|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{about|the numerical integration method|the neurological examination maneuver|Romberg's test}}
| | Have you been thinking "how do I speed up my computer" lately? Well odds are if you are reading this article; then you are experiencing one of many computer issues that thousands of people discover that they face regularly.<br><br>We all learn that the registry is the critical component of the Windows running program as it stores all information regarding the Dll files, programs found on the computer plus system settings. However, as days by, it's unavoidable which we can encounter registry matter due to a big amount of invalid, useless and unwanted entries.<br><br>Windows is surprisingly dumb. It only knows how to follow commands and instructions, meaning which whenever we install a system, which program has to tell Windows what to do. This is done by storing an "training file" inside the registry of the program. All the computer programs put these "manuals" into the registry, permitting a computer to run a broad array of programs. When you load up 1 of those programs, Windows just looks up the program file inside the registry, and carries out its instructions.<br><br>Analysis a files plus clean it up frequently. Destroy all of the unnecessary and unused files because they just jam the computer program. It will surely better the speed of your computer plus be careful that a computer do not afflicted by a virus. Remember always to update the antivirus software every time. If you never use the computer surprisingly usually, we can take a free antivirus.<br><br>Another thing you need to check is whether or not the [http://bestregistrycleanerfix.com/tune-up-utilities tuneup utilities] system which you are considering has the ability to detect files plus programs that are advantageous. One of the registry cleaner programs we would try is RegCure. It is helpful for speeding up and cleaning up problems on a computer.<br><br>Windows relies heavily on this database, storing everything from a latest emails to a Internet favorites in there. Because it's thus important, a computer is regularly adding and updating the files inside it. This is fine, however it can make your computer run slow, when your computer accidentally breaks its important registry files. This really is a especially prevalent problem, and actually makes a computer run slower every day. What occurs is that since the computer is continually utilizing 100's of registry files at when, it often gets confused plus create a few of them unreadable. This then makes a computer run slow, considering Windows takes longer to read the files it needs.<br><br>In alternative words, when a PC has any corrupt settings inside the registry database, these settings may create your computer run slower plus with a lot of mistakes. And unfortunately, it's the case that XP is prone to saving many settings from the registry inside the wrong means, making them unable to run correctly, slowing it down plus causing a lot of mistakes. Each time we use your PC, it has to read 100's of registry settings... plus there are frequently numerous files open at once which XP gets confuse plus saves many in the incorrect technique. Fixing these damaged settings may boost the speed of your program... and to do that, you need to look to utilize a 'registry cleaner'.<br><br>There is a lot a wise registry cleaner may do for the computer. It could check for and download updates for Windows, Java plus Adobe. Keeping changes current is an significant part of good computer health. It could moreover safeguard the personal and company privacy and a online security. |
| | |
| In [[numerical analysis]], '''Romberg's method''' {{Harv|Romberg|1955}} is used to estimate the [[Integral|definite integral]]
| |
| | |
| :<math> \int_a^b f(x) \, dx </math>
| |
| | |
| by applying [[Richardson extrapolation]] {{Harv|Richardson|1911}} repeatedly on the [[trapezium rule]] or the [[rectangle rule]] (midpoint rule). The estimates generate a triangular array. Romberg's method is a [[Newton–Cotes formulas|Newton–Cotes formula]] – it evaluates the integrand at equally-spaced points.
| |
| The integrand must have continuous derivatives, though fairly good results
| |
| may be obtained if only a few derivatives exist.
| |
| If it is possible to evaluate the integrand at unequally-spaced points, then other methods such as [[Gaussian quadrature]] and [[Clenshaw–Curtis quadrature]] are generally more accurate.
| |
| | |
| The method is named after [[Werner Romberg]] (1909–2003), who published the method in 1955.
| |
| | |
| == Method ==
| |
| | |
| The method can be defined inductively
| |
| | |
| :<math>R(0,0) = \frac{1}{2} (b-a) (f(a) + f(b))</math>
| |
| | |
| :<math>R(n,0) = \frac{1}{2} R(n-1,0) + h_n \sum_{k=1}^{2^{n-1}} f(a + (2k-1)h_n)</math>
| |
| | |
| :<math>R(n,m) = R(n,m-1) + \frac{1}{4^m-1} (R(n,m-1) - R(n-1,m-1))</math>
| |
| | |
| or
| |
| | |
| :<math>R(n,m) = \frac{1}{4^m-1} ( 4^m R(n,m-1) - R(n-1,m-1))</math>
| |
| | |
| where
| |
| | |
| :<math> n \ge m \, </math>
| |
| | |
| :<math> m \ge 1 \, </math>
| |
| | |
| :<math> h_n = \frac{b-a}{2^n}. </math>
| |
| | |
| In [[big O notation]], the error for ''R''(''n'', ''m'') is {{Harv|Mysovskikh|2002}}:
| |
| | |
| :<math> O\left(h_n^{2m+2}\right). \, </math>
| |
| | |
| The zeroeth extrapolation, ''R''(''n'', 0), is equivalent to the [[trapezoidal rule]] with 2<sup>''n''</sup> + 1 points; the first extrapolation, ''R''(''n'', 1), is equivalent to [[Simpson's rule]] with 2<sup>''n''</sup> + 1 points. The second extrapolation, ''R''(''n'', 2), is equivalent to [[Boole's rule]] with 2<sup>''n''</sup> + 1 points. Further extrapolations differ from Newton Cotes formulas. In particular further Romberg extrapolations expand on Boole's rule in very slight ways, modifying weights into ratios similar as in Boole's rule. In contrast, further Newton Cotes methods produce increasingly differing weights, eventually leading to large positive and negative weights. This is indicative of how large degree interpolating polynomial Newton Cotes methods fail to converge for many integrals, while Romberg integration is more stable.
| |
| | |
| When function evaluations are expensive, it may be preferable to replace the polynomial interpolation of Richardson with the rational interpolation proposed by {{Harvtxt|Bulirsch|Stoer|1967}}.
| |
| | |
| == A geometric example ==
| |
| To estimate the area under a curve the trapezoid rule is applied first to one-piece, then two, then four, and so on.
| |
| | |
| [[File:Freeform curve.gif|thumb|One-piece (click to enlarge)]] [[File:2 piece.gif|thumb|Two-piece]] [[File:4 piece.gif|thumb|Four-piece]] [[File:8 piece.gif|thumb|Eight-piece]]
| |
| | |
| After trapezoid rule estimates are obtained Richardson's Extrapolation is applied
| |
| | |
| *For the first iteration the two piece and one piece estimates are used in the formula (4 X (more accurate) - (less accurate))/3 The same formula is then used to compare the four piece and the two piece estimate, and likewise for the higher estimates
| |
| | |
| *For the second iteration the values of the first iteration are used in the formula (16(more accurate)-less accurate)/15
| |
| | |
| *The third iteration uses the next power of 4: (64 (More accurate) - less accurate)/63 on the values derived by the second iteration.
| |
| | |
| *The pattern is continued until there is one estimate.
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| |'''Number of pieces'''||'''Trapezoid estimates'''|| '''First iteration'''||'''second iteration||'''third iteration'''
| |
| |-
| |
| | || ||'''(4MA-LA)/3'''* || '''(16MA-LA)/15'''|| '''(64MA-LA)/63'''
| |
| |-
| |
| |1||0 || (4*480-0)/3 = 640|| (16*880-640)/15 =896 || (64*1015.11-896)/63 = 1017.002
| |
| |-
| |
| |2||480 || (4*780-480)/3 = 880 || (16*1006.67-880)/15 = 1015.11.. ||
| |
| |-
| |
| |4||780 || (4*950-780)/3 =1006.666.. || ||
| |
| |-
| |
| |8||950 || || ||
| |
| |-
| |
|
| |
| |-
| |
| |}
| |
| *MA stands for more accurate, LA stands for less accurate
| |
| | |
| == Example ==
| |
| | |
| As an example, the [[Gaussian function]] is integrated from 0 to 1, i.e. the [[error function]] erf(1) ≈ 0.842700792949715. The triangular array is calculated row by row and calculation is terminated if the two last entries in the last row differ less than 10<sup>−8</sup>.
| |
| | |
| <pre><nowiki> | |
| 0.77174333
| |
| 0.82526296 0.84310283
| |
| 0.83836778 0.84273605 0.84271160
| |
| 0.84161922 0.84270304 0.84270083 0.84270066
| |
| 0.84243051 0.84270093 0.84270079 0.84270079 0.84270079
| |
| </nowiki></pre>
| |
| | |
| The result in the lower right corner of the triangular array is accurate to the digits shown.
| |
| It is remarkable that this result is derived from the less accurate approximations
| |
| obtained by the trapezium rule in the first column of the triangular array.
| |
| | |
| == Implementation ==
| |
| Here is an example of a computer implementation of the Romberg method (in the [[C programming language]]). It needs one vector and one variable, as well as a sub-routine Trap:
| |
| | |
| <source lang=c>
| |
| #include <stdio.h>
| |
| #include <stdlib.h>
| |
| #include <math.h>
| |
| #define MAX 6
| |
| | |
| int main()
| |
| {
| |
| double s[MAX];
| |
| int i,k;
| |
| double var ;
| |
| for (i = 1; i< MAX; i++)
| |
| s[i] = 1;
| |
|
| |
| for (k=1; k< MAX; k++)
| |
| {
| |
| for (i=1; i <=k; i++)
| |
| {
| |
| if (i==1)
| |
| {
| |
| var = s[i];
| |
| s[i] = Trap(0, 1, pow(2, k-1)); // sub-routine Trap
| |
| } // integrated from 0 and 1
| |
| /* pow() is the number of subdivisions*/
| |
| else
| |
| {
| |
| s[k]= ( pow(4 , i-1)*s[i-1]-var )/(pow(4, i-1) - 1);
| |
|
| |
| var = s[i];
| |
| s[i]= s[k];
| |
| }
| |
| printf (" %f ", s[i]);
| |
| }
| |
| printf ("\n");
| |
| }
| |
| | |
| return 0;
| |
| }
| |
| </source>
| |
| | |
| == References ==
| |
| * {{citation|last1=Richardson|first1=L. F.|title=The Approximate Arithmetical Solution by Finite Differences of Physical Problems Involving Differential Equations, with an Application to the Stresses in a Masonry Dam|journal= Philosophical Transactions of the Royal Society A<!--, Containing Papers of a Mathematical or Physical Character--> |volume=210|issue=459-470|year=1911|pages=307–357|doi=10.1098/rsta.1911.0009|jstor=90994}}
| |
| * {{citation|last1=Romberg|first1=W.|title=Vereinfachte numerische Integration|journal=Det Kongelige Norske Videnskabers Selskab Forhandlinger|volume=28|year=1955|location=Trondheim|pages=30–36|issue=7}}
| |
| * {{citation|last=Thacher, Jr.|first=Henry C.|title=Remark on Algorithm 60: Romberg integration|journal=Communications of the ACM|volume=7|pages =420–421|date=July 1964|url=http://portal.acm.org/citation.cfm?id=364520.364542|doi=10.1145/364520.364542|issue=7}}
| |
| * {{citation|last1=Bauer|first1=F.L.|last2=Rutishauser|last3=Stiefel|first3=E.|title=New aspects in numerical quadrature|editor-last=Metropolis|editor-first=N. C., et al.|journal=Experimental Arithmetic, high-speed computing and mathematics, Proceedings of Symposia in Applied Mathematics|publisher=[[American Mathematical Society|AMS]]|year=1963|pages=199–218|first2=H.|issue=15}}
| |
| * {{citation|last1=Bulirsch|first1=Roland|last2=Stoer|first2=Josef|title= Handbook Series Numerical Integration. Numerical quadrature by extrapolation|journal=Numerische Mathematik|volume=9|year=1967|pages=271–278 |url=http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN362160546_0009}}
| |
| * {{citation|last=Mysovskikh|first=I.P.|contribution=Romberg method|editor-last=Hazewinkel|editor-first=Michiel|title=Encyclopaedia of Mathematics|publisher=[[Springer-Verlag]]|year=2002|isbn=1-4020-0609-8|url=http://eom.springer.de/r/r082570.htm}}
| |
| * {{Citation |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 4.3. Romberg Integration|chapter-url=http://apps.nrbook.com/empanel/index.html?pg=166}}
| |
| | |
| == External links ==
| |
| * [http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=34&objectType=file ROMBINT] – code for [[MATLAB]] (author: Martin Kacenak)
| |
| *[http://math.fullerton.edu/mathews/n2003/RombergMod.html Module for Romberg integration]
| |
| *[http://www.hvks.com/Numerical/webintegration.html Free online integration tool using Romberg, Fox–Romberg, Gauss–Legendre and other numerical methods]
| |
| | |
| [[Category:Numerical integration (quadrature)]]
| |
| [[Category:Articles with example C code]]
| |
Have you been thinking "how do I speed up my computer" lately? Well odds are if you are reading this article; then you are experiencing one of many computer issues that thousands of people discover that they face regularly.
We all learn that the registry is the critical component of the Windows running program as it stores all information regarding the Dll files, programs found on the computer plus system settings. However, as days by, it's unavoidable which we can encounter registry matter due to a big amount of invalid, useless and unwanted entries.
Windows is surprisingly dumb. It only knows how to follow commands and instructions, meaning which whenever we install a system, which program has to tell Windows what to do. This is done by storing an "training file" inside the registry of the program. All the computer programs put these "manuals" into the registry, permitting a computer to run a broad array of programs. When you load up 1 of those programs, Windows just looks up the program file inside the registry, and carries out its instructions.
Analysis a files plus clean it up frequently. Destroy all of the unnecessary and unused files because they just jam the computer program. It will surely better the speed of your computer plus be careful that a computer do not afflicted by a virus. Remember always to update the antivirus software every time. If you never use the computer surprisingly usually, we can take a free antivirus.
Another thing you need to check is whether or not the tuneup utilities system which you are considering has the ability to detect files plus programs that are advantageous. One of the registry cleaner programs we would try is RegCure. It is helpful for speeding up and cleaning up problems on a computer.
Windows relies heavily on this database, storing everything from a latest emails to a Internet favorites in there. Because it's thus important, a computer is regularly adding and updating the files inside it. This is fine, however it can make your computer run slow, when your computer accidentally breaks its important registry files. This really is a especially prevalent problem, and actually makes a computer run slower every day. What occurs is that since the computer is continually utilizing 100's of registry files at when, it often gets confused plus create a few of them unreadable. This then makes a computer run slow, considering Windows takes longer to read the files it needs.
In alternative words, when a PC has any corrupt settings inside the registry database, these settings may create your computer run slower plus with a lot of mistakes. And unfortunately, it's the case that XP is prone to saving many settings from the registry inside the wrong means, making them unable to run correctly, slowing it down plus causing a lot of mistakes. Each time we use your PC, it has to read 100's of registry settings... plus there are frequently numerous files open at once which XP gets confuse plus saves many in the incorrect technique. Fixing these damaged settings may boost the speed of your program... and to do that, you need to look to utilize a 'registry cleaner'.
There is a lot a wise registry cleaner may do for the computer. It could check for and download updates for Windows, Java plus Adobe. Keeping changes current is an significant part of good computer health. It could moreover safeguard the personal and company privacy and a online security.