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3057:{\displaystyle \mathbf {v} ^{H}\mathbf {x} =({\mathbf {u} \over \|\mathbf {u} \|})^{H}\mathbf {x} ={\mathbf {u} ^{H}\mathbf {x} \over \|\mathbf {u} \|}={({\mathbf {x} -\alpha \mathbf {e} _{1}})^{H}\mathbf {x} \over \|\mathbf {u} \|}={{\mathbf {x} ^{H}\mathbf {x} -\alpha ^{*}\mathbf {e} _{1}^{T}\mathbf {x} } \over \|\mathbf {u} \|}}
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the essential properties among which the QR-factorization has. Also, except for the uniqueness, QR factorization has good features even in non-square matrices, I guess. For example, we can recognize that an orthonormal basis of the column space of A (=QR) consists of the columns of Q associated with non-zero diagonal entries of R.
1256:{\displaystyle QR={\begin{pmatrix}6/7&-69/175&58/175\\3/7&158/175&-6/175\\-2/7&6/35&33/35\end{pmatrix}}\cdot {\begin{pmatrix}14&21&-14\\0&175&-70\\0&0&-35\end{pmatrix}}={\begin{pmatrix}12&-51&4\\6&167&-68\\-4&24&-41\end{pmatrix}}=A}
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That particular referenced book is not available to me, so I don't know what's written there. The argument by itself however is clearly wrong, as if $ A = QR$ , then $ \cond(A) = \cond(R)$ (at least in the 2-norm). Hence, the condition number of the linear system one must solve (now involving R) has
6149:
At the very beginning of the page, there is a mistake: "In linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix A into a product A = QR of an orthogonal matrix Q and an upper triangular matrix R." I think it's worth noting that Q is
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In our linear algebra course we used the term QR decomposition of a m×n matrix to mean Q (of dimensions m×n) with orthonormal columns and upper triangular R (of dimensions n×n). This would be A=Q₁R₁ in the article's section
Rectangular matrix. However, I don't know how to edit the section to mention
2017:
Hi, I am new to the
Knowledge community, as far as editing goes... this is my first so please let me know if this is against protocol. After studying numerical linear algebra using the textbook by Trefethen and Bau, I noticed that there is a crucial step in the Householder implementation of the QR
715:
I agree that it's possible to give only the general definition (complex rectangular matrices), and that the general definition is useful, but I think that it's better from an educational point of view to start with the simpler definition (real square matrices). However, I do not feel strongly about
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Somehow, I don't think that we need to start with a square matrix. When we solve an over-determined system of equations in order to minimize the unitarily invariant norm of residuals, we sometimes use the QR-factorization to reduce a computational complexity. That kind of application reveals one of
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I only had a look at the section "Using for solution to linear inverse problems" because I was going to point students to something for further reading on economy/deluxe decomposition, but that section seems to be full of mistake. I'm about to catch a flight but maybe I'll come back to this article
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The given dimensions of the matrices in the "Rectangular matrix" paragraph seem inconsistent to me. If, e.g., R is n by n as stated in the first sentence, then its bottom (m-n) rows are not zero. If "Q1 is m×n, Q2 is m×(m−n)", then Q should be m by m and not "an m×n unitary matrix Q". Am I wrong?
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The first paragraph states that QR decomposition has a lower condition number than direct matrix inverse. However, "condition number" is a property of the problem (solving a linear system) and not of the method. So AFAIK, this sentence has no sense. Somebody knows what is the actual reason why QR
5892:
The part about householder says: u = x - alpha * e1. In the example below, alpha is -14 and x is ^T. 12 - (-14) is 26. So according to the above, u should be ^T. However, it's ^T in the rest of the example, so alpha was added rather than subtracted. It still seems to work though, it seems to work
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Hi everyone.i had checked converting tranpose to transpose conjugate before and it doesn't work in householder reflectors algorithm(Q will be unitary but R won't be upper triangular).In Gram-Schmidt the algorithm have a constraint that "A" must be square(the author have mentioned it in the
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factorization does not say anything about when the method was first published. In lieu of date of first publication, we could cite the older method or approach (the proceeding popular technique) and maybe also cite one of the newer techniques. That way, the position of
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I readily agree that the pseudocode is much easier to understand than the explanation in the article. Indeed, I've never been happy how the recursion is explained in the article (I think I am myself to blame for this), so please do go ahead and improve the text. --
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factorization could be edited to say when QR factorization was first published. If someone published it even earlier, that is good, the
Knowledge page is openly editable by anyone, and we can amend the official date of publication and name of the author.
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5963:. If you are solving linear equations Ax=b with LUx = b versus QRx=b then the condition number of L and Q are relevant to the accuracy of the solution. L can have a large condition number; however, Q is orthogonal and thus has condition number 1.
2669:. I am not sure whether I have the privilege to change Jitse's note above, so I am appending this one. If it is acceptable under Knowledge guidelines to change Jitse's note, I hope someone will please do so, and then delete this note. Thanks.
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Using the other sign does not matter if you use exact computations, as in the example. I tried to clarify this. However, perhaps it is better to change the example to match the sign convention, or at least make a remark at that point. --
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We must guide people through the progression of different techniques for solving linear systems through time, so that students see the easy-to-understand stuff first, and they see the complicated cutting-edge research sometime later.
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Actually, I think that the algorithms all work for complex matrices if you replace the transpose with the conjugate transpose. For the first algorithm (Gram-Schmidt), this is mentioned in the section listed in the references. --
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decomposition in this article that makes this article less than optimal in numerical stability. The operation that I would like to verify is the last one in the following snippet of pseudo code from "Numerical Linear
Algebra":
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hi. it seems that the algorithms that are discussed in this page are only compatible with matrices that have real elements and it seems that they can't be modified easily to be compatible with complex matrices.please guide me.
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2389:-0.8548 0.1738 + 0.0515i 0.4789 - 0.0850i 0.1738 - 0.0515i 0.9823 -0.0425 + 0.0213i 0.4789 + 0.0850i -0.0425 - 0.0213i 0.8725
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every iteration of the loop, which adds to the numerical instability of the algorithm. The pseudo code proposed by
Trefethen and Bau reduces that by having a fresh copy of the original A every iteration.
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The example is correct. The QR decomposition is not unique. Q is orthogonal, R is upper triangular, and Q*R = A. What part worries you? You even used a calculator to prove the example was correct.
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1990:"We can similarly form Givens matrices G2 and G3, which will zero the sub-diagonal elements a21 and a32, forming a rectangular matrix" should this actually say triangular instead of rectangular? --
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PS:I apologize if the text has some mistakes gramatically or in vocabulary.I'm not a native speaker and i am not fluent and i am not familiar with the other languages of the the article. thank you.
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Apparently the way sign of alpha is computed is also different. Even with these 2 changes I didn't get the correct answer. Can anyone please add a example for complex matrix too? thanks.
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as it is clear,the elements (2,1) and (3,1) of the matrix "q1*a" are not zero.(against the algorithm).I continued the algorithm but i don't want to take your time more than this.finally:
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I'm just wondering why (in the householder-example) the resulting Q- and R-matrices don't multiply to A. Futhermore, the given Q and Q^t are not equal to I. Is the example correct?
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as.matrix(q) %*% as.matrix(r) V1 V2 V3 11.99996 -50.98650 4.002192 5.99998 166.97736 24.000335 3.99994 -67.98786 -40.998769 : -->
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It took me a while before I had time to go through it; sorry. It also was more complicated than I'd expected. Householder reflections behave a bit differently in complex spaces.
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2404:-0.8548 -0.2307 + 0.4265i 0.0231 - 0.1837i 0.1738 - 0.0515i -0.1566 - 0.0464i -0.5619 - 0.7904i 0.4789 + 0.0850i -0.4857 + 0.7088i 0.1520 + 0.0464i
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I'm trying to translate this article in french. I'm wondering about the example of
Householder reflections method and especially, the calcul for the length of the vector
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It would be interesting from a historical point of view (of just for curiosity's sake) to know who first used the letters "QR", and why (if any special reason exists).
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I know lots of people have vociferous opposition to free software in mathematics, so this is why I ask before I go ahead and do it. I don't want to start an edit war.
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hi there.I have performed QR decomposition via householder reflection algorithm for a non-square small-sized complex matrix.I tried to follow the algorithm exactly:
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615:. In the example which follows, however, the second matrix generated appears not to use the sign operation. Am I missing something, or should this be fixed?--
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Compared to the direct matrix inverse, inverse solutions using QR decomposition are more numerically stable as evidenced by their reduced condition numbers .
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In fact, Golub & Van Loan and Stoer & Bulirsch use a slightly different (but equivalent) approach. I added a sentence about it to the article. --
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Hello everyone. Is anyone interested in adding some information for FULL QR decomposition? Since currently, everything seem to be in reduced form.
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later, but since that part looks so weird, I am suspicious of everything. Maybe someone else can look at it if I get distracted and don't come back.
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matrix returned by Matlab has minus signs on the diagonal just like Octave. That's not so surprising because both of them use the same routines (
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The newer techniques run quickly on GPUs (graphics processing units), but are too complicated for most undergraduate students to understand.
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instead of Octave in the section where it mentions the numerical QR factorisation. I note that Octave's result is -Q and -R of Matlab's result.
2648:. Apparently, this is somewhere in Golub and Van Loan; I'll look it up and add to the Knowledge article. Let me know if you need some more help.
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you're considering is not square. Why do you think R is not upper triangular when doing QR via
Householder reflections? Where does it say that
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Can anybody say more about the statement? I'd actually be in favor of removing it completely, rather than having it here in its current form
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The reason that I say this is more accurate is because the one presented in this page requires that A be multiplied with the successive new Q
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Does someone know the total flop counts for the various QR algorithms (in particular, for a general m x n matrix, not necessarily square)?
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I guess
Knowledge doesn't have a specific policy on using non-free software for mathematics, but I thought it would be nice to mention
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5554:{\displaystyle {\text{cond}}(R_{1})=\|R_{1}\|_{2}\|R_{1}^{-1}\|_{2}={\sqrt {\lambda _{max}(\Gamma )\lambda _{max}(\Gamma ^{-1})}}=c}
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I'll answer my own question, thanks to Golub and Van Loan's "Matrix
Computations". The setup is: A is a m x n matrix, with m : -->
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q V1 V2 V3 1 -0.85714 0.3110 -0.4106 2 -0.42857 -0.8728 0.2335 3 -0.28571 0.3761 0.8814 : -->
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describes an algorithm for computing the eigenvalues of a matrix, which uses the QR decomposition. Those are different topics. --
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11.8198 - 4.0000i -1.7425 + 9.0803i 0.1775 + 0.3551i 8.1473 + 4.5214i -0.0000 + 1.0652i 2.1279 + 3.6281i
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4103:= n. I assume that if m < n, then everything still holds if you swap m and n. The flop counts for various algorithms are:
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He's right though. When you do it correctly, the signs change in the second Q matrix. The correct answer has signs flipped.
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Clearly the example is screwed up just after the first step (the first column of Q is correct as is the first element of R).
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after initializing it to A with every iteration of Q. Whereas the technique I had you can get a fresh copy of A every time.
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not changed. There might be still advantages due to more subtle issues (though I doubt that the advantage would be big).
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4057:) to do the factorization. I guess different versions of Matlab yield different results and that's why the article had
3390:{\displaystyle \alpha ^{*}x_{k}=(-e^{-i\arg x_{k}}\|\mathbf {x} \|)(|x_{k}|e^{i\arg x_{k}})=-\|\mathbf {x} \||x_{k}|}
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1575:{\displaystyle u=x-sgn(x_{1})*||x||*e_{1}=(12,6,-4)^{T}-(+1)*14*(1,0,0)^{T}=(12,6,-4)^{T}-(14,0,0)^{T}=(-2,6,-4)^{T}}
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is used instead. The reason should be made explicit. Then the text gives the equation for the
Householder Matrix as
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I and others prefer forcing the R factor diagonal to be positive real which reduces range of different solutions.
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r V1 V2 V3 1 -14 -8.4286 -2.0000 2 0 -187.1736 -35.1241 3 0 0.0000 -32.1761
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I don't think they should be merged. This page explains the QR decomposition and algorithms for computing it. The
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However, the current explanation loops over an n-by-n matrixm multiplication n times, making the given algorithm
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11.8198 - 4.0000i -1.7425 + 9.0803i 0.8315 - 0.6607i 0.3750 - 4.5215i -0.3873 + 0.1202i -7.9021 + 4.6355i
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to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the
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12.5300 -3.9904 + 7.6616i 0 -10.7413 0 0
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5734:{\displaystyle {\text{cond}}(\Gamma )={\sqrt {\lambda _{max}(\Gamma ^{2})\lambda _{max}(\Gamma ^{-}2)}}=c^{2}}
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You are right, the algorithm does not work for complex matrices. However, it's possible to fix it. Instead of
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http://web.archive.org/web/20061122183956/http://documents.wolfram.com/mathematica/functions/QRDecomposition
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2611:{\displaystyle Q=I-(1+\omega )vv^{*}\quad {\text{with}}\quad \omega ={\frac {\overline {v^{*}x}}{v^{*}x}}.}
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I'm not sure why Matlab is mentioned at all. It doesn't seem to add anything, so I removed that sentence.
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If you have discovered URLs which were erroneously considered dead by the bot, you can report them with
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Also, this link at the bottom of the page is bad -- maybe someone knows the new URL? The failed URL is
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6000:. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit
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with both. Still, why is the example inconsistent with the article? Might be worth explaining. Thanks!
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I'm afraid that I don't quite understand what you mean. What is the difference between the statement
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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2807:) is potentially not real (as stated in the same post). However, in this case it is always real:
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before doing mass systematic removals. This message is updated dynamically through the template
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For example, when studying algorithms for the maximum flow problem, we could say that...
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only orthogonal if it's square. Also it can be changed to "matrix with orthonormal columns Q"
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People are always publishing faster, more efficient approaches to solving these problems.
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800:{\displaystyle A={\begin{bmatrix}1&1&1\\0&0&1\\0&0&1\end{bmatrix}}}
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Usually, the old approaches were easy to understand, but take a long time to computer.
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factorization and what method of solving linear systems computationally was popular after
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Aha, I didn't notice that this was the point behind the edit I reverted. Better now? --
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3457:{\displaystyle w={{\mathbf {x} ^{H}\mathbf {v} } \over {\mathbf {v} ^{H}\mathbf {x} }}}
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If you found an error with any archives or the URLs themselves, you can fix them with
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In the section "Using for solution to linear inverse problems", there is the sentence
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http://web.archive.org/web/20081212221215/http://www.bluebit.gr:80/matrix-calculator/
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This statement should be made in reference to another method of solving Ax=b such as
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Is Someone Willing to Include a Year of Publication for a paper on QR Factorization?
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qr.R(qr(x)) V1 V2 V3 -14 -21 14 0 -175 70 0 0 -35
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Why isn't there a more simple example (easier numbers, these n/175 scares me ;-) )?
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At the end of the subsection "square matrix" there is the following statement: “If
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The old techniques for matrix decomposition are good for undergraduate students.
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The year of publication is useful for determining what method was popular before
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has to be square for Gram-Schmidt? Did you check whether that is in fact true? --
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Can you even form the Q matrix of qr factorization by householder reflections in
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Is there somebody to editorialize the definition without assuming the full rank?
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fourth came the Chen, Kyng, Liu, Peng, Gutenberg and Sachdeva's algorithm (2022)
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5155:, so not long ago. I've just undone it. Hope that section makes more sense now.
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5243:(more often referred to as an upper triangular matrix these days, it seems).
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So, if m << n, then it looks like Modified Gram-Schmidt is the winner.
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after trying to implement that I noticed that you have to do tempA = tempA*Q
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that QR decomposition can sometimes mean Q₁R₁. If I should edit it at all.
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The referenced text discusses this in terms of the condition numbers of
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Clarification needed on assumptions in ‘Cases and definitions’ Section
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3952:= #rows then every row except the first needs to be "triangularized".
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The link to Dirk Laurie's post above is dead. I found Dirk's post at
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Suggested fix for number of iterations using Householder reflections
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are real vectors then ω = 1 and we get the same algorithm as before.
2383:-0.9482+0.1683i 0.0842-0.0421i 0.2525
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x V1 V2 V3 1 12 -51 4 2 6 167 -68 3 -4 24 -41
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alfa=norm(a1)=sqrt(a1'*a1) ("'" means transpose conjugate)
6439:”. However, this is not generally true. It assumes that the first
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http://documents.wolfram.com/mathematica/functions/QRDecomposition
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paper referenced there says (bottom left of first page): "... the
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You are right. My claim was true only if A is full rank, I guess.
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Yep, you're right. Thanks for bringing this to our attention. --
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the dot-product, which is 2/14 (arguably, this can be written as
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I do not understand the example. The first step is still clear:
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Thanks for all the work ( I stumbled on this while implementing)
3775:{\displaystyle A={\begin{pmatrix}12&-51&4\end{pmatrix}}}
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I also performed QR decomposition in MATLAB and the result was:
909:. Could you please be more specific? What value do you take for
538:{\displaystyle \|a_{1}\|_{2}={\sqrt {12^{2}+6^{2}+(-4)^{2}}}=14}
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are linearly independent. Shouldn't this be stated explicitly?
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can potentially be more accurate than those relying on solving
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The reason that say that is because the operation you had was:
15:
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When you have finished reviewing my changes, please set the
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Remark about "Using for solution to linear inverse problems"
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Inconsistencies in matrix dimension for rectangular matrices
3572:, which I think should be merged in this page, and then the
4216:"Hessenberg QR via Givens" (i.e. A is in Hessenberg form):
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A, shouldn't it follow that the QR decomp is given by Q = Q
3173:{\displaystyle \alpha ^{*}\mathbf {e} _{1}^{T}\mathbf {x} }
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for additional information. I made the following changes:
3948:#columns every column needs to be "triangularized" in turn
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which is also real. In which case, the top and bottom of
2106:= sub-matrix of A starting at the k-th row and k-th column
4727:{\displaystyle A_{k+1}=A_{k}-2\cdot v_{k}\cdot v_{k}^{T}}
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Oh, you're right. I'm sorry, math is too hard for me :-)
428:{\displaystyle \|a_{1}\|={\sqrt {12^{2}+6^{2}+(-4)^{2}}}}
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linearly independent columns, More generally, the first
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is responsible for naming the orthogonal/unitary matrix
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5872:). The sentence in question really should be reworded.
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The example given is just wrong. Using R, I get this:
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You know you can just use that in your French version.
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4432:{\displaystyle Q=Q_{1}^{T}Q_{2}^{T}\cdots Q_{t}^{T},}
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3884:{\displaystyle A={\begin{pmatrix}12\\6\end{pmatrix}}}
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form an orthonormal basis for the span of the first
6375:factorization in time is available to the reader.
5056:
Testing the result given in the example gives this:
4086:
http://www.cs.ut.ee/~toomas_l/linalg/lin2/node5.html
101:, a collaborative effort to improve the coverage of
6125:
Alternative QR decomposition for rectangular matrix
6066:using the archive tool instructions below. Editors
4872:, yes you can do matrix multiplication faster than
2356:-10+4i 4-5i 2-i 8+6i 6 7i
719:I do not understand your last sentence. If we take
545:. So there is no problem. PierreLM April 15th 2005
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2635:q1 = eye(3) - (1 + conj(v1'*a1)/(v1'*a1)) * v1*v1'
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5215:hadn't been invented prior to its use in the QR
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3646:
1693:{\displaystyle ||u||={\sqrt {(}}56)\approx 7.48}
573:The section on the Householder method says that
6328:At the time of my writing, the this article on
5353:(in this article's notation; the original uses
2325:I'm afraid I don't understand you. I guess the
2156:in Trefethen and Bau's code, and the operation
593:should take its sign from the first element in
158:No time to fix this but there seem to be errors
6210:first came the Ford–Fulkerson algorithm (1956)
6052:This message was posted before February 2018.
1774:{\displaystyle ||(-1,3,-2)^{T}||={\sqrt {14}}}
6213:second came the Edmonds–Karp algorithm (1970)
5932:decomposition is more "numerically stable"?
5603:indicates the largest eigenvalue of a matrix
5207:article has some history in its lead and its
4647:Using Householder reflections O(n^3) incorect
2644:This formulation of the method is taken from
1266:No worries, happens to me all the time ;) --
8:
6216:third came the Preflow–push algorithm (1974)
5912:QR decomposition is basis-dependent, yes? --
5469:
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3092:{\displaystyle \mathbf {x} ^{H}\mathbf {x} }
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2800:{\displaystyle \mathbf {x} ^{H}\mathbf {v} }
2766:{\displaystyle \mathbf {v} ^{H}\mathbf {x} }
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6223:The Maximum Flow problem is not related to
5259:Stability of inverse using QR decomposition
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6249:However, I was hoping that the article on
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6014:http://www.bluebit.gr/matrix-calculator/
4865:{\displaystyle O(n^{3})\cdot n=O(n^{2})}
4358:Why writing transposed Qi instead of Qi?
3131:are always real, and the remaining term
2371:u1=a1-(alfa*e) e=(1,0,0)t
3969:In the Householder section, since R = Q
3782:. It neads no iterations and correctly
49:
2262:QR decomposition for complex matrices?
343:{\displaystyle \|a_{1}\|=12+6+|-4|=22}
273:{\displaystyle \|a_{1}\|=12+6+(-4)=14}
6041:to let others know (documentation at
5151:Well spotted. That was the result of
3568:There is a page on the same topic on
3464:are always the same. So I think the
2013:A more stable version of Householder?
893:When I calculate the products, I get
665:QR decomposition with rank deficiency
7:
5908:Is QR decomposition basis-dependent?
5888:Sign of alpha in householder example
4619:{\displaystyle Q_{i}^{T}=Q_{i}^{-1}}
4546:because it's symmetric, isn't it?!
95:This article is within the scope of
6180:It would be useful to know how old
38:It is of interest to the following
5701:
5669:
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5324:
2480:
1975:{\displaystyle 2/{\sqrt {14}}^{2}}
823:is the 3-by-3 identity matrix and
14:
6476:Mid-priority mathematics articles
5996:. Please take a moment to review
5596:{\displaystyle \lambda _{max}(M)}
3891:needs one iteration. Incorrectly
1881:suddenly appears. Textbooks give
115:Knowledge:WikiProject Mathematics
6471:Start-Class mathematics articles
4944:making the algorithm still : -->
4025:Mention Octave instead of Matlab
3447:
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3210:{\displaystyle \alpha ^{*}x_{k}}
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2491:{\displaystyle Q=I-2vv^{\top },}
601:
118:Template:WikiProject Mathematics
82:
72:
51:
20:
5219:, though I could be wrong. The
4539:{\displaystyle Q_{i}^{T}=Q_{i}}
3937:{\displaystyle t=\min(2,1)-1=0}
3828:{\displaystyle t=\min(1,2)-1=0}
2646:Dirk Laurie's post to NA Digest
2633:In your Matlab code, you write
2362:-10+4i 2-i 6
135:This article has been rated as
6246:factorization of matrices.
5800:
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5346:{\displaystyle \Gamma =A^{T}A}
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3180:has only one non-zero element
3124:{\displaystyle |\mathbf {u} |}
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1:
5979:19:36, 16 February 2016 (UTC)
5285:14:10, 10 December 2012 (UTC)
5165:15:28, 15 February 2012 (UTC)
5145:12:54, 15 February 2012 (UTC)
4641:14:05, 13 December 2016 (UTC)
4294:{\displaystyle 2n^{2}(m-n/3)}
4207:{\displaystyle 3n^{2}(m-n/3)}
4152:{\displaystyle 2n^{2}(m-n/2)}
3720:{\displaystyle t=\min(m-1,n)}
3673:{\displaystyle t=\min(m,n)-1}
3527:11:43, 30 December 2013 (UTC)
3506:19:07, 7 September 2017 (UTC)
2679:01:37, 8 September 2014 (UTC)
2651:PS: Your English is fine. --
2558:
2551:
2343:13:59, 28 February 2007 (UTC)
2317:13:25, 27 February 2007 (UTC)
2305:04:44, 15 February 2007 (UTC)
2287:11:59, 14 February 2007 (UTC)
658:16:37, 30 November 2016 (UTC)
173:18:06, 21 February 2020 (UTC)
109:and see a list of open tasks.
6457:02:54, 12 January 2024 (UTC)
6170:16:30, 18 October 2021 (UTC)
5820:{\displaystyle (A^{T}A)X=B'}
5097:19:24, 19 January 2012 (UTC)
5081:08:23, 19 January 2012 (UTC)
5041:Example is grossly incorrect
5035:22:51, 1 November 2010 (UTC)
4567:16:27, 6 December 2009 (UTC)
4071:16:58, 27 October 2007 (UTC)
4041:17:39, 26 October 2007 (UTC)
4019:16:58, 27 October 2007 (UTC)
3961:I am making the change-chad
3633:iterations of this process,
2584:
2252:17:36, 31 January 2007 (UTC)
2212:12:06, 24 January 2007 (UTC)
2123:06:58, 23 January 2007 (UTC)
2007:12:43, 22 October 2006 (UTC)
636:13:44, 27 January 2006 (UTC)
620:16:11, 25 January 2006 (UTC)
608:{\displaystyle \mathbf {x} }
5953:12:02, 17 August 2014 (UTC)
5903:00:42, 6 January 2014 (UTC)
4061:with the opposite sign. --
3496:can be replaced with 2. --
1997:Yes, indeed. Now fixed. --
1874:{\displaystyle 2/||v||^{2}}
1819:{\displaystyle Q=I-2vv^{T}}
1294:22:35, 11 August 2009 (UTC)
882:Householder-Example: QR=A ?
435:. PierreLM April 13th 2005
178:Householder-Example: QR=A ?
6492:
6145:Q is not always orthogonal
6083:(last update: 5 June 2024)
5989:Hello fellow Wikipedians,
5922:23:07, 25 April 2014 (UTC)
3603:03:35, 16 April 2007 (UTC)
3584:22:12, 15 April 2007 (UTC)
3576:should link to this page.
2697:07:19, 14 March 2007 (UTC)
2661:06:45, 10 March 2007 (UTC)
913:, and what do you get for
831:, but the column space of
6140:03:53, 17 June 2019 (UTC)
6120:10:21, 21 July 2016 (UTC)
5882:12:45, 18 June 2013 (UTC)
5253:12:22, 27 July 2012 (UTC)
5239:is an obvious name for a
5196:07:57, 27 July 2012 (UTC)
4467:{\displaystyle Q_{i}^{T}}
4352:19:10, 4 April 2008 (UTC)
4303:"Modified Gram-Schmidt":
4097:16:18, 4 April 2008 (UTC)
3557:22:11, 6 March 2007 (UTC)
2440:17:07, 5 March 2007 (UTC)
1631:{\displaystyle v=u/||u||}
134:
67:
46:
6398:16:50, 6 June 2023 (UTC)
5770:{\displaystyle R_{1}X=B}
5118:15:57, 17 May 2016 (UTC)
5012:{\displaystyle O(n^{3})}
4973:{\displaystyle O(n^{3})}
4937:{\displaystyle O(n^{2})}
4901:{\displaystyle O(n^{3})}
4801:{\displaystyle O(n^{4})}
4763:{\displaystyle O(n^{3})}
2187:{\displaystyle A=Q_{k}A}
1638:with the Euklidian norm
1276:13:54, 9 June 2006 (UTC)
927:12:59, 8 June 2006 (UTC)
443:13:30, 14 Apr 2005 (UTC)
141:project's priority scale
5985:External links modified
5241:right triangular matrix
5153:this edit of 5 February
4329:{\displaystyle 2mn^{2}}
3680:" I think it should be
870:21:38, 8 May 2006 (UTC)
849:05:00, 8 May 2006 (UTC)
703:02:21, 8 May 2006 (UTC)
689:01:45, 8 May 2006 (UTC)
674:14:03, 7 May 2006 (UTC)
586:{\displaystyle \alpha }
562:{\displaystyle \alpha }
98:WikiProject Mathematics
6369:
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4734:Then the algorithm is
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4239:{\displaystyle 2n^{2}}
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2639:q1 = eye(3) - 2*v1*v1'
2612:
2492:
2377:-22.53+4i 2-i 6
2312:"definition" section).
2188:
1976:
1937:
1936:{\displaystyle \odot }
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28:This article is rated
6370:
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6241:
6203:factorization is.
6198:
5867:
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5822:
5772:
5736:
5623:. Contrast this with
5618:
5598:
5556:
5388:
5368:
5348:
5312:
5310:{\displaystyle R_{1}}
5211:. It appears that QR
5014:
4975:
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4908:, but its still : -->
4903:
4867:
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4765:
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4541:
4496:
4494:{\displaystyle Q_{i}}
4469:
4434:
4331:
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4154:
3993:instead of just Q = Q
3951:If the #columns : -->
3939:
3886:
3830:
3777:
3722:
3675:
3628:
3613:The text has- "After
3543:comment was added by
3533:FULL QR decomposition
3491:
3489:{\displaystyle (1+w)}
3459:
3392:
3212:
3175:
3126:
3094:
3059:
2802:
2768:
2739:is only necessary if
2735:referred to above in
2730:
2728:{\displaystyle (1+w)}
2613:
2493:
2273:comment was added by
2242:comment was added by
2189:
2099:* = complex conjugate
1977:
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209:
207:{\displaystyle a_{1}}
6356:
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6064:regular verification
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4049:On my computer, the
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2386:q1=I(3*3)-2*v1*v1'=
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183:translation question
121:mathematics articles
6054:After February 2018
6033:parameter below to
5467:
5171:Origin of the name?
4723:
4615:
4594:
4522:
4474:instead of simply:
4463:
4425:
4407:
4392:
3965:Householder Problem
3164:
3032:
2641:and it should work.
1584:In the next step,
6368:{\displaystyle QR}
6365:
6344:{\displaystyle QR}
6341:
6315:{\displaystyle QR}
6312:
6292:{\displaystyle QR}
6289:
6265:{\displaystyle QR}
6262:
6239:{\displaystyle QR}
6236:
6196:{\displaystyle QR}
6193:
6108:InternetArchiveBot
6059:InternetArchiveBot
5865:{\displaystyle B'}
5862:
5837:
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4378:
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4291:
4248:"Fast Givens QR":
4236:
4204:
4149:
4106:"Householder QR":
3947:If the #rows : -->
3934:
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3486:
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3207:
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2797:
2773:(or its conjugate
2763:
2737:Dirk Laurie's post
2725:
2608:
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1992:User:Richard Giuly
1972:
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90:Mathematics portal
34:content assessment
6156:comment added by
6084:
5981:
5969:comment added by
5956:
5939:comment added by
5840:{\displaystyle B}
5716:
5633:
5616:{\displaystyle M}
5543:
5403:
5393:). Specifically,
5386:{\displaystyle A}
5366:{\displaystyle G}
5229:John G.F. Francis
5225:QR transformation
5199:
5182:comment added by
5135:comment added by
5120:
5108:comment added by
5071:comment added by
5025:comment added by
4643:
4631:comment added by
4570:
4553:comment added by
4077:total flop counts
3626:{\displaystyle t}
3560:
3517:comment added by
3452:
3052:
2978:
2913:
2859:
2603:
2587:
2556:
2502:you have to take
2290:
2255:
1964:
1911:
1769:
1676:
660:
648:comment added by
527:
423:
280:, I would prefer
155:
154:
151:
150:
147:
146:
6483:
6374:
6372:
6371:
6366:
6350:
6348:
6347:
6342:
6322:factorization.
6321:
6319:
6318:
6313:
6298:
6296:
6295:
6290:
6271:
6269:
6268:
6263:
6245:
6243:
6242:
6237:
6202:
6200:
6199:
6194:
6172:
6118:
6109:
6082:
6081:
6060:
6048:
5994:QR decomposition
5961:LU decomposition
5955:
5933:
5871:
5869:
5868:
5863:
5861:
5846:
5844:
5843:
5838:
5826:
5824:
5823:
5818:
5816:
5796:
5795:
5776:
5774:
5773:
5768:
5757:
5756:
5740:
5738:
5737:
5732:
5730:
5729:
5717:
5709:
5708:
5696:
5695:
5677:
5676:
5664:
5663:
5648:
5634:
5631:
5622:
5620:
5619:
5614:
5602:
5600:
5599:
5594:
5583:
5582:
5560:
5558:
5557:
5552:
5544:
5539:
5538:
5523:
5522:
5498:
5497:
5482:
5477:
5476:
5466:
5458:
5446:
5445:
5436:
5435:
5417:
5416:
5404:
5401:
5392:
5390:
5389:
5384:
5372:
5370:
5369:
5364:
5352:
5350:
5349:
5344:
5339:
5338:
5316:
5314:
5313:
5308:
5306:
5305:
5198:
5176:
5147:
5083:
5037:
5018:
5016:
5015:
5010:
5005:
5004:
4979:
4977:
4976:
4971:
4966:
4965:
4943:
4941:
4940:
4935:
4930:
4929:
4907:
4905:
4904:
4899:
4894:
4893:
4871:
4869:
4868:
4863:
4858:
4857:
4830:
4829:
4807:
4805:
4804:
4799:
4794:
4793:
4769:
4767:
4766:
4761:
4756:
4755:
4733:
4731:
4730:
4725:
4722:
4717:
4705:
4704:
4686:
4685:
4673:
4672:
4625:
4623:
4622:
4617:
4614:
4606:
4593:
4588:
4569:
4547:
4545:
4543:
4542:
4537:
4535:
4534:
4521:
4516:
4500:
4498:
4497:
4492:
4490:
4489:
4473:
4471:
4470:
4465:
4462:
4457:
4438:
4436:
4435:
4430:
4424:
4419:
4406:
4401:
4391:
4386:
4335:
4333:
4332:
4327:
4325:
4324:
4300:
4298:
4297:
4292:
4284:
4267:
4266:
4245:
4243:
4242:
4237:
4235:
4234:
4213:
4211:
4210:
4205:
4197:
4180:
4179:
4158:
4156:
4155:
4150:
4142:
4125:
4124:
3943:
3941:
3940:
3935:
3890:
3888:
3887:
3882:
3880:
3879:
3834:
3832:
3831:
3826:
3781:
3779:
3778:
3773:
3771:
3770:
3726:
3724:
3723:
3718:
3679:
3677:
3676:
3671:
3632:
3630:
3629:
3624:
3538:
3529:
3495:
3493:
3492:
3487:
3463:
3461:
3460:
3455:
3453:
3451:
3450:
3445:
3444:
3439:
3432:
3431:
3426:
3425:
3420:
3413:
3396:
3394:
3393:
3388:
3386:
3381:
3380:
3371:
3363:
3346:
3345:
3344:
3343:
3320:
3315:
3314:
3305:
3291:
3283:
3282:
3281:
3280:
3245:
3244:
3235:
3234:
3216:
3214:
3213:
3208:
3206:
3205:
3196:
3195:
3179:
3177:
3176:
3171:
3169:
3163:
3158:
3153:
3147:
3146:
3130:
3128:
3127:
3122:
3120:
3115:
3110:
3098:
3096:
3095:
3090:
3088:
3083:
3082:
3077:
3063:
3061:
3060:
3055:
3053:
3051:
3047:
3038:
3037:
3031:
3026:
3021:
3015:
3014:
3002:
2997:
2996:
2991:
2984:
2979:
2977:
2973:
2964:
2963:
2958:
2957:
2948:
2947:
2946:
2941:
2929:
2919:
2914:
2912:
2908:
2899:
2898:
2893:
2892:
2887:
2880:
2875:
2870:
2869:
2860:
2858:
2854:
2845:
2840:
2832:
2827:
2826:
2821:
2806:
2804:
2803:
2798:
2796:
2791:
2790:
2785:
2772:
2770:
2769:
2764:
2762:
2757:
2756:
2751:
2734:
2732:
2731:
2726:
2640:
2636:
2617:
2615:
2614:
2609:
2604:
2602:
2598:
2597:
2583:
2579:
2578:
2568:
2567:
2557:
2554:
2550:
2549:
2497:
2495:
2494:
2489:
2484:
2483:
2380:v1=u1/norm(u1)=
2268:
2237:
2193:
2191:
2190:
2185:
2180:
2179:
1981:
1979:
1978:
1973:
1971:
1970:
1965:
1960:
1957:
1942:
1940:
1939:
1934:
1922:
1920:
1919:
1914:
1912:
1910:
1903:
1902:
1889:
1880:
1878:
1877:
1872:
1870:
1869:
1864:
1858:
1850:
1845:
1840:
1825:
1823:
1822:
1817:
1815:
1814:
1780:
1778:
1777:
1772:
1770:
1765:
1760:
1755:
1750:
1749:
1716:
1711:
1699:
1697:
1696:
1691:
1677:
1672:
1667:
1662:
1654:
1649:
1637:
1635:
1634:
1629:
1627:
1622:
1614:
1609:
1604:
1581:
1579:
1578:
1573:
1571:
1570:
1534:
1533:
1503:
1502:
1469:
1468:
1417:
1416:
1383:
1382:
1370:
1365:
1357:
1352:
1341:
1340:
1262:
1260:
1259:
1254:
1246:
1245:
1168:
1167:
1093:
1092:
1082:
1069:
1056:
1038:
1022:
1009:
994:
981:
965:
806:
804:
803:
798:
796:
795:
614:
612:
611:
606:
604:
592:
590:
589:
584:
568:
566:
565:
560:
544:
542:
541:
536:
528:
526:
525:
504:
503:
491:
490:
481:
476:
475:
466:
465:
434:
432:
431:
426:
424:
422:
421:
400:
399:
387:
386:
377:
369:
368:
349:
347:
346:
341:
333:
322:
299:
298:
279:
277:
276:
271:
233:
232:
213:
211:
210:
205:
203:
202:
123:
122:
119:
116:
113:
92:
87:
86:
76:
69:
68:
63:
55:
48:
31:
25:
24:
16:
6491:
6490:
6486:
6485:
6484:
6482:
6481:
6480:
6461:
6460:
6405:
6354:
6353:
6330:
6329:
6301:
6300:
6278:
6277:
6251:
6250:
6225:
6224:
6182:
6181:
6178:
6151:
6147:
6127:
6112:
6107:
6075:
6068:have permission
6058:
6042:
6002:this simple FaQ
5987:
5934:
5929:
5910:
5890:
5854:
5849:
5848:
5829:
5828:
5809:
5787:
5779:
5778:
5748:
5743:
5742:
5721:
5700:
5681:
5668:
5649:
5625:
5624:
5605:
5604:
5568:
5563:
5562:
5527:
5508:
5483:
5468:
5437:
5427:
5408:
5395:
5394:
5375:
5374:
5355:
5354:
5330:
5319:
5318:
5297:
5292:
5291:
5261:
5209:history section
5177:
5173:
5137:195.176.179.210
5130:
5126:
5066:
5063:
5054:
5050:
5043:
5027:139.222.243.106
5020:
4996:
4985:
4984:
4957:
4946:
4945:
4921:
4910:
4909:
4885:
4874:
4873:
4849:
4821:
4810:
4809:
4785:
4774:
4773:
4747:
4736:
4735:
4696:
4677:
4658:
4653:
4652:
4649:
4575:
4574:
4548:
4526:
4503:
4502:
4481:
4476:
4475:
4444:
4443:
4367:
4366:
4360:
4344:131.215.105.118
4316:
4305:
4304:
4258:
4250:
4249:
4226:
4218:
4217:
4171:
4163:
4162:
4116:
4108:
4107:
4079:
4027:
4005:? -- anonymous
4004:
4000:
3996:
3992:
3988:
3984:
3980:
3976:
3972:
3967:
3893:
3892:
3874:
3873:
3867:
3866:
3856:
3844:
3843:
3784:
3783:
3765:
3764:
3759:
3751:
3741:
3729:
3728:
3682:
3681:
3635:
3634:
3615:
3614:
3611:
3566:
3539:—The preceding
3535:
3512:
3466:
3465:
3434:
3415:
3401:
3400:
3372:
3335:
3321:
3306:
3272:
3255:
3236:
3226:
3221:
3220:
3197:
3187:
3182:
3181:
3138:
3133:
3132:
3101:
3100:
3072:
3067:
3066:
3039:
3006:
2986:
2965:
2949:
2936:
2920:
2900:
2882:
2881:
2861:
2846:
2816:
2811:
2810:
2780:
2775:
2774:
2746:
2741:
2740:
2705:
2704:
2638:
2634:
2589:
2588:
2570:
2569:
2541:
2506:
2505:
2475:
2452:
2451:
2433:
2424:
2414:
2405:
2396:
2390:
2384:
2378:
2363:
2357:
2269:—The preceding
2264:
2238:—The preceding
2235:
2226:
2198:in the article?
2171:
2160:
2159:
2150:
2146:
2142:
2138:
2134:
2116:
2105:
2087:
2083:
2079:
2075:
2071:
2065:
2061:
2057:
2053:
2046:
2042:
2038:
2034:
2028:
2021:for k = 1 to n
2015:
1988:
1958:
1945:
1944:
1925:
1924:
1894:
1893:
1883:
1882:
1859:
1828:
1827:
1806:
1783:
1782:
1741:
1702:
1701:
1640:
1639:
1586:
1585:
1562:
1525:
1494:
1460:
1408:
1374:
1332:
1303:
1302:
1240:
1239:
1231:
1226:
1217:
1216:
1208:
1203:
1197:
1196:
1191:
1183:
1173:
1162:
1161:
1153:
1148:
1142:
1141:
1133:
1128:
1122:
1121:
1113:
1108:
1098:
1087:
1086:
1073:
1060:
1043:
1042:
1026:
1013:
999:
998:
985:
969:
951:
936:
935:
884:
790:
789:
784:
779:
773:
772:
767:
762:
756:
755:
750:
745:
735:
723:
722:
667:
595:
594:
575:
574:
571:
551:
550:
517:
495:
482:
467:
457:
449:
448:
413:
391:
378:
360:
352:
351:
290:
282:
281:
224:
216:
215:
194:
189:
188:
185:
180:
160:
120:
117:
114:
111:
110:
88:
81:
61:
32:on Knowledge's
29:
12:
11:
5:
6489:
6487:
6479:
6478:
6473:
6463:
6462:
6404:
6401:
6390:174.215.16.174
6364:
6361:
6340:
6337:
6311:
6308:
6288:
6285:
6261:
6258:
6235:
6232:
6221:
6220:
6217:
6214:
6211:
6192:
6189:
6177:
6174:
6146:
6143:
6126:
6123:
6102:
6101:
6094:
6027:
6026:
6018:Added archive
6016:
6008:Added archive
5986:
5983:
5928:
5925:
5909:
5906:
5889:
5886:
5885:
5884:
5860:
5857:
5836:
5815:
5812:
5808:
5805:
5802:
5799:
5794:
5790:
5786:
5766:
5763:
5760:
5755:
5751:
5728:
5724:
5720:
5715:
5712:
5707:
5703:
5699:
5694:
5691:
5688:
5684:
5680:
5675:
5671:
5667:
5662:
5659:
5656:
5652:
5646:
5643:
5640:
5637:
5612:
5592:
5589:
5586:
5581:
5578:
5575:
5571:
5550:
5547:
5542:
5537:
5534:
5530:
5526:
5521:
5518:
5515:
5511:
5507:
5504:
5501:
5496:
5493:
5490:
5486:
5480:
5475:
5471:
5465:
5462:
5457:
5453:
5449:
5444:
5440:
5434:
5430:
5426:
5423:
5420:
5415:
5411:
5407:
5382:
5362:
5342:
5337:
5333:
5329:
5326:
5304:
5300:
5260:
5257:
5256:
5255:
5221:Francis (1961)
5172:
5169:
5168:
5167:
5125:
5122:
5110:149.199.62.254
5100:
5099:
5073:67.160.196.149
5058:
5051:
5047:
5042:
5039:
5008:
5003:
4999:
4995:
4992:
4969:
4964:
4960:
4956:
4953:
4933:
4928:
4924:
4920:
4917:
4897:
4892:
4888:
4884:
4881:
4861:
4856:
4852:
4848:
4845:
4842:
4839:
4836:
4833:
4828:
4824:
4820:
4817:
4797:
4792:
4788:
4784:
4781:
4759:
4754:
4750:
4746:
4743:
4721:
4716:
4712:
4708:
4703:
4699:
4695:
4692:
4689:
4684:
4680:
4676:
4671:
4668:
4665:
4661:
4648:
4645:
4633:131.211.53.159
4613:
4610:
4605:
4601:
4597:
4592:
4587:
4583:
4533:
4529:
4525:
4520:
4515:
4511:
4488:
4484:
4461:
4456:
4452:
4440:
4439:
4428:
4423:
4418:
4414:
4410:
4405:
4400:
4396:
4390:
4385:
4381:
4377:
4374:
4359:
4356:
4355:
4354:
4339:
4338:
4337:
4336:
4323:
4319:
4315:
4312:
4301:
4290:
4287:
4283:
4279:
4276:
4273:
4270:
4265:
4261:
4257:
4246:
4233:
4229:
4225:
4214:
4203:
4200:
4196:
4192:
4189:
4186:
4183:
4178:
4174:
4170:
4159:
4148:
4145:
4141:
4137:
4134:
4131:
4128:
4123:
4119:
4115:
4078:
4075:
4074:
4073:
4047:
4026:
4023:
4022:
4021:
4002:
3998:
3994:
3990:
3986:
3982:
3978:
3974:
3970:
3966:
3963:
3959:
3958:
3954:
3953:
3949:
3933:
3930:
3927:
3924:
3921:
3918:
3915:
3912:
3909:
3906:
3903:
3900:
3878:
3872:
3869:
3868:
3865:
3862:
3861:
3859:
3854:
3851:
3841:
3840:
3824:
3821:
3818:
3815:
3812:
3809:
3806:
3803:
3800:
3797:
3794:
3791:
3769:
3763:
3760:
3758:
3755:
3752:
3750:
3747:
3746:
3744:
3739:
3736:
3716:
3713:
3710:
3707:
3704:
3701:
3698:
3695:
3692:
3689:
3669:
3666:
3663:
3660:
3657:
3654:
3651:
3648:
3645:
3642:
3622:
3610:
3607:
3606:
3605:
3565:
3562:
3534:
3531:
3509:
3508:
3485:
3482:
3479:
3476:
3473:
3449:
3443:
3438:
3430:
3424:
3419:
3411:
3408:
3397:
3385:
3379:
3375:
3370:
3366:
3362:
3358:
3355:
3352:
3349:
3342:
3338:
3334:
3331:
3328:
3324:
3319:
3313:
3309:
3304:
3300:
3297:
3294:
3290:
3286:
3279:
3275:
3271:
3268:
3265:
3262:
3258:
3254:
3251:
3248:
3243:
3239:
3233:
3229:
3218:
3204:
3200:
3194:
3190:
3168:
3162:
3157:
3152:
3145:
3141:
3119:
3114:
3109:
3087:
3081:
3076:
3064:
3050:
3046:
3042:
3036:
3030:
3025:
3020:
3013:
3009:
3005:
3001:
2995:
2990:
2982:
2976:
2972:
2968:
2962:
2956:
2952:
2945:
2940:
2935:
2932:
2928:
2923:
2917:
2911:
2907:
2903:
2897:
2891:
2886:
2878:
2874:
2868:
2864:
2857:
2853:
2849:
2844:
2838:
2835:
2831:
2825:
2820:
2808:
2795:
2789:
2784:
2761:
2755:
2750:
2724:
2721:
2718:
2715:
2712:
2700:
2699:
2684:
2683:
2682:
2681:
2649:
2642:
2631:
2622:Of course, if
2620:
2619:
2618:
2607:
2601:
2596:
2592:
2586:
2582:
2577:
2573:
2565:
2562:
2548:
2544:
2540:
2537:
2534:
2531:
2528:
2525:
2522:
2519:
2516:
2513:
2500:
2499:
2498:
2487:
2482:
2478:
2474:
2471:
2468:
2465:
2462:
2459:
2446:
2431:
2422:
2416:
2412:
2403:
2394:
2388:
2382:
2376:
2361:
2355:
2348:
2347:
2346:
2345:
2320:
2319:
2308:
2307:
2263:
2260:
2259:
2258:
2257:
2256:
2244:144.118.52.106
2233:
2230:
2229:
2228:
2224:
2215:
2214:
2199:
2196:
2195:
2194:
2183:
2178:
2174:
2170:
2167:
2154:
2153:
2152:
2148:
2144:
2140:
2136:
2132:
2114:
2111:
2110:
2109:sign = sign of
2107:
2103:
2100:
2090:
2089:
2085:
2081:
2077:
2073:
2069:
2066:
2063:
2059:
2055:
2051:
2048:
2044:
2040:
2036:
2032:
2029:
2026:
2014:
2011:
2010:
2009:
1987:
1984:
1969:
1963:
1956:
1952:
1932:
1909:
1906:
1901:
1897:
1892:
1868:
1863:
1857:
1853:
1849:
1844:
1839:
1835:
1813:
1809:
1805:
1802:
1799:
1796:
1793:
1790:
1768:
1763:
1759:
1754:
1748:
1744:
1740:
1737:
1734:
1731:
1728:
1725:
1722:
1719:
1715:
1710:
1700:. Apparently,
1689:
1686:
1683:
1680:
1675:
1670:
1666:
1661:
1657:
1653:
1648:
1626:
1621:
1617:
1613:
1608:
1603:
1599:
1596:
1593:
1569:
1565:
1561:
1558:
1555:
1552:
1549:
1546:
1543:
1540:
1537:
1532:
1528:
1524:
1521:
1518:
1515:
1512:
1509:
1506:
1501:
1497:
1493:
1490:
1487:
1484:
1481:
1478:
1475:
1472:
1467:
1463:
1459:
1456:
1453:
1450:
1447:
1444:
1441:
1438:
1435:
1432:
1429:
1426:
1423:
1420:
1415:
1411:
1407:
1404:
1401:
1398:
1395:
1392:
1389:
1386:
1381:
1377:
1373:
1369:
1364:
1360:
1356:
1351:
1347:
1344:
1339:
1335:
1331:
1328:
1325:
1322:
1319:
1316:
1313:
1310:
1299:
1298:
1297:
1296:
1279:
1278:
1252:
1249:
1244:
1238:
1235:
1232:
1230:
1227:
1225:
1222:
1219:
1218:
1215:
1212:
1209:
1207:
1204:
1202:
1199:
1198:
1195:
1192:
1190:
1187:
1184:
1182:
1179:
1178:
1176:
1171:
1166:
1160:
1157:
1154:
1152:
1149:
1147:
1144:
1143:
1140:
1137:
1134:
1132:
1129:
1127:
1124:
1123:
1120:
1117:
1114:
1112:
1109:
1107:
1104:
1103:
1101:
1096:
1091:
1085:
1081:
1077:
1074:
1072:
1068:
1064:
1061:
1059:
1055:
1051:
1048:
1045:
1044:
1041:
1037:
1033:
1030:
1027:
1025:
1021:
1017:
1014:
1012:
1008:
1004:
1001:
1000:
997:
993:
989:
986:
984:
980:
976:
973:
970:
968:
964:
960:
957:
956:
954:
949:
946:
943:
930:
929:
883:
880:
879:
878:
877:
876:
875:
874:
873:
872:
856:
855:
854:
853:
852:
851:
809:
808:
807:
794:
788:
785:
783:
780:
778:
775:
774:
771:
768:
766:
763:
761:
758:
757:
754:
751:
749:
746:
744:
741:
740:
738:
733:
730:
717:
708:
707:
706:
705:
692:
691:
666:
663:
662:
661:
639:
638:
603:
582:
570:
569:in Householder
558:
547:
534:
531:
524:
520:
516:
513:
510:
507:
502:
498:
494:
489:
485:
479:
474:
470:
464:
460:
456:
447:I'm so sorry,
445:
444:
420:
416:
412:
409:
406:
403:
398:
394:
390:
385:
381:
375:
372:
367:
363:
359:
339:
336:
332:
328:
325:
321:
317:
314:
311:
308:
305:
302:
297:
293:
289:
269:
266:
263:
260:
257:
254:
251:
248:
245:
242:
239:
236:
231:
227:
223:
201:
197:
184:
181:
179:
176:
159:
156:
153:
152:
149:
148:
145:
144:
133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
6488:
6477:
6474:
6472:
6469:
6468:
6466:
6459:
6458:
6454:
6450:
6446:
6442:
6438:
6434:
6430:
6426:
6422:
6418:
6414:
6410:
6402:
6400:
6399:
6395:
6391:
6385:
6382:
6379:
6376:
6362:
6359:
6338:
6335:
6326:
6323:
6309:
6306:
6286:
6283:
6274:
6259:
6256:
6247:
6233:
6230:
6218:
6215:
6212:
6209:
6208:
6207:
6204:
6190:
6187:
6175:
6173:
6171:
6167:
6163:
6159:
6155:
6144:
6142:
6141:
6137:
6133:
6124:
6122:
6121:
6116:
6111:
6110:
6099:
6095:
6092:
6088:
6087:
6086:
6079:
6073:
6069:
6065:
6061:
6055:
6050:
6046:
6040:
6036:
6032:
6025:
6021:
6017:
6015:
6011:
6007:
6006:
6005:
6003:
5999:
5995:
5990:
5984:
5982:
5980:
5976:
5972:
5971:128.61.55.207
5968:
5962:
5957:
5954:
5950:
5946:
5942:
5938:
5926:
5924:
5923:
5919:
5915:
5907:
5905:
5904:
5900:
5896:
5887:
5883:
5879:
5875:
5858:
5855:
5834:
5813:
5810:
5806:
5803:
5797:
5792:
5788:
5764:
5761:
5758:
5753:
5749:
5726:
5722:
5718:
5710:
5705:
5692:
5689:
5686:
5682:
5673:
5660:
5657:
5654:
5650:
5644:
5610:
5587:
5579:
5576:
5573:
5569:
5548:
5545:
5535:
5532:
5519:
5516:
5513:
5509:
5494:
5491:
5488:
5484:
5478:
5473:
5463:
5460:
5455:
5451:
5442:
5432:
5428:
5421:
5413:
5409:
5380:
5360:
5340:
5335:
5331:
5327:
5302:
5298:
5289:
5288:
5287:
5286:
5282:
5278:
5274:
5271:
5267:
5266:
5258:
5254:
5250:
5246:
5242:
5238:
5234:
5230:
5226:
5222:
5218:
5214:
5213:decomposition
5210:
5206:
5202:
5201:
5200:
5197:
5193:
5189:
5185:
5181:
5170:
5166:
5162:
5158:
5154:
5150:
5149:
5148:
5146:
5142:
5138:
5134:
5123:
5121:
5119:
5115:
5111:
5107:
5098:
5094:
5090:
5086:
5085:
5084:
5082:
5078:
5074:
5070:
5057:
5046:
5040:
5038:
5036:
5032:
5028:
5024:
5001:
4997:
4990:
4981:
4962:
4958:
4951:
4926:
4922:
4915:
4890:
4886:
4879:
4854:
4850:
4843:
4840:
4837:
4834:
4826:
4822:
4815:
4790:
4786:
4779:
4770:
4752:
4748:
4741:
4719:
4714:
4710:
4706:
4701:
4697:
4693:
4690:
4687:
4682:
4678:
4674:
4669:
4666:
4663:
4659:
4646:
4644:
4642:
4638:
4634:
4630:
4611:
4608:
4603:
4599:
4595:
4590:
4585:
4581:
4571:
4568:
4564:
4560:
4556:
4552:
4531:
4527:
4523:
4518:
4513:
4509:
4501:? After all
4486:
4482:
4459:
4454:
4450:
4426:
4421:
4416:
4412:
4408:
4403:
4398:
4394:
4388:
4383:
4379:
4375:
4372:
4365:
4364:
4363:
4357:
4353:
4349:
4345:
4341:
4340:
4321:
4317:
4313:
4310:
4302:
4285:
4281:
4277:
4274:
4271:
4263:
4259:
4255:
4247:
4231:
4227:
4223:
4215:
4198:
4194:
4190:
4187:
4184:
4176:
4172:
4168:
4161:"Givens QR":
4160:
4143:
4139:
4135:
4132:
4129:
4121:
4117:
4113:
4105:
4104:
4101:
4100:
4099:
4098:
4094:
4090:
4087:
4082:
4076:
4072:
4068:
4064:
4060:
4056:
4052:
4048:
4045:
4044:
4043:
4042:
4039:
4034:
4032:
4024:
4020:
4016:
4012:
4008:
4007:
4006:
3964:
3962:
3956:
3955:
3950:
3946:
3945:
3944:
3931:
3928:
3925:
3922:
3916:
3913:
3910:
3901:
3898:
3876:
3870:
3863:
3857:
3852:
3849:
3838:
3837:
3836:
3822:
3819:
3816:
3813:
3807:
3804:
3801:
3792:
3789:
3767:
3761:
3756:
3753:
3748:
3742:
3737:
3734:
3711:
3708:
3705:
3702:
3699:
3690:
3687:
3667:
3664:
3658:
3655:
3652:
3643:
3640:
3620:
3608:
3604:
3600:
3596:
3592:
3588:
3587:
3586:
3585:
3582:
3577:
3575:
3571:
3563:
3561:
3558:
3554:
3550:
3546:
3542:
3532:
3530:
3528:
3524:
3520:
3519:192.54.222.11
3516:
3507:
3503:
3499:
3480:
3477:
3474:
3441:
3422:
3409:
3406:
3398:
3377:
3373:
3353:
3350:
3340:
3336:
3332:
3329:
3326:
3322:
3311:
3307:
3277:
3273:
3269:
3266:
3263:
3260:
3256:
3252:
3246:
3241:
3237:
3231:
3227:
3219:
3202:
3198:
3192:
3188:
3160:
3155:
3143:
3139:
3079:
3065:
3028:
3023:
3011:
3007:
3003:
2993:
2980:
2954:
2943:
2933:
2930:
2915:
2889:
2876:
2866:
2833:
2823:
2809:
2787:
2753:
2738:
2719:
2716:
2713:
2702:
2701:
2698:
2694:
2690:
2686:
2685:
2680:
2676:
2672:
2668:
2664:
2663:
2662:
2658:
2654:
2650:
2647:
2643:
2632:
2629:
2625:
2621:
2605:
2599:
2594:
2590:
2580:
2575:
2571:
2563:
2560:
2546:
2542:
2538:
2532:
2529:
2526:
2520:
2517:
2514:
2511:
2504:
2503:
2501:
2485:
2476:
2472:
2469:
2466:
2463:
2460:
2457:
2450:
2449:
2447:
2444:
2443:
2442:
2441:
2438:
2430:
2427:
2421:
2418:
2411:
2408:
2402:
2399:
2393:
2387:
2381:
2375:
2372:
2369:
2366:
2360:
2354:
2351:
2344:
2340:
2336:
2332:
2328:
2324:
2323:
2322:
2321:
2318:
2315:
2310:
2309:
2306:
2302:
2298:
2293:
2292:
2291:
2288:
2284:
2280:
2276:
2272:
2261:
2253:
2249:
2245:
2241:
2231:
2222:
2221:
2219:
2218:
2217:
2216:
2213:
2209:
2205:
2200:
2197:
2181:
2176:
2172:
2168:
2165:
2158:
2157:
2155:
2130:
2129:
2127:
2126:
2125:
2124:
2121:
2108:
2101:
2098:
2097:
2096:
2093:
2067:
2049:
2030:
2024:
2023:
2022:
2019:
2012:
2008:
2004:
2000:
1996:
1995:
1994:
1993:
1985:
1983:
1967:
1961:
1954:
1950:
1930:
1907:
1904:
1899:
1895:
1890:
1866:
1851:
1837:
1833:
1811:
1807:
1803:
1800:
1797:
1794:
1791:
1788:
1766:
1761:
1746:
1738:
1735:
1732:
1729:
1726:
1723:
1720:
1687:
1684:
1678:
1668:
1655:
1615:
1601:
1597:
1594:
1591:
1582:
1567:
1559:
1556:
1553:
1550:
1547:
1544:
1541:
1535:
1530:
1522:
1519:
1516:
1513:
1510:
1504:
1499:
1491:
1488:
1485:
1482:
1479:
1476:
1470:
1465:
1457:
1454:
1451:
1448:
1445:
1439:
1436:
1433:
1427:
1424:
1418:
1413:
1405:
1402:
1399:
1396:
1393:
1390:
1384:
1379:
1375:
1371:
1358:
1345:
1337:
1333:
1326:
1323:
1320:
1317:
1314:
1311:
1308:
1295:
1291:
1287:
1286:84.72.103.163
1283:
1282:
1281:
1280:
1277:
1273:
1269:
1265:
1264:
1263:
1250:
1247:
1242:
1236:
1233:
1228:
1223:
1220:
1213:
1210:
1205:
1200:
1193:
1188:
1185:
1180:
1174:
1169:
1164:
1158:
1155:
1150:
1145:
1138:
1135:
1130:
1125:
1118:
1115:
1110:
1105:
1099:
1094:
1089:
1083:
1079:
1075:
1070:
1066:
1062:
1057:
1053:
1049:
1046:
1039:
1035:
1031:
1028:
1023:
1019:
1015:
1010:
1006:
1002:
995:
991:
987:
982:
978:
974:
971:
966:
962:
958:
952:
947:
944:
941:
933:
928:
924:
920:
916:
912:
908:
904:
900:
896:
892:
891:
890:
887:
886:Hello there,
881:
871:
868:
864:
863:
862:
861:
860:
859:
858:
857:
850:
846:
842:
838:
834:
830:
826:
822:
818:
814:
810:
792:
786:
781:
776:
769:
764:
759:
752:
747:
742:
736:
731:
728:
721:
720:
718:
714:
713:
712:
711:
710:
709:
704:
701:
696:
695:
694:
693:
690:
686:
682:
678:
677:
676:
675:
672:
664:
659:
655:
651:
647:
641:
640:
637:
633:
629:
624:
623:
622:
621:
618:
580:
556:
546:
532:
529:
522:
514:
511:
505:
500:
496:
492:
487:
483:
477:
472:
462:
458:
442:
438:
437:
436:
418:
410:
407:
401:
396:
392:
388:
383:
379:
373:
365:
361:
337:
334:
326:
323:
315:
312:
309:
306:
303:
295:
291:
267:
264:
258:
255:
249:
246:
243:
240:
237:
229:
225:
214:. I see that
199:
195:
182:
177:
175:
174:
170:
166:
157:
142:
138:
132:
129:
128:
125:
108:
104:
100:
99:
91:
85:
80:
78:
75:
71:
70:
66:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
6444:
6440:
6436:
6432:
6431:for any 1 ≤
6428:
6424:
6420:
6416:
6412:
6408:
6406:
6386:
6383:
6380:
6377:
6327:
6324:
6275:
6248:
6222:
6205:
6179:
6152:— Preceding
6148:
6128:
6106:
6103:
6078:source check
6057:
6051:
6038:
6034:
6030:
6028:
5991:
5988:
5965:— Preceding
5958:
5935:— Preceding
5930:
5914:VictorPorton
5911:
5891:
5275:
5272:
5268:
5264:
5262:
5236:
5232:
5224:
5216:
5212:
5205:QR algorithm
5178:— Preceding
5174:
5131:— Preceding
5127:
5104:— Preceding
5101:
5067:— Preceding
5064:
5055:
5044:
4982:
4771:
4650:
4627:— Preceding
4572:
4441:
4361:
4083:
4080:
4063:Jitse Niesen
4058:
4050:
4035:
4028:
4011:Jitse Niesen
3968:
3960:
3842:
3612:
3595:Jitse Niesen
3591:QR algorithm
3578:
3574:QR algorithm
3570:QR algorithm
3567:
3536:
3513:— Preceding
3510:
2689:Jitse Niesen
2653:Jitse Niesen
2627:
2623:
2434:
2428:
2425:
2419:
2415:
2409:
2406:
2400:
2397:
2391:
2385:
2379:
2373:
2370:
2367:
2364:
2358:
2352:
2349:
2335:Jitse Niesen
2330:
2326:
2297:Jitse Niesen
2265:
2204:Jitse Niesen
2112:
2094:
2092://---------
2091:
2020:
2016:
1999:Jitse Niesen
1989:
1583:
1300:
1268:Jitse Niesen
934:
931:
919:Jitse Niesen
914:
910:
906:
902:
898:
894:
888:
885:
841:Jitse Niesen
836:
832:
828:
824:
820:
816:
812:
681:Jitse Niesen
668:
644:— Preceding
628:Jitse Niesen
572:
446:
186:
161:
137:Mid-priority
136:
96:
62:Mid‑priority
40:WikiProjects
6443:columns of
6427:columns of
6419:columns of
6132:Michal Grňo
6045:Sourcecheck
5373:instead of
5089:JackSchmidt
5021:—Preceding
4549:—Preceding
4362:Why in the
3727:. Consider
3564:QR alorithm
2671:Mike Mannon
2637:instead of
2368:alfa=12.53
2359:a1=A(:,1)=
2095:Notation:
650:50.40.70.99
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
6465:Categories
6115:Report bug
5895:85.2.5.221
5827:(for some
4555:Milimetr88
4442:there is:
4031:GNU Octave
2667:netlib.org
6098:this tool
6091:this tool
5217:algorithm
3498:Rogueigor
2227:= I - ...
441:Dysprosia
6449:Simonedl
6166:contribs
6158:Astadnik
6154:unsigned
6104:Cheers.—
5967:unsigned
5949:contribs
5941:Alexshtf
5937:unsigned
5561:, where
5277:Wmmrmchl
5192:contribs
5180:unsigned
5133:unsigned
5106:unsigned
5069:unsigned
5023:unsigned
4629:unsigned
4563:contribs
4551:unsigned
3553:contribs
3541:unsigned
3515:unsigned
2283:contribs
2271:unsigned
2240:unsigned
2035:= sign(x
646:unsigned
617:rlhatton
549:Sign of
6031:checked
5998:my edit
5874:Aldebrn
5317:versus
5019:time?
3581:Fbianco
2437:Sina kh
2314:Sina kh
2275:Sina kh
2149:k:m,k:n
2137:k:m,k:n
2133:k:m,k:n
2104:k:m,k:n
2086:k:m,k:n
2074:k:m,k:n
2070:k:m,k:n
1923:, with
139:on the
6039:failed
5235:. The
5184:Boriaj
4573:No...
4089:Lavaka
4055:LAPACK
3545:Yongke
2392:q1*a=
2120:Jasper
2039:)||x||
1986:typo ?
819:where
165:Loisel
36:scale.
5059:: -->
5052:: -->
5048:: -->
3989:... Q
2058:/ ||v
2027:k:m,k
2025:x = A
917:? --
867:S. Jo
839:. --
811:then
716:this.
700:S. Jo
671:S. Jo
6453:talk
6411:has
6394:talk
6162:talk
6136:talk
6035:true
5975:talk
5945:talk
5918:talk
5899:talk
5878:talk
5847:and
5632:cond
5402:cond
5281:talk
5249:talk
5245:Qwfp
5203:The
5188:talk
5161:talk
5157:Qwfp
5141:talk
5114:talk
5093:talk
5077:talk
5031:talk
4637:talk
4559:talk
4348:talk
4093:talk
4067:talk
4038:Swap
4015:talk
4001:...Q
3973:...Q
3599:talk
3549:talk
3523:talk
3502:talk
3099:and
2703:The
2693:talk
2675:talk
2657:talk
2626:and
2555:with
2426:and
2407:and
2374:u1=
2339:talk
2301:talk
2279:talk
2248:talk
2208:talk
2139:- 2v
2076:- 2v
2003:talk
1688:7.48
1290:talk
1272:talk
923:talk
901:and
845:talk
685:talk
654:talk
632:talk
169:talk
6072:RfC
6049:).
6037:or
6022:to
6012:to
4980:).
3905:min
3839:But
3835:.
3796:min
3694:min
3647:min
3330:arg
3267:arg
2429:R=
2420:Q=
2410:R=
2401:Q=
2353:A=
2135:= A
2072:= A
2054:= v
2047:+ x
1982:).
1206:167
1131:175
1040:175
1024:175
1016:158
996:175
983:175
350:or
131:Mid
6467::
6455:)
6435:≤
6396:)
6168:)
6164:•
6138:)
6085:.
6080:}}
6076:{{
6047:}}
6043:{{
5977:)
5951:)
5947:•
5920:)
5901:)
5880:)
5706:−
5702:Γ
5683:λ
5670:Γ
5651:λ
5639:Γ
5570:λ
5533:−
5529:Γ
5510:λ
5503:Γ
5485:λ
5470:‖
5461:−
5448:‖
5439:‖
5425:‖
5325:Γ
5283:)
5251:)
5194:)
5190:•
5163:)
5143:)
5116:)
5095:)
5079:)
5033:)
4835:⋅
4707:⋅
4694:⋅
4688:−
4639:)
4609:−
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4561:•
4409:⋯
4350:)
4275:−
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4095:)
4069:)
4017:)
3923:−
3864:12
3814:−
3757:51
3754:−
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3579:--
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3333:
3293:‖
3285:‖
3270:
3261:−
3253:−
3232:∗
3228:α
3193:∗
3189:α
3144:∗
3140:α
3049:‖
3041:‖
3012:∗
3008:α
3004:−
2975:‖
2967:‖
2934:α
2931:−
2910:‖
2902:‖
2856:‖
2848:‖
2695:)
2677:)
2659:)
2595:∗
2585:¯
2576:∗
2561:ω
2547:∗
2533:ω
2521:−
2481:⊤
2467:−
2341:)
2303:)
2285:)
2281:•
2250:)
2210:)
2147:*A
2143:(v
2118:--
2084:*A
2080:(v
2062:||
2005:)
1962:14
1931:⊙
1905:⊙
1798:−
1767:14
1736:−
1721:−
1685:≈
1679:56
1557:−
1542:−
1511:14
1505:−
1489:−
1477:12
1440:∗
1437:14
1434:∗
1419:−
1403:−
1391:12
1372:∗
1346:∗
1318:−
1292:)
1274:)
1237:41
1234:−
1229:24
1221:−
1214:68
1211:−
1189:51
1186:−
1181:12
1159:35
1156:−
1139:70
1136:−
1119:14
1116:−
1111:21
1106:14
1095:⋅
1084:35
1076:33
1071:35
1047:−
1029:−
988:58
975:69
972:−
925:)
915:QQ
905:=
903:QQ
897:=
895:QR
847:)
827:=
817:QR
815:=
687:)
656:)
634:)
581:α
557:α
533:14
512:−
484:12
469:‖
455:‖
408:−
380:12
371:‖
358:‖
338:22
324:−
307:12
301:‖
288:‖
268:14
256:−
241:12
235:‖
222:‖
171:)
6451:(
6445:A
6441:k
6437:n
6433:k
6429:A
6425:k
6421:Q
6417:k
6413:n
6409:A
6392:(
6363:R
6360:Q
6339:R
6336:Q
6310:R
6307:Q
6287:R
6284:Q
6260:R
6257:Q
6234:R
6231:Q
6191:R
6188:Q
6160:(
6134:(
6117:)
6113:(
6100:.
6093:.
5973:(
5943:(
5916:(
5897:(
5876:(
5859:′
5856:B
5835:B
5814:′
5811:B
5807:=
5804:X
5801:)
5798:A
5793:T
5789:A
5785:(
5765:B
5762:=
5759:X
5754:1
5750:R
5727:2
5723:c
5719:=
5714:)
5711:2
5698:(
5693:x
5690:a
5687:m
5679:)
5674:2
5666:(
5661:x
5658:a
5655:m
5645:=
5642:)
5636:(
5611:M
5591:)
5588:M
5585:(
5580:x
5577:a
5574:m
5549:c
5546:=
5541:)
5536:1
5525:(
5520:x
5517:a
5514:m
5506:)
5500:(
5495:x
5492:a
5489:m
5479:=
5474:2
5464:1
5456:1
5452:R
5443:2
5433:1
5429:R
5422:=
5419:)
5414:1
5410:R
5406:(
5381:A
5361:G
5341:A
5336:T
5332:A
5328:=
5303:1
5299:R
5279:(
5247:(
5237:R
5233:Q
5186:(
5159:(
5139:(
5112:(
5091:(
5075:(
5029:(
5007:)
5002:3
4998:n
4994:(
4991:O
4968:)
4963:3
4959:n
4955:(
4952:O
4932:)
4927:2
4923:n
4919:(
4916:O
4896:)
4891:3
4887:n
4883:(
4880:O
4860:)
4855:2
4851:n
4847:(
4844:O
4841:=
4838:n
4832:)
4827:3
4823:n
4819:(
4816:O
4808:(
4796:)
4791:4
4787:n
4783:(
4780:O
4758:)
4753:3
4749:n
4745:(
4742:O
4720:T
4715:k
4711:v
4702:k
4698:v
4691:2
4683:k
4679:A
4675:=
4670:1
4667:+
4664:k
4660:A
4635:(
4612:1
4604:i
4600:Q
4596:=
4591:T
4586:i
4582:Q
4557:(
4532:i
4528:Q
4524:=
4519:T
4514:i
4510:Q
4487:i
4483:Q
4460:T
4455:i
4451:Q
4427:,
4422:T
4417:t
4413:Q
4404:T
4399:2
4395:Q
4389:T
4384:1
4380:Q
4376:=
4373:Q
4346:(
4322:2
4318:n
4314:m
4311:2
4289:)
4286:3
4282:/
4278:n
4272:m
4269:(
4264:2
4260:n
4256:2
4232:2
4228:n
4224:2
4202:)
4199:3
4195:/
4191:n
4185:m
4182:(
4177:2
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4147:)
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4140:/
4136:n
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4127:(
4122:2
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4114:2
4091:(
4065:(
4059:R
4051:R
4013:(
4003:m
3999:2
3997:Q
3995:1
3991:m
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3971:m
3932:0
3929:=
3926:1
3920:)
3917:1
3914:,
3911:2
3908:(
3902:=
3899:t
3877:)
3871:6
3858:(
3853:=
3850:A
3823:0
3820:=
3817:1
3811:)
3808:2
3805:,
3802:1
3799:(
3793:=
3790:t
3768:)
3762:4
3743:(
3738:=
3735:A
3715:)
3712:n
3709:,
3706:1
3700:m
3697:(
3691:=
3688:t
3668:1
3662:)
3659:n
3656:,
3653:m
3650:(
3644:=
3641:t
3621:t
3597:(
3559:.
3547:(
3521:(
3500:(
3484:)
3481:w
3478:+
3475:1
3472:(
3448:x
3442:H
3437:v
3429:v
3423:H
3418:x
3410:=
3407:w
3384:|
3378:k
3374:x
3369:|
3361:x
3351:=
3348:)
3341:k
3337:x
3327:i
3323:e
3318:|
3312:k
3308:x
3303:|
3299:(
3296:)
3289:x
3278:k
3274:x
3264:i
3257:e
3250:(
3247:=
3242:k
3238:x
3217::
3203:k
3199:x
3167:x
3161:T
3156:1
3151:e
3118:|
3113:u
3108:|
3086:x
3080:H
3075:x
3045:u
3035:x
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2994:H
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2981:=
2971:u
2961:x
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2951:)
2944:1
2939:e
2927:x
2922:(
2916:=
2906:u
2896:x
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2877:=
2873:x
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2863:)
2852:u
2843:u
2837:(
2834:=
2830:x
2824:H
2819:v
2794:v
2788:H
2783:x
2760:x
2754:H
2749:v
2723:)
2720:w
2717:+
2714:1
2711:(
2691:(
2673:(
2655:(
2628:x
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2606:.
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2581:x
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2564:=
2543:v
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2530:+
2527:1
2524:(
2518:I
2515:=
2512:Q
2486:,
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2458:Q
2337:(
2331:A
2327:A
2299:(
2289:.
2277:(
2254:.
2246:(
2234:n
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2206:(
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2169:=
2166:A
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2145:k
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2082:k
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2001:(
1968:2
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1838:/
1834:2
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1789:Q
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1758:|
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1747:T
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1739:2
1733:,
1730:3
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1724:1
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1598:u
1595:=
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1560:4
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1551:6
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1545:2
1539:(
1536:=
1531:T
1527:)
1523:0
1520:,
1517:0
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1508:(
1500:T
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1492:4
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1474:(
1471:=
1466:T
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1458:0
1455:,
1452:0
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1446:1
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1428:1
1425:+
1422:(
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1380:1
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1330:(
1327:n
1324:g
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1312:=
1309:u
1288:(
1270:(
1251:A
1248:=
1243:)
1224:4
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1194:4
1175:(
1170:=
1165:)
1151:0
1146:0
1126:0
1100:(
1090:)
1080:/
1067:/
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1011:7
1007:/
1003:3
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979:/
967:7
963:/
959:6
953:(
948:=
945:R
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921:(
911:Q
907:I
899:A
843:(
837:Q
833:A
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813:A
793:]
787:1
782:0
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737:[
732:=
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683:(
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473:2
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304:=
296:1
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265:=
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259:4
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238:=
230:1
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167:(
143:.
42::
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