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In the [[mathematics|mathematical]] subfield of [[numerical analysis]] the '''de Boor's algorithm''' is a fast and [[numerically stable]] [[algorithm]] for evaluating [[spline curve]]s in [[B-spline]] form. It is a generalization of the [[de Casteljau's algorithm]] for [[Bézier curve]]s. The algorithm was devised by [[Carl R. de Boor]].  Simplified, potentially faster variants of the de Boor algorithm have been created but they suffer from comparatively lower stability.<ref>{{cite journal |last=Lee |first=E. T. Y. |date=December 1982 |title=A Simplified B-Spline Computation Routine |journal=Computing |volume=29 |issue=4 |pages=365–371 |publisher=Springer-Verlag|doi=10.1007/BF02246763}}</ref><ref>{{cite journal | author = Lee, E. T. Y. | journal = Computing | issue = 3 | pages = 229–238 | publisher = Springer-Verlag | doi=10.1007/BF02240069|title = Comments on some B-spline algorithms | volume = 36 | year = 1986}}</ref>
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== Introduction ==
 
The general setting is as follows. We would like to construct a curve whose shape is described by a sequence of ''p'' points <math>\mathbf{d}_0, \mathbf{d}_1, \dots, \mathbf{d}_{p-1}</math>, which plays the role of a ''control  polygon''. The curve can be described as a function <math> \mathbf{s}(x)</math> of one parameter ''x''. To pass through the sequence of points, the curve must satisfy <math>\mathbf{s}(u_0)=\mathbf{d}_0, \dots,
\mathbf{s}(u_{p-1})=\mathbf{d}_{p-1}</math>. But this is not quite the case: in general we are satisfied that the curve "approximates" the control polygon. We assume that ''u<sub>0</sub>, ..., u<sub>p-1</sub>'' are given to us along with <math>\mathbf{d}_0, \mathbf{d}_1, \dots, \mathbf{d}_{p-1}</math>.
 
One approach to solve this problem is by [[spline (mathematics)|spline]]s. A spline is a curve that is a piecewise ''n<sup>th</sup>'' degree polynomial. This means that, on any interval ''<nowiki>[</nowiki>u<sub>i</sub>, u<sub>i+1</sub>)'', the curve must be equal to a polynomial of degree at most ''n''. It may be equal to different polynomials on different intervals. The polynomials must be ''synchronized'': when the polynomials from intervals  ''<nowiki>[</nowiki>u<sub>i-1</sub>, u<sub>i</sub>)'' and ''<nowiki>[</nowiki>u<sub>i</sub>, u<sub>i+1</sub>)'' meet at the point ''u<sub>i</sub>'', they must have the same value at this point and their derivatives must be equal (to ensure that the curve is smooth).
 
De Boor's algorithm is an algorithm which, given ''u<sub>0</sub>, ..., u<sub>p-1</sub>'' and <math>\mathbf{d}_0, \mathbf{d}_1, \dots, \mathbf{d}_{p-1}</math>, finds the value of spline curve <math>\mathbf{s}(x)</math> at a point ''x''. It uses [[Big O notation|O]](n<sup>2</sup>) operations. Notice that the running time of the algorithm depends only on degree ''n'' and not on the number of points ''p''.
 
== Outline of the algorithm==
Suppose we want to evaluate the spline curve for a parameter value <math> x \in [u_{\ell},u_{\ell+1}] </math>.
We can express the curve as
 
:<math> \mathbf{s}(x) = \sum_{i=0}^{p-1} \mathbf{d}_i N_i^n(x) , </math>
where<ref>http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/B-spline/bspline-basis.html</ref>
<math>N_i^n(x)=\frac{x-u_i}{u_{i+n}-u_i}N_i^{n-1}(x) + \frac{u_{i+n+1}-x}{u_{i+n+1}-u_{i+1}}N_{i+1}^{n-1}(x) ,</math>
and
<math>N_i^0(x)=\left\{\begin{matrix} 1, & \mbox{if }x \in [u_{i},u_{i+1}) \\ 0, & \mbox{otherwise } \end{matrix}\right.</math>
 
 
Due to the spline locality property,  
:<math> \mathbf{s}(x) = \sum_{i=\ell-n}^{\ell} \mathbf{d}_i N_i^n(x) </math>
So the value <math>\mathbf{s}(x)</math> is determined by the control points <math> \mathbf{d}_{\ell-n},\mathbf{d}_{\ell-n+1},\dots,\mathbf{d}_{\ell} </math>; the other control points <math>\mathbf{d}_i</math> have no influence. De Boor's algorithm, described in the next section, is a procedure which efficiently calculates the expression for <math> \mathbf{s}(x) </math>.
 
== The algorithm ==
 
Suppose <math> x \in [u_{\ell},u_{\ell+1}) </math> and <math> \mathbf{d}_i^{[0]} = \mathbf{d}_i </math> for <math>i = \ell-n, \dots, \ell</math>.
Now calculate
:<math> \mathbf{d}_i^{[k]} = (1-\alpha_{k,i}) \mathbf{d}_{i-1}^{[k-1]} + \alpha_{k,i} \mathbf{d}_i^{[k-1]}; \qquad k=1,\dots,n; \quad i=\ell-n+k,\dots,\ell </math>
with
:<math> \alpha_{k,i} = \frac{x-u_i}{u_{i+n+1-k}-u_i}. </math>
Then <math> \mathbf{s}(x) = \mathbf{d}_{\ell}^{[n]} </math>.
 
== See also ==
*[[De Casteljau's algorithm]]
*[[Bézier curve]]
*[[NURBS]]
 
== External links ==
*[http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/de-Boor.html  De Boor's Algorithm]
*[http://www.css.taylor.edu/~btoll/s99/424/res/ucdavis/CAGDNotes/Deboor-Cox-Calculation/Deboor-Cox-Calculation.html The DeBoor-Cox Calculation]
 
== References ==
<references/>
 
[[Category:Numerical analysis]]
[[Category:Splines]]
[[Category:Interpolation]]

Latest revision as of 21:20, 11 June 2014

A lagging computer is absolutely annoying plus can be very a headache. Almost every individual that utilizes a computer faces this problem some time or the different. If the computer furthermore suffers within the same issue, there are it difficult to continue working as normal. In such a condition, the thought, "what could I do to create my PC run quicker?" is recurring and infuriating. There's a answer, but!

Registry is not moreover important to fast computer boot up, and important to the total performance of a computer. If you have a registry error, we might face blue screen, freezing or even crash. It's mandatory to regularly clean up the invalid, lost, junk registry keys to keep the computer healthy plus running swiftly.

When you compare registry products you want a swiftly acting registry cleaning. It's no good spending hours and the PC waiting for the registry cleaning to complete its task. You wish the cleaner to complete its task inside minutes.

Paid registry cleaners on the alternative hand, I have found, are usually cheap. They offer standard, free updates or at least inexpensive changes. This follows because the software manufacturer must confirm their product is most effective in staying before its competitors.

Another usual cause of PC slow down is a corrupt registry. The registry is a 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 simple with the utilize of a system plus tuneup utilities 2014.

Windows relies heavily on this database, storing everything from a latest emails to your Internet favorites in there. Because it's so crucial, a computer is constantly adding plus updating the files inside it. This is okay, nevertheless it could make a computer run slow, when the computer accidentally breaks its important registry files. This is a extremely popular issue, and really makes a computer run slower every day. What arises is that since a computer is frequently using 100's of registry files at when, it sometimes gets confused plus create some of them unreadable. This then makes your computer run slow, considering Windows takes longer to read the files it needs.

To accelerate your computer, we just should be able to get rid of all these junk files, permitting the computer to find exactly what it wants, when it wants. Luckily, there's a tool which allows you to do this conveniently and immediately. It's a tool called a 'registry cleaner'.

If you like to have a computer with quick running speed, you'd better install a good registry cleaner to clean the useless files for we. As long as we take care of the computer, it may keep in good condition.

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