equal, the three means are always equal to one another; e.g. the harmonic, geometric, and arithmetic means of {2, 2, 2} are all 2.)
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| It is the special case ''M''<sub>−1</sub> of the [[power mean]].
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| Since the harmonic mean of a list of numbers tends strongly toward the least elements of the list, it tends (compared to the arithmetic mean) to mitigate the impact of large outliers and aggravate the impact of small ones.
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| The arithmetic mean is often mistakenly used in places calling for the harmonic mean.<ref>*''Statistical Analysis'', Ya-lun Chou, Holt International, 1969, ISBN 0030730953</ref> In the speed example [[#Examples|below]] for instance the arithmetic mean 50 is incorrect, and too big.
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| The harmonic mean is related to the other Pythagorean means, as seen in the third formula in the above equation. This is noticed if we interpret the denominator to be the arithmetic mean of the product of numbers ''n'' times but each time we omit the ''j''th term. That is, for the first term we multiply all ''n'' numbers but omit the first, for the second we multiply all ''n'' numbers but omit the second and so on. The numerator, excluding the ''n'', which goes with the arithmetic mean, is the geometric mean to the power ''n''. Thus the ''n''th harmonic mean is related to the ''n''th geometric and arithmetic means. | |
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| If a set of non-identical numbers is subjected to a [[mean-preserving spread]] — that is, two or more elements of the set are "spread apart" from each other while leaving the arithmetic mean unchanged — then the harmonic mean always decreases.<ref>Mitchell, Douglas W., "More on spreads and non-arithmetic means," ''[[The Mathematical Gazette]]'' 88, March 2004, 142-144.</ref>
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| ==Weighted harmonic mean==
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| If a set of [[weight function|weights]] <math>w_1</math>, ..., <math>w_n</math> is associated to the dataset <math>x_1</math>, ..., <math>x_n</math>, the '''weighted harmonic mean''' is defined by
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| :<math>\frac{\sum_{i=1}^n w_i }{ \sum_{i=1}^n \frac{w_i}{x_i}}. </math>
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| The harmonic mean as defined is the special case where all of the weights are equal to 1, and is equivalent to any weighted harmonic mean where all weights are equal | |
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| ==Examples==
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| ===In physics===
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| In certain situations, especially many situations involving [[rate (mathematics)|rate]]s and [[ratio]]s, the harmonic mean provides the truest [[average]]. For instance, if a vehicle travels a certain distance at a speed ''x'' (e.g. 60 kilometres per hour) and then the same distance again at a speed ''y'' (e.g. 40 kilometres per hour), then its average speed is the harmonic mean of ''x'' and ''y'' (48 kilometres per hour), and its total travel time is the same as if it had traveled the whole distance at that average speed. However, if the vehicle travels for a certain amount of ''time'' at a speed ''x'' and then the same amount of time at a speed ''y'', then its average speed is the [[arithmetic mean]] of ''x'' and ''y'', which in the above example is 50 kilometres per hour. The same principle applies to more than two segments: given a series of sub-trips at different speeds, if each sub-trip covers the same ''distance'', then the average speed is the ''harmonic'' mean of all the sub-trip speeds, and if each sub-trip takes the same amount of ''time'', then the average speed is the ''arithmetic'' mean of all the sub-trip speeds. (If neither is the case, then a [[weighted harmonic mean]] or [[weighted arithmetic mean]] is needed.)
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| Similarly, if one connects two electrical [[resistor]]s in parallel, one having resistance ''x'' (e.g. 60[[Ohm (unit)|Ω]]) and one having resistance ''y'' (e.g. 40Ω), then the effect is the same as if one had used two resistors with the same resistance, both equal to the harmonic mean of ''x'' and ''y'' (48Ω): the equivalent resistance in either case is 24Ω (one-half of the harmonic mean). However, if one connects the resistors in series, then the average resistance is the arithmetic mean of ''x'' and ''y'' (with total resistance equal to the sum of x and y). And, as with previous example, the same principle applies when more than two resistors are connected, provided that all are in parallel or all are in series.
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| ===In other sciences===
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| In [[computer science]], specifically [[information retrieval]] and [[machine learning]], the harmonic mean of the [[precision (information retrieval)|precision]] and the [[recall (information retrieval)|recall]] is often used as an aggregated performance score for the evaluation of algorithms and systems: the [[F-score]] (or F-measure).
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| An interesting consequence arises from basic algebra in problems of working together. As an example, if a gas-powered pump can drain a pool in 4 hours and a battery-powered pump can drain the same pool in 6 hours, then it will take both pumps (6 · 4)/(6 + 4), which is equal to 2.4 hours, to drain the pool together. Interestingly, this is one-half of the harmonic mean of 6 and 4.
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| In [[hydrology]] the harmonic mean is used to average [[hydraulic conductivity]] values for flow that is perpendicular to layers (e.g. geologic or soil) while flow parallel to layers uses the arithmetic mean. This apparent difference in averaging is explained by the fact that hydrology uses conductivity, which is the inverse of resistivity.
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| In [[sabermetrics]], the [[Power-speed number]] of a player is the harmonic mean of his [[home run]] and [[stolen base]] totals.
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| In [[population genetics]] the harmonic mean is used when calculating the effects of fluctuations in generation size on the effective breeding population. This is to take into account the fact that a very small generation is effectively like a [[bottleneck]] and means that a very small number of individuals are contributing disproportionately to the [[gene pool]] which can result in higher levels of [[inbreeding]].
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| When considering [[fuel economy in automobiles]] two measures are commonly used – miles per gallon (mpg), and litres per 100 km. As the dimensions of these quantities are the inverse of each other (one is distance per volume, the other volume per distance) when taking the mean value of the fuel-economy of a range of cars one measure will produce the harmonic mean of the other – i.e. converting the mean value of fuel economy expressed in litres per 100 km to miles per gallon will produce the harmonic mean of the fuel economy expressed in miles-per-gallon. | |
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| ===In finance===
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| The harmonic mean is the preferable method for averaging multiples, such as the [[price/earning ratio]], in which price is in the numerator. If these ratios are averaged using an arithmetic mean (a common error), high data points are given greater weights than low data points. The harmonic mean, on the other hand, gives equal weight to each data point.<ref>"Fairness Opinions: Common Errors and Omissions", ''The Handbook of Business Valuation and Intellectual Property Analysis'', McGraw Hill, 2004. ISBN 0071429670</ref>
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| ===In geometry===
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| In any [[triangle]], the radius of the [[Incircle and excircles of a triangle|incircle]] is one-third the harmonic mean of the [[Altitude (triangle)|altitudes]].
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| For any point P on the [[minor arc]] BC of the [[circumcircle]] of an [[equilateral triangle]] ABC, with distances ''q'' and ''t'' from B and C respectively, and with the intersection of PA and BC being at a distance ''y'' from point P, we have that ''y'' is half the harmonic mean of ''q'' and ''t''.<ref>Posamentier, Alfred S., and Salkind, Charles T., ''Challenging Problems in Geometry'', second edition, Dover Publ. Co., 1996, p 172.</ref>
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| In a [[right triangle]] with legs ''a'' and ''b'' and [[Altitude (triangle)|altitude]] ''h'' from the [[hypotenuse]] to the right angle, ''h''<sup>2</sup> is half the harmonic mean of ''a''<sup>2</sup> and ''b''<sup>2</sup>.<ref>Voles, Roger, "Integer solutions of <math>a^{-2}+b^{-2}=d^{-2}</math>," ''Mathematical Gazette'' 83, July 1999, 269–271.</ref><ref>Richinick, Jennifer, "The upside-down Pythagorean Theorem," ''Mathematical Gazette'' 92, July 2008, 313–;317.</ref>
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| Let ''t'' and ''s'' (''t'' > ''s'') be the sides of the two inscribed squares in a right triangle with hypotenuse ''c''. Then ''s''<sup>2</sup> equals half the harmonic mean of ''c''<sup>2</sup> and ''t''<sup>2</sup>.
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| Let a [[trapezoid]] have vertices A, B, C, and D in sequence and have parallel sides AB and CD. Let E be the intersection of the [[diagonal]]s, and let F be on side DA and G be on side BC such that FEG is parallel to AB and CD. Then FG is the harmonic mean of AB and DC. (This is provable using similar triangles.)
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| [[File:CrossedLadders.png|thumb|right|Crossed ladders. ''h'' is half the harmonic mean of ''A'' and ''B'']]
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| In the [[crossed ladders problem]], two ladders lie oppositely across an alley, each with feet at the base of one sidewall, with one leaning against a wall at height ''A'' and the other leaning against the opposite wall at height ''B'', as shown. The ladders cross at a height of ''h'' above the alley floor. Then ''h'' is half the harmonic mean of ''A'' and ''B''. This result still holds if the walls are slanted but still parallel and the "heights" ''A'', ''B'', and ''h'' are measured as distances from the floor along lines parallel to the walls.
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| In an [[ellipse]], the [[Kepler's laws of planetary motion#Mathematics of the three laws|semi-latus rectum]] (the distance from a focus to the ellipse along a line parallel to the minor axis) is the harmonic mean of the maximum and minimum distances of the ellipse from a focus.
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| ===In trigonometry===
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| In the case of the [[Trigonometric identities#Double-, triple-, and half-angle formulae|double-angle tangent identity]], if the [[Trigonometric functions#Sine, cosine, and tangent|tangent]] of an angle ''A'' is given as ''a''/''b'', then the tangent of 2''A'' is the product of (1) the harmonic mean of the numerator and denominator of tan ''A'' and (2) the reciprocal of (the denominator minus the numerator of tan ''A'').
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| In general the double-angle formula can be written as
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| :<math>\tan 2A = \frac{2 a b}{a + b} * \frac{1}{b - a}</math>
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| if <math>\tan A = \frac{a}{b}</math> and <math>a</math> and <math>b</math> are real numbers.
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| For example, if
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| :<math>\tan A = \frac{3}{7},</math>
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| then the most familiar form of the double-angle formula is
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| :<math>\tan 2A = \frac{2 * \frac{3}{7}}{1 - (\frac{3}{7})^2}= \frac{21}{20};</math>
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| but this can also be written as
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| :<math>\frac{2 * 3 * 7}{3 + 7} * \frac{1}{7 - 3} = \frac{21}{20}.</math>
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| ==Harmonic mean of two numbers==
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| For the special case of just two numbers <math>x_1</math> and <math>x_2</math>, the harmonic mean can be written
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| :<math>H = \frac{2 x_1 x_2}{x_1 + x_2}.</math>
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| In this special case, the harmonic mean is related to the [[arithmetic mean]] <math>A = \frac{x_1 + x_2}{2}</math>
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| and the [[geometric mean]] <math>G = \sqrt{x_1 x_2},</math> by
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| :<math>H = \frac {G^2} {A}.</math>
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| So <math>G = \sqrt{A H}</math>, meaning the two numbers' geometric mean equals the geometric mean of their arithmetic and harmonic means.
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| As noted above this relationship between the three Pythagorean means is not limited to ''n'' equals 1 or 2; there is a relationship for all ''n''. However it should be noted that for ''n'' equals 1 all means are equal and for ''n'' equals 2 we have the above relationship between the means. For arbitrary ''n'' ≥ 2 we may generalize this formula, as noted above, by interpreting the third equation for the harmonic mean differently. The generalized relationship was already explained above. If one carefully observes the third equation one will notice it also works for ''n'' = 1. That is, it predicts the equivalence between the harmonic and geometric means but it falls short by not predicting the equivalence between the harmonic and arithmetic means.
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| The general formula, which can be derived from the third formula for the harmonic mean by the reinterpretation as explained in relationship with other means, is
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| :<math> H(x_1, \ldots , x_n)= \frac{(G(x_1, \ldots , x_n))^n}{A(x_2x_3 \cdots x_n, x_1x_3 \cdots x_n, \ldots , x_1x_2 \cdots x_{n-1})} = \frac{(G(x_1, \ldots , x_n))^n}{A\left( \frac{\prod_{i=1}^n x_i}{x_1}, \frac{\prod_{i=1}^n x_i}{x_2}, \ldots , \frac{\prod_{i=1}^n x_i}{x_n} \right)} .</math>
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| Notice that for <math>n=2</math> we have
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| :<math>H(x_1, x_2)= \frac{(G(x_1, x_2))^2}{A(x_2, x_1)}=\frac{(G(x_2, x_1))^2}{A(x_2, x_1)}</math>
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| where we used the fact that the arithmetic mean evaluates to the same number independent of the order of the terms. This equation can be reduced to the original equation if we reinterpret this result in terms of the operators themselves. If we do this we get the symbolic equation
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| :<math>H = \frac {G^2} {A}</math>
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| because each function was evaluated at
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| :<math> (x_1, x_2).</math>
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| ==See also==
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| {{Portal box|Statistics|Mathematics}}
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| * [[Generalized mean]]
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| * [[Harmonic number]]
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| * [[Rate (mathematics)]]
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| * [[Weighted mean]]
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| ==References==
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| <references/>
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| ==External links==
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| * {{Mathworld |id=HarmonicMean |title=Harmonic Mean}}
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| * [http://www.cut-the-knot.org/arithmetic/HarmonicMean.shtml Averages, Arithmetic and Harmonic Means] at [[cut-the-knot]]
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| {{Statistics|descriptive}}
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| {{DEFAULTSORT:Harmonic Mean}}
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| [[Category:Means]]
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A lagging computer is really annoying and is quite a headache. Almost each person who utilizes a computer faces this issue some time or the alternative. If a computer additionally suffers within the same issue, you will find it difficult to continue working because normal. In such a condition, the thought, "what must I do to create my PC run faster?" is repeated and infuriating. There's a solution, yet!
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Another common cause of PC slow down is a corrupt registry. The registry is a very important component of computers running on Windows platform. When this gets corrupted a PC will slowdown, or worse, not begin at all. Fixing the registry is easy with the employ of the system and registry reviver.
S/w associated error handling - If the blue screen physical memory dump happens after the installation of s/w application or perhaps a driver it may be which there is system incompatibility. By booting into safe mode and removing the software you can immediately fix this error. You can also try out a "program restore" to revert to an earlier state.
The initially reason the computer will be slow is because it demands more RAM. You'll notice this problem right away, specifically in the event you have less than a gig of RAM. Most unique computers come with a least which much. While Microsoft says Windows XP will run on 128 MB, it and Vista want at least a gig to run smoothly and enable you to run multiple programs at when. Fortunately, the cost of RAM has dropped significantly, plus you can get a gig installed for $100 or less.
What I would recommend is to look on your for registry cleaners. You are able to do this with a Google search. If you find treatments, look for reviews plus reviews regarding the product. Next you are able to see how others like the product, and how well it works.