14430:
16894:
13707:
16209:
15769:
14425:{\displaystyle \lambda _{1}-r(p(A)v_{1})=\lambda _{1}-{\frac {v_{1}^{*}\sum _{k=1}^{n}d_{k}p^{*}(\lambda _{k})\lambda _{k}p(\lambda _{k})z_{k}}{v_{1}^{*}\sum _{k=1}^{n}d_{k}p^{*}(\lambda _{k})p(\lambda _{k})z_{k}}}=\lambda _{1}-{\frac {\sum _{k=1}^{n}|d_{k}|^{2}\lambda _{k}p(\lambda _{k})^{*}p(\lambda _{k})}{\sum _{k=1}^{n}|d_{k}|^{2}p(\lambda _{k})^{*}p(\lambda _{k})}}={\frac {\sum _{k=1}^{n}|d_{k}|^{2}(\lambda _{1}-\lambda _{k})\left|p(\lambda _{k})\right|^{2}}{\sum _{k=1}^{n}|d_{k}|^{2}\left|p(\lambda _{k})\right|^{2}}}.}
16889:{\displaystyle {\begin{aligned}\lambda _{1}-\theta _{1}&\leqslant {\frac {(\lambda _{1}-\lambda _{n})\left(1-|d_{1}|^{2}\right)}{c_{m-1}(2\rho +1)^{2}|d_{1}|^{2}}}\\&={\frac {1-|d_{1}|^{2}}{|d_{1}|^{2}}}(\lambda _{1}-\lambda _{n}){\frac {1}{\cosh ^{2}((m-1)\operatorname {arcosh} (1+2\rho ))}}\\&={\frac {1-|d_{1}|^{2}}{|d_{1}|^{2}}}(\lambda _{1}-\lambda _{n}){\frac {4}{\left(R^{m-1}+R^{-(m-1)}\right)^{2}}}\\&\leqslant 4{\frac {1-|d_{1}|^{2}}{|d_{1}|^{2}}}(\lambda _{1}-\lambda _{n})R^{-2(m-1)}\end{aligned}}}
1664:
15174:
1405:
13401:
15764:{\displaystyle \lambda _{1}-\theta _{1}\leqslant \lambda _{1}-r(p(A)v_{1})={\frac {\sum _{k=2}^{n}|d_{k}|^{2}(\lambda _{1}-\lambda _{k})|p(\lambda _{k})|^{2}}{\sum _{k=1}^{n}|d_{k}|^{2}|p(\lambda _{k})|^{2}}}\leqslant {\frac {\sum _{k=2}^{n}|d_{k}|^{2}(\lambda _{1}-\lambda _{k})}{|d_{1}|^{2}|p(\lambda _{1})|^{2}}}\leqslant {\frac {(\lambda _{1}-\lambda _{n})\sum _{k=2}^{n}|d_{k}|^{2}}{|p(\lambda _{1})|^{2}|d_{1}|^{2}}}.}
1659:{\displaystyle T={\begin{pmatrix}\alpha _{1}&\beta _{2}&&&&0\\\beta _{2}&\alpha _{2}&\beta _{3}&&&\\&\beta _{3}&\alpha _{3}&\ddots &&\\&&\ddots &\ddots &\beta _{m-1}&\\&&&\beta _{m-1}&\alpha _{m-1}&\beta _{m}\\0&&&&\beta _{m}&\alpha _{m}\\\end{pmatrix}}}
12838:
16003:
17487:
9539:
185:, the reduced number of vectors (i.e. it should be selected to be approximately 1.5 times the number of accurate eigenvalues desired). Soon thereafter their work was followed by Paige, who also provided an error analysis. In 1988, Ojalvo produced a more detailed history of this algorithm and an efficient eigenvalue error test.
156:
In 1970, Ojalvo and Newman showed how to make the method numerically stable and applied it to the solution of very large engineering structures subjected to dynamic loading. This was achieved using a method for purifying the
Lanczos vectors (i.e. by repeatedly reorthogonalizing each newly generated
18062:
basis, and the eigenvalues/vectors solved are good approximations to those of the original matrix. However, in practice (as the calculations are performed in floating point arithmetic where inaccuracy is inevitable), the orthogonality is quickly lost and in some cases the new vector could even be
17985:
Stability means how much the algorithm will be affected (i.e. will it produce the approximate result close to the original one) if there are small numerical errors introduced and accumulated. Numerical stability is the central criterion for judging the usefulness of implementing an algorithm on a
3306:
have been favoured since the 1950s. During the 1960s the
Lanczos algorithm was disregarded. Interest in it was rejuvenated by the Kaniel–Paige convergence theory and the development of methods to prevent numerical instability, but the Lanczos algorithm remains the alternative algorithm that one
13396:{\displaystyle r(p(A)v_{1})={\frac {(p(A)v_{1})^{*}Ap(A)v_{1}}{(p(A)v_{1})^{*}p(A)v_{1}}}={\frac {v_{1}^{*}p(A)^{*}Ap(A)v_{1}}{v_{1}^{*}p(A)^{*}p(A)v_{1}}}={\frac {v_{1}^{*}p^{*}(A^{*})Ap(A)v_{1}}{v_{1}^{*}p^{*}(A^{*})p(A)v_{1}}}={\frac {v_{1}^{*}p^{*}(A)Ap(A)v_{1}}{v_{1}^{*}p^{*}(A)p(A)v_{1}}}}
7581:
can be computed, so nothing was lost by switching vectors. (Indeed, it turns out that the data collected here give significantly better approximations of the largest eigenvalue than one gets from an equal number of iterations in the power method, although that is not necessarily obvious at this
2692:
2364:
would make eigenvectors and eigenvalues apparent from inspection, the same is not true for the tridiagonalization performed by the
Lanczos algorithm; nontrivial additional steps are needed to compute even a single eigenvalue or eigenvector. Nonetheless, applying the Lanczos algorithm is often a
18085:
Variations on the
Lanczos algorithm exist where the vectors involved are tall, narrow matrices instead of vectors and the normalizing constants are small square matrices. These are called "block" Lanczos algorithms and can be much faster on computers with large numbers of registers and long
5863:
15780:
17306:
10676:
11366:
depleted of some eigencomponent will delay convergence to the corresponding eigenvalue, and even though this just comes out as a constant factor in the error bounds, depletion remains undesirable. One common technique for avoiding being consistently hit by it is to pick
9387:
8215:
14977:
11596:
The fact that the
Lanczos algorithm is coordinate-agnostic – operations only look at inner products of vectors, never at individual elements of vectors – makes it easy to construct examples with known eigenstructure to run the algorithm on: make
18094:. This has led into a number of other restarted variations such as restarted Lanczos bidiagonalization. Another successful restarted variation is the Thick-Restart Lanczos method, which has been implemented in a software package called TRLan.
7325:
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2514:
8388:
18063:
linearly dependent on the set that is already constructed. As a result, some of the eigenvalues of the resultant tridiagonal matrix may not be approximations to the original matrix. Therefore, the
Lanczos algorithm is not very stable.
16198:
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11004:
1804:
164:
In their original work, these authors also suggested how to select a starting vector (i.e. use a random-number generator to select each element of the starting vector) and suggested an empirically determined method for determining
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8792:
is also lower
Hessenberg, so it must in fact be tridiagional. Being Hermitian, its main diagonal is real, and since its first subdiagonal is real by construction, the same is true for its first superdiagonal. Therefore,
8963:
9230:
6011:
15998:{\displaystyle p(\lambda _{1})=c_{m-1}\left({\frac {2\lambda _{1}-\lambda _{2}-\lambda _{n}}{\lambda _{2}-\lambda _{n}}}\right)=c_{m-1}\left(2{\frac {\lambda _{1}-\lambda _{2}}{\lambda _{2}-\lambda _{n}}}+1\right);}
16214:
17684:
7962:
8747:
7429:
1329:
18089:
Many implementations of the
Lanczos algorithm restart after a certain number of iterations. One of the most influential restarted variations is the implicitly restarted Lanczos method, which is implemented in
17248:
13673:
4256:
6545:
5121:
17587:
4320:
17482:{\displaystyle {\frac {\lambda _{2}}{\lambda _{1}}}={\frac {\lambda _{2}}{\lambda _{2}+(\lambda _{1}-\lambda _{2})}}={\frac {1}{1+{\frac {\lambda _{1}-\lambda _{2}}{\lambda _{2}}}}}={\frac {1}{1+2\rho }}.}
5612:
12783:
11337:
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6173:
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The coefficients need not both be real, but the phase is of little importance. Nor need the composants for other eigenvectors have completely disappeared, but they shrink at least as fast as that for
3959:
566:
The
Lanczos iteration is prone to numerical instability. When executed in non-exact arithmetic, additional measures (as outlined in later sections) should be taken to ensure validity of the results.
5342:
17030:
17822:
9534:{\displaystyle {\begin{aligned}r(x_{1})&\leqslant r(x_{2})\leqslant \cdots \leqslant \lambda _{\max }\\r(y_{1})&\geqslant r(y_{2})\geqslant \cdots \geqslant \lambda _{\min }\end{aligned}}}
161:
previously generated ones) to any degree of accuracy, which when not performed, produced a series of vectors that were highly contaminated by those associated with the lowest natural frequencies.
6971:
3302:
Though the eigenproblem is often the motivation for applying the
Lanczos algorithm, the operation the algorithm primarily performs is tridiagonalization of a matrix, for which numerically stable
9392:
2519:
816:
12303:
12058:
10524:
9379:
1842:
There are in principle four ways to write the iteration procedure. Paige and other works show that the above order of operations is the most numerically stable. In practice the initial vector
18120:; since the set of people interested in large sparse matrices over finite fields and the set of people interested in large eigenvalue problems scarcely overlap, this is often also called the
18066:
Users of this algorithm must be able to find and remove those "spurious" eigenvalues. Practical implementations of the Lanczos algorithm go in three directions to fight this stability issue:
10156:
3294:
The combination of good performance for sparse matrices and the ability to compute several (without computing all) eigenvalues are the main reasons for choosing to use the Lanczos algorithm.
1231:
747:
11992:
7135:
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4915:
18153:
is the only large-scale linear operation. Since weighted-term text retrieval engines implement just this operation, the Lanczos algorithm can be applied efficiently to text documents (see
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609:
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11103:
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914:
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by construction is orthogonal to this subspace, this inner product must be zero. (This is essentially also the reason why sequences of orthogonal polynomials can always be given a
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A critique that can be raised against this method is that it is wasteful: it spends a lot of work (the matrix–vector products in step 2.1) extracting information from the matrix
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1400:
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to also be independent normally distributed stochastic variables from the same normal distribution (since the change of coordinates is unitary), and after rescaling the vector
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11926:
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1900:
859:
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469:
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2740:
For tridiagonal matrices, there exist a number of specialised algorithms, often with better computational complexity than general-purpose algorithms. For example, if
2288:
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2019:
18311:
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12810:
11642:
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11171:
10814:
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2114:
1964:
1867:
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3361:(and has no problem with it being known only implicitly), whereas raw Householder wants to modify the matrix during the computation (although that can be avoided).
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1932:
18151:
16105:
14459:
12682:
8244:
2084:
495:
290:
10362:
4067:, having each new iteration overwrite the results from the previous one. It may be desirable to instead keep all the intermediate results and organise the data.
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3198:
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3131:
3107:
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1984:
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18791:
Chen, HY; Atkinson, W.A.; Wortis, R. (July 2011). "Disorder-induced zero-bias anomaly in the Anderson-Hubbard model: Numerical and analytical calculations".
13439:
11761:
2086:; the Lanczos algorithm can be very fast for sparse matrices. Schemes for improving numerical stability are typically judged against this high performance.
16014:
1676:
2687:{\displaystyle {\begin{aligned}Ay&=AVx\\&=VTV^{*}Vx\\&=VTIx\\&=VTx\\&=V(\lambda x)\\&=\lambda Vx\\&=\lambda y.\end{aligned}}}
10816:
cannot converge slower than that of the power method, and will achieve more by approximating both eigenvalue extremes. For the subproblem of optimising
19015:
12785:. The polynomial we want will turn out to have real coefficients, but for the moment we should allow also for complex coefficients, and we will write
2317:, although some schemes for improving the numerical stability would need it later on. Sometimes the subsequent Lanczos vectors are recomputed from
18584:
Coakley, Ed S.; Rokhlin, Vladimir (2013). "A fast divide-and-conquer algorithm for computing the spectra of real symmetric tridiagonal matrices".
8874:
9181:
5926:
17879:, and performs like the power method would with an eigengap twice as large; a notable improvement. The more challenging case is however that of
18648:
17598:
7887:
8638:
7360:
3109:
is very sparse with all nonzero elements in highly predictable positions, it permits compact storage with excellent performance vis-à-vis
19136:
17135:
13576:
10919:
for this vector space. Thus we are again led to the problem of iteratively computing such a basis for the sequence of Krylov subspaces.
4188:
3157:
in general may have complex elements and eigenvectors, so real arithmetic is sufficient for finding the eigenvectors and eigenvalues of
7589:. The Lanczos algorithm then arises as the simplification one gets from eliminating calculation steps that turn out to be trivial when
17498:
5858:{\displaystyle Au_{j}=\|u_{j+1}'\|u_{j+1}=u_{j+1}'=w_{j+1}+\sum _{k=1}^{j}g_{k,j}v_{k}=\|w_{j+1}\|v_{j+1}+\sum _{k=1}^{j}g_{k,j}v_{k}}
19161:
4261:
1239:
10671:{\displaystyle r(x_{j})=\max _{z\in {\mathcal {L}}_{j}}r(z)\qquad {\text{and}}\qquad r(y_{j})=\min _{z\in {\mathcal {L}}_{j}}r(z).}
12687:
6445:
5021:
3011:
just as for the divide-and-conquer algorithm (though the constant factor may be different); since the eigenvectors together have
2866:
11269:
9731:
6175:, which is cancelled out by the orthogonalisation process. Thus the same basis for the chain of Krylov subspaces is computed by
6120:
5162:
19156:
19008:
18227:
2975:, are known to converge faster for tridiagonal matrices than for general matrices. Asymptotic complexity of tridiagonal QR is
18984:
18684:
12080:. The convergence for the Lanczos algorithm is often orders of magnitude faster than that for the power iteration algorithm.
18204:
contains several routines for the solution of large scale linear systems and eigenproblems which use the Lanczos algorithm.
16975:
17766:
6890:
3454:. Each factor is however determined by a single vector, so the storage requirements are the same for both algorithms, and
16972:, but since the power method primarily is sensitive to the quotient between absolute values of the eigenvalues, we need
12268:
12017:
10472:
9327:
4040:, but pays attention only to the very last result; implementations typically use the same variable for all the vectors
3582:
18242:
10106:
19182:
19115:
19001:
18931:
18892:
18623:
18568:
18535:
11951:
11518:
9821:
so the directions of interest are easy enough to compute in matrix arithmetic, but if one wishes to improve on both
3876:
19146:
18109:
7992:
5262:
755:
19073:
18766:
18421:
Ojalvo, I. U.; Newman, M. (1970). "Vibration modes of large structures by an automatic matrix-reduction method".
17996:
17911:
5347:
1176:
692:
149:. Although computationally efficient in principle, the method as initially formulated was not useful, due to its
17089:
10161:
14985:
8210:{\displaystyle h_{j-1,j}=(Av_{j-1})^{*}v_{j}={\overline {v_{j}^{*}Av_{j-1}}}={\overline {h_{j,j-1}}}=h_{j,j-1}}
7070:
6665:
6372:
4850:
572:
18384:"An iteration method for the solution of the eigenvalue problem of linear differential and integral operators"
14511:
11055:
10207:
959:
18866:
18234:
11553:
10870:
3698:
The power method for finding the eigenvalue of largest magnitude and a corresponding eigenvector of a matrix
3457:
18844:
Shimizu, Noritaka (21 October 2013). "Nuclear shell-model code for massive parallel computation, "KSHELL"".
18263:
library incorporates a large scale parallel implementation of the Lanczos algorithm (in C++) for multicore.
14645:
867:
19105:
18316:
18181:
15052:
14795:
but grows rapidly outside it. With some scaling of the argument, we can have it map all eigenvalues except
12684:; the coefficients of that polynomial are simply the coefficients in the linear combination of the vectors
11468:
10927:
When analysing the dynamics of the algorithm, it is convenient to take the eigenvalues and eigenvectors of
7006:
6553:
4509:
4430:
3387:
77:
18701:
14972:{\displaystyle p(x)=c_{m-1}\left({\frac {2x-\lambda _{2}-\lambda _{n}}{\lambda _{2}-\lambda _{n}}}\right)}
11176:
1070:
18669:
ARPACK Users Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods
11009:
10731:
9281:
9235:
8419:
7538:
7434:
6800:
5501:
4972:
4108:
10839:
10055:
4325:
2237:
is computed. Hence one may use the same storage for all three. Likewise, if only the tridiagonal matrix
1359:
19059:
18252:
A Matlab implementation of the Lanczos algorithm (note precision issues) is available as a part of the
12384:
12189:
9544:
The question then arises how to choose the subspaces so that these sequences converge at optimal rate.
3303:
11524:
9686:
9092:
9018:
8971:
4159:
3819:
926:
19110:
17692:
9600:
6620:
6279:
6232:
6068:
6016:
5429:
4920:
4757:
4710:
2794:
9992:
can be linearly independent vectors (indeed, are close to orthogonal), one cannot in general expect
9068:
19024:
18173:
18154:
17086:
to be the dominant one. Under that constraint, the case that most favours the power method is that
11231:
11108:
6334:
5223:
4812:
4610:
4101:. One way of stating that without introducing sets into the algorithm is to claim that it computes
3364:
Each iteration of the Lanczos algorithm produces another column of the final transformation matrix
2790:
1127:
643:
17062:
17035:
15105:
15025:
14798:
14484:
12142:
11904:
11850:
10316:
4479:(and in the case that there is such a dependence then one may continue the sequence by picking as
3763:
3240:
is of a manageable size will still allow finding the more extreme eigenvalues and eigenvectors of
2431:
1902:
and indicators of numerical imprecision being included as additional loop termination conditions.
1872:
826:
295:
Strictly speaking, the algorithm does not need access to the explicit matrix, but only a function
18260:
18103:
17882:
12589:{\displaystyle \operatorname {span} \left\{v_{1},Av_{1},A^{2}v_{1},\ldots ,A^{m-1}v_{1}\right\},}
12426:
12357:
12221:
12086:
11877:
11823:
11644:
has enough nonzero elements, the algorithm will output a general tridiagonal symmetric matrix as
7320:{\displaystyle v_{j+1}^{*}w_{j+1}'=v_{j+1}^{*}w_{j+1}=\|w_{j+1}\|v_{j+1}^{*}v_{j+1}=\|w_{j+1}\|.}
4667:
4363:
3727:
3588:
1809:
298:
20:
17950:
17827:
7842:{\displaystyle h_{k,j}=v_{k}^{*}w_{j+1}'=v_{k}^{*}Av_{j}=v_{k}^{*}A^{*}v_{j}=(Av_{k})^{*}v_{j}.}
3615:
2931:
324:
that computes the product of the matrix by an arbitrary vector. This function is called at most
18949:
12602:
11715:
11617:
a diagonal matrix with the desired eigenvalues on the diagonal; as long as the starting vector
8576:
8537:
8498:
3041:
Even algorithms whose convergence rates are unaffected by unitary transformations, such as the
2763:
2361:
2024:
524:
448:
409:
355:
223:
83:
18969:
18919:
17853:
17737:
8383:{\displaystyle h_{j,j}=(Av_{j})^{*}v_{j}={\overline {v_{j}^{*}Av_{j}}}={\overline {h_{j,j}}},}
7855:
5126:
3642:
3513:
3335:
2978:
2892:
2800:
2371:
19069:
18508:
Ojalvo, I. U. (1988). "Origins and advantages of Lanczos vectors for large dynamic systems".
18201:
12110:
come from the above interpretation of eigenvalues as extreme values of the Rayleigh quotient
10280:
10247:
9908:
8465:
7612:
7485:
6854:
6747:
2260:
2207:
34:
16922:
16193:{\displaystyle R=e^{\operatorname {arcosh} (1+2\rho )}=1+2\rho +2{\sqrt {\rho ^{2}+\rho }},}
10025:
9995:
9878:
6976:
3263:
2123:
1989:
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17256:
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14592:
13409:
12788:
11620:
11441:
11370:
11342:
11149:
11146:
It is also convenient to fix a notation for the coefficients of the initial Lanczos vector
10792:
10765:
10704:
10459:{\displaystyle {\mathcal {L}}_{j}=\operatorname {span} (x_{1},Ax_{1},\ldots ,A^{j-1}x_{1})}
9968:
9941:
9851:
9824:
9639:
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1937:
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1023:
18957:
IJCAI'01 Proceedings of the 17th International Joint Conference on Artificial Intelligence
18702:"Computing smallest singular triplets with implicitly restarted Lanczos bidiagonalization"
18481:
Paige, C. C. (1972). "Computational Variants of the Lanczos Method for the Eigenproblem".
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2463:
1908:
8:
19043:
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18185:
18135:
16952:
For comparison, one may consider how the convergence rate of the power method depends on
14438:
12661:
11395:
10999:{\displaystyle \lambda _{1}\geqslant \lambda _{2}\geqslant \dotsb \geqslant \lambda _{n}}
8223:
3384:, whereas an iteration of Householder produces another factor in a unitary factorisation
2063:
474:
269:
150:
18814:
18436:
11006:
be the eigenvalues (these are known to all be real, and thus possible to order) and let
3049:, may enjoy low-level performance benefits from being applied to the tridiagonal matrix
18845:
18826:
18800:
18177:
18076:
After the good and "spurious" eigenvalues are all identified, remove the spurious ones.
17283:
16902:
15151:
15131:
14708:
14625:
14574:
14554:
14464:
13681:
13560:{\displaystyle Av_{1}=A\sum _{k=1}^{n}d_{k}z_{k}=\sum _{k=1}^{n}d_{k}\lambda _{k}z_{k}}
12815:
12641:
12456:
12308:
12248:
12169:
12063:
11997:
11931:
11741:
11695:
11675:
11647:
11600:
11421:
11401:
10930:
10819:
10684:
10086:
9666:
9577:
9161:
9141:
9121:
9045:
8998:
8844:
8816:
8796:
8775:
8755:
8611:
8399:
7648:
7592:
7518:
6780:
6209:
5481:
4647:
4592:
4023:
3970:
3701:
3437:
3367:
3344:
3317:
3243:
3223:
3203:
3183:
3160:
3140:
3116:
3092:
3072:
3052:
2872:
2743:
2719:
2699:
2492:
2411:
2391:
2240:
1969:
1339:
1050:
617:
500:
403:
381:
327:
249:
203:
168:
132:
112:
59:
27:
18223:
14825:
14763:
14728:
11813:{\displaystyle \theta _{1}\geqslant \theta _{2}\geqslant \dots \geqslant \theta _{m}.}
10947:
as given, even though they are not explicitly known to the user. To fix notation, let
18980:
18945:
18927:
18830:
18680:
18619:
18564:
18531:
18463:
17592:
so the above bound for the Lanczos algorithm convergence rate should be compared to
8866:
8629:
7586:
4589:, since this sequence of vectors is by design meant to converge to an eigenvector of
3675:
scalar quantities computed that each depend on the previous quantity in the sequence.
3287:
3110:
3046:
49:
18112:
published an algorithm, based on the Lanczos algorithm, for finding elements of the
16081:{\displaystyle \rho ={\frac {\lambda _{1}-\lambda _{2}}{\lambda _{2}-\lambda _{n}}}}
4014:
approaches the normed eigenvector corresponding to the largest magnitude eigenvalue.
1799:{\displaystyle Av_{j}=w_{j}'=\beta _{j+1}v_{j+1}+\alpha _{j}v_{j}+\beta _{j}v_{j-1}}
19079:
18818:
18747:
18716:
18672:
18639:
18593:
18490:
18440:
18398:
8862:
198:
106:
45:
3290:
scheme for Hermitian matrices, that emphasises preserving the extreme eigenvalues.
19120:
18896:
18383:
18253:
18189:
18113:
14461:
term vanishes in the numerator, but not in the denominator. Thus if one can pick
4586:
4098:
3693:
1986:
is the average number of nonzero elements in a row. The total complexity is thus
53:
18720:
18644:"An Implicitly Restarted Lanczos Method for Large Symmetric Eigenvalue Problems"
19038:
18822:
18597:
18558:
18158:
18157:). Eigenvectors are also important for large-scale ranking methods such as the
14591:
has coefficients, this may seem a tall order, but one way to meet it is to use
917:
612:
518:
399:
18751:
18494:
19176:
19084:
18911:
18467:
18460:
The computation of eigenvalues and eigenvectors of very large sparse matrices
18162:
2352:
The Lanczos algorithm is most often brought up in the context of finding the
18676:
14508:
but small at all other eigenvalues, one will get a tight bound on the error
8958:{\displaystyle r(x)={\frac {x^{*}Ax}{x^{*}x}},\qquad x\in \mathbb {C} ^{n}.}
18993:
18423:
18245:
which is also a wrapper for the SSEUPD and DSEUPD functions functions from
9225:{\displaystyle {\mathcal {L}}_{1}\subset {\mathcal {L}}_{2}\subset \cdots }
6006:{\displaystyle u_{j}-v_{j}\in \operatorname {span} (v_{1},\dotsc ,v_{j-1})}
3042:
2972:
18889:
18267:
3684:
There are several lines of reasoning which lead to the Lanczos algorithm.
18554:
18403:
16099:) is thus of key importance for the convergence rate here. Also writing
3134:
2696:
Thus the Lanczos algorithm transforms the eigendecomposition problem for
2357:
1934:
arithmetical operations. The matrix–vector multiplication can be done in
18218:
built-in. Both stored and implicit matrices can be analyzed through the
18211:
12812:
for the polynomial obtained by complex conjugating all coefficients of
8220:
since the latter is real on account of being the norm of a vector. For
3334:
being a sparse matrix, whereas Householder does not, and will generate
2353:
18770:
18736:"Thick-Restart Lanczos Method for Large Symmetric Eigenvalue Problems"
17679:{\displaystyle \lambda _{1}-u^{*}Au=(\lambda _{1}-\lambda _{2})t^{2},}
7957:{\displaystyle Av_{k}\in \operatorname {span} (v_{1},\ldots ,v_{j-1})}
4613:, to instead produce an orthonormal basis of these Krylov subspaces.
4070:
One piece of information that trivially is available from the vectors
19100:
18941:
18132:
Lanczos algorithms are very attractive because the multiplication by
18735:
18699:
17129:, so consider that. Late in the power method, the iteration vector:
14435:
A key difference between numerator and denominator here is that the
8836:
8742:{\displaystyle H^{*}=\left(V^{*}AV\right)^{*}=V^{*}A^{*}V=V^{*}AV=H}
7424:{\displaystyle u_{j}\in \operatorname {span} (v_{1},\ldots ,v_{j}),}
18444:
18257:
18166:
17734:
for each iteration. The difference thus boils down to that between
16092:
11550:. This makes it possible to bound the probability that for example
9595:
1905:
Not counting the matrix–vector multiplication, each iteration does
18850:
18805:
18666:
18638:
17243:{\displaystyle u=(1-t^{2})^{1/2}z_{1}+tz_{2}\approx z_{1}+tz_{2},}
13668:{\displaystyle q(A)v_{1}=\sum _{k=1}^{n}d_{k}q(\lambda _{k})z_{k}}
4251:{\displaystyle Ax\in \operatorname {span} (v_{1},\dotsc ,v_{j+1})}
17973:
region is where the Lanczos algorithm convergence-wise makes the
10052:
to be parallel. It is not necessary to increase the dimension of
8841:
One way of characterising the eigenvectors of a Hermitian matrix
17582:{\displaystyle u^{*}Au=(1-t^{2})\lambda _{1}+t^{2}\lambda _{2},}
10313:
In other words, we can start with some arbitrary initial vector
19151:
19141:
18926:. Baltimore: Johns Hopkins University Press. pp. 470–507.
18246:
18215:
18207:
18091:
4315:{\displaystyle x\in \operatorname {span} (v_{1},\dotsc ,v_{j})}
1324:{\displaystyle w_{j}=w_{j}'-\alpha _{j}v_{j}-\beta _{j}v_{j-1}}
18528:
Lanczos Algorithms for Large Symmetric Eigenvalue Computations
3549:
Householder is numerically stable, whereas raw Lanczos is not.
2928:
The Fast Multipole Method can compute all eigenvalues in just
2365:
significant step forward in computing the eigendecomposition.
18238:
18117:
14642:
Chebyshev polynomial of the first kind (that which satisfies
11394:
by first drawing the elements randomly according to the same
18790:
12778:{\displaystyle v_{1},Av_{1},A^{2}v_{1},\ldots ,A^{m-1}v_{1}}
6540:{\displaystyle w_{j+1}=w_{j+1}'-\sum _{k=1}^{j}h_{k,j}v_{k}}
5116:{\displaystyle w_{j+1}=u_{j+1}'-\sum _{k=1}^{j}g_{k,j}v_{k}}
4609:. To avoid that, one can combine the power iteration with a
3137:
matrix with all eigenvectors and eigenvalues real, whereas
18864:
12832:. In this parametrisation of the Krylov subspace, we have
11820:
By convergence is primarily understood the convergence of
11332:{\displaystyle \textstyle v_{1}=\sum _{k=1}^{n}d_{k}z_{k}}
9811:{\displaystyle \nabla r(x)={\frac {2}{x^{*}x}}(Ax-r(x)x),}
9574:, the optimal direction in which to seek larger values of
6168:{\displaystyle \operatorname {span} (v_{1},\dotsc ,v_{j})}
5210:{\displaystyle \operatorname {span} (v_{1},\dotsc ,v_{j})}
18510:
Proc. 6th Modal Analysis Conference (IMAC), Kissimmee, FL
9663:
the optimal direction in which to seek smaller values of
5869:
Here it may be observed that we do not actually need the
15022:, use instead the largest eigenvalue strictly less than
9875:
then there are two new directions to take into account:
2971:
Some general eigendecomposition algorithms, notably the
2869:
can be used to compute the entire eigendecomposition of
1869:
may be taken as another argument of the procedure, with
18391:
Journal of Research of the National Bureau of Standards
18073:
Recover the orthogonality after the basis is generated.
11738:
real symmetric matrix, that similarly to the above has
11438:. Prior to the rescaling, this causes the coefficients
76:"most useful" (tending towards extreme highest/lowest)
17989:
For the Lanczos algorithm, it can be proved that with
17025:{\displaystyle |\lambda _{n}|\leqslant |\lambda _{2}|}
11273:
3639:). Householder is less parallel, having a sequence of
1420:
18764:
18733:
18618:(3. ed.). Baltimore: Johns Hopkins Univ. Press.
18319:
18292:
18138:
17999:
17953:
17914:
17885:
17856:
17830:
17817:{\displaystyle R=1+2\rho +2{\sqrt {\rho ^{2}+\rho }}}
17769:
17740:
17695:
17601:
17501:
17309:
17286:
17259:
17138:
17092:
17065:
17038:
16978:
16958:
16925:
16905:
16212:
16108:
16017:
15783:
15177:
15154:
15134:
15108:
15055:
15028:
14988:
14866:
14828:
14801:
14766:
14731:
14711:
14648:
14628:
14601:
14577:
14557:
14514:
14487:
14467:
14441:
13710:
13684:
13579:
13442:
13412:
12841:
12818:
12791:
12690:
12664:
12644:
12605:
12482:
12459:
12429:
12387:
12360:
12331:
12311:
12271:
12251:
12224:
12192:
12172:
12145:
12116:
12089:
12066:
12020:
12000:
11954:
11948:
grows, and secondarily the convergence of some range
11934:
11907:
11880:
11853:
11826:
11764:
11744:
11718:
11698:
11678:
11650:
11623:
11603:
11556:
11527:
11471:
11444:
11424:
11404:
11373:
11345:
11272:
11234:
11179:
11152:
11111:
11058:
11012:
10953:
10933:
10873:
10842:
10822:
10795:
10768:
10734:
10707:
10687:
10535:
10475:
10365:
10319:
10283:
10250:
10210:
10164:
10109:
10089:
10058:
10028:
9998:
9971:
9944:
9911:
9881:
9854:
9827:
9734:
9689:
9669:
9642:
9603:
9580:
9553:
9390:
9330:
9284:
9238:
9184:
9164:
9144:
9124:
9095:
9071:
9048:
9021:
9001:
8974:
8877:
8847:
8819:
8799:
8778:
8758:
8641:
8614:
8579:
8540:
8501:
8468:
8422:
8402:
8255:
8226:
8036:
8001:
7970:
7890:
7858:
7674:
7651:
7615:
7595:
7541:
7521:
7488:
7437:
7363:
7336:
7146:
7073:
7009:
6979:
6893:
6857:
6803:
6783:
6750:
6668:
6623:
6556:
6448:
6375:
6337:
6282:
6235:
6212:
6185:
6123:
6071:
6019:
5929:
5902:
5875:
5615:
5585:
5558:
5504:
5484:
5432:
5350:
5265:
5226:
5165:
5129:
5024:
4975:
4923:
4853:
4815:
4760:
4713:
4670:
4650:
4623:
4595:
4564:
4512:
4485:
4433:
4406:
4366:
4328:
4264:
4191:
4162:
4111:
4076:
4046:
4026:
3993:
3973:
3879:
3822:
3792:
3766:
3730:
3704:
3645:
3618:
3591:
3558:
3516:
3460:
3440:
3390:
3370:
3347:
3320:
3266:
3246:
3226:
3206:
3186:
3163:
3143:
3119:
3095:
3075:
3055:
3017:
2981:
2934:
2895:
2875:
2839:
2803:
2766:
2746:
2722:
2702:
2517:
2495:
2466:
2434:
2414:
2394:
2374:
2323:
2296:
2263:
2243:
2210:
2183:
2156:
2126:
2095:
2066:
2027:
1992:
1972:
1940:
1911:
1875:
1848:
1812:
1679:
1408:
1362:
1342:
1242:
1179:
1130:
1073:
1053:
1026:
962:
929:
870:
829:
758:
695:
646:
620:
575:
527:
503:
477:
451:
412:
384:
358:
330:
301:
272:
252:
226:
206:
171:
135:
115:
86:
62:
17253:
where each new iteration effectively multiplies the
6966:{\displaystyle Av_{j}=\sum _{k=1}^{j+1}h_{k,j}v_{k}}
3310:
Aspects in which the two algorithms differ include:
3297:
18249:which use the Implicitly Restarted Lanczos Method.
17947:is an even larger improvement on the eigengap; the
16899:The convergence rate is thus controlled chiefly by
18354:
18305:
18270:library also implements a Lanczos-like algorithm.
18145:
18050:
17965:
17939:
17900:
17871:
17842:
17816:
17755:
17726:
17678:
17581:
17481:
17292:
17272:
17242:
17121:
17078:
17051:
17024:
16964:
16941:
16911:
16888:
16192:
16080:
15997:
15763:
15160:
15140:
15120:
15094:
15041:
15014:
14971:
14849:
14814:
14787:
14752:
14717:
14697:
14634:
14614:
14583:
14563:
14540:
14500:
14473:
14453:
14424:
13690:
13667:
13559:
13425:
13395:
12824:
12804:
12777:
12676:
12650:
12630:
12588:
12465:
12442:
12415:
12373:
12346:
12317:
12297:
12257:
12237:
12210:
12178:
12158:
12131:
12102:
12072:
12052:
12006:
11986:
11940:
11920:
11893:
11866:
11839:
11812:
11750:
11730:
11704:
11684:
11667:
11656:
11636:
11609:
11585:
11542:
11509:
11457:
11430:
11410:
11386:
11358:
11331:
11258:
11220:
11165:
11135:
11097:
11044:
10998:
10939:
10911:
10859:
10828:
10808:
10781:
10754:
10720:
10693:
10670:
10518:
10458:
10348:
10302:
10269:
10236:
10196:
10150:
10095:
10075:
10044:
10014:
9984:
9957:
9930:
9897:
9867:
9840:
9810:
9717:
9675:
9655:
9628:
9586:
9566:
9533:
9373:
9316:
9270:
9224:
9170:
9150:
9130:
9110:
9081:
9054:
9034:
9007:
8987:
8957:
8853:
8825:
8805:
8784:
8764:
8741:
8620:
8600:
8565:
8526:
8487:
8454:
8408:
8382:
8238:
8209:
8019:
7983:
7956:
7876:
7841:
7657:
7634:
7601:
7573:
7527:
7507:
7474:
7423:
7349:
7319:
7129:
7059:
6991:
6965:
6876:
6835:
6789:
6769:
6733:
6654:
6606:
6539:
6431:
6361:
6320:
6265:
6218:
6198:
6167:
6109:
6057:
6005:
5915:
5888:
5857:
5598:
5571:
5536:
5490:
5470:
5415:
5336:
5251:
5209:
5151:
5115:
5007:
4961:
4909:
4839:
4798:
4743:
4696:
4656:
4636:
4601:
4577:
4550:
4498:
4471:
4419:
4392:
4349:
4314:
4250:
4177:
4148:
4089:
4059:
4032:
4006:
3979:
3953:
3863:
3805:
3778:
3749:
3710:
3667:
3631:
3604:
3573:
3538:
3502:
3446:
3426:
3376:
3353:
3341:Lanczos works throughout with the original matrix
3326:
3278:
3252:
3232:
3212:
3192:
3169:
3149:
3125:
3101:
3081:
3061:
3030:
3003:
2958:
2917:
2881:
2854:
2825:
2778:
2752:
2728:
2708:
2686:
2501:
2481:
2452:
2420:
2400:
2380:
2347:
2336:
2309:
2282:
2249:
2229:
2196:
2169:
2142:
2108:
2078:
2052:
2013:
1978:
1958:
1926:
1894:
1861:
1830:
1798:
1658:
1394:
1348:
1323:
1225:
1162:
1111:
1059:
1039:
1009:
948:
908:
853:
810:
741:
678:
626:
603:
552:
509:
489:
463:
437:
390:
370:
336:
316:
284:
258:
238:
212:
177:
141:
121:
98:
68:
18700:E. Kokiopoulou; C. Bekas; E. Gallopoulos (2004).
14725:), we have a polynomial which stays in the range
12298:{\displaystyle \lambda _{1}\geqslant \theta _{1}}
12053:{\displaystyle \lambda _{1},\ldots ,\lambda _{k}}
10922:
10519:{\displaystyle x_{j},y_{j}\in {\mathcal {L}}_{j}}
9374:{\displaystyle x_{j},y_{j}\in {\mathcal {L}}_{j}}
8837:Simultaneous approximation of extreme eigenvalues
7357:that were eliminated from this recursion satisfy
7067:may seem a bit odd, but fits the general pattern
5552:The relation between the power iteration vectors
3286:region, the Lanczos algorithm can be viewed as a
33:For the approximation of the gamma function, see
19174:
18740:SIAM Journal on Matrix Analysis and Applications
18525:
18097:
17492:The estimate of the largest eigenvalue is then
13433:as a linear combination of eigenvectors, we get
11052:be an orthonormal set of eigenvectors such that
10867:, it is convenient to have an orthonormal basis
10627:
10559:
10151:{\displaystyle \{{\mathcal {L}}_{j}\}_{j=1}^{m}}
9522:
9454:
9027:
8980:
3687:
2257:is sought, then the raw iteration does not need
18667:R. B. Lehoucq; D. C. Sorensen; C. Yang (1998).
18560:Numerical Methods for Large Eigenvalue Problems
12325:in that Krylov subspace provides a lower bound
12265:-dimensional Krylov subspace, we trivially get
11987:{\displaystyle \theta _{1},\ldots ,\theta _{k}}
10158:are taken to be Krylov subspaces, because then
3307:tries only if Householder is not satisfactory.
129:is often but not necessarily much smaller than
18609:
18607:
18583:
3954:{\displaystyle u_{j+1}=u_{j+1}'/\|u_{j+1}'\|.}
3679:
19009:
18940:
18910:
18649:Electronic Transactions on Numerical Analysis
18614:Golub, Gene H.; Van Loan, Charles F. (1996).
18613:
12423:is small then this provides a tight bound on
5337:{\displaystyle u_{j+1}=u_{j+1}'/\|u_{j+1}'\|}
811:{\displaystyle w_{1}=w_{1}'-\alpha _{1}v_{1}}
19023:
18975:. In E. Pavarini; E. Koch; S. Zhang (eds.).
18420:
10906:
10874:
10762:it follows that an iteration to produce the
10128:
10110:
8495:can be identified as elements of the matrix
7452:
7438:
7311:
7292:
7249:
7230:
7054:
7035:
6601:
6582:
5786:
5767:
5654:
5632:
5410:
5391:
5331:
5309:
4585:vectors is however likely to be numerically
4506:an arbitrary vector linearly independent of
4126:
4112:
3945:
3923:
2833:operations, and evaluating it at a point in
903:
884:
18950:"Link Analysis, Eigenvectors and Stability"
18604:
18586:Applied and Computational Harmonic Analysis
18553:
18051:{\displaystyle v_{1},v_{2},\cdots ,v_{m+1}}
17940:{\displaystyle R\approx 1+2{\sqrt {\rho }}}
12381:, so if a point can be exhibited for which
5416:{\displaystyle v_{j+1}=w_{j+1}/\|w_{j+1}\|}
1226:{\displaystyle \alpha _{j}=w_{j}'^{*}v_{j}}
742:{\displaystyle \alpha _{1}=w_{1}'^{*}v_{1}}
19016:
19002:
18970:"Exact Diagonalization and Lanczos Method"
18549:
18547:
18254:Gaussian Belief Propagation Matlab Package
17122:{\displaystyle \lambda _{n}=-\lambda _{2}}
11692:iteration steps of the Lanczos algorithm,
10197:{\displaystyle Az\in {\mathcal {L}}_{j+1}}
19:For the null space-finding algorithm, see
18967:
18849:
18804:
18402:
18142:
15015:{\displaystyle \lambda _{2}=\lambda _{1}}
12599:so any element of it can be expressed as
12195:
11530:
9178:. Repeating that for an increasing chain
9118:it can be feasible to locate the maximum
9098:
8942:
7130:{\displaystyle h_{k,j}=v_{k}^{*}w_{j+1}'}
6734:{\displaystyle v_{j+1}=w_{j+1}/h_{j+1,j}}
6432:{\displaystyle h_{k,j}=v_{k}^{*}w_{j+1}'}
4910:{\displaystyle g_{k,j}=v_{k}^{*}u_{j+1}'}
4165:
3038:elements, this is asymptotically optimal.
604:{\displaystyle v_{1}\in \mathbb {C} ^{n}}
591:
18521:
18519:
18124:without causing unreasonable confusion.
14541:{\displaystyle \lambda _{1}-\theta _{1}}
11098:{\displaystyle Az_{k}=\lambda _{k}z_{k}}
10237:{\displaystyle z\in {\mathcal {L}}_{j},}
8833:of the Lanczos algorithm specification.
2716:into the eigendecomposition problem for
1047:an arbitrary vector with Euclidean norm
1010:{\displaystyle v_{j}=w_{j-1}/\beta _{j}}
246:, and optionally a number of iterations
19147:Basic Linear Algebra Subprograms (BLAS)
18843:
18544:
18381:
16919:, since this bound shrinks by a factor
11586:{\displaystyle |d_{1}|<\varepsilon }
10912:{\displaystyle \{v_{1},\ldots ,v_{j}\}}
8813:is a real, symmetric matrix—the matrix
7609:is Hermitian—in particular most of the
3503:{\displaystyle V=Q_{1}Q_{2}\dots Q_{n}}
2867:divide-and-conquer eigenvalue algorithm
19175:
18507:
17980:
14698:{\displaystyle c_{k}(\cos x)=\cos(kx)}
8968:In particular, the largest eigenvalue
6777:an arbitrary vector of Euclidean norm
5478:an arbitrary vector of Euclidean norm
3552:Lanczos is highly parallel, with only
909:{\displaystyle \beta _{j}=\|w_{j-1}\|}
18997:
18516:
18480:
18457:
18355:{\displaystyle u\approx z_{1}+tz_{2}}
15095:{\displaystyle |p(\lambda _{k})|^{2}}
11510:{\displaystyle (d_{1},\dotsc ,d_{n})}
11173:with respect to this eigenbasis; let
7060:{\displaystyle h_{j+1,j}=\|w_{j+1}\|}
6607:{\displaystyle h_{j+1,j}=\|w_{j+1}\|}
6013:and therefore the difference between
4551:{\displaystyle u_{1},\dotsc ,u_{j-1}}
4472:{\displaystyle u_{1},\dotsc ,u_{j-1}}
3427:{\displaystyle Q_{1}Q_{2}\dots Q_{n}}
2360:of a matrix, but whereas an ordinary
18977:Many-Body Methods for Real Materials
18642:; L. Reichel; D.C. Sorensen (1994).
18416:
18414:
18182:strongly correlated electron systems
18172:Lanczos algorithms are also used in
11874:(and the symmetrical convergence of
11418:and then rescale the vector to norm
11221:{\displaystyle d_{k}=z_{k}^{*}v_{1}}
7330:Because the power iteration vectors
1112:{\displaystyle v_{1},\dots ,v_{j-1}}
637:Abbreviated initial iteration step:
16095:to the diameter of the rest of the
11045:{\displaystyle z_{1},\dotsc ,z_{n}}
10755:{\displaystyle {\mathcal {L}}_{j},}
9317:{\displaystyle y_{1},y_{2},\dotsc }
9271:{\displaystyle x_{1},x_{2},\ldots }
9232:produces two sequences of vectors:
8455:{\displaystyle v_{1},\ldots ,v_{m}}
7574:{\displaystyle u_{1},\ldots ,u_{m}}
7475:{\displaystyle \{v_{j}\}_{j=1}^{m}}
6836:{\displaystyle v_{1},\dotsc ,v_{j}}
5537:{\displaystyle v_{1},\dotsc ,v_{j}}
5008:{\displaystyle v_{1},\dotsc ,v_{j}}
4149:{\displaystyle \{v_{j}\}_{j=1}^{m}}
2786:tridiagonal symmetric matrix then:
13:
18904:
18195:
18070:Prevent the loss of orthogonality,
10860:{\displaystyle {\mathcal {L}}_{j}}
10846:
10738:
10640:
10572:
10505:
10369:
10220:
10177:
10116:
10076:{\displaystyle {\mathcal {L}}_{j}}
10062:
9735:
9693:
9604:
9360:
9205:
9188:
9074:
9065:Within a low-dimensional subspace
7642:coefficients turn out to be zero.
4969:with respect to the basis vectors
4350:{\displaystyle 1\leqslant j<m;}
2489:is a corresponding eigenvector of
1395:{\displaystyle v_{1},\dots ,v_{m}}
26:For the interpolation method, see
14:
19194:
18411:
17977:improvement on the power method.
12416:{\displaystyle \lambda _{1}-r(x)}
12211:{\displaystyle \mathbb {C} ^{n},}
9683:is that of the negative gradient
3298:Application to tridiagonalization
18865:The Numerical Algorithms Group.
18765:Kesheng Wu; Horst Simon (2001).
18734:Kesheng Wu; Horst Simon (2000).
17850:region, the latter is more like
11543:{\displaystyle \mathbb {C} ^{n}}
9718:{\displaystyle -\nabla r(y_{j})}
9111:{\displaystyle \mathbb {C} ^{n}}
9035:{\displaystyle \lambda _{\min }}
8988:{\displaystyle \lambda _{\max }}
8772:is Hermitian. This implies that
7515:contain enough information from
4917:. (These are the coordinates of
4178:{\displaystyle \mathbb {C} ^{n}}
3864:{\displaystyle u_{j+1}'=Au_{j}.}
3069:rather than the original matrix
949:{\displaystyle \beta _{j}\neq 0}
188:
16:Numerical eigenvalue calculation
18883:
18858:
18837:
18784:
18758:
18727:
18693:
18660:
18632:
18280:
18127:
17727:{\displaystyle (1+2\rho )^{-2}}
14571:has many more eigenvalues than
11668:Kaniel–Paige convergence theory
10603:
10597:
9629:{\displaystyle \nabla r(x_{j})}
8933:
6655:{\displaystyle h_{j+1,j}\neq 0}
6321:{\displaystyle w_{j+1}'=Av_{j}}
6266:{\displaystyle j=1,\dotsc ,m-1}
6110:{\displaystyle w_{j+1}'=Av_{j}}
6058:{\displaystyle u_{j+1}'=Au_{j}}
5471:{\displaystyle u_{j+1}=v_{j+1}}
4962:{\displaystyle Au_{j}=u_{j+1}'}
4799:{\displaystyle u_{j+1}'=Au_{j}}
4744:{\displaystyle j=1,\dotsc ,m-1}
4360:this is trivially satisfied by
2348:Application to the eigenproblem
18577:
18501:
18474:
18462:(Ph.D. thesis). U. of London.
18451:
18375:
18116:of a large sparse matrix over
17712:
17696:
17660:
17634:
17540:
17521:
17389:
17363:
17165:
17145:
17018:
17003:
16995:
16980:
16877:
16865:
16851:
16825:
16812:
16796:
16783:
16767:
16726:
16714:
16673:
16647:
16634:
16618:
16605:
16589:
16563:
16560:
16545:
16536:
16524:
16521:
16499:
16473:
16460:
16444:
16431:
16415:
16382:
16366:
16356:
16340:
16307:
16291:
16276:
16250:
16141:
16126:
16091:(i.e., the ratio of the first
15800:
15787:
15745:
15729:
15718:
15713:
15700:
15693:
15680:
15664:
15639:
15613:
15594:
15589:
15576:
15569:
15558:
15542:
15536:
15510:
15500:
15484:
15443:
15438:
15425:
15418:
15407:
15391:
15357:
15352:
15339:
15332:
15328:
15302:
15292:
15276:
15245:
15232:
15226:
15220:
15082:
15077:
15064:
15057:
14876:
14870:
14844:
14829:
14782:
14767:
14747:
14732:
14692:
14683:
14671:
14659:
14402:
14389:
14370:
14354:
14316:
14303:
14291:
14265:
14255:
14239:
14205:
14192:
14180:
14166:
14153:
14137:
14110:
14097:
14085:
14071:
14048:
14032:
13975:
13962:
13956:
13943:
13872:
13859:
13843:
13830:
13752:
13739:
13733:
13727:
13652:
13639:
13589:
13583:
13377:
13371:
13365:
13359:
13319:
13313:
13304:
13298:
13251:
13245:
13239:
13226:
13186:
13180:
13171:
13158:
13111:
13105:
13093:
13086:
13053:
13047:
13032:
13025:
12985:
12979:
12967:
12953:
12947:
12941:
12926:
12920:
12905:
12891:
12885:
12879:
12870:
12857:
12851:
12845:
12615:
12609:
12410:
12404:
12341:
12335:
12126:
12120:
11573:
11558:
11504:
11472:
10923:Convergence and other dynamics
10662:
10656:
10620:
10607:
10594:
10588:
10552:
10539:
10453:
10389:
9802:
9796:
9790:
9775:
9747:
9741:
9712:
9699:
9623:
9610:
9505:
9492:
9479:
9466:
9437:
9424:
9411:
9398:
9082:{\displaystyle {\mathcal {L}}}
8887:
8881:
8292:
8275:
8085:
8062:
7993:three-term recurrence relation
7951:
7913:
7817:
7800:
7415:
7383:
6162:
6130:
6000:
5962:
5204:
5172:
4309:
4277:
4245:
4207:
3662:
3649:
3568:
3562:
3533:
3520:
2998:
2985:
2953:
2938:
2912:
2899:
2849:
2843:
2820:
2807:
2639:
2630:
2047:
2031:
2008:
1996:
1966:arithmetical operations where
1953:
1944:
1921:
1915:
305:
1:
18369:
18098:Nullspace over a finite field
18080:
17689:which shrinks by a factor of
15049:), then the maximal value of
13406:Using now the expression for
11259:{\displaystyle k=1,\dotsc ,n}
11136:{\displaystyle k=1,\dotsc ,n}
6797:that is orthogonal to all of
6362:{\displaystyle k=1,\dotsc ,j}
5498:that is orthogonal to all of
5252:{\displaystyle w_{j+1}\neq 0}
4840:{\displaystyle k=1,\dotsc ,j}
3688:A more provident power method
3200:is very large, then reducing
1163:{\displaystyle w_{j}'=Av_{j}}
1067:that is orthogonal to all of
679:{\displaystyle w_{1}'=Av_{1}}
17079:{\displaystyle \lambda _{2}}
17052:{\displaystyle \lambda _{1}}
15121:{\displaystyle k\geqslant 2}
15042:{\displaystyle \lambda _{1}}
14815:{\displaystyle \lambda _{1}}
14501:{\displaystyle \lambda _{1}}
12245:is merely the maximum on an
12159:{\displaystyle \lambda _{1}}
11921:{\displaystyle \lambda _{n}}
11867:{\displaystyle \lambda _{1}}
10356:construct the vector spaces
10349:{\displaystyle x_{1}=y_{1},}
10244:thus in particular for both
9015:and the smallest eigenvalue
8372:
8346:
8177:
8145:
3779:{\displaystyle j\geqslant 1}
2453:{\displaystyle Tx=\lambda x}
2177:is computed, and the vector
1895:{\displaystyle \beta _{j}=0}
854:{\displaystyle j=2,\dots ,m}
611:be an arbitrary vector with
78:eigenvalues and eigenvectors
7:
18871:NAG Library Manual, Mark 23
18721:10.1016/j.apnum.2003.11.011
18526:Cullum; Willoughby (1985).
17901:{\displaystyle \rho \ll 1,}
12443:{\displaystyle \theta _{1}}
12374:{\displaystyle \theta _{1}}
12238:{\displaystyle \theta _{1}}
12166:is a priori the maximum of
12103:{\displaystyle \theta _{1}}
11894:{\displaystyle \theta _{m}}
11840:{\displaystyle \theta _{1}}
8416:is the matrix with columns
7585:This last procedure is the
5579:and the orthogonal vectors
5123:. (Cancel the component of
4697:{\displaystyle v_{1}=u_{1}}
4427:is linearly independent of
4393:{\displaystyle v_{j}=u_{j}}
3750:{\displaystyle u_{1}\neq 0}
3680:Derivation of the algorithm
3605:{\displaystyle \alpha _{j}}
3314:Lanczos takes advantage of
3304:Householder transformations
2362:diagonalization of a matrix
1831:{\displaystyle 2<j<m}
1356:be the matrix with columns
317:{\displaystyle v\mapsto Av}
10:
19199:
19060:System of linear equations
18823:10.1103/PhysRevB.84.045113
18598:10.1016/j.acha.2012.06.003
18169:algorithm used by Google.
18101:
17966:{\displaystyle \rho \gg 1}
17843:{\displaystyle \rho \gg 1}
16949:for each extra iteration.
8393:meaning this is real too.
6851:A priori the coefficients
4558:). A basis containing the
3691:
3632:{\displaystyle \beta _{j}}
2959:{\displaystyle O(m\log m)}
2509:with the same eigenvalue:
32:
25:
18:
19129:
19111:Cache-oblivious algorithm
19093:
19052:
19031:
18752:10.1137/S0895479898334605
18362:describes the worst case.
18243:scipy.sparse.linalg.eigsh
17032:for the eigengap between
15148:and the minimal value is
12631:{\displaystyle p(A)v_{1}}
11731:{\displaystyle m\times m}
9042:is the global minimum of
8995:is the global maximum of
8601:{\displaystyle k>j+1;}
8566:{\displaystyle h_{k,j}=0}
8527:{\displaystyle H=V^{*}AV}
5896:vectors to compute these
2795:characteristic polynomial
2779:{\displaystyle m\times m}
2053:{\displaystyle O(dn^{2})}
553:{\displaystyle A=VTV^{*}}
464:{\displaystyle m\times m}
438:{\displaystyle T=V^{*}AV}
371:{\displaystyle n\times m}
239:{\displaystyle n\times n}
99:{\displaystyle n\times n}
52:that is an adaptation of
19183:Numerical linear algebra
19162:General purpose software
19025:Numerical linear algebra
18867:"Keyword Index: Lanczos"
18767:"TRLan software package"
18273:
18176:as a method for solving
18174:condensed matter physics
18155:latent semantic indexing
17986:computer with roundoff.
17872:{\displaystyle 1+4\rho }
17756:{\displaystyle 1+2\rho }
12305:. Conversely, any point
10701:th power method iterate
7877:{\displaystyle k<j-1}
5152:{\displaystyle u_{j+1}'}
3786:(until the direction of
3668:{\displaystyle O(n^{2})}
3539:{\displaystyle O(n^{3})}
3004:{\displaystyle O(m^{2})}
2918:{\displaystyle O(m^{2})}
2826:{\displaystyle O(m^{2})}
2381:{\displaystyle \lambda }
18677:10.1137/1.9780898719628
18495:10.1093/imamat/10.3.373
18261:collaborative filtering
18122:block Lanczos algorithm
18104:Block Lanczos algorithm
10303:{\displaystyle z=y_{j}}
10270:{\displaystyle z=x_{j}}
9931:{\displaystyle Ay_{j};}
8488:{\displaystyle h_{k,j}}
7635:{\displaystyle h_{k,j}}
7508:{\displaystyle h_{k,j}}
6877:{\displaystyle h_{k,j}}
6770:{\displaystyle v_{j+1}}
2283:{\displaystyle v_{j-1}}
2230:{\displaystyle v_{j+1}}
21:block Lanczos algorithm
18483:J. Inst. Maths Applics
18356:
18307:
18147:
18052:
17967:
17941:
17902:
17873:
17844:
17818:
17757:
17728:
17680:
17583:
17483:
17294:
17274:
17244:
17123:
17080:
17053:
17026:
16966:
16943:
16942:{\displaystyle R^{-2}}
16913:
16890:
16194:
16082:
15999:
15765:
15662:
15482:
15389:
15274:
15162:
15142:
15122:
15096:
15043:
15016:
14973:
14851:
14816:
14789:
14760:on the known interval
14754:
14719:
14699:
14636:
14616:
14585:
14565:
14542:
14502:
14475:
14455:
14426:
14352:
14237:
14135:
14030:
13922:
13809:
13692:
13669:
13625:
13561:
13526:
13482:
13427:
13397:
12826:
12806:
12779:
12678:
12652:
12632:
12590:
12467:
12444:
12417:
12375:
12348:
12319:
12299:
12259:
12239:
12212:
12180:
12160:
12133:
12104:
12074:
12054:
12014:to their counterparts
12008:
11988:
11942:
11922:
11895:
11868:
11841:
11814:
11752:
11732:
11706:
11686:
11658:
11638:
11611:
11587:
11544:
11521:on the unit sphere in
11511:
11459:
11432:
11412:
11388:
11360:
11333:
11307:
11260:
11222:
11167:
11137:
11099:
11046:
11000:
10941:
10913:
10861:
10830:
10810:
10783:
10756:
10722:
10695:
10672:
10520:
10460:
10350:
10304:
10271:
10238:
10198:
10152:
10097:
10077:
10046:
10045:{\displaystyle Ay_{j}}
10016:
10015:{\displaystyle Ax_{j}}
9986:
9959:
9932:
9899:
9898:{\displaystyle Ax_{j}}
9869:
9842:
9812:
9719:
9677:
9657:
9630:
9588:
9568:
9535:
9375:
9318:
9272:
9226:
9172:
9152:
9132:
9112:
9083:
9056:
9036:
9009:
8989:
8959:
8855:
8827:
8807:
8786:
8766:
8743:
8622:
8602:
8567:
8528:
8489:
8456:
8410:
8384:
8240:
8211:
8021:
7985:
7958:
7878:
7843:
7659:
7636:
7603:
7575:
7529:
7509:
7476:
7425:
7351:
7321:
7131:
7061:
6993:
6992:{\displaystyle j<m}
6967:
6936:
6878:
6837:
6791:
6771:
6735:
6656:
6608:
6541:
6510:
6433:
6363:
6322:
6267:
6220:
6200:
6169:
6111:
6059:
6007:
5917:
5890:
5859:
5828:
5737:
5600:
5573:
5538:
5492:
5472:
5417:
5338:
5253:
5211:
5153:
5117:
5086:
5009:
4963:
4911:
4841:
4800:
4745:
4698:
4658:
4638:
4603:
4579:
4552:
4500:
4473:
4421:
4394:
4351:
4316:
4252:
4179:
4150:
4091:
4061:
4034:
4008:
3981:
3955:
3865:
3807:
3780:
3751:
3712:
3669:
3633:
3606:
3575:
3540:
3504:
3448:
3428:
3378:
3355:
3328:
3280:
3279:{\displaystyle m\ll n}
3254:
3234:
3214:
3194:
3171:
3151:
3127:
3103:
3083:
3063:
3032:
3005:
2960:
2919:
2883:
2856:
2827:
2780:
2754:
2730:
2710:
2688:
2503:
2483:
2454:
2422:
2402:
2382:
2338:
2311:
2290:after having computed
2284:
2251:
2231:
2198:
2171:
2144:
2143:{\displaystyle w_{j}'}
2110:
2080:
2054:
2015:
2014:{\displaystyle O(dmn)}
1980:
1960:
1928:
1896:
1863:
1832:
1800:
1660:
1396:
1350:
1325:
1227:
1164:
1113:
1061:
1041:
1011:
950:
910:
855:
812:
743:
680:
628:
605:
554:
511:
491:
465:
439:
406:real symmetric matrix
392:
372:
338:
318:
286:
260:
240:
214:
179:
143:
123:
100:
70:
19157:Specialized libraries
19070:Matrix multiplication
19065:Matrix decompositions
18458:Paige, C. C. (1971).
18357:
18308:
18306:{\displaystyle z_{2}}
18148:
18053:
17993:, the set of vectors
17968:
17942:
17903:
17874:
17845:
17819:
17758:
17729:
17681:
17584:
17484:
17295:
17275:
17273:{\displaystyle z_{2}}
17245:
17124:
17081:
17054:
17027:
16967:
16965:{\displaystyle \rho }
16944:
16914:
16891:
16203:we may conclude that
16195:
16083:
16000:
15766:
15642:
15462:
15369:
15254:
15163:
15143:
15123:
15097:
15044:
15017:
14974:
14852:
14817:
14790:
14755:
14720:
14700:
14637:
14617:
14615:{\displaystyle c_{k}}
14593:Chebyshev polynomials
14586:
14566:
14543:
14503:
14476:
14456:
14427:
14332:
14217:
14115:
14010:
13902:
13789:
13693:
13670:
13605:
13562:
13506:
13462:
13428:
13426:{\displaystyle v_{1}}
13398:
12827:
12807:
12805:{\displaystyle p^{*}}
12780:
12679:
12653:
12633:
12591:
12468:
12445:
12418:
12376:
12349:
12320:
12300:
12260:
12240:
12213:
12181:
12161:
12134:
12105:
12075:
12055:
12009:
11989:
11943:
11923:
11896:
11869:
11842:
11815:
11753:
11733:
11707:
11687:
11659:
11639:
11637:{\displaystyle v_{1}}
11612:
11588:
11545:
11512:
11460:
11458:{\displaystyle d_{k}}
11433:
11413:
11389:
11387:{\displaystyle v_{1}}
11361:
11359:{\displaystyle v_{1}}
11334:
11287:
11261:
11223:
11168:
11166:{\displaystyle v_{1}}
11138:
11100:
11047:
11001:
10942:
10914:
10862:
10831:
10811:
10809:{\displaystyle y_{j}}
10784:
10782:{\displaystyle x_{j}}
10757:
10723:
10721:{\displaystyle u_{j}}
10696:
10673:
10521:
10461:
10351:
10305:
10272:
10239:
10199:
10153:
10098:
10078:
10047:
10017:
9987:
9985:{\displaystyle y_{j}}
9960:
9958:{\displaystyle x_{j}}
9933:
9900:
9870:
9868:{\displaystyle y_{j}}
9843:
9841:{\displaystyle x_{j}}
9813:
9720:
9678:
9658:
9656:{\displaystyle y_{j}}
9631:
9589:
9569:
9567:{\displaystyle x_{j}}
9536:
9376:
9319:
9273:
9227:
9173:
9153:
9133:
9113:
9084:
9057:
9037:
9010:
8990:
8960:
8856:
8828:
8808:
8787:
8767:
8744:
8623:
8603:
8568:
8529:
8490:
8457:
8411:
8385:
8241:
8212:
8022:
8020:{\displaystyle k=j-1}
7986:
7984:{\displaystyle v_{j}}
7959:
7879:
7844:
7660:
7637:
7604:
7576:
7530:
7510:
7477:
7426:
7352:
7350:{\displaystyle u_{j}}
7322:
7132:
7062:
6994:
6968:
6910:
6879:
6838:
6792:
6772:
6736:
6657:
6609:
6542:
6490:
6434:
6364:
6323:
6268:
6221:
6201:
6199:{\displaystyle v_{1}}
6179:Pick a random vector
6170:
6112:
6060:
6008:
5918:
5916:{\displaystyle v_{j}}
5891:
5889:{\displaystyle u_{j}}
5860:
5808:
5717:
5601:
5599:{\displaystyle v_{j}}
5574:
5572:{\displaystyle u_{j}}
5539:
5493:
5473:
5418:
5339:
5254:
5212:
5154:
5118:
5066:
5010:
4964:
4912:
4842:
4801:
4746:
4699:
4659:
4639:
4637:{\displaystyle u_{1}}
4617:Pick a random vector
4604:
4580:
4578:{\displaystyle u_{j}}
4553:
4501:
4499:{\displaystyle v_{j}}
4474:
4422:
4420:{\displaystyle u_{j}}
4395:
4352:
4317:
4253:
4180:
4151:
4092:
4090:{\displaystyle u_{j}}
4062:
4060:{\displaystyle u_{j}}
4035:
4009:
4007:{\displaystyle u_{j}}
3982:
3956:
3866:
3808:
3806:{\displaystyle u_{j}}
3781:
3752:
3724:Pick a random vector
3713:
3670:
3634:
3607:
3585:(the computations of
3576:
3541:
3505:
3449:
3429:
3379:
3356:
3329:
3281:
3255:
3235:
3215:
3195:
3172:
3152:
3128:
3104:
3084:
3064:
3033:
3031:{\displaystyle m^{2}}
3006:
2961:
2920:
2884:
2857:
2828:
2793:allows computing the
2781:
2755:
2731:
2711:
2689:
2504:
2484:
2455:
2423:
2403:
2383:
2339:
2337:{\displaystyle v_{1}}
2312:
2310:{\displaystyle w_{j}}
2285:
2252:
2232:
2199:
2197:{\displaystyle w_{j}}
2172:
2170:{\displaystyle w_{j}}
2145:
2111:
2109:{\displaystyle v_{j}}
2081:
2055:
2016:
1981:
1961:
1959:{\displaystyle O(dn)}
1929:
1897:
1864:
1862:{\displaystyle v_{1}}
1833:
1801:
1661:
1397:
1351:
1326:
1228:
1165:
1114:
1062:
1042:
1040:{\displaystyle v_{j}}
1012:
951:
911:
856:
813:
744:
681:
629:
606:
555:
512:
492:
466:
440:
393:
373:
339:
319:
287:
261:
241:
215:
180:
151:numerical instability
144:
124:
101:
71:
35:Lanczos approximation
18916:Van Loan, Charles F.
18746:(2). SIAM: 602–616.
18404:10.6028/jres.045.026
18382:Lanczos, C. (1950).
18317:
18290:
18136:
18086:memory-fetch times.
17997:
17951:
17912:
17883:
17854:
17828:
17767:
17738:
17693:
17599:
17499:
17307:
17284:
17257:
17136:
17090:
17063:
17036:
16976:
16956:
16923:
16903:
16210:
16106:
16015:
15781:
15175:
15152:
15132:
15106:
15053:
15026:
14986:
14864:
14826:
14799:
14764:
14729:
14709:
14646:
14626:
14599:
14575:
14555:
14512:
14485:
14465:
14439:
13708:
13682:
13577:
13570:and more generally
13440:
13410:
12839:
12816:
12789:
12688:
12662:
12642:
12638:for some polynomial
12603:
12480:
12473:Krylov subspace is
12457:
12427:
12385:
12358:
12347:{\displaystyle r(x)}
12329:
12309:
12269:
12249:
12222:
12190:
12170:
12143:
12132:{\displaystyle r(x)}
12114:
12087:
12064:
12018:
11998:
11952:
11932:
11905:
11878:
11851:
11824:
11762:
11742:
11716:
11696:
11676:
11648:
11621:
11601:
11554:
11525:
11519:uniform distribution
11469:
11442:
11422:
11402:
11371:
11343:
11339:. A starting vector
11270:
11232:
11177:
11150:
11109:
11056:
11010:
10951:
10931:
10871:
10840:
10820:
10793:
10766:
10732:
10705:
10685:
10533:
10473:
10363:
10317:
10281:
10248:
10208:
10162:
10107:
10087:
10056:
10026:
9996:
9969:
9942:
9909:
9879:
9852:
9825:
9732:
9687:
9667:
9640:
9636:, and likewise from
9601:
9578:
9551:
9388:
9328:
9282:
9236:
9182:
9162:
9142:
9122:
9093:
9069:
9046:
9019:
8999:
8972:
8875:
8845:
8817:
8797:
8776:
8756:
8639:
8612:
8577:
8538:
8499:
8466:
8420:
8400:
8396:More abstractly, if
8253:
8224:
8034:
7999:
7968:
7888:
7856:
7672:
7649:
7613:
7593:
7539:
7519:
7486:
7435:
7361:
7334:
7144:
7071:
7007:
6977:
6891:
6855:
6801:
6781:
6748:
6666:
6621:
6554:
6446:
6373:
6335:
6280:
6233:
6210:
6183:
6121:
6069:
6017:
5927:
5900:
5873:
5613:
5583:
5556:
5502:
5482:
5430:
5348:
5263:
5224:
5163:
5127:
5022:
4973:
4921:
4851:
4813:
4758:
4711:
4668:
4648:
4621:
4611:Gram–Schmidt process
4593:
4562:
4510:
4483:
4431:
4404:
4364:
4326:
4262:
4189:
4160:
4109:
4074:
4044:
4024:
3991:
3971:
3877:
3820:
3790:
3764:
3728:
3702:
3643:
3616:
3589:
3574:{\displaystyle O(n)}
3556:
3514:
3458:
3438:
3388:
3368:
3345:
3318:
3264:
3244:
3224:
3204:
3184:
3161:
3141:
3117:
3093:
3073:
3053:
3015:
2979:
2932:
2893:
2873:
2855:{\displaystyle O(m)}
2837:
2801:
2791:continuant recursion
2764:
2744:
2720:
2700:
2515:
2493:
2482:{\displaystyle y=Vx}
2464:
2432:
2412:
2392:
2388:is an eigenvalue of
2372:
2321:
2294:
2261:
2241:
2208:
2181:
2154:
2124:
2093:
2064:
2025:
1990:
1970:
1938:
1927:{\displaystyle O(n)}
1909:
1873:
1846:
1810:
1677:
1406:
1360:
1340:
1240:
1177:
1128:
1071:
1051:
1024:
960:
927:
868:
827:
756:
693:
644:
618:
573:
525:
501:
475:
449:
410:
382:
356:
328:
299:
270:
250:
224:
204:
169:
133:
113:
84:
60:
19044:Numerical stability
18944:; Zheng, Alice X.;
18924:Matrix Computations
18815:2011PhRvB..84d5113C
18616:Matrix computations
18512:. pp. 489–494.
18437:1970AIAAJ...8.1234N
18146:{\displaystyle A\,}
17981:Numerical stability
14454:{\displaystyle k=1}
13901:
13788:
13678:for any polynomial
13348:
13287:
13215:
13147:
13082:
13021:
12677:{\displaystyle m-1}
11396:normal distribution
11207:
10147:
8331:
8239:{\displaystyle k=j}
8124:
7776:
7745:
7727:
7708:
7665:is Hermitian then
7471:
7272:
7210:
7186:
7167:
7126:
7107:
6486:
6428:
6409:
6301:
6090:
6038:
5694:
5653:
5330:
5303:
5148:
5062:
4958:
4906:
4887:
4779:
4145:
3944:
3917:
3841:
3813:has converged) do:
3510:can be computed in
2139:
2079:{\displaystyle m=n}
1708:
1268:
1212:
1143:
784:
728:
659:
490:{\displaystyle m=n}
285:{\displaystyle m=n}
18968:Erik Koch (2019).
18946:Jordan, Michael I.
18895:2011-03-14 at the
18352:
18303:
18143:
18048:
17963:
17937:
17898:
17869:
17840:
17814:
17753:
17724:
17676:
17579:
17479:
17290:
17270:
17240:
17119:
17076:
17049:
17022:
16962:
16939:
16909:
16886:
16884:
16190:
16078:
15995:
15761:
15158:
15138:
15118:
15092:
15039:
15012:
14969:
14847:
14812:
14785:
14750:
14715:
14695:
14632:
14612:
14581:
14561:
14538:
14498:
14471:
14451:
14422:
13887:
13774:
13688:
13665:
13557:
13423:
13393:
13334:
13273:
13201:
13133:
13068:
13007:
12822:
12802:
12775:
12674:
12658:of degree at most
12648:
12628:
12586:
12463:
12440:
12413:
12371:
12344:
12315:
12295:
12255:
12235:
12208:
12176:
12156:
12129:
12100:
12070:
12050:
12004:
11994:of eigenvalues of
11984:
11938:
11918:
11891:
11864:
11837:
11810:
11748:
11728:
11702:
11682:
11654:
11634:
11607:
11583:
11540:
11507:
11455:
11428:
11408:
11384:
11356:
11329:
11328:
11256:
11218:
11193:
11163:
11133:
11095:
11042:
10996:
10937:
10909:
10857:
10826:
10806:
10779:
10752:
10718:
10691:
10668:
10652:
10584:
10516:
10456:
10346:
10300:
10267:
10234:
10194:
10148:
10127:
10093:
10073:
10042:
10012:
9982:
9955:
9928:
9895:
9865:
9838:
9808:
9715:
9673:
9653:
9626:
9584:
9564:
9531:
9529:
9371:
9314:
9268:
9222:
9168:
9148:
9128:
9108:
9079:
9052:
9032:
9005:
8985:
8955:
8851:
8823:
8803:
8782:
8762:
8739:
8618:
8598:
8563:
8524:
8485:
8452:
8406:
8380:
8317:
8236:
8207:
8110:
8017:
7981:
7954:
7874:
7839:
7762:
7731:
7709:
7694:
7655:
7632:
7599:
7571:
7525:
7505:
7472:
7451:
7421:
7347:
7317:
7252:
7190:
7168:
7147:
7127:
7108:
7093:
7057:
6989:
6963:
6874:
6833:
6787:
6767:
6744:otherwise pick as
6731:
6652:
6604:
6537:
6468:
6429:
6410:
6395:
6359:
6318:
6283:
6263:
6216:
6206:of Euclidean norm
6196:
6165:
6107:
6072:
6055:
6020:
6003:
5913:
5886:
5855:
5676:
5635:
5596:
5569:
5534:
5488:
5468:
5426:otherwise pick as
5413:
5334:
5312:
5285:
5249:
5207:
5149:
5130:
5113:
5044:
5005:
4959:
4940:
4907:
4888:
4873:
4837:
4796:
4761:
4741:
4694:
4654:
4644:of Euclidean norm
4634:
4599:
4575:
4548:
4496:
4469:
4417:
4390:
4347:
4312:
4248:
4175:
4146:
4125:
4087:
4057:
4030:
4004:
3977:
3951:
3926:
3899:
3861:
3823:
3803:
3776:
3747:
3708:
3665:
3629:
3602:
3571:
3536:
3500:
3444:
3424:
3374:
3351:
3324:
3276:
3250:
3230:
3210:
3190:
3167:
3147:
3123:
3099:
3079:
3059:
3028:
3001:
2956:
2915:
2879:
2852:
2823:
2776:
2750:
2726:
2706:
2684:
2682:
2499:
2479:
2450:
2418:
2398:
2378:
2334:
2307:
2280:
2247:
2227:
2204:is not used after
2194:
2167:
2150:is not used after
2140:
2127:
2106:
2076:
2050:
2011:
1976:
1956:
1924:
1892:
1859:
1828:
1796:
1696:
1656:
1650:
1392:
1346:
1321:
1256:
1223:
1193:
1160:
1131:
1109:
1057:
1037:
1007:
946:
906:
851:
808:
772:
739:
709:
676:
647:
624:
601:
550:
507:
487:
461:
435:
388:
368:
334:
314:
282:
256:
236:
210:
175:
139:
119:
96:
66:
28:Lanczos resampling
19170:
19169:
18986:978-3-95806-400-3
18920:"Lanczos Methods"
18793:Physical Review B
18709:Appl. Numer. Math
18686:978-0-89871-407-4
17935:
17812:
17474:
17450:
17447:
17393:
17332:
17293:{\displaystyle t}
16912:{\displaystyle R}
16823:
16742:
16645:
16567:
16471:
16393:
16185:
16076:
15979:
15894:
15756:
15605:
15454:
15161:{\displaystyle 0}
15141:{\displaystyle 1}
14963:
14718:{\displaystyle x}
14635:{\displaystyle k}
14584:{\displaystyle p}
14564:{\displaystyle A}
14474:{\displaystyle p}
14417:
14209:
13989:
13691:{\displaystyle q}
13391:
13265:
13125:
12999:
12825:{\displaystyle p}
12651:{\displaystyle p}
12466:{\displaystyle m}
12318:{\displaystyle x}
12258:{\displaystyle m}
12179:{\displaystyle r}
12073:{\displaystyle A}
12007:{\displaystyle T}
11941:{\displaystyle m}
11751:{\displaystyle m}
11705:{\displaystyle T}
11685:{\displaystyle m}
11657:{\displaystyle T}
11610:{\displaystyle A}
11431:{\displaystyle 1}
11411:{\displaystyle 0}
10940:{\displaystyle A}
10829:{\displaystyle r}
10694:{\displaystyle j}
10626:
10601:
10558:
10103:on every step if
10096:{\displaystyle 2}
9773:
9676:{\displaystyle r}
9587:{\displaystyle r}
9171:{\displaystyle r}
9151:{\displaystyle y}
9131:{\displaystyle x}
9055:{\displaystyle r}
9008:{\displaystyle r}
8928:
8867:Rayleigh quotient
8863:stationary points
8854:{\displaystyle A}
8826:{\displaystyle T}
8806:{\displaystyle H}
8785:{\displaystyle H}
8765:{\displaystyle H}
8621:{\displaystyle H}
8462:then the numbers
8409:{\displaystyle V}
8375:
8349:
8180:
8148:
7658:{\displaystyle A}
7645:Elementarily, if
7602:{\displaystyle A}
7587:Arnoldi iteration
7528:{\displaystyle A}
7482:and coefficients
6790:{\displaystyle 1}
6219:{\displaystyle 1}
5491:{\displaystyle 1}
4657:{\displaystyle 1}
4602:{\displaystyle A}
4033:{\displaystyle A}
3980:{\displaystyle j}
3711:{\displaystyle A}
3447:{\displaystyle V}
3377:{\displaystyle V}
3354:{\displaystyle A}
3327:{\displaystyle A}
3288:lossy compression
3253:{\displaystyle A}
3233:{\displaystyle T}
3213:{\displaystyle m}
3193:{\displaystyle n}
3170:{\displaystyle T}
3150:{\displaystyle A}
3126:{\displaystyle T}
3102:{\displaystyle T}
3082:{\displaystyle A}
3062:{\displaystyle T}
3047:inverse iteration
2882:{\displaystyle T}
2753:{\displaystyle T}
2729:{\displaystyle T}
2709:{\displaystyle A}
2502:{\displaystyle A}
2428:its eigenvector (
2421:{\displaystyle x}
2401:{\displaystyle T}
2250:{\displaystyle T}
1979:{\displaystyle d}
1349:{\displaystyle V}
1060:{\displaystyle 1}
627:{\displaystyle 1}
510:{\displaystyle V}
391:{\displaystyle V}
337:{\displaystyle m}
266:(as default, let
259:{\displaystyle m}
213:{\displaystyle A}
178:{\displaystyle m}
142:{\displaystyle n}
122:{\displaystyle m}
69:{\displaystyle m}
50:Cornelius Lanczos
42:Lanczos algorithm
19190:
19080:Matrix splitting
19018:
19011:
19004:
18995:
18994:
18990:
18974:
18964:
18954:
18937:
18899:
18887:
18881:
18880:
18878:
18877:
18862:
18856:
18855:
18853:
18841:
18835:
18834:
18808:
18788:
18782:
18781:
18779:
18778:
18769:. Archived from
18762:
18756:
18755:
18731:
18725:
18724:
18706:
18697:
18691:
18690:
18664:
18658:
18657:
18636:
18630:
18629:
18611:
18602:
18601:
18581:
18575:
18574:
18551:
18542:
18541:
18523:
18514:
18513:
18505:
18499:
18498:
18478:
18472:
18471:
18455:
18449:
18448:
18431:(7): 1234–1239.
18418:
18409:
18408:
18406:
18388:
18379:
18363:
18361:
18359:
18358:
18353:
18351:
18350:
18335:
18334:
18312:
18310:
18309:
18304:
18302:
18301:
18284:
18184:, as well as in
18152:
18150:
18149:
18144:
18110:Peter Montgomery
18057:
18055:
18054:
18049:
18047:
18046:
18022:
18021:
18009:
18008:
17991:exact arithmetic
17972:
17970:
17969:
17964:
17946:
17944:
17943:
17938:
17936:
17931:
17907:
17905:
17904:
17899:
17878:
17876:
17875:
17870:
17849:
17847:
17846:
17841:
17823:
17821:
17820:
17815:
17813:
17805:
17804:
17795:
17762:
17760:
17759:
17754:
17733:
17731:
17730:
17725:
17723:
17722:
17685:
17683:
17682:
17677:
17672:
17671:
17659:
17658:
17646:
17645:
17624:
17623:
17611:
17610:
17588:
17586:
17585:
17580:
17575:
17574:
17565:
17564:
17552:
17551:
17539:
17538:
17511:
17510:
17488:
17486:
17485:
17480:
17475:
17473:
17456:
17451:
17449:
17448:
17446:
17445:
17436:
17435:
17434:
17422:
17421:
17411:
17399:
17394:
17392:
17388:
17387:
17375:
17374:
17359:
17358:
17348:
17347:
17338:
17333:
17331:
17330:
17321:
17320:
17311:
17299:
17297:
17296:
17291:
17279:
17277:
17276:
17271:
17269:
17268:
17249:
17247:
17246:
17241:
17236:
17235:
17220:
17219:
17207:
17206:
17191:
17190:
17181:
17180:
17176:
17163:
17162:
17128:
17126:
17125:
17120:
17118:
17117:
17102:
17101:
17085:
17083:
17082:
17077:
17075:
17074:
17058:
17056:
17055:
17050:
17048:
17047:
17031:
17029:
17028:
17023:
17021:
17016:
17015:
17006:
16998:
16993:
16992:
16983:
16971:
16969:
16968:
16963:
16948:
16946:
16945:
16940:
16938:
16937:
16918:
16916:
16915:
16910:
16895:
16893:
16892:
16887:
16885:
16881:
16880:
16850:
16849:
16837:
16836:
16824:
16822:
16821:
16820:
16815:
16809:
16808:
16799:
16793:
16792:
16791:
16786:
16780:
16779:
16770:
16758:
16747:
16743:
16741:
16740:
16735:
16731:
16730:
16729:
16702:
16701:
16677:
16672:
16671:
16659:
16658:
16646:
16644:
16643:
16642:
16637:
16631:
16630:
16621:
16615:
16614:
16613:
16608:
16602:
16601:
16592:
16580:
16572:
16568:
16566:
16517:
16516:
16503:
16498:
16497:
16485:
16484:
16472:
16470:
16469:
16468:
16463:
16457:
16456:
16447:
16441:
16440:
16439:
16434:
16428:
16427:
16418:
16406:
16398:
16394:
16392:
16391:
16390:
16385:
16379:
16378:
16369:
16364:
16363:
16339:
16338:
16322:
16321:
16317:
16316:
16315:
16310:
16304:
16303:
16294:
16275:
16274:
16262:
16261:
16248:
16239:
16238:
16226:
16225:
16199:
16197:
16196:
16191:
16186:
16178:
16177:
16168:
16145:
16144:
16087:
16085:
16084:
16079:
16077:
16075:
16074:
16073:
16061:
16060:
16050:
16049:
16048:
16036:
16035:
16025:
16004:
16002:
16001:
15996:
15991:
15987:
15980:
15978:
15977:
15976:
15964:
15963:
15953:
15952:
15951:
15939:
15938:
15928:
15918:
15917:
15899:
15895:
15893:
15892:
15891:
15879:
15878:
15868:
15867:
15866:
15854:
15853:
15841:
15840:
15827:
15821:
15820:
15799:
15798:
15770:
15768:
15767:
15762:
15757:
15755:
15754:
15753:
15748:
15742:
15741:
15732:
15727:
15726:
15721:
15712:
15711:
15696:
15690:
15689:
15688:
15683:
15677:
15676:
15667:
15661:
15656:
15638:
15637:
15625:
15624:
15611:
15606:
15604:
15603:
15602:
15597:
15588:
15587:
15572:
15567:
15566:
15561:
15555:
15554:
15545:
15539:
15535:
15534:
15522:
15521:
15509:
15508:
15503:
15497:
15496:
15487:
15481:
15476:
15460:
15455:
15453:
15452:
15451:
15446:
15437:
15436:
15421:
15416:
15415:
15410:
15404:
15403:
15394:
15388:
15383:
15367:
15366:
15365:
15360:
15351:
15350:
15335:
15327:
15326:
15314:
15313:
15301:
15300:
15295:
15289:
15288:
15279:
15273:
15268:
15252:
15244:
15243:
15213:
15212:
15200:
15199:
15187:
15186:
15167:
15165:
15164:
15159:
15147:
15145:
15144:
15139:
15127:
15125:
15124:
15119:
15101:
15099:
15098:
15093:
15091:
15090:
15085:
15076:
15075:
15060:
15048:
15046:
15045:
15040:
15038:
15037:
15021:
15019:
15018:
15013:
15011:
15010:
14998:
14997:
14978:
14976:
14975:
14970:
14968:
14964:
14962:
14961:
14960:
14948:
14947:
14937:
14936:
14935:
14923:
14922:
14903:
14897:
14896:
14856:
14854:
14853:
14850:{\displaystyle }
14848:
14821:
14819:
14818:
14813:
14811:
14810:
14794:
14792:
14791:
14788:{\displaystyle }
14786:
14759:
14757:
14756:
14753:{\displaystyle }
14751:
14724:
14722:
14721:
14716:
14704:
14702:
14701:
14696:
14658:
14657:
14641:
14639:
14638:
14633:
14621:
14619:
14618:
14613:
14611:
14610:
14590:
14588:
14587:
14582:
14570:
14568:
14567:
14562:
14547:
14545:
14544:
14539:
14537:
14536:
14524:
14523:
14507:
14505:
14504:
14499:
14497:
14496:
14480:
14478:
14477:
14472:
14460:
14458:
14457:
14452:
14431:
14429:
14428:
14423:
14418:
14416:
14415:
14414:
14409:
14405:
14401:
14400:
14379:
14378:
14373:
14367:
14366:
14357:
14351:
14346:
14330:
14329:
14328:
14323:
14319:
14315:
14314:
14290:
14289:
14277:
14276:
14264:
14263:
14258:
14252:
14251:
14242:
14236:
14231:
14215:
14210:
14208:
14204:
14203:
14188:
14187:
14178:
14177:
14162:
14161:
14156:
14150:
14149:
14140:
14134:
14129:
14113:
14109:
14108:
14093:
14092:
14083:
14082:
14067:
14066:
14057:
14056:
14051:
14045:
14044:
14035:
14029:
14024:
14008:
14003:
14002:
13990:
13988:
13987:
13986:
13974:
13973:
13955:
13954:
13942:
13941:
13932:
13931:
13921:
13916:
13900:
13895:
13885:
13884:
13883:
13871:
13870:
13855:
13854:
13842:
13841:
13829:
13828:
13819:
13818:
13808:
13803:
13787:
13782:
13772:
13767:
13766:
13751:
13750:
13720:
13719:
13697:
13695:
13694:
13689:
13674:
13672:
13671:
13666:
13664:
13663:
13651:
13650:
13635:
13634:
13624:
13619:
13601:
13600:
13566:
13564:
13563:
13558:
13556:
13555:
13546:
13545:
13536:
13535:
13525:
13520:
13502:
13501:
13492:
13491:
13481:
13476:
13455:
13454:
13432:
13430:
13429:
13424:
13422:
13421:
13402:
13400:
13399:
13394:
13392:
13390:
13389:
13388:
13358:
13357:
13347:
13342:
13332:
13331:
13330:
13297:
13296:
13286:
13281:
13271:
13266:
13264:
13263:
13262:
13238:
13237:
13225:
13224:
13214:
13209:
13199:
13198:
13197:
13170:
13169:
13157:
13156:
13146:
13141:
13131:
13126:
13124:
13123:
13122:
13101:
13100:
13081:
13076:
13066:
13065:
13064:
13040:
13039:
13020:
13015:
13005:
13000:
12998:
12997:
12996:
12975:
12974:
12965:
12964:
12939:
12938:
12937:
12913:
12912:
12903:
12902:
12877:
12869:
12868:
12831:
12829:
12828:
12823:
12811:
12809:
12808:
12803:
12801:
12800:
12784:
12782:
12781:
12776:
12774:
12773:
12764:
12763:
12739:
12738:
12729:
12728:
12716:
12715:
12700:
12699:
12683:
12681:
12680:
12675:
12657:
12655:
12654:
12649:
12637:
12635:
12634:
12629:
12627:
12626:
12595:
12593:
12592:
12587:
12582:
12578:
12577:
12576:
12567:
12566:
12542:
12541:
12532:
12531:
12519:
12518:
12503:
12502:
12472:
12470:
12469:
12464:
12449:
12447:
12446:
12441:
12439:
12438:
12422:
12420:
12419:
12414:
12397:
12396:
12380:
12378:
12377:
12372:
12370:
12369:
12353:
12351:
12350:
12345:
12324:
12322:
12321:
12316:
12304:
12302:
12301:
12296:
12294:
12293:
12281:
12280:
12264:
12262:
12261:
12256:
12244:
12242:
12241:
12236:
12234:
12233:
12217:
12215:
12214:
12209:
12204:
12203:
12198:
12186:on the whole of
12185:
12183:
12182:
12177:
12165:
12163:
12162:
12157:
12155:
12154:
12138:
12136:
12135:
12130:
12109:
12107:
12106:
12101:
12099:
12098:
12079:
12077:
12076:
12071:
12059:
12057:
12056:
12051:
12049:
12048:
12030:
12029:
12013:
12011:
12010:
12005:
11993:
11991:
11990:
11985:
11983:
11982:
11964:
11963:
11947:
11945:
11944:
11939:
11927:
11925:
11924:
11919:
11917:
11916:
11900:
11898:
11897:
11892:
11890:
11889:
11873:
11871:
11870:
11865:
11863:
11862:
11846:
11844:
11843:
11838:
11836:
11835:
11819:
11817:
11816:
11811:
11806:
11805:
11787:
11786:
11774:
11773:
11757:
11755:
11754:
11749:
11737:
11735:
11734:
11729:
11711:
11709:
11708:
11703:
11691:
11689:
11688:
11683:
11663:
11661:
11660:
11655:
11643:
11641:
11640:
11635:
11633:
11632:
11616:
11614:
11613:
11608:
11592:
11590:
11589:
11584:
11576:
11571:
11570:
11561:
11549:
11547:
11546:
11541:
11539:
11538:
11533:
11516:
11514:
11513:
11508:
11503:
11502:
11484:
11483:
11464:
11462:
11461:
11456:
11454:
11453:
11437:
11435:
11434:
11429:
11417:
11415:
11414:
11409:
11393:
11391:
11390:
11385:
11383:
11382:
11365:
11363:
11362:
11357:
11355:
11354:
11338:
11336:
11335:
11330:
11327:
11326:
11317:
11316:
11306:
11301:
11283:
11282:
11265:
11263:
11262:
11257:
11227:
11225:
11224:
11219:
11217:
11216:
11206:
11201:
11189:
11188:
11172:
11170:
11169:
11164:
11162:
11161:
11142:
11140:
11139:
11134:
11104:
11102:
11101:
11096:
11094:
11093:
11084:
11083:
11071:
11070:
11051:
11049:
11048:
11043:
11041:
11040:
11022:
11021:
11005:
11003:
11002:
10997:
10995:
10994:
10976:
10975:
10963:
10962:
10946:
10944:
10943:
10938:
10918:
10916:
10915:
10910:
10905:
10904:
10886:
10885:
10866:
10864:
10863:
10858:
10856:
10855:
10850:
10849:
10835:
10833:
10832:
10827:
10815:
10813:
10812:
10807:
10805:
10804:
10788:
10786:
10785:
10780:
10778:
10777:
10761:
10759:
10758:
10753:
10748:
10747:
10742:
10741:
10727:
10725:
10724:
10719:
10717:
10716:
10700:
10698:
10697:
10692:
10677:
10675:
10674:
10669:
10651:
10650:
10649:
10644:
10643:
10619:
10618:
10602:
10599:
10583:
10582:
10581:
10576:
10575:
10551:
10550:
10525:
10523:
10522:
10517:
10515:
10514:
10509:
10508:
10498:
10497:
10485:
10484:
10465:
10463:
10462:
10457:
10452:
10451:
10442:
10441:
10417:
10416:
10401:
10400:
10379:
10378:
10373:
10372:
10355:
10353:
10352:
10347:
10342:
10341:
10329:
10328:
10309:
10307:
10306:
10301:
10299:
10298:
10276:
10274:
10273:
10268:
10266:
10265:
10243:
10241:
10240:
10235:
10230:
10229:
10224:
10223:
10203:
10201:
10200:
10195:
10193:
10192:
10181:
10180:
10157:
10155:
10154:
10149:
10146:
10141:
10126:
10125:
10120:
10119:
10102:
10100:
10099:
10094:
10082:
10080:
10079:
10074:
10072:
10071:
10066:
10065:
10051:
10049:
10048:
10043:
10041:
10040:
10021:
10019:
10018:
10013:
10011:
10010:
9991:
9989:
9988:
9983:
9981:
9980:
9964:
9962:
9961:
9956:
9954:
9953:
9937:
9935:
9934:
9929:
9924:
9923:
9904:
9902:
9901:
9896:
9894:
9893:
9874:
9872:
9871:
9866:
9864:
9863:
9847:
9845:
9844:
9839:
9837:
9836:
9817:
9815:
9814:
9809:
9774:
9772:
9768:
9767:
9754:
9724:
9722:
9721:
9716:
9711:
9710:
9682:
9680:
9679:
9674:
9662:
9660:
9659:
9654:
9652:
9651:
9635:
9633:
9632:
9627:
9622:
9621:
9593:
9591:
9590:
9585:
9573:
9571:
9570:
9565:
9563:
9562:
9540:
9538:
9537:
9532:
9530:
9526:
9525:
9504:
9503:
9478:
9477:
9458:
9457:
9436:
9435:
9410:
9409:
9380:
9378:
9377:
9372:
9370:
9369:
9364:
9363:
9353:
9352:
9340:
9339:
9323:
9321:
9320:
9315:
9307:
9306:
9294:
9293:
9277:
9275:
9274:
9269:
9261:
9260:
9248:
9247:
9231:
9229:
9228:
9223:
9215:
9214:
9209:
9208:
9198:
9197:
9192:
9191:
9177:
9175:
9174:
9169:
9157:
9155:
9154:
9149:
9137:
9135:
9134:
9129:
9117:
9115:
9114:
9109:
9107:
9106:
9101:
9088:
9086:
9085:
9080:
9078:
9077:
9061:
9059:
9058:
9053:
9041:
9039:
9038:
9033:
9031:
9030:
9014:
9012:
9011:
9006:
8994:
8992:
8991:
8986:
8984:
8983:
8964:
8962:
8961:
8956:
8951:
8950:
8945:
8929:
8927:
8923:
8922:
8912:
8905:
8904:
8894:
8860:
8858:
8857:
8852:
8832:
8830:
8829:
8824:
8812:
8810:
8809:
8804:
8791:
8789:
8788:
8783:
8771:
8769:
8768:
8763:
8748:
8746:
8745:
8740:
8726:
8725:
8710:
8709:
8700:
8699:
8687:
8686:
8681:
8677:
8670:
8669:
8651:
8650:
8630:upper Hessenberg
8627:
8625:
8624:
8619:
8607:
8605:
8604:
8599:
8572:
8570:
8569:
8564:
8556:
8555:
8533:
8531:
8530:
8525:
8517:
8516:
8494:
8492:
8491:
8486:
8484:
8483:
8461:
8459:
8458:
8453:
8451:
8450:
8432:
8431:
8415:
8413:
8412:
8407:
8389:
8387:
8386:
8381:
8376:
8371:
8370:
8355:
8350:
8345:
8344:
8343:
8330:
8325:
8315:
8310:
8309:
8300:
8299:
8290:
8289:
8271:
8270:
8245:
8243:
8242:
8237:
8216:
8214:
8213:
8208:
8206:
8205:
8181:
8176:
8175:
8154:
8149:
8144:
8143:
8142:
8123:
8118:
8108:
8103:
8102:
8093:
8092:
8083:
8082:
8058:
8057:
8026:
8024:
8023:
8018:
7990:
7988:
7987:
7982:
7980:
7979:
7963:
7961:
7960:
7955:
7950:
7949:
7925:
7924:
7903:
7902:
7883:
7881:
7880:
7875:
7848:
7846:
7845:
7840:
7835:
7834:
7825:
7824:
7815:
7814:
7796:
7795:
7786:
7785:
7775:
7770:
7758:
7757:
7744:
7739:
7723:
7707:
7702:
7690:
7689:
7664:
7662:
7661:
7656:
7641:
7639:
7638:
7633:
7631:
7630:
7608:
7606:
7605:
7600:
7580:
7578:
7577:
7572:
7570:
7569:
7551:
7550:
7534:
7532:
7531:
7526:
7514:
7512:
7511:
7506:
7504:
7503:
7481:
7479:
7478:
7473:
7470:
7465:
7450:
7449:
7430:
7428:
7427:
7422:
7414:
7413:
7395:
7394:
7373:
7372:
7356:
7354:
7353:
7348:
7346:
7345:
7326:
7324:
7323:
7318:
7310:
7309:
7288:
7287:
7271:
7266:
7248:
7247:
7226:
7225:
7209:
7204:
7182:
7166:
7161:
7136:
7134:
7133:
7128:
7122:
7106:
7101:
7089:
7088:
7066:
7064:
7063:
7058:
7053:
7052:
7031:
7030:
6998:
6996:
6995:
6990:
6972:
6970:
6969:
6964:
6962:
6961:
6952:
6951:
6935:
6924:
6906:
6905:
6883:
6881:
6880:
6875:
6873:
6872:
6842:
6840:
6839:
6834:
6832:
6831:
6813:
6812:
6796:
6794:
6793:
6788:
6776:
6774:
6773:
6768:
6766:
6765:
6740:
6738:
6737:
6732:
6730:
6729:
6708:
6703:
6702:
6684:
6683:
6661:
6659:
6658:
6653:
6645:
6644:
6613:
6611:
6610:
6605:
6600:
6599:
6578:
6577:
6546:
6544:
6543:
6538:
6536:
6535:
6526:
6525:
6509:
6504:
6482:
6464:
6463:
6438:
6436:
6435:
6430:
6424:
6408:
6403:
6391:
6390:
6368:
6366:
6365:
6360:
6327:
6325:
6324:
6319:
6317:
6316:
6297:
6272:
6270:
6269:
6264:
6225:
6223:
6222:
6217:
6205:
6203:
6202:
6197:
6195:
6194:
6174:
6172:
6171:
6166:
6161:
6160:
6142:
6141:
6116:
6114:
6113:
6108:
6106:
6105:
6086:
6064:
6062:
6061:
6056:
6054:
6053:
6034:
6012:
6010:
6009:
6004:
5999:
5998:
5974:
5973:
5952:
5951:
5939:
5938:
5922:
5920:
5919:
5914:
5912:
5911:
5895:
5893:
5892:
5887:
5885:
5884:
5864:
5862:
5861:
5856:
5854:
5853:
5844:
5843:
5827:
5822:
5804:
5803:
5785:
5784:
5763:
5762:
5753:
5752:
5736:
5731:
5713:
5712:
5690:
5672:
5671:
5649:
5628:
5627:
5605:
5603:
5602:
5597:
5595:
5594:
5578:
5576:
5575:
5570:
5568:
5567:
5543:
5541:
5540:
5535:
5533:
5532:
5514:
5513:
5497:
5495:
5494:
5489:
5477:
5475:
5474:
5469:
5467:
5466:
5448:
5447:
5422:
5420:
5419:
5414:
5409:
5408:
5390:
5385:
5384:
5366:
5365:
5343:
5341:
5340:
5335:
5326:
5308:
5299:
5281:
5280:
5258:
5256:
5255:
5250:
5242:
5241:
5216:
5214:
5213:
5208:
5203:
5202:
5184:
5183:
5158:
5156:
5155:
5150:
5144:
5122:
5120:
5119:
5114:
5112:
5111:
5102:
5101:
5085:
5080:
5058:
5040:
5039:
5014:
5012:
5011:
5006:
5004:
5003:
4985:
4984:
4968:
4966:
4965:
4960:
4954:
4936:
4935:
4916:
4914:
4913:
4908:
4902:
4886:
4881:
4869:
4868:
4846:
4844:
4843:
4838:
4805:
4803:
4802:
4797:
4795:
4794:
4775:
4750:
4748:
4747:
4742:
4703:
4701:
4700:
4695:
4693:
4692:
4680:
4679:
4663:
4661:
4660:
4655:
4643:
4641:
4640:
4635:
4633:
4632:
4608:
4606:
4605:
4600:
4584:
4582:
4581:
4576:
4574:
4573:
4557:
4555:
4554:
4549:
4547:
4546:
4522:
4521:
4505:
4503:
4502:
4497:
4495:
4494:
4478:
4476:
4475:
4470:
4468:
4467:
4443:
4442:
4426:
4424:
4423:
4418:
4416:
4415:
4399:
4397:
4396:
4391:
4389:
4388:
4376:
4375:
4356:
4354:
4353:
4348:
4321:
4319:
4318:
4313:
4308:
4307:
4289:
4288:
4257:
4255:
4254:
4249:
4244:
4243:
4219:
4218:
4184:
4182:
4181:
4176:
4174:
4173:
4168:
4155:
4153:
4152:
4147:
4144:
4139:
4124:
4123:
4099:Krylov subspaces
4096:
4094:
4093:
4088:
4086:
4085:
4066:
4064:
4063:
4058:
4056:
4055:
4039:
4037:
4036:
4031:
4013:
4011:
4010:
4005:
4003:
4002:
3986:
3984:
3983:
3978:
3960:
3958:
3957:
3952:
3940:
3922:
3913:
3895:
3894:
3870:
3868:
3867:
3862:
3857:
3856:
3837:
3812:
3810:
3809:
3804:
3802:
3801:
3785:
3783:
3782:
3777:
3756:
3754:
3753:
3748:
3740:
3739:
3717:
3715:
3714:
3709:
3674:
3672:
3671:
3666:
3661:
3660:
3638:
3636:
3635:
3630:
3628:
3627:
3611:
3609:
3608:
3603:
3601:
3600:
3580:
3578:
3577:
3572:
3545:
3543:
3542:
3537:
3532:
3531:
3509:
3507:
3506:
3501:
3499:
3498:
3486:
3485:
3476:
3475:
3453:
3451:
3450:
3445:
3433:
3431:
3430:
3425:
3423:
3422:
3410:
3409:
3400:
3399:
3383:
3381:
3380:
3375:
3360:
3358:
3357:
3352:
3333:
3331:
3330:
3325:
3285:
3283:
3282:
3277:
3259:
3257:
3256:
3251:
3239:
3237:
3236:
3231:
3219:
3217:
3216:
3211:
3199:
3197:
3196:
3191:
3176:
3174:
3173:
3168:
3156:
3154:
3153:
3148:
3132:
3130:
3129:
3124:
3108:
3106:
3105:
3100:
3088:
3086:
3085:
3080:
3068:
3066:
3065:
3060:
3037:
3035:
3034:
3029:
3027:
3026:
3010:
3008:
3007:
3002:
2997:
2996:
2965:
2963:
2962:
2957:
2924:
2922:
2921:
2916:
2911:
2910:
2888:
2886:
2885:
2880:
2861:
2859:
2858:
2853:
2832:
2830:
2829:
2824:
2819:
2818:
2785:
2783:
2782:
2777:
2759:
2757:
2756:
2751:
2735:
2733:
2732:
2727:
2715:
2713:
2712:
2707:
2693:
2691:
2690:
2685:
2683:
2664:
2645:
2620:
2601:
2579:
2569:
2568:
2547:
2508:
2506:
2505:
2500:
2488:
2486:
2485:
2480:
2459:
2457:
2456:
2451:
2427:
2425:
2424:
2419:
2407:
2405:
2404:
2399:
2387:
2385:
2384:
2379:
2343:
2341:
2340:
2335:
2333:
2332:
2316:
2314:
2313:
2308:
2306:
2305:
2289:
2287:
2286:
2281:
2279:
2278:
2256:
2254:
2253:
2248:
2236:
2234:
2233:
2228:
2226:
2225:
2203:
2201:
2200:
2195:
2193:
2192:
2176:
2174:
2173:
2168:
2166:
2165:
2149:
2147:
2146:
2141:
2135:
2115:
2113:
2112:
2107:
2105:
2104:
2085:
2083:
2082:
2077:
2059:
2057:
2056:
2051:
2046:
2045:
2020:
2018:
2017:
2012:
1985:
1983:
1982:
1977:
1965:
1963:
1962:
1957:
1933:
1931:
1930:
1925:
1901:
1899:
1898:
1893:
1885:
1884:
1868:
1866:
1865:
1860:
1858:
1857:
1837:
1835:
1834:
1829:
1805:
1803:
1802:
1797:
1795:
1794:
1779:
1778:
1766:
1765:
1756:
1755:
1743:
1742:
1727:
1726:
1704:
1692:
1691:
1665:
1663:
1662:
1657:
1655:
1654:
1647:
1646:
1635:
1634:
1624:
1623:
1622:
1613:
1612:
1601:
1600:
1583:
1582:
1566:
1565:
1564:
1561:
1559:
1558:
1532:
1531:
1528:
1527:
1520:
1519:
1508:
1507:
1497:
1494:
1493:
1492:
1490:
1489:
1478:
1477:
1466:
1465:
1448:
1447:
1446:
1444:
1443:
1432:
1431:
1401:
1399:
1398:
1393:
1391:
1390:
1372:
1371:
1355:
1353:
1352:
1347:
1330:
1328:
1327:
1322:
1320:
1319:
1304:
1303:
1291:
1290:
1281:
1280:
1264:
1252:
1251:
1232:
1230:
1229:
1224:
1222:
1221:
1211:
1210:
1201:
1189:
1188:
1169:
1167:
1166:
1161:
1159:
1158:
1139:
1118:
1116:
1115:
1110:
1108:
1107:
1083:
1082:
1066:
1064:
1063:
1058:
1046:
1044:
1043:
1038:
1036:
1035:
1016:
1014:
1013:
1008:
1006:
1005:
996:
991:
990:
972:
971:
955:
953:
952:
947:
939:
938:
915:
913:
912:
907:
902:
901:
880:
879:
860:
858:
857:
852:
817:
815:
814:
809:
807:
806:
797:
796:
780:
768:
767:
748:
746:
745:
740:
738:
737:
727:
726:
717:
705:
704:
685:
683:
682:
677:
675:
674:
655:
633:
631:
630:
625:
610:
608:
607:
602:
600:
599:
594:
585:
584:
559:
557:
556:
551:
549:
548:
516:
514:
513:
508:
496:
494:
493:
488:
470:
468:
467:
462:
444:
442:
441:
436:
428:
427:
397:
395:
394:
389:
377:
375:
374:
369:
343:
341:
340:
335:
323:
321:
320:
315:
291:
289:
288:
283:
265:
263:
262:
257:
245:
243:
242:
237:
219:
217:
216:
211:
199:Hermitian matrix
184:
182:
181:
176:
148:
146:
145:
140:
128:
126:
125:
120:
107:Hermitian matrix
105:
103:
102:
97:
75:
73:
72:
67:
46:iterative method
19198:
19197:
19193:
19192:
19191:
19189:
19188:
19187:
19173:
19172:
19171:
19166:
19125:
19121:Multiprocessing
19089:
19085:Sparse problems
19048:
19027:
19022:
18987:
18972:
18952:
18934:
18907:
18905:Further reading
18902:
18897:Wayback Machine
18888:
18884:
18875:
18873:
18863:
18859:
18842:
18838:
18789:
18785:
18776:
18774:
18763:
18759:
18732:
18728:
18704:
18698:
18694:
18687:
18665:
18661:
18637:
18633:
18626:
18612:
18605:
18582:
18578:
18571:
18552:
18545:
18538:
18530:. Vol. 1.
18524:
18517:
18506:
18502:
18479:
18475:
18456:
18452:
18419:
18412:
18386:
18380:
18376:
18372:
18367:
18366:
18346:
18342:
18330:
18326:
18318:
18315:
18314:
18297:
18293:
18291:
18288:
18287:
18285:
18281:
18276:
18198:
18196:Implementations
18190:nuclear physics
18137:
18134:
18133:
18130:
18106:
18100:
18083:
18036:
18032:
18017:
18013:
18004:
18000:
17998:
17995:
17994:
17983:
17952:
17949:
17948:
17930:
17913:
17910:
17909:
17884:
17881:
17880:
17855:
17852:
17851:
17829:
17826:
17825:
17800:
17796:
17794:
17768:
17765:
17764:
17739:
17736:
17735:
17715:
17711:
17694:
17691:
17690:
17667:
17663:
17654:
17650:
17641:
17637:
17619:
17615:
17606:
17602:
17600:
17597:
17596:
17570:
17566:
17560:
17556:
17547:
17543:
17534:
17530:
17506:
17502:
17500:
17497:
17496:
17460:
17455:
17441:
17437:
17430:
17426:
17417:
17413:
17412:
17410:
17403:
17398:
17383:
17379:
17370:
17366:
17354:
17350:
17349:
17343:
17339:
17337:
17326:
17322:
17316:
17312:
17310:
17308:
17305:
17304:
17285:
17282:
17281:
17264:
17260:
17258:
17255:
17254:
17231:
17227:
17215:
17211:
17202:
17198:
17186:
17182:
17172:
17168:
17164:
17158:
17154:
17137:
17134:
17133:
17113:
17109:
17097:
17093:
17091:
17088:
17087:
17070:
17066:
17064:
17061:
17060:
17043:
17039:
17037:
17034:
17033:
17017:
17011:
17007:
17002:
16994:
16988:
16984:
16979:
16977:
16974:
16973:
16957:
16954:
16953:
16930:
16926:
16924:
16921:
16920:
16904:
16901:
16900:
16883:
16882:
16858:
16854:
16845:
16841:
16832:
16828:
16816:
16811:
16810:
16804:
16800:
16795:
16794:
16787:
16782:
16781:
16775:
16771:
16766:
16759:
16757:
16745:
16744:
16736:
16710:
16706:
16691:
16687:
16686:
16682:
16681:
16676:
16667:
16663:
16654:
16650:
16638:
16633:
16632:
16626:
16622:
16617:
16616:
16609:
16604:
16603:
16597:
16593:
16588:
16581:
16579:
16570:
16569:
16512:
16508:
16507:
16502:
16493:
16489:
16480:
16476:
16464:
16459:
16458:
16452:
16448:
16443:
16442:
16435:
16430:
16429:
16423:
16419:
16414:
16407:
16405:
16396:
16395:
16386:
16381:
16380:
16374:
16370:
16365:
16359:
16355:
16328:
16324:
16323:
16311:
16306:
16305:
16299:
16295:
16290:
16283:
16279:
16270:
16266:
16257:
16253:
16249:
16247:
16240:
16234:
16230:
16221:
16217:
16213:
16211:
16208:
16207:
16173:
16169:
16167:
16119:
16115:
16107:
16104:
16103:
16069:
16065:
16056:
16052:
16051:
16044:
16040:
16031:
16027:
16026:
16024:
16016:
16013:
16012:
15972:
15968:
15959:
15955:
15954:
15947:
15943:
15934:
15930:
15929:
15927:
15923:
15919:
15907:
15903:
15887:
15883:
15874:
15870:
15869:
15862:
15858:
15849:
15845:
15836:
15832:
15828:
15826:
15822:
15810:
15806:
15794:
15790:
15782:
15779:
15778:
15749:
15744:
15743:
15737:
15733:
15728:
15722:
15717:
15716:
15707:
15703:
15692:
15691:
15684:
15679:
15678:
15672:
15668:
15663:
15657:
15646:
15633:
15629:
15620:
15616:
15612:
15610:
15598:
15593:
15592:
15583:
15579:
15568:
15562:
15557:
15556:
15550:
15546:
15541:
15540:
15530:
15526:
15517:
15513:
15504:
15499:
15498:
15492:
15488:
15483:
15477:
15466:
15461:
15459:
15447:
15442:
15441:
15432:
15428:
15417:
15411:
15406:
15405:
15399:
15395:
15390:
15384:
15373:
15368:
15361:
15356:
15355:
15346:
15342:
15331:
15322:
15318:
15309:
15305:
15296:
15291:
15290:
15284:
15280:
15275:
15269:
15258:
15253:
15251:
15239:
15235:
15208:
15204:
15195:
15191:
15182:
15178:
15176:
15173:
15172:
15153:
15150:
15149:
15133:
15130:
15129:
15107:
15104:
15103:
15086:
15081:
15080:
15071:
15067:
15056:
15054:
15051:
15050:
15033:
15029:
15027:
15024:
15023:
15006:
15002:
14993:
14989:
14987:
14984:
14983:
14956:
14952:
14943:
14939:
14938:
14931:
14927:
14918:
14914:
14904:
14902:
14898:
14886:
14882:
14865:
14862:
14861:
14827:
14824:
14823:
14806:
14802:
14800:
14797:
14796:
14765:
14762:
14761:
14730:
14727:
14726:
14710:
14707:
14706:
14653:
14649:
14647:
14644:
14643:
14627:
14624:
14623:
14622:for the degree
14606:
14602:
14600:
14597:
14596:
14576:
14573:
14572:
14556:
14553:
14552:
14532:
14528:
14519:
14515:
14513:
14510:
14509:
14492:
14488:
14486:
14483:
14482:
14481:to be large at
14466:
14463:
14462:
14440:
14437:
14436:
14410:
14396:
14392:
14385:
14381:
14380:
14374:
14369:
14368:
14362:
14358:
14353:
14347:
14336:
14331:
14324:
14310:
14306:
14299:
14295:
14294:
14285:
14281:
14272:
14268:
14259:
14254:
14253:
14247:
14243:
14238:
14232:
14221:
14216:
14214:
14199:
14195:
14183:
14179:
14173:
14169:
14157:
14152:
14151:
14145:
14141:
14136:
14130:
14119:
14114:
14104:
14100:
14088:
14084:
14078:
14074:
14062:
14058:
14052:
14047:
14046:
14040:
14036:
14031:
14025:
14014:
14009:
14007:
13998:
13994:
13982:
13978:
13969:
13965:
13950:
13946:
13937:
13933:
13927:
13923:
13917:
13906:
13896:
13891:
13886:
13879:
13875:
13866:
13862:
13850:
13846:
13837:
13833:
13824:
13820:
13814:
13810:
13804:
13793:
13783:
13778:
13773:
13771:
13762:
13758:
13746:
13742:
13715:
13711:
13709:
13706:
13705:
13683:
13680:
13679:
13659:
13655:
13646:
13642:
13630:
13626:
13620:
13609:
13596:
13592:
13578:
13575:
13574:
13551:
13547:
13541:
13537:
13531:
13527:
13521:
13510:
13497:
13493:
13487:
13483:
13477:
13466:
13450:
13446:
13441:
13438:
13437:
13417:
13413:
13411:
13408:
13407:
13384:
13380:
13353:
13349:
13343:
13338:
13333:
13326:
13322:
13292:
13288:
13282:
13277:
13272:
13270:
13258:
13254:
13233:
13229:
13220:
13216:
13210:
13205:
13200:
13193:
13189:
13165:
13161:
13152:
13148:
13142:
13137:
13132:
13130:
13118:
13114:
13096:
13092:
13077:
13072:
13067:
13060:
13056:
13035:
13031:
13016:
13011:
13006:
13004:
12992:
12988:
12970:
12966:
12960:
12956:
12940:
12933:
12929:
12908:
12904:
12898:
12894:
12878:
12876:
12864:
12860:
12840:
12837:
12836:
12817:
12814:
12813:
12796:
12792:
12790:
12787:
12786:
12769:
12765:
12753:
12749:
12734:
12730:
12724:
12720:
12711:
12707:
12695:
12691:
12689:
12686:
12685:
12663:
12660:
12659:
12643:
12640:
12639:
12622:
12618:
12604:
12601:
12600:
12572:
12568:
12556:
12552:
12537:
12533:
12527:
12523:
12514:
12510:
12498:
12494:
12493:
12489:
12481:
12478:
12477:
12458:
12455:
12454:
12434:
12430:
12428:
12425:
12424:
12392:
12388:
12386:
12383:
12382:
12365:
12361:
12359:
12356:
12355:
12330:
12327:
12326:
12310:
12307:
12306:
12289:
12285:
12276:
12272:
12270:
12267:
12266:
12250:
12247:
12246:
12229:
12225:
12223:
12220:
12219:
12199:
12194:
12193:
12191:
12188:
12187:
12171:
12168:
12167:
12150:
12146:
12144:
12141:
12140:
12115:
12112:
12111:
12094:
12090:
12088:
12085:
12084:
12083:The bounds for
12065:
12062:
12061:
12044:
12040:
12025:
12021:
12019:
12016:
12015:
11999:
11996:
11995:
11978:
11974:
11959:
11955:
11953:
11950:
11949:
11933:
11930:
11929:
11912:
11908:
11906:
11903:
11902:
11885:
11881:
11879:
11876:
11875:
11858:
11854:
11852:
11849:
11848:
11831:
11827:
11825:
11822:
11821:
11801:
11797:
11782:
11778:
11769:
11765:
11763:
11760:
11759:
11743:
11740:
11739:
11717:
11714:
11713:
11697:
11694:
11693:
11677:
11674:
11673:
11670:
11649:
11646:
11645:
11628:
11624:
11622:
11619:
11618:
11602:
11599:
11598:
11572:
11566:
11562:
11557:
11555:
11552:
11551:
11534:
11529:
11528:
11526:
11523:
11522:
11498:
11494:
11479:
11475:
11470:
11467:
11466:
11449:
11445:
11443:
11440:
11439:
11423:
11420:
11419:
11403:
11400:
11399:
11378:
11374:
11372:
11369:
11368:
11350:
11346:
11344:
11341:
11340:
11322:
11318:
11312:
11308:
11302:
11291:
11278:
11274:
11271:
11268:
11267:
11233:
11230:
11229:
11212:
11208:
11202:
11197:
11184:
11180:
11178:
11175:
11174:
11157:
11153:
11151:
11148:
11147:
11110:
11107:
11106:
11089:
11085:
11079:
11075:
11066:
11062:
11057:
11054:
11053:
11036:
11032:
11017:
11013:
11011:
11008:
11007:
10990:
10986:
10971:
10967:
10958:
10954:
10952:
10949:
10948:
10932:
10929:
10928:
10925:
10900:
10896:
10881:
10877:
10872:
10869:
10868:
10851:
10845:
10844:
10843:
10841:
10838:
10837:
10821:
10818:
10817:
10800:
10796:
10794:
10791:
10790:
10773:
10769:
10767:
10764:
10763:
10743:
10737:
10736:
10735:
10733:
10730:
10729:
10712:
10708:
10706:
10703:
10702:
10686:
10683:
10682:
10645:
10639:
10638:
10637:
10630:
10614:
10610:
10598:
10577:
10571:
10570:
10569:
10562:
10546:
10542:
10534:
10531:
10530:
10510:
10504:
10503:
10502:
10493:
10489:
10480:
10476:
10474:
10471:
10470:
10447:
10443:
10431:
10427:
10412:
10408:
10396:
10392:
10374:
10368:
10367:
10366:
10364:
10361:
10360:
10337:
10333:
10324:
10320:
10318:
10315:
10314:
10294:
10290:
10282:
10279:
10278:
10261:
10257:
10249:
10246:
10245:
10225:
10219:
10218:
10217:
10209:
10206:
10205:
10182:
10176:
10175:
10174:
10163:
10160:
10159:
10142:
10131:
10121:
10115:
10114:
10113:
10108:
10105:
10104:
10088:
10085:
10084:
10067:
10061:
10060:
10059:
10057:
10054:
10053:
10036:
10032:
10027:
10024:
10023:
10006:
10002:
9997:
9994:
9993:
9976:
9972:
9970:
9967:
9966:
9949:
9945:
9943:
9940:
9939:
9919:
9915:
9910:
9907:
9906:
9889:
9885:
9880:
9877:
9876:
9859:
9855:
9853:
9850:
9849:
9832:
9828:
9826:
9823:
9822:
9763:
9759:
9758:
9753:
9733:
9730:
9729:
9706:
9702:
9688:
9685:
9684:
9668:
9665:
9664:
9647:
9643:
9641:
9638:
9637:
9617:
9613:
9602:
9599:
9598:
9594:is that of the
9579:
9576:
9575:
9558:
9554:
9552:
9549:
9548:
9528:
9527:
9521:
9517:
9499:
9495:
9482:
9473:
9469:
9460:
9459:
9453:
9449:
9431:
9427:
9414:
9405:
9401:
9391:
9389:
9386:
9385:
9365:
9359:
9358:
9357:
9348:
9344:
9335:
9331:
9329:
9326:
9325:
9302:
9298:
9289:
9285:
9283:
9280:
9279:
9256:
9252:
9243:
9239:
9237:
9234:
9233:
9210:
9204:
9203:
9202:
9193:
9187:
9186:
9185:
9183:
9180:
9179:
9163:
9160:
9159:
9143:
9140:
9139:
9123:
9120:
9119:
9102:
9097:
9096:
9094:
9091:
9090:
9073:
9072:
9070:
9067:
9066:
9047:
9044:
9043:
9026:
9022:
9020:
9017:
9016:
9000:
8997:
8996:
8979:
8975:
8973:
8970:
8969:
8946:
8941:
8940:
8918:
8914:
8913:
8900:
8896:
8895:
8893:
8876:
8873:
8872:
8846:
8843:
8842:
8839:
8818:
8815:
8814:
8798:
8795:
8794:
8777:
8774:
8773:
8757:
8754:
8753:
8721:
8717:
8705:
8701:
8695:
8691:
8682:
8665:
8661:
8660:
8656:
8655:
8646:
8642:
8640:
8637:
8636:
8613:
8610:
8609:
8578:
8575:
8574:
8545:
8541:
8539:
8536:
8535:
8512:
8508:
8500:
8497:
8496:
8473:
8469:
8467:
8464:
8463:
8446:
8442:
8427:
8423:
8421:
8418:
8417:
8401:
8398:
8397:
8360:
8356:
8354:
8339:
8335:
8326:
8321:
8316:
8314:
8305:
8301:
8295:
8291:
8285:
8281:
8260:
8256:
8254:
8251:
8250:
8225:
8222:
8221:
8189:
8185:
8159:
8155:
8153:
8132:
8128:
8119:
8114:
8109:
8107:
8098:
8094:
8088:
8084:
8072:
8068:
8041:
8037:
8035:
8032:
8031:
8000:
7997:
7996:
7975:
7971:
7969:
7966:
7965:
7939:
7935:
7920:
7916:
7898:
7894:
7889:
7886:
7885:
7857:
7854:
7853:
7830:
7826:
7820:
7816:
7810:
7806:
7791:
7787:
7781:
7777:
7771:
7766:
7753:
7749:
7740:
7735:
7713:
7703:
7698:
7679:
7675:
7673:
7670:
7669:
7650:
7647:
7646:
7620:
7616:
7614:
7611:
7610:
7594:
7591:
7590:
7565:
7561:
7546:
7542:
7540:
7537:
7536:
7520:
7517:
7516:
7493:
7489:
7487:
7484:
7483:
7466:
7455:
7445:
7441:
7436:
7433:
7432:
7409:
7405:
7390:
7386:
7368:
7364:
7362:
7359:
7358:
7341:
7337:
7335:
7332:
7331:
7299:
7295:
7277:
7273:
7267:
7256:
7237:
7233:
7215:
7211:
7205:
7194:
7172:
7162:
7151:
7145:
7142:
7141:
7112:
7102:
7097:
7078:
7074:
7072:
7069:
7068:
7042:
7038:
7014:
7010:
7008:
7005:
7004:
7003:the definition
6978:
6975:
6974:
6957:
6953:
6941:
6937:
6925:
6914:
6901:
6897:
6892:
6889:
6888:
6862:
6858:
6856:
6853:
6852:
6827:
6823:
6808:
6804:
6802:
6799:
6798:
6782:
6779:
6778:
6755:
6751:
6749:
6746:
6745:
6713:
6709:
6704:
6692:
6688:
6673:
6669:
6667:
6664:
6663:
6628:
6624:
6622:
6619:
6618:
6589:
6585:
6561:
6557:
6555:
6552:
6551:
6531:
6527:
6515:
6511:
6505:
6494:
6472:
6453:
6449:
6447:
6444:
6443:
6414:
6404:
6399:
6380:
6376:
6374:
6371:
6370:
6336:
6333:
6332:
6312:
6308:
6287:
6281:
6278:
6277:
6234:
6231:
6230:
6211:
6208:
6207:
6190:
6186:
6184:
6181:
6180:
6156:
6152:
6137:
6133:
6122:
6119:
6118:
6101:
6097:
6076:
6070:
6067:
6066:
6049:
6045:
6024:
6018:
6015:
6014:
5988:
5984:
5969:
5965:
5947:
5943:
5934:
5930:
5928:
5925:
5924:
5907:
5903:
5901:
5898:
5897:
5880:
5876:
5874:
5871:
5870:
5849:
5845:
5833:
5829:
5823:
5812:
5793:
5789:
5774:
5770:
5758:
5754:
5742:
5738:
5732:
5721:
5702:
5698:
5680:
5661:
5657:
5639:
5623:
5619:
5614:
5611:
5610:
5590:
5586:
5584:
5581:
5580:
5563:
5559:
5557:
5554:
5553:
5528:
5524:
5509:
5505:
5503:
5500:
5499:
5483:
5480:
5479:
5456:
5452:
5437:
5433:
5431:
5428:
5427:
5398:
5394:
5386:
5374:
5370:
5355:
5351:
5349:
5346:
5345:
5316:
5304:
5289:
5270:
5266:
5264:
5261:
5260:
5231:
5227:
5225:
5222:
5221:
5198:
5194:
5179:
5175:
5164:
5161:
5160:
5134:
5128:
5125:
5124:
5107:
5103:
5091:
5087:
5081:
5070:
5048:
5029:
5025:
5023:
5020:
5019:
4999:
4995:
4980:
4976:
4974:
4971:
4970:
4944:
4931:
4927:
4922:
4919:
4918:
4892:
4882:
4877:
4858:
4854:
4852:
4849:
4848:
4814:
4811:
4810:
4790:
4786:
4765:
4759:
4756:
4755:
4712:
4709:
4708:
4688:
4684:
4675:
4671:
4669:
4666:
4665:
4649:
4646:
4645:
4628:
4624:
4622:
4619:
4618:
4594:
4591:
4590:
4587:ill-conditioned
4569:
4565:
4563:
4560:
4559:
4536:
4532:
4517:
4513:
4511:
4508:
4507:
4490:
4486:
4484:
4481:
4480:
4457:
4453:
4438:
4434:
4432:
4429:
4428:
4411:
4407:
4405:
4402:
4401:
4384:
4380:
4371:
4367:
4365:
4362:
4361:
4327:
4324:
4323:
4303:
4299:
4284:
4280:
4263:
4260:
4259:
4233:
4229:
4214:
4210:
4190:
4187:
4186:
4169:
4164:
4163:
4161:
4158:
4157:
4140:
4129:
4119:
4115:
4110:
4107:
4106:
4081:
4077:
4075:
4072:
4071:
4051:
4047:
4045:
4042:
4041:
4025:
4022:
4021:
3998:
3994:
3992:
3989:
3988:
3972:
3969:
3968:
3930:
3918:
3903:
3884:
3880:
3878:
3875:
3874:
3852:
3848:
3827:
3821:
3818:
3817:
3797:
3793:
3791:
3788:
3787:
3765:
3762:
3761:
3735:
3731:
3729:
3726:
3725:
3703:
3700:
3699:
3696:
3694:Power iteration
3690:
3682:
3656:
3652:
3644:
3641:
3640:
3623:
3619:
3617:
3614:
3613:
3596:
3592:
3590:
3587:
3586:
3583:synchronisation
3557:
3554:
3553:
3527:
3523:
3515:
3512:
3511:
3494:
3490:
3481:
3477:
3471:
3467:
3459:
3456:
3455:
3439:
3436:
3435:
3418:
3414:
3405:
3401:
3395:
3391:
3389:
3386:
3385:
3369:
3366:
3365:
3346:
3343:
3342:
3319:
3316:
3315:
3300:
3265:
3262:
3261:
3245:
3242:
3241:
3225:
3222:
3221:
3205:
3202:
3201:
3185:
3182:
3181:
3162:
3159:
3158:
3142:
3139:
3138:
3118:
3115:
3114:
3094:
3091:
3090:
3074:
3071:
3070:
3054:
3051:
3050:
3022:
3018:
3016:
3013:
3012:
2992:
2988:
2980:
2977:
2976:
2933:
2930:
2929:
2906:
2902:
2894:
2891:
2890:
2874:
2871:
2870:
2838:
2835:
2834:
2814:
2810:
2802:
2799:
2798:
2765:
2762:
2761:
2745:
2742:
2741:
2721:
2718:
2717:
2701:
2698:
2697:
2681:
2680:
2662:
2661:
2643:
2642:
2618:
2617:
2599:
2598:
2577:
2576:
2564:
2560:
2545:
2544:
2528:
2518:
2516:
2513:
2512:
2494:
2491:
2490:
2465:
2462:
2461:
2433:
2430:
2429:
2413:
2410:
2409:
2393:
2390:
2389:
2373:
2370:
2369:
2350:
2328:
2324:
2322:
2319:
2318:
2301:
2297:
2295:
2292:
2291:
2268:
2264:
2262:
2259:
2258:
2242:
2239:
2238:
2215:
2211:
2209:
2206:
2205:
2188:
2184:
2182:
2179:
2178:
2161:
2157:
2155:
2152:
2151:
2131:
2125:
2122:
2121:
2118:Lanczos vectors
2100:
2096:
2094:
2091:
2090:
2065:
2062:
2061:
2041:
2037:
2026:
2023:
2022:
1991:
1988:
1987:
1971:
1968:
1967:
1939:
1936:
1935:
1910:
1907:
1906:
1880:
1876:
1874:
1871:
1870:
1853:
1849:
1847:
1844:
1843:
1811:
1808:
1807:
1784:
1780:
1774:
1770:
1761:
1757:
1751:
1747:
1732:
1728:
1716:
1712:
1700:
1687:
1683:
1678:
1675:
1674:
1649:
1648:
1642:
1638:
1636:
1630:
1626:
1621:
1615:
1614:
1608:
1604:
1602:
1590:
1586:
1584:
1572:
1568:
1562:
1560:
1548:
1544:
1542:
1537:
1529:
1526:
1521:
1515:
1511:
1509:
1503:
1499:
1495:
1491:
1485:
1481:
1479:
1473:
1469:
1467:
1461:
1457:
1454:
1453:
1445:
1439:
1435:
1433:
1427:
1423:
1416:
1415:
1407:
1404:
1403:
1386:
1382:
1367:
1363:
1361:
1358:
1357:
1341:
1338:
1337:
1309:
1305:
1299:
1295:
1286:
1282:
1276:
1272:
1260:
1247:
1243:
1241:
1238:
1237:
1217:
1213:
1206:
1202:
1197:
1184:
1180:
1178:
1175:
1174:
1154:
1150:
1135:
1129:
1126:
1125:
1097:
1093:
1078:
1074:
1072:
1069:
1068:
1052:
1049:
1048:
1031:
1027:
1025:
1022:
1021:
1001:
997:
992:
980:
976:
967:
963:
961:
958:
957:
934:
930:
928:
925:
924:
891:
887:
875:
871:
869:
866:
865:
828:
825:
824:
802:
798:
792:
788:
776:
763:
759:
757:
754:
753:
733:
729:
722:
718:
713:
700:
696:
694:
691:
690:
670:
666:
651:
645:
642:
641:
619:
616:
615:
595:
590:
589:
580:
576:
574:
571:
570:
544:
540:
526:
523:
522:
502:
499:
498:
476:
473:
472:
450:
447:
446:
423:
419:
411:
408:
407:
383:
380:
379:
357:
354:
353:
329:
326:
325:
300:
297:
296:
271:
268:
267:
251:
248:
247:
225:
222:
221:
205:
202:
201:
191:
170:
167:
166:
134:
131:
130:
114:
111:
110:
85:
82:
81:
61:
58:
57:
38:
31:
24:
17:
12:
11:
5:
19196:
19186:
19185:
19168:
19167:
19165:
19164:
19159:
19154:
19149:
19144:
19139:
19133:
19131:
19127:
19126:
19124:
19123:
19118:
19113:
19108:
19103:
19097:
19095:
19091:
19090:
19088:
19087:
19082:
19077:
19067:
19062:
19056:
19054:
19050:
19049:
19047:
19046:
19041:
19039:Floating point
19035:
19033:
19029:
19028:
19021:
19020:
19013:
19006:
18998:
18992:
18991:
18985:
18965:
18938:
18932:
18912:Golub, Gene H.
18906:
18903:
18901:
18900:
18882:
18857:
18836:
18783:
18757:
18726:
18692:
18685:
18659:
18631:
18624:
18603:
18592:(3): 379–414.
18576:
18569:
18557:(1992-06-22).
18543:
18536:
18515:
18500:
18489:(3): 373–381.
18473:
18450:
18445:10.2514/3.5878
18410:
18397:(4): 255–282.
18373:
18371:
18368:
18365:
18364:
18349:
18345:
18341:
18338:
18333:
18329:
18325:
18322:
18300:
18296:
18278:
18277:
18275:
18272:
18233:Similarly, in
18197:
18194:
18159:HITS algorithm
18141:
18129:
18126:
18102:Main article:
18099:
18096:
18082:
18079:
18078:
18077:
18074:
18071:
18058:constructs an
18045:
18042:
18039:
18035:
18031:
18028:
18025:
18020:
18016:
18012:
18007:
18003:
17982:
17979:
17962:
17959:
17956:
17934:
17929:
17926:
17923:
17920:
17917:
17897:
17894:
17891:
17888:
17868:
17865:
17862:
17859:
17839:
17836:
17833:
17811:
17808:
17803:
17799:
17793:
17790:
17787:
17784:
17781:
17778:
17775:
17772:
17752:
17749:
17746:
17743:
17721:
17718:
17714:
17710:
17707:
17704:
17701:
17698:
17687:
17686:
17675:
17670:
17666:
17662:
17657:
17653:
17649:
17644:
17640:
17636:
17633:
17630:
17627:
17622:
17618:
17614:
17609:
17605:
17590:
17589:
17578:
17573:
17569:
17563:
17559:
17555:
17550:
17546:
17542:
17537:
17533:
17529:
17526:
17523:
17520:
17517:
17514:
17509:
17505:
17490:
17489:
17478:
17472:
17469:
17466:
17463:
17459:
17454:
17444:
17440:
17433:
17429:
17425:
17420:
17416:
17409:
17406:
17402:
17397:
17391:
17386:
17382:
17378:
17373:
17369:
17365:
17362:
17357:
17353:
17346:
17342:
17336:
17329:
17325:
17319:
17315:
17289:
17267:
17263:
17251:
17250:
17239:
17234:
17230:
17226:
17223:
17218:
17214:
17210:
17205:
17201:
17197:
17194:
17189:
17185:
17179:
17175:
17171:
17167:
17161:
17157:
17153:
17150:
17147:
17144:
17141:
17116:
17112:
17108:
17105:
17100:
17096:
17073:
17069:
17046:
17042:
17020:
17014:
17010:
17005:
17001:
16997:
16991:
16987:
16982:
16961:
16936:
16933:
16929:
16908:
16897:
16896:
16879:
16876:
16873:
16870:
16867:
16864:
16861:
16857:
16853:
16848:
16844:
16840:
16835:
16831:
16827:
16819:
16814:
16807:
16803:
16798:
16790:
16785:
16778:
16774:
16769:
16765:
16762:
16756:
16753:
16750:
16748:
16746:
16739:
16734:
16728:
16725:
16722:
16719:
16716:
16713:
16709:
16705:
16700:
16697:
16694:
16690:
16685:
16680:
16675:
16670:
16666:
16662:
16657:
16653:
16649:
16641:
16636:
16629:
16625:
16620:
16612:
16607:
16600:
16596:
16591:
16587:
16584:
16578:
16575:
16573:
16571:
16565:
16562:
16559:
16556:
16553:
16550:
16547:
16544:
16541:
16538:
16535:
16532:
16529:
16526:
16523:
16520:
16515:
16511:
16506:
16501:
16496:
16492:
16488:
16483:
16479:
16475:
16467:
16462:
16455:
16451:
16446:
16438:
16433:
16426:
16422:
16417:
16413:
16410:
16404:
16401:
16399:
16397:
16389:
16384:
16377:
16373:
16368:
16362:
16358:
16354:
16351:
16348:
16345:
16342:
16337:
16334:
16331:
16327:
16320:
16314:
16309:
16302:
16298:
16293:
16289:
16286:
16282:
16278:
16273:
16269:
16265:
16260:
16256:
16252:
16246:
16243:
16241:
16237:
16233:
16229:
16224:
16220:
16216:
16215:
16201:
16200:
16189:
16184:
16181:
16176:
16172:
16166:
16163:
16160:
16157:
16154:
16151:
16148:
16143:
16140:
16137:
16134:
16131:
16128:
16125:
16122:
16118:
16114:
16111:
16089:
16088:
16072:
16068:
16064:
16059:
16055:
16047:
16043:
16039:
16034:
16030:
16023:
16020:
16008:the quantity
16006:
16005:
15994:
15990:
15986:
15983:
15975:
15971:
15967:
15962:
15958:
15950:
15946:
15942:
15937:
15933:
15926:
15922:
15916:
15913:
15910:
15906:
15902:
15898:
15890:
15886:
15882:
15877:
15873:
15865:
15861:
15857:
15852:
15848:
15844:
15839:
15835:
15831:
15825:
15819:
15816:
15813:
15809:
15805:
15802:
15797:
15793:
15789:
15786:
15772:
15771:
15760:
15752:
15747:
15740:
15736:
15731:
15725:
15720:
15715:
15710:
15706:
15702:
15699:
15695:
15687:
15682:
15675:
15671:
15666:
15660:
15655:
15652:
15649:
15645:
15641:
15636:
15632:
15628:
15623:
15619:
15615:
15609:
15601:
15596:
15591:
15586:
15582:
15578:
15575:
15571:
15565:
15560:
15553:
15549:
15544:
15538:
15533:
15529:
15525:
15520:
15516:
15512:
15507:
15502:
15495:
15491:
15486:
15480:
15475:
15472:
15469:
15465:
15458:
15450:
15445:
15440:
15435:
15431:
15427:
15424:
15420:
15414:
15409:
15402:
15398:
15393:
15387:
15382:
15379:
15376:
15372:
15364:
15359:
15354:
15349:
15345:
15341:
15338:
15334:
15330:
15325:
15321:
15317:
15312:
15308:
15304:
15299:
15294:
15287:
15283:
15278:
15272:
15267:
15264:
15261:
15257:
15250:
15247:
15242:
15238:
15234:
15231:
15228:
15225:
15222:
15219:
15216:
15211:
15207:
15203:
15198:
15194:
15190:
15185:
15181:
15157:
15137:
15117:
15114:
15111:
15089:
15084:
15079:
15074:
15070:
15066:
15063:
15059:
15036:
15032:
15009:
15005:
15001:
14996:
14992:
14980:
14979:
14967:
14959:
14955:
14951:
14946:
14942:
14934:
14930:
14926:
14921:
14917:
14913:
14910:
14907:
14901:
14895:
14892:
14889:
14885:
14881:
14878:
14875:
14872:
14869:
14846:
14843:
14840:
14837:
14834:
14831:
14809:
14805:
14784:
14781:
14778:
14775:
14772:
14769:
14749:
14746:
14743:
14740:
14737:
14734:
14714:
14694:
14691:
14688:
14685:
14682:
14679:
14676:
14673:
14670:
14667:
14664:
14661:
14656:
14652:
14631:
14609:
14605:
14580:
14560:
14535:
14531:
14527:
14522:
14518:
14495:
14491:
14470:
14450:
14447:
14444:
14433:
14432:
14421:
14413:
14408:
14404:
14399:
14395:
14391:
14388:
14384:
14377:
14372:
14365:
14361:
14356:
14350:
14345:
14342:
14339:
14335:
14327:
14322:
14318:
14313:
14309:
14305:
14302:
14298:
14293:
14288:
14284:
14280:
14275:
14271:
14267:
14262:
14257:
14250:
14246:
14241:
14235:
14230:
14227:
14224:
14220:
14213:
14207:
14202:
14198:
14194:
14191:
14186:
14182:
14176:
14172:
14168:
14165:
14160:
14155:
14148:
14144:
14139:
14133:
14128:
14125:
14122:
14118:
14112:
14107:
14103:
14099:
14096:
14091:
14087:
14081:
14077:
14073:
14070:
14065:
14061:
14055:
14050:
14043:
14039:
14034:
14028:
14023:
14020:
14017:
14013:
14006:
14001:
13997:
13993:
13985:
13981:
13977:
13972:
13968:
13964:
13961:
13958:
13953:
13949:
13945:
13940:
13936:
13930:
13926:
13920:
13915:
13912:
13909:
13905:
13899:
13894:
13890:
13882:
13878:
13874:
13869:
13865:
13861:
13858:
13853:
13849:
13845:
13840:
13836:
13832:
13827:
13823:
13817:
13813:
13807:
13802:
13799:
13796:
13792:
13786:
13781:
13777:
13770:
13765:
13761:
13757:
13754:
13749:
13745:
13741:
13738:
13735:
13732:
13729:
13726:
13723:
13718:
13714:
13687:
13676:
13675:
13662:
13658:
13654:
13649:
13645:
13641:
13638:
13633:
13629:
13623:
13618:
13615:
13612:
13608:
13604:
13599:
13595:
13591:
13588:
13585:
13582:
13568:
13567:
13554:
13550:
13544:
13540:
13534:
13530:
13524:
13519:
13516:
13513:
13509:
13505:
13500:
13496:
13490:
13486:
13480:
13475:
13472:
13469:
13465:
13461:
13458:
13453:
13449:
13445:
13420:
13416:
13404:
13403:
13387:
13383:
13379:
13376:
13373:
13370:
13367:
13364:
13361:
13356:
13352:
13346:
13341:
13337:
13329:
13325:
13321:
13318:
13315:
13312:
13309:
13306:
13303:
13300:
13295:
13291:
13285:
13280:
13276:
13269:
13261:
13257:
13253:
13250:
13247:
13244:
13241:
13236:
13232:
13228:
13223:
13219:
13213:
13208:
13204:
13196:
13192:
13188:
13185:
13182:
13179:
13176:
13173:
13168:
13164:
13160:
13155:
13151:
13145:
13140:
13136:
13129:
13121:
13117:
13113:
13110:
13107:
13104:
13099:
13095:
13091:
13088:
13085:
13080:
13075:
13071:
13063:
13059:
13055:
13052:
13049:
13046:
13043:
13038:
13034:
13030:
13027:
13024:
13019:
13014:
13010:
13003:
12995:
12991:
12987:
12984:
12981:
12978:
12973:
12969:
12963:
12959:
12955:
12952:
12949:
12946:
12943:
12936:
12932:
12928:
12925:
12922:
12919:
12916:
12911:
12907:
12901:
12897:
12893:
12890:
12887:
12884:
12881:
12875:
12872:
12867:
12863:
12859:
12856:
12853:
12850:
12847:
12844:
12821:
12799:
12795:
12772:
12768:
12762:
12759:
12756:
12752:
12748:
12745:
12742:
12737:
12733:
12727:
12723:
12719:
12714:
12710:
12706:
12703:
12698:
12694:
12673:
12670:
12667:
12647:
12625:
12621:
12617:
12614:
12611:
12608:
12597:
12596:
12585:
12581:
12575:
12571:
12565:
12562:
12559:
12555:
12551:
12548:
12545:
12540:
12536:
12530:
12526:
12522:
12517:
12513:
12509:
12506:
12501:
12497:
12492:
12488:
12485:
12462:
12453:The dimension
12437:
12433:
12412:
12409:
12406:
12403:
12400:
12395:
12391:
12368:
12364:
12343:
12340:
12337:
12334:
12314:
12292:
12288:
12284:
12279:
12275:
12254:
12232:
12228:
12207:
12202:
12197:
12175:
12153:
12149:
12128:
12125:
12122:
12119:
12097:
12093:
12069:
12047:
12043:
12039:
12036:
12033:
12028:
12024:
12003:
11981:
11977:
11973:
11970:
11967:
11962:
11958:
11937:
11915:
11911:
11888:
11884:
11861:
11857:
11834:
11830:
11809:
11804:
11800:
11796:
11793:
11790:
11785:
11781:
11777:
11772:
11768:
11747:
11727:
11724:
11721:
11701:
11681:
11669:
11666:
11653:
11631:
11627:
11606:
11582:
11579:
11575:
11569:
11565:
11560:
11537:
11532:
11506:
11501:
11497:
11493:
11490:
11487:
11482:
11478:
11474:
11452:
11448:
11427:
11407:
11381:
11377:
11353:
11349:
11325:
11321:
11315:
11311:
11305:
11300:
11297:
11294:
11290:
11286:
11281:
11277:
11255:
11252:
11249:
11246:
11243:
11240:
11237:
11215:
11211:
11205:
11200:
11196:
11192:
11187:
11183:
11160:
11156:
11132:
11129:
11126:
11123:
11120:
11117:
11114:
11092:
11088:
11082:
11078:
11074:
11069:
11065:
11061:
11039:
11035:
11031:
11028:
11025:
11020:
11016:
10993:
10989:
10985:
10982:
10979:
10974:
10970:
10966:
10961:
10957:
10936:
10924:
10921:
10908:
10903:
10899:
10895:
10892:
10889:
10884:
10880:
10876:
10854:
10848:
10825:
10803:
10799:
10776:
10772:
10751:
10746:
10740:
10715:
10711:
10690:
10679:
10678:
10667:
10664:
10661:
10658:
10655:
10648:
10642:
10636:
10633:
10629:
10625:
10622:
10617:
10613:
10609:
10606:
10596:
10593:
10590:
10587:
10580:
10574:
10568:
10565:
10561:
10557:
10554:
10549:
10545:
10541:
10538:
10513:
10507:
10501:
10496:
10492:
10488:
10483:
10479:
10469:and then seek
10467:
10466:
10455:
10450:
10446:
10440:
10437:
10434:
10430:
10426:
10423:
10420:
10415:
10411:
10407:
10404:
10399:
10395:
10391:
10388:
10385:
10382:
10377:
10371:
10345:
10340:
10336:
10332:
10327:
10323:
10297:
10293:
10289:
10286:
10264:
10260:
10256:
10253:
10233:
10228:
10222:
10216:
10213:
10191:
10188:
10185:
10179:
10173:
10170:
10167:
10145:
10140:
10137:
10134:
10130:
10124:
10118:
10112:
10092:
10070:
10064:
10039:
10035:
10031:
10009:
10005:
10001:
9979:
9975:
9952:
9948:
9927:
9922:
9918:
9914:
9892:
9888:
9884:
9862:
9858:
9835:
9831:
9819:
9818:
9807:
9804:
9801:
9798:
9795:
9792:
9789:
9786:
9783:
9780:
9777:
9771:
9766:
9762:
9757:
9752:
9749:
9746:
9743:
9740:
9737:
9714:
9709:
9705:
9701:
9698:
9695:
9692:
9672:
9650:
9646:
9625:
9620:
9616:
9612:
9609:
9606:
9583:
9561:
9557:
9542:
9541:
9524:
9520:
9516:
9513:
9510:
9507:
9502:
9498:
9494:
9491:
9488:
9485:
9483:
9481:
9476:
9472:
9468:
9465:
9462:
9461:
9456:
9452:
9448:
9445:
9442:
9439:
9434:
9430:
9426:
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9415:
9413:
9408:
9404:
9400:
9397:
9394:
9393:
9368:
9362:
9356:
9351:
9347:
9343:
9338:
9334:
9313:
9310:
9305:
9301:
9297:
9292:
9288:
9267:
9264:
9259:
9255:
9251:
9246:
9242:
9221:
9218:
9213:
9207:
9201:
9196:
9190:
9167:
9147:
9127:
9105:
9100:
9076:
9051:
9029:
9025:
9004:
8982:
8978:
8966:
8965:
8954:
8949:
8944:
8939:
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8932:
8926:
8921:
8917:
8911:
8908:
8903:
8899:
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8802:
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8716:
8713:
8708:
8704:
8698:
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8690:
8685:
8680:
8676:
8673:
8668:
8664:
8659:
8654:
8649:
8645:
8617:
8597:
8594:
8591:
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8585:
8582:
8562:
8559:
8554:
8551:
8548:
8544:
8523:
8520:
8515:
8511:
8507:
8504:
8482:
8479:
8476:
8472:
8449:
8445:
8441:
8438:
8435:
8430:
8426:
8405:
8391:
8390:
8379:
8374:
8369:
8366:
8363:
8359:
8353:
8348:
8342:
8338:
8334:
8329:
8324:
8320:
8313:
8308:
8304:
8298:
8294:
8288:
8284:
8280:
8277:
8274:
8269:
8266:
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8259:
8235:
8232:
8229:
8218:
8217:
8204:
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8141:
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8135:
8131:
8127:
8122:
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8113:
8106:
8101:
8097:
8091:
8087:
8081:
8078:
8075:
8071:
8067:
8064:
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8056:
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8050:
8047:
8044:
8040:
8016:
8013:
8010:
8007:
8004:
7978:
7974:
7953:
7948:
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7942:
7938:
7934:
7931:
7928:
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7919:
7915:
7912:
7909:
7906:
7901:
7897:
7893:
7873:
7870:
7867:
7864:
7861:
7850:
7849:
7838:
7833:
7829:
7823:
7819:
7813:
7809:
7805:
7802:
7799:
7794:
7790:
7784:
7780:
7774:
7769:
7765:
7761:
7756:
7752:
7748:
7743:
7738:
7734:
7730:
7726:
7722:
7719:
7716:
7712:
7706:
7701:
7697:
7693:
7688:
7685:
7682:
7678:
7654:
7629:
7626:
7623:
7619:
7598:
7568:
7564:
7560:
7557:
7554:
7549:
7545:
7524:
7502:
7499:
7496:
7492:
7469:
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7461:
7458:
7454:
7448:
7444:
7440:
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7408:
7404:
7401:
7398:
7393:
7389:
7385:
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7379:
7376:
7371:
7367:
7344:
7340:
7328:
7327:
7316:
7313:
7308:
7305:
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7298:
7294:
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7286:
7283:
7280:
7276:
7270:
7265:
7262:
7259:
7255:
7251:
7246:
7243:
7240:
7236:
7232:
7229:
7224:
7221:
7218:
7214:
7208:
7203:
7200:
7197:
7193:
7189:
7185:
7181:
7178:
7175:
7171:
7165:
7160:
7157:
7154:
7150:
7125:
7121:
7118:
7115:
7111:
7105:
7100:
7096:
7092:
7087:
7084:
7081:
7077:
7056:
7051:
7048:
7045:
7041:
7037:
7034:
7029:
7026:
7023:
7020:
7017:
7013:
7001:
7000:
6988:
6985:
6982:
6960:
6956:
6950:
6947:
6944:
6940:
6934:
6931:
6928:
6923:
6920:
6917:
6913:
6909:
6904:
6900:
6896:
6871:
6868:
6865:
6861:
6849:
6848:
6847:
6846:
6845:
6844:
6830:
6826:
6822:
6819:
6816:
6811:
6807:
6786:
6764:
6761:
6758:
6754:
6728:
6725:
6722:
6719:
6716:
6712:
6707:
6701:
6698:
6695:
6691:
6687:
6682:
6679:
6676:
6672:
6651:
6648:
6643:
6640:
6637:
6634:
6631:
6627:
6615:
6603:
6598:
6595:
6592:
6588:
6584:
6581:
6576:
6573:
6570:
6567:
6564:
6560:
6548:
6534:
6530:
6524:
6521:
6518:
6514:
6508:
6503:
6500:
6497:
6493:
6489:
6485:
6481:
6478:
6475:
6471:
6467:
6462:
6459:
6456:
6452:
6440:
6427:
6423:
6420:
6417:
6413:
6407:
6402:
6398:
6394:
6389:
6386:
6383:
6379:
6358:
6355:
6352:
6349:
6346:
6343:
6340:
6329:
6315:
6311:
6307:
6304:
6300:
6296:
6293:
6290:
6286:
6262:
6259:
6256:
6253:
6250:
6247:
6244:
6241:
6238:
6227:
6215:
6193:
6189:
6164:
6159:
6155:
6151:
6148:
6145:
6140:
6136:
6132:
6129:
6126:
6104:
6100:
6096:
6093:
6089:
6085:
6082:
6079:
6075:
6052:
6048:
6044:
6041:
6037:
6033:
6030:
6027:
6023:
6002:
5997:
5994:
5991:
5987:
5983:
5980:
5977:
5972:
5968:
5964:
5961:
5958:
5955:
5950:
5946:
5942:
5937:
5933:
5910:
5906:
5883:
5879:
5867:
5866:
5852:
5848:
5842:
5839:
5836:
5832:
5826:
5821:
5818:
5815:
5811:
5807:
5802:
5799:
5796:
5792:
5788:
5783:
5780:
5777:
5773:
5769:
5766:
5761:
5757:
5751:
5748:
5745:
5741:
5735:
5730:
5727:
5724:
5720:
5716:
5711:
5708:
5705:
5701:
5697:
5693:
5689:
5686:
5683:
5679:
5675:
5670:
5667:
5664:
5660:
5656:
5652:
5648:
5645:
5642:
5638:
5634:
5631:
5626:
5622:
5618:
5593:
5589:
5566:
5562:
5550:
5549:
5548:
5547:
5546:
5545:
5531:
5527:
5523:
5520:
5517:
5512:
5508:
5487:
5465:
5462:
5459:
5455:
5451:
5446:
5443:
5440:
5436:
5412:
5407:
5404:
5401:
5397:
5393:
5389:
5383:
5380:
5377:
5373:
5369:
5364:
5361:
5358:
5354:
5333:
5329:
5325:
5322:
5319:
5315:
5311:
5307:
5302:
5298:
5295:
5292:
5288:
5284:
5279:
5276:
5273:
5269:
5248:
5245:
5240:
5237:
5234:
5230:
5218:
5206:
5201:
5197:
5193:
5190:
5187:
5182:
5178:
5174:
5171:
5168:
5147:
5143:
5140:
5137:
5133:
5110:
5106:
5100:
5097:
5094:
5090:
5084:
5079:
5076:
5073:
5069:
5065:
5061:
5057:
5054:
5051:
5047:
5043:
5038:
5035:
5032:
5028:
5016:
5002:
4998:
4994:
4991:
4988:
4983:
4979:
4957:
4953:
4950:
4947:
4943:
4939:
4934:
4930:
4926:
4905:
4901:
4898:
4895:
4891:
4885:
4880:
4876:
4872:
4867:
4864:
4861:
4857:
4836:
4833:
4830:
4827:
4824:
4821:
4818:
4807:
4793:
4789:
4785:
4782:
4778:
4774:
4771:
4768:
4764:
4740:
4737:
4734:
4731:
4728:
4725:
4722:
4719:
4716:
4705:
4691:
4687:
4683:
4678:
4674:
4653:
4631:
4627:
4598:
4572:
4568:
4545:
4542:
4539:
4535:
4531:
4528:
4525:
4520:
4516:
4493:
4489:
4466:
4463:
4460:
4456:
4452:
4449:
4446:
4441:
4437:
4414:
4410:
4387:
4383:
4379:
4374:
4370:
4358:
4357:
4346:
4343:
4340:
4337:
4334:
4331:
4311:
4306:
4302:
4298:
4295:
4292:
4287:
4283:
4279:
4276:
4273:
4270:
4267:
4247:
4242:
4239:
4236:
4232:
4228:
4225:
4222:
4217:
4213:
4209:
4206:
4203:
4200:
4197:
4194:
4172:
4167:
4156:of a basis of
4143:
4138:
4135:
4132:
4128:
4122:
4118:
4114:
4097:is a chain of
4084:
4080:
4054:
4050:
4029:
4018:
4017:
4016:
4015:
4001:
3997:
3976:
3964:
3963:
3962:
3961:
3950:
3947:
3943:
3939:
3936:
3933:
3929:
3925:
3921:
3916:
3912:
3909:
3906:
3902:
3898:
3893:
3890:
3887:
3883:
3871:
3860:
3855:
3851:
3847:
3844:
3840:
3836:
3833:
3830:
3826:
3800:
3796:
3775:
3772:
3769:
3758:
3746:
3743:
3738:
3734:
3707:
3692:Main article:
3689:
3686:
3681:
3678:
3677:
3676:
3664:
3659:
3655:
3651:
3648:
3626:
3622:
3599:
3595:
3570:
3567:
3564:
3561:
3550:
3547:
3535:
3530:
3526:
3522:
3519:
3497:
3493:
3489:
3484:
3480:
3474:
3470:
3466:
3463:
3443:
3421:
3417:
3413:
3408:
3404:
3398:
3394:
3373:
3362:
3350:
3339:
3323:
3299:
3296:
3292:
3291:
3275:
3272:
3269:
3249:
3229:
3209:
3189:
3178:
3166:
3146:
3122:
3098:
3078:
3058:
3039:
3025:
3021:
3000:
2995:
2991:
2987:
2984:
2969:
2968:
2967:
2955:
2952:
2949:
2946:
2943:
2940:
2937:
2926:
2914:
2909:
2905:
2901:
2898:
2878:
2863:
2851:
2848:
2845:
2842:
2822:
2817:
2813:
2809:
2806:
2775:
2772:
2769:
2749:
2725:
2705:
2679:
2676:
2673:
2670:
2667:
2665:
2663:
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2657:
2654:
2651:
2648:
2646:
2644:
2641:
2638:
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2623:
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2619:
2616:
2613:
2610:
2607:
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2600:
2597:
2594:
2591:
2588:
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2582:
2580:
2578:
2575:
2572:
2567:
2563:
2559:
2556:
2553:
2550:
2548:
2546:
2543:
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2534:
2531:
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2478:
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2449:
2446:
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2437:
2417:
2397:
2377:
2349:
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2327:
2304:
2300:
2277:
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2271:
2267:
2246:
2224:
2221:
2218:
2214:
2191:
2187:
2164:
2160:
2138:
2134:
2130:
2120:. The vector
2103:
2099:
2075:
2072:
2069:
2049:
2044:
2040:
2036:
2033:
2030:
2010:
2007:
2004:
2001:
1998:
1995:
1975:
1955:
1952:
1949:
1946:
1943:
1923:
1920:
1917:
1914:
1891:
1888:
1883:
1879:
1856:
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1839:
1827:
1824:
1821:
1818:
1815:
1793:
1790:
1787:
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1773:
1769:
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1750:
1746:
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1711:
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1563:
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1514:
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1468:
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1452:
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1438:
1434:
1430:
1426:
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1414:
1411:
1389:
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1271:
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1263:
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1234:
1220:
1216:
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1200:
1196:
1192:
1187:
1183:
1171:
1157:
1153:
1149:
1146:
1142:
1138:
1134:
1122:
1121:
1120:
1106:
1103:
1100:
1096:
1092:
1089:
1086:
1081:
1077:
1056:
1034:
1030:
1004:
1000:
995:
989:
986:
983:
979:
975:
970:
966:
945:
942:
937:
933:
921:
918:Euclidean norm
905:
900:
897:
894:
890:
886:
883:
878:
874:
850:
847:
844:
841:
838:
835:
832:
821:
820:
819:
805:
801:
795:
791:
787:
783:
779:
775:
771:
766:
762:
750:
736:
732:
725:
721:
716:
712:
708:
703:
699:
687:
673:
669:
665:
662:
658:
654:
650:
635:
623:
613:Euclidean norm
598:
593:
588:
583:
579:
561:
547:
543:
539:
536:
533:
530:
506:
486:
483:
480:
460:
457:
454:
434:
431:
426:
422:
418:
415:
402:columns and a
387:
367:
364:
361:
347:
346:
345:
333:
313:
310:
307:
304:
281:
278:
275:
255:
235:
232:
229:
209:
190:
187:
174:
138:
118:
95:
92:
89:
65:
15:
9:
6:
4:
3:
2:
19195:
19184:
19181:
19180:
19178:
19163:
19160:
19158:
19155:
19153:
19150:
19148:
19145:
19143:
19140:
19138:
19135:
19134:
19132:
19128:
19122:
19119:
19117:
19114:
19112:
19109:
19107:
19104:
19102:
19099:
19098:
19096:
19092:
19086:
19083:
19081:
19078:
19075:
19071:
19068:
19066:
19063:
19061:
19058:
19057:
19055:
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19045:
19042:
19040:
19037:
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19034:
19030:
19026:
19019:
19014:
19012:
19007:
19005:
19000:
18999:
18996:
18988:
18982:
18978:
18971:
18966:
18962:
18958:
18951:
18947:
18943:
18942:Ng, Andrew Y.
18939:
18935:
18933:0-8018-5414-8
18929:
18925:
18921:
18917:
18913:
18909:
18908:
18898:
18894:
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18847:
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18824:
18820:
18816:
18812:
18807:
18802:
18799:(4): 045113.
18798:
18794:
18787:
18773:on 2007-07-01
18772:
18768:
18761:
18753:
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18682:
18678:
18674:
18670:
18663:
18655:
18651:
18650:
18645:
18641:
18635:
18627:
18625:0-8018-5413-X
18621:
18617:
18610:
18608:
18599:
18595:
18591:
18587:
18580:
18572:
18570:0-470-21820-7
18566:
18562:
18561:
18556:
18550:
18548:
18539:
18537:0-8176-3058-9
18533:
18529:
18522:
18520:
18511:
18504:
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18492:
18488:
18484:
18477:
18469:
18465:
18461:
18454:
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18434:
18430:
18426:
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18417:
18415:
18405:
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18392:
18385:
18378:
18374:
18347:
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18336:
18331:
18327:
18323:
18320:
18298:
18294:
18283:
18279:
18271:
18269:
18264:
18262:
18259:
18255:
18250:
18248:
18244:
18240:
18236:
18231:
18229:
18225:
18221:
18217:
18213:
18209:
18205:
18203:
18193:
18191:
18187:
18183:
18179:
18175:
18170:
18168:
18164:
18163:Jon Kleinberg
18161:developed by
18160:
18156:
18139:
18125:
18123:
18119:
18115:
18111:
18105:
18095:
18093:
18087:
18075:
18072:
18069:
18068:
18067:
18064:
18061:
18043:
18040:
18037:
18033:
18029:
18026:
18023:
18018:
18014:
18010:
18005:
18001:
17992:
17987:
17978:
17976:
17960:
17957:
17954:
17932:
17927:
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17921:
17918:
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17895:
17892:
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17863:
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17857:
17837:
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17809:
17806:
17801:
17797:
17791:
17788:
17785:
17782:
17779:
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17773:
17770:
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17708:
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17699:
17673:
17668:
17664:
17655:
17651:
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17638:
17631:
17628:
17625:
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17616:
17612:
17607:
17603:
17595:
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17593:
17576:
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17567:
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17557:
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17548:
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17524:
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17512:
17507:
17503:
17495:
17494:
17493:
17476:
17470:
17467:
17464:
17461:
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17438:
17431:
17427:
17423:
17418:
17414:
17407:
17404:
17400:
17395:
17384:
17380:
17376:
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17360:
17355:
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17344:
17340:
17334:
17327:
17323:
17317:
17313:
17303:
17302:
17301:
17287:
17265:
17261:
17237:
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17224:
17221:
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17212:
17208:
17203:
17199:
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17187:
17183:
17177:
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17159:
17155:
17151:
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17142:
17139:
17132:
17131:
17130:
17114:
17110:
17106:
17103:
17098:
17094:
17071:
17067:
17044:
17040:
17012:
17008:
16999:
16989:
16985:
16959:
16950:
16934:
16931:
16927:
16906:
16874:
16871:
16868:
16862:
16859:
16855:
16846:
16842:
16838:
16833:
16829:
16817:
16805:
16801:
16788:
16776:
16772:
16763:
16760:
16754:
16751:
16749:
16737:
16732:
16723:
16720:
16717:
16711:
16707:
16703:
16698:
16695:
16692:
16688:
16683:
16678:
16668:
16664:
16660:
16655:
16651:
16639:
16627:
16623:
16610:
16598:
16594:
16585:
16582:
16576:
16574:
16557:
16554:
16551:
16548:
16542:
16539:
16533:
16530:
16527:
16518:
16513:
16509:
16504:
16494:
16490:
16486:
16481:
16477:
16465:
16453:
16449:
16436:
16424:
16420:
16411:
16408:
16402:
16400:
16387:
16375:
16371:
16360:
16352:
16349:
16346:
16343:
16335:
16332:
16329:
16325:
16318:
16312:
16300:
16296:
16287:
16284:
16280:
16271:
16267:
16263:
16258:
16254:
16244:
16242:
16235:
16231:
16227:
16222:
16218:
16206:
16205:
16204:
16187:
16182:
16179:
16174:
16170:
16164:
16161:
16158:
16155:
16152:
16149:
16146:
16138:
16135:
16132:
16129:
16123:
16120:
16116:
16112:
16109:
16102:
16101:
16100:
16098:
16094:
16070:
16066:
16062:
16057:
16053:
16045:
16041:
16037:
16032:
16028:
16021:
16018:
16011:
16010:
16009:
15992:
15988:
15984:
15981:
15973:
15969:
15965:
15960:
15956:
15948:
15944:
15940:
15935:
15931:
15924:
15920:
15914:
15911:
15908:
15904:
15900:
15896:
15888:
15884:
15880:
15875:
15871:
15863:
15859:
15855:
15850:
15846:
15842:
15837:
15833:
15829:
15823:
15817:
15814:
15811:
15807:
15803:
15795:
15791:
15784:
15777:
15776:
15775:
15758:
15750:
15738:
15734:
15723:
15708:
15704:
15697:
15685:
15673:
15669:
15658:
15653:
15650:
15647:
15643:
15634:
15630:
15626:
15621:
15617:
15607:
15599:
15584:
15580:
15573:
15563:
15551:
15547:
15531:
15527:
15523:
15518:
15514:
15505:
15493:
15489:
15478:
15473:
15470:
15467:
15463:
15456:
15448:
15433:
15429:
15422:
15412:
15400:
15396:
15385:
15380:
15377:
15374:
15370:
15362:
15347:
15343:
15336:
15323:
15319:
15315:
15310:
15306:
15297:
15285:
15281:
15270:
15265:
15262:
15259:
15255:
15248:
15240:
15236:
15229:
15223:
15217:
15214:
15209:
15205:
15201:
15196:
15192:
15188:
15183:
15179:
15171:
15170:
15169:
15155:
15135:
15115:
15112:
15109:
15087:
15072:
15068:
15061:
15034:
15030:
15007:
15003:
14999:
14994:
14990:
14965:
14957:
14953:
14949:
14944:
14940:
14932:
14928:
14924:
14919:
14915:
14911:
14908:
14905:
14899:
14893:
14890:
14887:
14883:
14879:
14873:
14867:
14860:
14859:
14858:
14841:
14838:
14835:
14832:
14807:
14803:
14779:
14776:
14773:
14770:
14744:
14741:
14738:
14735:
14712:
14689:
14686:
14680:
14677:
14674:
14668:
14665:
14662:
14654:
14650:
14629:
14607:
14603:
14594:
14578:
14558:
14549:
14533:
14529:
14525:
14520:
14516:
14493:
14489:
14468:
14448:
14445:
14442:
14419:
14411:
14406:
14397:
14393:
14386:
14382:
14375:
14363:
14359:
14348:
14343:
14340:
14337:
14333:
14325:
14320:
14311:
14307:
14300:
14296:
14286:
14282:
14278:
14273:
14269:
14260:
14248:
14244:
14233:
14228:
14225:
14222:
14218:
14211:
14200:
14196:
14189:
14184:
14174:
14170:
14163:
14158:
14146:
14142:
14131:
14126:
14123:
14120:
14116:
14105:
14101:
14094:
14089:
14079:
14075:
14068:
14063:
14059:
14053:
14041:
14037:
14026:
14021:
14018:
14015:
14011:
14004:
13999:
13995:
13991:
13983:
13979:
13970:
13966:
13959:
13951:
13947:
13938:
13934:
13928:
13924:
13918:
13913:
13910:
13907:
13903:
13897:
13892:
13888:
13880:
13876:
13867:
13863:
13856:
13851:
13847:
13838:
13834:
13825:
13821:
13815:
13811:
13805:
13800:
13797:
13794:
13790:
13784:
13779:
13775:
13768:
13763:
13759:
13755:
13747:
13743:
13736:
13730:
13724:
13721:
13716:
13712:
13704:
13703:
13702:
13699:
13685:
13660:
13656:
13647:
13643:
13636:
13631:
13627:
13621:
13616:
13613:
13610:
13606:
13602:
13597:
13593:
13586:
13580:
13573:
13572:
13571:
13552:
13548:
13542:
13538:
13532:
13528:
13522:
13517:
13514:
13511:
13507:
13503:
13498:
13494:
13488:
13484:
13478:
13473:
13470:
13467:
13463:
13459:
13456:
13451:
13447:
13443:
13436:
13435:
13434:
13418:
13414:
13385:
13381:
13374:
13368:
13362:
13354:
13350:
13344:
13339:
13335:
13327:
13323:
13316:
13310:
13307:
13301:
13293:
13289:
13283:
13278:
13274:
13267:
13259:
13255:
13248:
13242:
13234:
13230:
13221:
13217:
13211:
13206:
13202:
13194:
13190:
13183:
13177:
13174:
13166:
13162:
13153:
13149:
13143:
13138:
13134:
13127:
13119:
13115:
13108:
13102:
13097:
13089:
13083:
13078:
13073:
13069:
13061:
13057:
13050:
13044:
13041:
13036:
13028:
13022:
13017:
13012:
13008:
13001:
12993:
12989:
12982:
12976:
12971:
12961:
12957:
12950:
12944:
12934:
12930:
12923:
12917:
12914:
12909:
12899:
12895:
12888:
12882:
12873:
12865:
12861:
12854:
12848:
12842:
12835:
12834:
12833:
12819:
12797:
12793:
12770:
12766:
12760:
12757:
12754:
12750:
12746:
12743:
12740:
12735:
12731:
12725:
12721:
12717:
12712:
12708:
12704:
12701:
12696:
12692:
12671:
12668:
12665:
12645:
12623:
12619:
12612:
12606:
12583:
12579:
12573:
12569:
12563:
12560:
12557:
12553:
12549:
12546:
12543:
12538:
12534:
12528:
12524:
12520:
12515:
12511:
12507:
12504:
12499:
12495:
12490:
12486:
12483:
12476:
12475:
12474:
12460:
12451:
12435:
12431:
12407:
12401:
12398:
12393:
12389:
12366:
12362:
12338:
12332:
12312:
12290:
12286:
12282:
12277:
12273:
12252:
12230:
12226:
12205:
12200:
12173:
12151:
12147:
12123:
12117:
12095:
12091:
12081:
12067:
12045:
12041:
12037:
12034:
12031:
12026:
12022:
12001:
11979:
11975:
11971:
11968:
11965:
11960:
11956:
11935:
11913:
11909:
11886:
11882:
11859:
11855:
11832:
11828:
11807:
11802:
11798:
11794:
11791:
11788:
11783:
11779:
11775:
11770:
11766:
11745:
11725:
11722:
11719:
11699:
11679:
11665:
11651:
11629:
11625:
11604:
11594:
11580:
11577:
11567:
11563:
11535:
11520:
11499:
11495:
11491:
11488:
11485:
11480:
11476:
11450:
11446:
11425:
11405:
11397:
11379:
11375:
11351:
11347:
11323:
11319:
11313:
11309:
11303:
11298:
11295:
11292:
11288:
11284:
11279:
11275:
11253:
11250:
11247:
11244:
11241:
11238:
11235:
11213:
11209:
11203:
11198:
11194:
11190:
11185:
11181:
11158:
11154:
11144:
11130:
11127:
11124:
11121:
11118:
11115:
11112:
11090:
11086:
11080:
11076:
11072:
11067:
11063:
11059:
11037:
11033:
11029:
11026:
11023:
11018:
11014:
10991:
10987:
10983:
10980:
10977:
10972:
10968:
10964:
10959:
10955:
10934:
10920:
10901:
10897:
10893:
10890:
10887:
10882:
10878:
10852:
10823:
10801:
10797:
10774:
10770:
10749:
10744:
10713:
10709:
10688:
10665:
10659:
10653:
10646:
10634:
10631:
10623:
10615:
10611:
10604:
10591:
10585:
10578:
10566:
10563:
10555:
10547:
10543:
10536:
10529:
10528:
10527:
10511:
10499:
10494:
10490:
10486:
10481:
10477:
10448:
10444:
10438:
10435:
10432:
10428:
10424:
10421:
10418:
10413:
10409:
10405:
10402:
10397:
10393:
10386:
10383:
10380:
10375:
10359:
10358:
10357:
10343:
10338:
10334:
10330:
10325:
10321:
10311:
10295:
10291:
10287:
10284:
10262:
10258:
10254:
10251:
10231:
10226:
10214:
10211:
10189:
10186:
10183:
10171:
10168:
10165:
10143:
10138:
10135:
10132:
10122:
10090:
10068:
10037:
10033:
10029:
10007:
10003:
9999:
9977:
9973:
9950:
9946:
9925:
9920:
9916:
9912:
9890:
9886:
9882:
9860:
9856:
9833:
9829:
9805:
9799:
9793:
9787:
9784:
9781:
9778:
9769:
9764:
9760:
9755:
9750:
9744:
9738:
9728:
9727:
9726:
9725:. In general
9707:
9703:
9696:
9690:
9670:
9648:
9644:
9618:
9614:
9607:
9597:
9581:
9559:
9555:
9545:
9518:
9514:
9511:
9508:
9500:
9496:
9489:
9486:
9484:
9474:
9470:
9463:
9450:
9446:
9443:
9440:
9432:
9428:
9421:
9418:
9416:
9406:
9402:
9395:
9384:
9383:
9382:
9366:
9354:
9349:
9345:
9341:
9336:
9332:
9311:
9308:
9303:
9299:
9295:
9290:
9286:
9265:
9262:
9257:
9253:
9249:
9244:
9240:
9219:
9216:
9211:
9199:
9194:
9165:
9145:
9125:
9103:
9063:
9049:
9023:
9002:
8976:
8952:
8947:
8937:
8934:
8930:
8924:
8919:
8915:
8909:
8906:
8901:
8897:
8890:
8884:
8878:
8871:
8870:
8869:
8868:
8864:
8848:
8834:
8820:
8800:
8779:
8759:
8736:
8733:
8730:
8727:
8722:
8718:
8714:
8711:
8706:
8702:
8696:
8692:
8688:
8683:
8678:
8674:
8671:
8666:
8662:
8657:
8652:
8647:
8643:
8635:
8634:
8633:
8631:
8615:
8595:
8592:
8589:
8586:
8583:
8580:
8560:
8557:
8552:
8549:
8546:
8542:
8521:
8518:
8513:
8509:
8505:
8502:
8480:
8477:
8474:
8470:
8447:
8443:
8439:
8436:
8433:
8428:
8424:
8403:
8394:
8377:
8367:
8364:
8361:
8357:
8351:
8340:
8336:
8332:
8327:
8322:
8318:
8311:
8306:
8302:
8296:
8286:
8282:
8278:
8272:
8267:
8264:
8261:
8257:
8249:
8248:
8247:
8233:
8230:
8227:
8202:
8199:
8196:
8193:
8190:
8186:
8182:
8172:
8169:
8166:
8163:
8160:
8156:
8150:
8139:
8136:
8133:
8129:
8125:
8120:
8115:
8111:
8104:
8099:
8095:
8089:
8079:
8076:
8073:
8069:
8065:
8059:
8054:
8051:
8048:
8045:
8042:
8038:
8030:
8029:
8028:
8014:
8011:
8008:
8005:
8002:
7994:
7976:
7972:
7946:
7943:
7940:
7936:
7932:
7929:
7926:
7921:
7917:
7910:
7907:
7904:
7899:
7895:
7891:
7884:we know that
7871:
7868:
7865:
7862:
7859:
7836:
7831:
7827:
7821:
7811:
7807:
7803:
7797:
7792:
7788:
7782:
7778:
7772:
7767:
7763:
7759:
7754:
7750:
7746:
7741:
7736:
7732:
7728:
7724:
7720:
7717:
7714:
7710:
7704:
7699:
7695:
7691:
7686:
7683:
7680:
7676:
7668:
7667:
7666:
7652:
7643:
7627:
7624:
7621:
7617:
7596:
7588:
7583:
7566:
7562:
7558:
7555:
7552:
7547:
7543:
7522:
7500:
7497:
7494:
7490:
7467:
7462:
7459:
7456:
7446:
7442:
7418:
7410:
7406:
7402:
7399:
7396:
7391:
7387:
7380:
7377:
7374:
7369:
7365:
7342:
7338:
7314:
7306:
7303:
7300:
7296:
7289:
7284:
7281:
7278:
7274:
7268:
7263:
7260:
7257:
7253:
7244:
7241:
7238:
7234:
7227:
7222:
7219:
7216:
7212:
7206:
7201:
7198:
7195:
7191:
7187:
7183:
7179:
7176:
7173:
7169:
7163:
7158:
7155:
7152:
7148:
7140:
7139:
7138:
7123:
7119:
7116:
7113:
7109:
7103:
7098:
7094:
7090:
7085:
7082:
7079:
7075:
7049:
7046:
7043:
7039:
7032:
7027:
7024:
7021:
7018:
7015:
7011:
6986:
6983:
6980:
6958:
6954:
6948:
6945:
6942:
6938:
6932:
6929:
6926:
6921:
6918:
6915:
6911:
6907:
6902:
6898:
6894:
6887:
6886:
6885:
6869:
6866:
6863:
6859:
6828:
6824:
6820:
6817:
6814:
6809:
6805:
6784:
6762:
6759:
6756:
6752:
6743:
6742:
6726:
6723:
6720:
6717:
6714:
6710:
6705:
6699:
6696:
6693:
6689:
6685:
6680:
6677:
6674:
6670:
6649:
6646:
6641:
6638:
6635:
6632:
6629:
6625:
6616:
6596:
6593:
6590:
6586:
6579:
6574:
6571:
6568:
6565:
6562:
6558:
6549:
6532:
6528:
6522:
6519:
6516:
6512:
6506:
6501:
6498:
6495:
6491:
6487:
6483:
6479:
6476:
6473:
6469:
6465:
6460:
6457:
6454:
6450:
6441:
6425:
6421:
6418:
6415:
6411:
6405:
6400:
6396:
6392:
6387:
6384:
6381:
6377:
6356:
6353:
6350:
6347:
6344:
6341:
6338:
6330:
6313:
6309:
6305:
6302:
6298:
6294:
6291:
6288:
6284:
6275:
6274:
6260:
6257:
6254:
6251:
6248:
6245:
6242:
6239:
6236:
6228:
6213:
6191:
6187:
6178:
6177:
6176:
6157:
6153:
6149:
6146:
6143:
6138:
6134:
6127:
6124:
6102:
6098:
6094:
6091:
6087:
6083:
6080:
6077:
6073:
6050:
6046:
6042:
6039:
6035:
6031:
6028:
6025:
6021:
5995:
5992:
5989:
5985:
5981:
5978:
5975:
5970:
5966:
5959:
5956:
5953:
5948:
5944:
5940:
5935:
5931:
5908:
5904:
5881:
5877:
5850:
5846:
5840:
5837:
5834:
5830:
5824:
5819:
5816:
5813:
5809:
5805:
5800:
5797:
5794:
5790:
5781:
5778:
5775:
5771:
5764:
5759:
5755:
5749:
5746:
5743:
5739:
5733:
5728:
5725:
5722:
5718:
5714:
5709:
5706:
5703:
5699:
5695:
5691:
5687:
5684:
5681:
5677:
5673:
5668:
5665:
5662:
5658:
5650:
5646:
5643:
5640:
5636:
5629:
5624:
5620:
5616:
5609:
5608:
5607:
5591:
5587:
5564:
5560:
5529:
5525:
5521:
5518:
5515:
5510:
5506:
5485:
5463:
5460:
5457:
5453:
5449:
5444:
5441:
5438:
5434:
5425:
5424:
5405:
5402:
5399:
5395:
5387:
5381:
5378:
5375:
5371:
5367:
5362:
5359:
5356:
5352:
5327:
5323:
5320:
5317:
5313:
5305:
5300:
5296:
5293:
5290:
5286:
5282:
5277:
5274:
5271:
5267:
5246:
5243:
5238:
5235:
5232:
5228:
5219:
5199:
5195:
5191:
5188:
5185:
5180:
5176:
5169:
5166:
5145:
5141:
5138:
5135:
5131:
5108:
5104:
5098:
5095:
5092:
5088:
5082:
5077:
5074:
5071:
5067:
5063:
5059:
5055:
5052:
5049:
5045:
5041:
5036:
5033:
5030:
5026:
5017:
5000:
4996:
4992:
4989:
4986:
4981:
4977:
4955:
4951:
4948:
4945:
4941:
4937:
4932:
4928:
4924:
4903:
4899:
4896:
4893:
4889:
4883:
4878:
4874:
4870:
4865:
4862:
4859:
4855:
4834:
4831:
4828:
4825:
4822:
4819:
4816:
4808:
4791:
4787:
4783:
4780:
4776:
4772:
4769:
4766:
4762:
4753:
4752:
4738:
4735:
4732:
4729:
4726:
4723:
4720:
4717:
4714:
4706:
4689:
4685:
4681:
4676:
4672:
4651:
4629:
4625:
4616:
4615:
4614:
4612:
4596:
4588:
4570:
4566:
4543:
4540:
4537:
4533:
4529:
4526:
4523:
4518:
4514:
4491:
4487:
4464:
4461:
4458:
4454:
4450:
4447:
4444:
4439:
4435:
4412:
4408:
4385:
4381:
4377:
4372:
4368:
4344:
4341:
4338:
4335:
4332:
4329:
4304:
4300:
4296:
4293:
4290:
4285:
4281:
4274:
4271:
4268:
4265:
4240:
4237:
4234:
4230:
4226:
4223:
4220:
4215:
4211:
4204:
4201:
4198:
4195:
4192:
4170:
4141:
4136:
4133:
4130:
4120:
4116:
4104:
4103:
4102:
4100:
4082:
4078:
4068:
4052:
4048:
4027:
3999:
3995:
3974:
3967:In the large
3966:
3965:
3948:
3941:
3937:
3934:
3931:
3927:
3919:
3914:
3910:
3907:
3904:
3900:
3896:
3891:
3888:
3885:
3881:
3872:
3858:
3853:
3849:
3845:
3842:
3838:
3834:
3831:
3828:
3824:
3815:
3814:
3798:
3794:
3773:
3770:
3767:
3759:
3744:
3741:
3736:
3732:
3723:
3722:
3721:
3720:
3719:
3705:
3695:
3685:
3657:
3653:
3646:
3624:
3620:
3597:
3593:
3584:
3565:
3559:
3551:
3548:
3528:
3524:
3517:
3495:
3491:
3487:
3482:
3478:
3472:
3468:
3464:
3461:
3441:
3419:
3415:
3411:
3406:
3402:
3396:
3392:
3371:
3363:
3348:
3340:
3337:
3321:
3313:
3312:
3311:
3308:
3305:
3295:
3289:
3273:
3270:
3267:
3247:
3227:
3207:
3187:
3179:
3164:
3144:
3136:
3120:
3112:
3096:
3076:
3056:
3048:
3044:
3040:
3023:
3019:
2993:
2989:
2982:
2974:
2970:
2950:
2947:
2944:
2941:
2935:
2927:
2907:
2903:
2896:
2876:
2868:
2864:
2846:
2840:
2815:
2811:
2804:
2796:
2792:
2788:
2787:
2773:
2770:
2767:
2747:
2739:
2738:
2737:
2723:
2703:
2694:
2677:
2674:
2671:
2668:
2666:
2658:
2655:
2652:
2649:
2647:
2636:
2633:
2627:
2624:
2622:
2614:
2611:
2608:
2605:
2603:
2595:
2592:
2589:
2586:
2583:
2581:
2573:
2570:
2565:
2561:
2557:
2554:
2551:
2549:
2541:
2538:
2535:
2532:
2530:
2525:
2522:
2510:
2496:
2476:
2473:
2470:
2467:
2447:
2444:
2441:
2438:
2435:
2415:
2395:
2375:
2366:
2363:
2359:
2355:
2345:
2344:when needed.
2329:
2325:
2302:
2298:
2275:
2272:
2269:
2265:
2244:
2222:
2219:
2216:
2212:
2189:
2185:
2162:
2158:
2136:
2132:
2128:
2119:
2101:
2097:
2087:
2073:
2070:
2067:
2042:
2038:
2034:
2028:
2005:
2002:
1999:
1993:
1973:
1950:
1947:
1941:
1918:
1912:
1903:
1889:
1886:
1881:
1877:
1854:
1850:
1825:
1822:
1819:
1816:
1813:
1791:
1788:
1785:
1781:
1775:
1771:
1767:
1762:
1758:
1752:
1748:
1744:
1739:
1736:
1733:
1729:
1723:
1720:
1717:
1713:
1709:
1705:
1701:
1697:
1693:
1688:
1684:
1680:
1673:
1670:
1651:
1643:
1639:
1631:
1627:
1618:
1609:
1605:
1597:
1594:
1591:
1587:
1579:
1576:
1573:
1569:
1555:
1552:
1549:
1545:
1539:
1534:
1523:
1516:
1512:
1504:
1500:
1486:
1482:
1474:
1470:
1462:
1458:
1450:
1440:
1436:
1428:
1424:
1417:
1412:
1409:
1387:
1383:
1379:
1376:
1373:
1368:
1364:
1343:
1335:
1316:
1313:
1310:
1306:
1300:
1296:
1292:
1287:
1283:
1277:
1273:
1269:
1265:
1261:
1257:
1253:
1248:
1244:
1235:
1218:
1214:
1207:
1203:
1198:
1194:
1190:
1185:
1181:
1172:
1155:
1151:
1147:
1144:
1140:
1136:
1132:
1123:
1104:
1101:
1098:
1094:
1090:
1087:
1084:
1079:
1075:
1054:
1032:
1028:
1020:else pick as
1019:
1018:
1002:
998:
993:
987:
984:
981:
977:
973:
968:
964:
943:
940:
935:
931:
922:
919:
898:
895:
892:
888:
881:
876:
872:
863:
862:
848:
845:
842:
839:
836:
833:
830:
822:
803:
799:
793:
789:
785:
781:
777:
773:
769:
764:
760:
751:
734:
730:
723:
719:
714:
710:
706:
701:
697:
688:
671:
667:
663:
660:
656:
652:
648:
639:
638:
636:
621:
614:
596:
586:
581:
577:
568:
567:
565:
562:
545:
541:
537:
534:
531:
528:
520:
504:
484:
481:
478:
458:
455:
452:
432:
429:
424:
420:
416:
413:
405:
401:
385:
365:
362:
359:
351:
348:
331:
311:
308:
302:
294:
293:
279:
276:
273:
253:
233:
230:
227:
207:
200:
196:
193:
192:
189:The algorithm
186:
172:
162:
160:
154:
152:
136:
116:
108:
93:
90:
87:
79:
63:
55:
54:power methods
51:
47:
43:
36:
29:
22:
19032:Key concepts
18976:
18960:
18956:
18923:
18885:
18874:. Retrieved
18870:
18860:
18839:
18796:
18792:
18786:
18775:. Retrieved
18771:the original
18760:
18743:
18739:
18729:
18712:
18708:
18695:
18668:
18662:
18653:
18647:
18634:
18615:
18589:
18585:
18579:
18559:
18527:
18509:
18503:
18486:
18482:
18476:
18459:
18453:
18428:
18424:AIAA Journal
18422:
18394:
18390:
18377:
18282:
18265:
18251:
18241:package has
18232:
18219:
18206:
18199:
18178:Hamiltonians
18171:
18131:
18128:Applications
18121:
18107:
18088:
18084:
18065:
18059:
17990:
17988:
17984:
17974:
17688:
17591:
17491:
17252:
16951:
16898:
16202:
16090:
16007:
15774:Furthermore
15773:
14981:
14550:
14434:
13700:
13677:
13569:
13405:
12598:
12452:
12082:
11758:eigenvalues
11671:
11595:
11517:will have a
11145:
10926:
10680:
10468:
10312:
9820:
9546:
9543:
9138:and minimum
9064:
8967:
8840:
8751:
8395:
8392:
8219:
7964:, and since
7851:
7644:
7584:
7535:that all of
7431:the vectors
7329:
7002:
6850:
5868:
5551:
4359:
4069:
4019:
3697:
3683:
3309:
3301:
3293:
3113:. Likewise,
3043:power method
2973:QR algorithm
2695:
2511:
2367:
2358:eigenvectors
2351:
2117:
2089:The vectors
2088:
1904:
1841:
1671:
563:
349:
194:
163:
158:
157:vector with
155:
56:to find the
41:
39:
18640:D. Calvetti
18555:Yousef Saad
18202:NAG Library
18186:shell model
18060:orthonormal
17280:-amplitude
10728:belongs to
10526:such that
8752:the matrix
8608:the matrix
5159:that is in
4400:as long as
3718:is roughly
2966:operations.
2925:operations.
2862:operations.
2354:eigenvalues
2116:are called
956:, then let
404:tridiagonal
400:orthonormal
48:devised by
19074:algorithms
18979:. Jülich.
18963:: 903–910.
18876:2012-02-09
18777:2007-06-30
18370:References
18222:function (
18214:come with
18212:GNU Octave
18081:Variations
14595:. Writing
11398:with mean
11266:, so that
10681:Since the
9324:such that
8246:one gets
8027:one gets
5923:, because
4258:for every
4185:such that
3581:points of
19101:CPU cache
18851:1310.5431
18831:118722138
18806:1012.1031
18715:: 39–61.
18468:654214109
18324:≈
18188:codes in
18165:, or the
18114:nullspace
18108:In 1995,
18027:⋯
17958:≫
17955:ρ
17933:ρ
17919:≈
17908:in which
17890:≪
17887:ρ
17867:ρ
17835:≫
17832:ρ
17824:. In the
17810:ρ
17798:ρ
17786:ρ
17751:ρ
17717:−
17709:ρ
17652:λ
17648:−
17639:λ
17621:∗
17613:−
17604:λ
17568:λ
17545:λ
17528:−
17508:∗
17471:ρ
17439:λ
17428:λ
17424:−
17415:λ
17381:λ
17377:−
17368:λ
17352:λ
17341:λ
17324:λ
17314:λ
17209:≈
17152:−
17111:λ
17107:−
17095:λ
17068:λ
17041:λ
17009:λ
17000:⩽
16986:λ
16960:ρ
16932:−
16872:−
16860:−
16843:λ
16839:−
16830:λ
16764:−
16752:⩽
16721:−
16712:−
16696:−
16665:λ
16661:−
16652:λ
16586:−
16558:ρ
16543:
16531:−
16519:
16491:λ
16487:−
16478:λ
16412:−
16347:ρ
16333:−
16288:−
16268:λ
16264:−
16255:λ
16245:⩽
16232:θ
16228:−
16219:λ
16183:ρ
16171:ρ
16159:ρ
16139:ρ
16124:
16067:λ
16063:−
16054:λ
16042:λ
16038:−
16029:λ
16019:ρ
15970:λ
15966:−
15957:λ
15945:λ
15941:−
15932:λ
15912:−
15885:λ
15881:−
15872:λ
15860:λ
15856:−
15847:λ
15843:−
15834:λ
15815:−
15792:λ
15705:λ
15644:∑
15631:λ
15627:−
15618:λ
15608:⩽
15581:λ
15528:λ
15524:−
15515:λ
15464:∑
15457:⩽
15430:λ
15371:∑
15344:λ
15320:λ
15316:−
15307:λ
15256:∑
15215:−
15206:λ
15202:⩽
15193:θ
15189:−
15180:λ
15113:⩾
15069:λ
15031:λ
15004:λ
14991:λ
14982:(in case
14954:λ
14950:−
14941:λ
14929:λ
14925:−
14916:λ
14912:−
14891:−
14833:−
14804:λ
14771:−
14736:−
14681:
14666:
14530:θ
14526:−
14517:λ
14490:λ
14394:λ
14334:∑
14308:λ
14283:λ
14279:−
14270:λ
14219:∑
14197:λ
14185:∗
14171:λ
14117:∑
14102:λ
14090:∗
14076:λ
14060:λ
14012:∑
14005:−
13996:λ
13967:λ
13948:λ
13939:∗
13904:∑
13898:∗
13864:λ
13848:λ
13835:λ
13826:∗
13791:∑
13785:∗
13769:−
13760:λ
13722:−
13713:λ
13644:λ
13607:∑
13539:λ
13508:∑
13464:∑
13355:∗
13345:∗
13294:∗
13284:∗
13235:∗
13222:∗
13212:∗
13167:∗
13154:∗
13144:∗
13098:∗
13079:∗
13037:∗
13018:∗
12972:∗
12910:∗
12798:∗
12758:−
12744:…
12669:−
12561:−
12547:…
12487:
12432:θ
12399:−
12390:λ
12363:θ
12287:θ
12283:⩾
12274:λ
12227:θ
12148:λ
12092:θ
12042:λ
12035:…
12023:λ
11976:θ
11969:…
11957:θ
11910:λ
11883:θ
11856:λ
11829:θ
11799:θ
11795:⩾
11792:⋯
11789:⩾
11780:θ
11776:⩾
11767:θ
11723:×
11581:ε
11489:…
11289:∑
11248:…
11204:∗
11125:…
11077:λ
11027:…
10988:λ
10984:⩾
10981:⋯
10978:⩾
10969:λ
10965:⩾
10956:λ
10891:…
10635:∈
10567:∈
10500:∈
10436:−
10422:…
10387:
10215:∈
10172:∈
9785:−
9765:∗
9736:∇
9694:∇
9691:−
9605:∇
9519:λ
9515:⩾
9512:⋯
9509:⩾
9487:⩾
9451:λ
9447:⩽
9444:⋯
9441:⩽
9419:⩽
9355:∈
9312:…
9266:…
9220:⋯
9217:⊂
9200:⊂
9024:λ
8977:λ
8938:∈
8920:∗
8902:∗
8723:∗
8707:∗
8697:∗
8684:∗
8667:∗
8648:∗
8632:. Since
8514:∗
8437:…
8373:¯
8347:¯
8328:∗
8297:∗
8200:−
8178:¯
8170:−
8146:¯
8137:−
8121:∗
8090:∗
8077:−
8046:−
8012:−
7944:−
7930:…
7911:
7905:∈
7869:−
7822:∗
7783:∗
7773:∗
7742:∗
7705:∗
7556:…
7400:…
7381:
7375:∈
7312:‖
7293:‖
7269:∗
7250:‖
7231:‖
7207:∗
7164:∗
7104:∗
7055:‖
7036:‖
6912:∑
6818:…
6662:then let
6647:≠
6602:‖
6583:‖
6492:∑
6488:−
6406:∗
6351:…
6258:−
6249:…
6147:…
6128:
5993:−
5979:…
5960:
5954:∈
5941:−
5810:∑
5787:‖
5768:‖
5719:∑
5655:‖
5633:‖
5519:…
5411:‖
5392:‖
5332:‖
5310:‖
5259:then let
5244:≠
5189:…
5170:
5068:∑
5064:−
4990:…
4884:∗
4829:…
4736:−
4727:…
4541:−
4527:…
4462:−
4448:…
4333:⩽
4294:…
4275:
4269:∈
4224:…
4205:
4199:∈
4105:a subset
3946:‖
3924:‖
3771:⩾
3742:≠
3621:β
3594:α
3488:…
3412:…
3271:≪
3260:; in the
2948:
2771:×
2672:λ
2653:λ
2634:λ
2566:∗
2445:λ
2376:λ
2273:−
1878:β
1789:−
1772:β
1749:α
1714:β
1640:α
1628:β
1606:β
1595:−
1588:α
1577:−
1570:β
1553:−
1546:β
1540:⋱
1535:⋱
1524:⋱
1513:α
1501:β
1483:β
1471:α
1459:β
1437:β
1425:α
1377:…
1314:−
1297:β
1293:−
1274:α
1270:−
1208:∗
1182:α
1102:−
1088:…
999:β
985:−
941:≠
932:β
904:‖
896:−
885:‖
873:β
843:…
790:α
786:−
724:∗
698:α
587:∈
546:∗
456:×
425:∗
363:×
306:↦
231:×
91:×
19177:Category
19130:Software
19094:Hardware
19053:Problems
18948:(2001).
18918:(1996).
18893:Archived
18890:GraphLab
18671:. SIAM.
18258:GraphLab
18167:PageRank
17975:smallest
16097:spectrum
16093:eigengap
14705:for all
12218:whereas
12139:. Since
11228:for all
11105:for all
10836:on some
10204:for all
9596:gradient
7725:′
7582:point.)
7184:′
7124:′
6973:for all
6884:satisfy
6484:′
6426:′
6331:For all
6299:′
6088:′
6036:′
5692:′
5651:′
5606:is that
5328:′
5301:′
5146:′
5060:′
4956:′
4904:′
4809:For all
4777:′
4322:and all
3942:′
3915:′
3839:′
3220:so that
3089:. Since
2460:), then
2137:′
1706:′
1266:′
1204:′
1141:′
782:′
720:′
657:′
445:of size
220:of size
109:, where
18811:Bibcode
18656:: 1–21.
18433:Bibcode
8865:of the
7995:.) For
7137:since
3987:limit,
3336:fill-in
3111:caching
564:Warning
519:unitary
497:, then
378:matrix
19152:LAPACK
19142:MATLAB
18983:
18930:
18829:
18683:
18622:
18567:
18534:
18466:
18268:PRIMME
18256:. The
18247:ARPACK
18237:, the
18235:Python
18228:Octave
18224:Matlab
18220:eigs()
18216:ARPACK
18208:MATLAB
18092:ARPACK
16540:arcosh
16121:arcosh
14857:. Let
14551:Since
11712:is an
11672:After
9938:since
8861:is as
8534:, and
6117:is in
4664:. Let
2760:is an
2408:, and
1402:. Let
916:(also
521:, and
350:Output
344:times.
80:of an
44:is an
19137:ATLAS
18973:(PDF)
18953:(PDF)
18846:arXiv
18827:S2CID
18801:arXiv
18705:(PDF)
18387:(PDF)
18313:, so
18274:Notes
18239:SciPy
18118:GF(2)
15168:, so
14822:into
13701:Thus
11928:) as
9547:From
3546:time.
3133:is a
2021:, or
471:. If
398:with
195:Input
19116:SIMD
18981:ISBN
18928:ISBN
18681:ISBN
18620:ISBN
18565:ISBN
18532:ISBN
18464:OCLC
18266:The
18210:and
18200:The
17763:and
17059:and
16510:cosh
15102:for
12484:span
12354:for
11578:<
10789:and
10384:span
10277:and
10022:and
9965:and
9905:and
9848:and
9381:and
9278:and
8584:>
8573:for
7908:span
7863:<
7852:For
7378:span
6984:<
6550:Let
6442:Let
6369:let
6276:Let
6273:do:
6229:For
6125:span
6065:and
5957:span
5344:and
5167:span
5018:Let
4847:let
4754:Let
4751:do:
4707:For
4339:<
4272:span
4202:span
3873:Let
3816:Let
3760:For
3612:and
3135:real
3045:and
2865:The
2789:The
2356:and
1823:<
1817:<
1806:for
1672:Note
1336:Let
1236:Let
1173:Let
1124:Let
864:Let
861:do:
823:For
752:Let
689:Let
640:Let
569:Let
40:The
19106:TLB
18819:doi
18748:doi
18717:doi
18673:doi
18594:doi
18491:doi
18441:doi
18399:doi
18230:).
18180:of
17300:by
15128:is
14678:cos
14663:cos
12060:of
11901:to
11847:to
10628:min
10600:and
10560:max
10083:by
9523:min
9455:max
9158:of
9089:of
9028:min
8981:max
8628:is
6617:If
5220:If
3434:of
3180:If
2945:log
2889:in
2797:in
2368:If
2060:if
923:If
517:is
352:an
292:).
159:all
19179::
18959:.
18955:.
18922:.
18914:;
18869:.
18825:.
18817:.
18809:.
18797:84
18795:.
18744:22
18742:.
18738:.
18713:49
18711:.
18707:.
18679:.
18652:.
18646:.
18606:^
18590:34
18588:.
18563:.
18546:^
18518:^
18487:10
18485:.
18439:.
18427:.
18413:^
18395:45
18393:.
18389:.
18192:.
14548:.
13698:.
12450:.
11664:.
11593:.
11143:.
10310:.
9062:.
6741:,
5423:,
5217:.)
5015:.)
2736:.
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920:).
197:a
153:.
19076:)
19072:(
19017:e
19010:t
19003:v
18989:.
18961:2
18936:.
18879:.
18854:.
18848::
18833:.
18821::
18813::
18803::
18780:.
18754:.
18750::
18723:.
18719::
18689:.
18675::
18654:2
18628:.
18600:.
18596::
18573:.
18540:.
18497:.
18493::
18470:.
18447:.
18443::
18435::
18429:8
18407:.
18401::
18348:2
18344:z
18340:t
18337:+
18332:1
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18299:2
18295:z
18226:/
18140:A
18044:1
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18038:m
18034:v
18030:,
18024:,
18019:2
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18011:,
18006:1
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17925:+
17922:1
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17838:1
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17789:+
17783:2
17780:+
17777:1
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17771:R
17748:2
17745:+
17742:1
17720:2
17713:)
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17703:+
17700:1
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17674:,
17669:2
17665:t
17661:)
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17635:(
17632:=
17629:u
17626:A
17617:u
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17577:,
17572:2
17562:2
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17549:1
17541:)
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17532:t
17525:1
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17519:=
17516:u
17513:A
17504:u
17477:.
17468:2
17465:+
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17432:2
17419:1
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17405:1
17401:1
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17390:)
17385:2
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17364:(
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17345:2
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17328:1
17318:2
17288:t
17266:2
17262:z
17238:,
17233:2
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17222:+
17217:1
17213:z
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17200:z
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17140:u
17115:2
17104:=
17099:n
17072:2
17045:1
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17013:2
17004:|
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16990:n
16981:|
16935:2
16928:R
16907:R
16878:)
16875:1
16869:m
16866:(
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16847:n
16834:1
16826:(
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16813:|
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16802:d
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16789:2
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16738:2
16733:)
16727:)
16724:1
16718:m
16715:(
16708:R
16704:+
16699:1
16693:m
16689:R
16684:(
16679:4
16674:)
16669:n
16656:1
16648:(
16640:2
16635:|
16628:1
16624:d
16619:|
16611:2
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16599:1
16595:d
16590:|
16583:1
16577:=
16564:)
16561:)
16555:2
16552:+
16549:1
16546:(
16537:)
16534:1
16528:m
16525:(
16522:(
16514:2
16505:1
16500:)
16495:n
16482:1
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16466:2
16461:|
16454:1
16450:d
16445:|
16437:2
16432:|
16425:1
16421:d
16416:|
16409:1
16403:=
16388:2
16383:|
16376:1
16372:d
16367:|
16361:2
16357:)
16353:1
16350:+
16344:2
16341:(
16336:1
16330:m
16326:c
16319:)
16313:2
16308:|
16301:1
16297:d
16292:|
16285:1
16281:(
16277:)
16272:n
16259:1
16251:(
16236:1
16223:1
16188:,
16180:+
16175:2
16165:2
16162:+
16156:2
16153:+
16150:1
16147:=
16142:)
16136:2
16133:+
16130:1
16127:(
16117:e
16113:=
16110:R
16071:n
16058:2
16046:2
16033:1
16022:=
15993:;
15989:)
15985:1
15982:+
15974:n
15961:2
15949:2
15936:1
15925:2
15921:(
15915:1
15909:m
15905:c
15901:=
15897:)
15889:n
15876:2
15864:n
15851:2
15838:1
15830:2
15824:(
15818:1
15812:m
15808:c
15804:=
15801:)
15796:1
15788:(
15785:p
15759:.
15751:2
15746:|
15739:1
15735:d
15730:|
15724:2
15719:|
15714:)
15709:1
15701:(
15698:p
15694:|
15686:2
15681:|
15674:k
15670:d
15665:|
15659:n
15654:2
15651:=
15648:k
15640:)
15635:n
15622:1
15614:(
15600:2
15595:|
15590:)
15585:1
15577:(
15574:p
15570:|
15564:2
15559:|
15552:1
15548:d
15543:|
15537:)
15532:k
15519:1
15511:(
15506:2
15501:|
15494:k
15490:d
15485:|
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15474:2
15471:=
15468:k
15449:2
15444:|
15439:)
15434:k
15426:(
15423:p
15419:|
15413:2
15408:|
15401:k
15397:d
15392:|
15386:n
15381:1
15378:=
15375:k
15363:2
15358:|
15353:)
15348:k
15340:(
15337:p
15333:|
15329:)
15324:k
15311:1
15303:(
15298:2
15293:|
15286:k
15282:d
15277:|
15271:n
15266:2
15263:=
15260:k
15249:=
15246:)
15241:1
15237:v
15233:)
15230:A
15227:(
15224:p
15221:(
15218:r
15210:1
15197:1
15184:1
15156:0
15136:1
15116:2
15110:k
15088:2
15083:|
15078:)
15073:k
15065:(
15062:p
15058:|
15035:1
15008:1
15000:=
14995:2
14966:)
14958:n
14945:2
14933:n
14920:2
14909:x
14906:2
14900:(
14894:1
14888:m
14884:c
14880:=
14877:)
14874:x
14871:(
14868:p
14845:]
14842:1
14839:,
14836:1
14830:[
14808:1
14783:]
14780:1
14777:,
14774:1
14768:[
14748:]
14745:1
14742:,
14739:1
14733:[
14713:x
14693:)
14690:x
14687:k
14684:(
14675:=
14672:)
14669:x
14660:(
14655:k
14651:c
14630:k
14608:k
14604:c
14579:p
14559:A
14534:1
14521:1
14494:1
14469:p
14449:1
14446:=
14443:k
14420:.
14412:2
14407:|
14403:)
14398:k
14390:(
14387:p
14383:|
14376:2
14371:|
14364:k
14360:d
14355:|
14349:n
14344:1
14341:=
14338:k
14326:2
14321:|
14317:)
14312:k
14304:(
14301:p
14297:|
14292:)
14287:k
14274:1
14266:(
14261:2
14256:|
14249:k
14245:d
14240:|
14234:n
14229:1
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14223:k
14212:=
14206:)
14201:k
14193:(
14190:p
14181:)
14175:k
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14164:p
14159:2
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14143:d
14138:|
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14127:1
14124:=
14121:k
14111:)
14106:k
14098:(
14095:p
14086:)
14080:k
14072:(
14069:p
14064:k
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14038:d
14033:|
14027:n
14022:1
14019:=
14016:k
14000:1
13992:=
13984:k
13980:z
13976:)
13971:k
13963:(
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13952:k
13944:(
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13868:k
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13839:k
13831:(
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13640:(
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13317:A
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13299:(
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13227:(
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13184:A
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13159:(
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13087:(
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13051:A
13048:(
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13042:A
13033:)
13029:A
13026:(
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13013:1
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13002:=
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12983:A
12980:(
12977:p
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12951:A
12948:(
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11242:1
11239:=
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11191:=
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11119:1
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11073:=
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11030:,
11024:,
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10710:u
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10666:.
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10660:z
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10624:=
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10616:j
10612:y
10608:(
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10595:)
10592:z
10589:(
10586:r
10579:j
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10564:z
10556:=
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10544:x
10540:(
10537:r
10512:j
10506:L
10495:j
10491:y
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10439:1
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10425:,
10419:,
10414:1
10410:x
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10398:1
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10390:(
10381:=
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10232:,
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10190:1
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10111:{
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9791:(
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8953:.
8948:n
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8935:x
8931:,
8925:x
8916:x
8910:x
8907:A
8898:x
8891:=
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8885:x
8882:(
8879:r
8849:A
8821:T
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8780:H
8760:H
8737:H
8734:=
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8715:=
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8703:A
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8689:=
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8675:V
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8658:(
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8596:;
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8561:0
8558:=
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8550:,
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8522:V
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8506:=
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8478:,
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8440:,
8434:,
8429:1
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8378:,
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8352:=
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8319:v
8312:=
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8112:v
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8080:1
8074:j
8070:v
8066:A
8063:(
8060:=
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8052:,
8049:1
8043:j
8039:h
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8006:=
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7947:1
7941:j
7937:v
7933:,
7927:,
7922:1
7918:v
7914:(
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7866:j
7860:k
7837:.
7832:j
7828:v
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7812:k
7808:v
7804:A
7801:(
7798:=
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7789:v
7779:A
7768:k
7764:v
7760:=
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7751:v
7747:A
7737:k
7733:v
7729:=
7721:1
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7715:j
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7700:k
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7692:=
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7684:,
7681:k
7677:h
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7628:j
7625:,
7622:k
7618:h
7597:A
7567:m
7563:u
7559:,
7553:,
7548:1
7544:u
7523:A
7501:j
7498:,
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7491:h
7468:m
7463:1
7460:=
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7453:}
7447:j
7443:v
7439:{
7419:,
7416:)
7411:j
7407:v
7403:,
7397:,
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7384:(
7370:j
7366:u
7343:j
7339:u
7315:.
7307:1
7304:+
7301:j
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7290:=
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7282:+
7279:j
7275:v
7264:1
7261:+
7258:j
7254:v
7245:1
7242:+
7239:j
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7228:=
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7220:+
7217:j
7213:w
7202:1
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7188:=
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7149:v
7120:1
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7114:j
7110:w
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7095:v
7091:=
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7083:,
7080:k
7076:h
7050:1
7047:+
7044:j
7040:w
7033:=
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7025:,
7022:1
7019:+
7016:j
7012:h
6999:;
6987:m
6981:j
6959:k
6955:v
6949:j
6946:,
6943:k
6939:h
6933:1
6930:+
6927:j
6922:1
6919:=
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6908:=
6903:j
6899:v
6895:A
6870:j
6867:,
6864:k
6860:h
6843:.
6829:j
6825:v
6821:,
6815:,
6810:1
6806:v
6785:1
6763:1
6760:+
6757:j
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6724:,
6721:1
6718:+
6715:j
6711:h
6706:/
6700:1
6697:+
6694:j
6690:w
6686:=
6681:1
6678:+
6675:j
6671:v
6650:0
6642:j
6639:,
6636:1
6633:+
6630:j
6626:h
6614:.
6597:1
6594:+
6591:j
6587:w
6580:=
6575:j
6572:,
6569:1
6566:+
6563:j
6559:h
6547:.
6533:k
6529:v
6523:j
6520:,
6517:k
6513:h
6507:j
6502:1
6499:=
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6480:1
6477:+
6474:j
6470:w
6466:=
6461:1
6458:+
6455:j
6451:w
6439:.
6422:1
6419:+
6416:j
6412:w
6401:k
6397:v
6393:=
6388:j
6385:,
6382:k
6378:h
6357:j
6354:,
6348:,
6345:1
6342:=
6339:k
6328:.
6314:j
6310:v
6306:A
6303:=
6295:1
6292:+
6289:j
6285:w
6261:1
6255:m
6252:,
6246:,
6243:1
6240:=
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6226:.
6214:1
6192:1
6188:v
6163:)
6158:j
6154:v
6150:,
6144:,
6139:1
6135:v
6131:(
6103:j
6099:v
6095:A
6092:=
6084:1
6081:+
6078:j
6074:w
6051:j
6047:u
6043:A
6040:=
6032:1
6029:+
6026:j
6022:u
6001:)
5996:1
5990:j
5986:v
5982:,
5976:,
5971:1
5967:v
5963:(
5949:j
5945:v
5936:j
5932:u
5909:j
5905:v
5882:j
5878:u
5865:.
5851:k
5847:v
5841:j
5838:,
5835:k
5831:g
5825:j
5820:1
5817:=
5814:k
5806:+
5801:1
5798:+
5795:j
5791:v
5782:1
5779:+
5776:j
5772:w
5765:=
5760:k
5756:v
5750:j
5747:,
5744:k
5740:g
5734:j
5729:1
5726:=
5723:k
5715:+
5710:1
5707:+
5704:j
5700:w
5696:=
5688:1
5685:+
5682:j
5678:u
5674:=
5669:1
5666:+
5663:j
5659:u
5647:1
5644:+
5641:j
5637:u
5630:=
5625:j
5621:u
5617:A
5592:j
5588:v
5565:j
5561:u
5544:.
5530:j
5526:v
5522:,
5516:,
5511:1
5507:v
5486:1
5464:1
5461:+
5458:j
5454:v
5450:=
5445:1
5442:+
5439:j
5435:u
5406:1
5403:+
5400:j
5396:w
5388:/
5382:1
5379:+
5376:j
5372:w
5368:=
5363:1
5360:+
5357:j
5353:v
5324:1
5321:+
5318:j
5314:u
5306:/
5297:1
5294:+
5291:j
5287:u
5283:=
5278:1
5275:+
5272:j
5268:u
5247:0
5239:1
5236:+
5233:j
5229:w
5205:)
5200:j
5196:v
5192:,
5186:,
5181:1
5177:v
5173:(
5142:1
5139:+
5136:j
5132:u
5109:k
5105:v
5099:j
5096:,
5093:k
5089:g
5083:j
5078:1
5075:=
5072:k
5056:1
5053:+
5050:j
5046:u
5042:=
5037:1
5034:+
5031:j
5027:w
5001:j
4997:v
4993:,
4987:,
4982:1
4978:v
4952:1
4949:+
4946:j
4942:u
4938:=
4933:j
4929:u
4925:A
4900:1
4897:+
4894:j
4890:u
4879:k
4875:v
4871:=
4866:j
4863:,
4860:k
4856:g
4835:j
4832:,
4826:,
4823:1
4820:=
4817:k
4806:.
4792:j
4788:u
4784:A
4781:=
4773:1
4770:+
4767:j
4763:u
4739:1
4733:m
4730:,
4724:,
4721:1
4718:=
4715:j
4704:.
4690:1
4686:u
4682:=
4677:1
4673:v
4652:1
4630:1
4626:u
4597:A
4571:j
4567:u
4544:1
4538:j
4534:u
4530:,
4524:,
4519:1
4515:u
4492:j
4488:v
4465:1
4459:j
4455:u
4451:,
4445:,
4440:1
4436:u
4413:j
4409:u
4386:j
4382:u
4378:=
4373:j
4369:v
4345:;
4342:m
4336:j
4330:1
4310:)
4305:j
4301:v
4297:,
4291:,
4286:1
4282:v
4278:(
4266:x
4246:)
4241:1
4238:+
4235:j
4231:v
4227:,
4221:,
4216:1
4212:v
4208:(
4196:x
4193:A
4171:n
4166:C
4142:m
4137:1
4134:=
4131:j
4127:}
4121:j
4117:v
4113:{
4083:j
4079:u
4053:j
4049:u
4028:A
4000:j
3996:u
3975:j
3949:.
3938:1
3935:+
3932:j
3928:u
3920:/
3911:1
3908:+
3905:j
3901:u
3897:=
3892:1
3889:+
3886:j
3882:u
3859:.
3854:j
3850:u
3846:A
3843:=
3835:1
3832:+
3829:j
3825:u
3799:j
3795:u
3774:1
3768:j
3757:.
3745:0
3737:1
3733:u
3706:A
3663:)
3658:2
3654:n
3650:(
3647:O
3625:j
3598:j
3569:)
3566:n
3563:(
3560:O
3534:)
3529:3
3525:n
3521:(
3518:O
3496:n
3492:Q
3483:2
3479:Q
3473:1
3469:Q
3465:=
3462:V
3442:V
3420:n
3416:Q
3407:2
3403:Q
3397:1
3393:Q
3372:V
3349:A
3338:.
3322:A
3274:n
3268:m
3248:A
3228:T
3208:m
3188:n
3177:.
3165:T
3145:A
3121:T
3097:T
3077:A
3057:T
3024:2
3020:m
2999:)
2994:2
2990:m
2986:(
2983:O
2954:)
2951:m
2942:m
2939:(
2936:O
2913:)
2908:2
2904:m
2900:(
2897:O
2877:T
2850:)
2847:m
2844:(
2841:O
2821:)
2816:2
2812:m
2808:(
2805:O
2774:m
2768:m
2748:T
2724:T
2704:A
2678:.
2675:y
2669:=
2659:x
2656:V
2650:=
2640:)
2637:x
2631:(
2628:V
2625:=
2615:x
2612:T
2609:V
2606:=
2596:x
2593:I
2590:T
2587:V
2584:=
2574:x
2571:V
2562:V
2558:T
2555:V
2552:=
2542:x
2539:V
2536:A
2533:=
2526:y
2523:A
2497:A
2477:x
2474:V
2471:=
2468:y
2448:x
2442:=
2439:x
2436:T
2416:x
2396:T
2330:1
2326:v
2303:j
2299:w
2276:1
2270:j
2266:v
2245:T
2223:1
2220:+
2217:j
2213:v
2190:j
2186:w
2163:j
2159:w
2133:j
2129:w
2102:j
2098:v
2074:n
2071:=
2068:m
2048:)
2043:2
2039:n
2035:d
2032:(
2029:O
2009:)
2006:n
2003:m
2000:d
1997:(
1994:O
1974:d
1954:)
1951:n
1948:d
1945:(
1942:O
1922:)
1919:n
1916:(
1913:O
1890:0
1887:=
1882:j
1855:1
1851:v
1838:.
1826:m
1820:j
1814:2
1792:1
1786:j
1782:v
1776:j
1768:+
1763:j
1759:v
1753:j
1745:+
1740:1
1737:+
1734:j
1730:v
1724:1
1721:+
1718:j
1710:=
1702:j
1698:w
1694:=
1689:j
1685:v
1681:A
1666:.
1652:)
1644:m
1632:m
1619:0
1610:m
1598:1
1592:m
1580:1
1574:m
1556:1
1550:m
1517:3
1505:3
1487:3
1475:2
1463:2
1451:0
1441:2
1429:1
1418:(
1413:=
1410:T
1388:m
1384:v
1380:,
1374:,
1369:1
1365:v
1344:V
1331:.
1317:1
1311:j
1307:v
1301:j
1288:j
1284:v
1278:j
1262:j
1258:w
1254:=
1249:j
1245:w
1233:.
1219:j
1215:v
1199:j
1195:w
1191:=
1186:j
1170:.
1156:j
1152:v
1148:A
1145:=
1137:j
1133:w
1119:.
1105:1
1099:j
1095:v
1091:,
1085:,
1080:1
1076:v
1055:1
1033:j
1029:v
1003:j
994:/
988:1
982:j
978:w
974:=
969:j
965:v
944:0
936:j
899:1
893:j
889:w
882:=
877:j
849:m
846:,
840:,
837:2
834:=
831:j
818:.
804:1
800:v
794:1
778:1
774:w
770:=
765:1
761:w
749:.
735:1
731:v
715:1
711:w
707:=
702:1
686:.
672:1
668:v
664:A
661:=
653:1
649:w
634:.
622:1
597:n
592:C
582:1
578:v
560:.
542:V
538:T
535:V
532:=
529:A
505:V
485:n
482:=
479:m
459:m
453:m
433:V
430:A
421:V
417:=
414:T
386:V
366:m
360:n
332:m
312:v
309:A
303:v
280:n
277:=
274:m
254:m
234:n
228:n
208:A
173:m
137:n
117:m
94:n
88:n
64:m
37:.
30:.
23:.
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