9917:
9185:
9912:{\displaystyle {\begin{aligned}\mathbf {A} =\mathbf {LDL} ^{\mathrm {T} }&={\begin{pmatrix}\mathbf {I} &0&0\\\mathbf {L} _{21}&\mathbf {I} &0\\\mathbf {L} _{31}&\mathbf {L} _{32}&\mathbf {I} \\\end{pmatrix}}{\begin{pmatrix}\mathbf {D} _{1}&0&0\\0&\mathbf {D} _{2}&0\\0&0&\mathbf {D} _{3}\\\end{pmatrix}}{\begin{pmatrix}\mathbf {I} &\mathbf {L} _{21}^{\mathrm {T} }&\mathbf {L} _{31}^{\mathrm {T} }\\0&\mathbf {I} &\mathbf {L} _{32}^{\mathrm {T} }\\0&0&\mathbf {I} \\\end{pmatrix}}\\&={\begin{pmatrix}\mathbf {D} _{1}&&(\mathrm {symmetric} )\\\mathbf {L} _{21}\mathbf {D} _{1}&\mathbf {L} _{21}\mathbf {D} _{1}\mathbf {L} _{21}^{\mathrm {T} }+\mathbf {D} _{2}&\\\mathbf {L} _{31}\mathbf {D} _{1}&\mathbf {L} _{31}\mathbf {D} _{1}\mathbf {L} _{21}^{\mathrm {T} }+\mathbf {L} _{32}\mathbf {D} _{2}&\mathbf {L} _{31}\mathbf {D} _{1}\mathbf {L} _{31}^{\mathrm {T} }+\mathbf {L} _{32}\mathbf {D} _{2}\mathbf {L} _{32}^{\mathrm {T} }+\mathbf {D} _{3}\end{pmatrix}},\end{aligned}}}
12566:
12099:
8575:
7976:
5624:
12561:{\displaystyle {\begin{aligned}\mathbf {S} _{11}&=\mathbf {L} _{11},\\\mathbf {S} _{12}&=\mathbf {L} _{11}^{\mathrm {T} }\setminus \mathbf {A} _{12},\\\mathbf {S} _{13}&=\mathbf {L} _{13},\\\mathbf {S} _{22}&=\mathrm {chol} \left(\mathbf {A} _{22}-\mathbf {S} _{12}^{\mathrm {T} }\mathbf {S} _{12}\right),\\\mathbf {S} _{23}&=\mathbf {S} _{22}^{\mathrm {T} }\setminus \left(\mathbf {A} _{23}-\mathbf {S} _{12}^{\mathrm {T} }\mathbf {S} _{13}\right),\\\mathbf {S} _{33}&=\mathrm {chol} \left(\mathbf {L} _{33}^{\mathrm {T} }\mathbf {L} _{33}-\mathbf {S} _{23}^{\mathrm {T} }\mathbf {S} _{23}\right).\end{aligned}}}
1860:
5124:
8570:{\displaystyle {\begin{aligned}\mathbf {A} =\mathbf {LDL} ^{\mathrm {T} }&={\begin{pmatrix}1&0&0\\L_{21}&1&0\\L_{31}&L_{32}&1\\\end{pmatrix}}{\begin{pmatrix}D_{1}&0&0\\0&D_{2}&0\\0&0&D_{3}\\\end{pmatrix}}{\begin{pmatrix}1&L_{21}&L_{31}\\0&1&L_{32}\\0&0&1\\\end{pmatrix}}\\&={\begin{pmatrix}D_{1}&&(\mathrm {symmetric} )\\L_{21}D_{1}&L_{21}^{2}D_{1}+D_{2}&\\L_{31}D_{1}&L_{31}L_{21}D_{1}+L_{32}D_{2}&L_{31}^{2}D_{1}+L_{32}^{2}D_{2}+D_{3}.\end{pmatrix}}.\end{aligned}}}
1844:
5619:{\displaystyle {\begin{aligned}\mathbf {A} =\mathbf {LL} ^{T}&={\begin{pmatrix}L_{11}&0&0\\L_{21}&L_{22}&0\\L_{31}&L_{32}&L_{33}\\\end{pmatrix}}{\begin{pmatrix}L_{11}&L_{21}&L_{31}\\0&L_{22}&L_{32}\\0&0&L_{33}\end{pmatrix}}\\&={\begin{pmatrix}L_{11}^{2}&&({\text{symmetric}})\\L_{21}L_{11}&L_{21}^{2}+L_{22}^{2}&\\L_{31}L_{11}&L_{31}L_{21}+L_{32}L_{22}&L_{31}^{2}+L_{32}^{2}+L_{33}^{2}\end{pmatrix}},\end{aligned}}}
12782:
11742:
1544:
15226:
1535:
13404:
12576:
11536:
5910:
15022:
7955:, in which case the algorithm cannot continue. However, this can only happen if the matrix is very ill-conditioned. One way to address this is to add a diagonal correction matrix to the matrix being decomposed in an attempt to promote the positive-definiteness. While this might lessen the accuracy of the decomposition, it can be very favorable for other reasons; for example, when performing
11972:
12945:
1302:
13197:
1839:{\displaystyle {\begin{aligned}{\begin{pmatrix}4&12&-16\\12&37&-43\\-16&-43&98\\\end{pmatrix}}&={\begin{pmatrix}1&0&0\\3&1&0\\-4&5&1\\\end{pmatrix}}{\begin{pmatrix}4&0&0\\0&1&0\\0&0&9\\\end{pmatrix}}{\begin{pmatrix}1&3&-4\\0&1&5\\0&0&1\\\end{pmatrix}}.\end{aligned}}}
5633:
14376:
992:
4978:
4668:
12777:{\displaystyle {\begin{aligned}{\tilde {\mathbf {A} }}&={\begin{pmatrix}\mathbf {A} _{11}&\mathbf {A} _{12}&\mathbf {A} _{13}\\\mathbf {A} _{12}^{\mathrm {T} }&\mathbf {A} _{22}&\mathbf {A} _{23}\\\mathbf {A} _{13}^{\mathrm {T} }&\mathbf {A} _{23}^{\mathrm {T} }&\mathbf {A} _{33}\\\end{pmatrix}}\end{aligned}}}
11737:{\displaystyle {\begin{aligned}{\tilde {\mathbf {A} }}&={\begin{pmatrix}\mathbf {A} _{11}&\mathbf {A} _{12}&\mathbf {A} _{13}\\\mathbf {A} _{12}^{\mathrm {T} }&\mathbf {A} _{22}&\mathbf {A} _{23}\\\mathbf {A} _{13}^{\mathrm {T} }&\mathbf {A} _{23}^{\mathrm {T} }&\mathbf {A} _{33}\\\end{pmatrix}}\end{aligned}}}
13190:
11811:
12787:
458:
4452:
10211:
16638:
15221:{\displaystyle \mathbf {A} ={\begin{bmatrix}\mathbf {A} _{11}&\mathbf {A} _{12}&\mathbf {A} _{13}&\;\\\mathbf {A} _{12}^{*}&\mathbf {A} _{22}&\mathbf {A} _{23}&\;\\\mathbf {A} _{13}^{*}&\mathbf {A} _{23}^{*}&\mathbf {A} _{33}&\;\\\;&\;&\;&\ddots \end{bmatrix}}}
13047:
11383:
14218:
16634:
16167:
14929:
806:
4788:
4494:
1530:{\displaystyle {\begin{aligned}{\begin{pmatrix}4&12&-16\\12&37&-43\\-16&-43&98\\\end{pmatrix}}={\begin{pmatrix}2&0&0\\6&1&0\\-8&5&3\\\end{pmatrix}}{\begin{pmatrix}2&6&-8\\0&1&5\\0&0&3\\\end{pmatrix}}.\end{aligned}}}
13399:{\displaystyle {\begin{aligned}\mathbf {L} _{11}&=\mathbf {S} _{11},\\\mathbf {L} _{13}&=\mathbf {S} _{13},\\\mathbf {L} _{33}&=\mathrm {chol} \left(\mathbf {S} _{33}^{\mathrm {T} }\mathbf {S} _{33}+\mathbf {S} _{23}^{\mathrm {T} }\mathbf {S} _{23}\right).\end{aligned}}}
13076:
10050:
5905:{\displaystyle {\begin{aligned}\mathbf {L} ={\begin{pmatrix}{\sqrt {A_{11}}}&0&0\\A_{21}/L_{11}&{\sqrt {A_{22}-L_{21}^{2}}}&0\\A_{31}/L_{11}&\left(A_{32}-L_{31}L_{21}\right)/L_{22}&{\sqrt {A_{33}-L_{31}^{2}-L_{32}^{2}}}\end{pmatrix}}\end{aligned}}}
326:
11476:
6515:
4287:
9164:
6203:
8849:
7796:
system of linear equations. If the LU decomposition is used, then the algorithm is unstable unless some sort of pivoting strategy is used. In the latter case, the error depends on the so-called growth factor of the matrix, which is usually (but not always) small.
10054:
13907:
13649:
15793:
Benoit (1924). "Note sur une méthode de résolution des équations normales provenant de l'application de la méthode des moindres carrés à un système d'équations linéaires en nombre inférieur à celui des inconnues (Procédé du
Commandant Cholesky)".
11967:{\displaystyle {\begin{aligned}{\tilde {\mathbf {S} }}&={\begin{pmatrix}\mathbf {S} _{11}&\mathbf {S} _{12}&\mathbf {S} _{13}\\0&\mathbf {S} _{22}&\mathbf {S} _{23}\\0&0&\mathbf {S} _{33}\\\end{pmatrix}}.\end{aligned}}}
12952:
12940:{\displaystyle {\begin{aligned}{\tilde {\mathbf {S} }}&={\begin{pmatrix}\mathbf {S} _{11}&\mathbf {S} _{12}&\mathbf {S} _{13}\\0&\mathbf {S} _{22}&\mathbf {S} _{23}\\0&0&\mathbf {S} _{33}\\\end{pmatrix}}\end{aligned}}}
616:
3608:(BFGS). Loss of the positive-definite condition through round-off error is avoided if rather than updating an approximation to the inverse of the Hessian, one updates the Cholesky decomposition of an approximation of the Hessian matrix itself.
13706:
11284:
13502:
has a
Cholesky decomposition. This result can be extended to the positive semi-definite case by a limiting argument. The argument is not fully constructive, i.e., it gives no explicit numerical algorithms for computing Cholesky factors.
4783:
10407:
9180:* factorization is known to be unstable without careful pivoting; specifically, the elements of the factorization can grow arbitrarily. A possible improvement is to perform the factorization on block sub-matrices, commonly 2 × 2:
5104:
15354:
13465:
7914:
2222:
11205:
11112:
10580:
6346:
14807:
9926:
6039:
4153:
12570:
These formulas may be used to determine the
Cholesky factor after the insertion of rows or columns in any position, if the row and column dimensions are appropriately set (including to zero). The inverse problem,
11390:
6350:
14371:{\displaystyle \langle \mathbf {A} x,y\rangle =\left\langle \lim \mathbf {A} _{k}x,y\right\rangle =\langle \lim \mathbf {L} _{k}\mathbf {L} _{k}^{*}x,y\rangle =\langle \mathbf {L} \mathbf {L} ^{*}x,y\rangle \,.}
9007:
9003:
2844:
13800:
3341:
6043:
987:{\displaystyle \mathbf {A} =\mathbf {LDL} ^{*}=\mathbf {L} \mathbf {D} ^{1/2}\left(\mathbf {D} ^{1/2}\right)^{*}\mathbf {L} ^{*}=\mathbf {L} \mathbf {D} ^{1/2}\left(\mathbf {L} \mathbf {D} ^{1/2}\right)^{*}.}
15284:
8697:
7946:
One concern with the
Cholesky decomposition to be aware of is the use of square roots. If the matrix being factorized is positive definite as required, the numbers under the square roots are always positive
8693:
15433:
9190:
7981:
5129:
4973:{\displaystyle \mathbf {A} ^{(i+1)}={\begin{pmatrix}\mathbf {I} _{i-1}&0&0\\0&1&0\\0&0&\mathbf {B} ^{(i)}-{\frac {1}{a_{i,i}}}\mathbf {b} _{i}\mathbf {b} _{i}^{*}\end{pmatrix}}.}
4663:{\displaystyle \mathbf {L} _{i}:={\begin{pmatrix}\mathbf {I} _{i-1}&0&0\\0&{\sqrt {a_{i,i}}}&0\\0&{\frac {1}{\sqrt {a_{i,i}}}}\mathbf {b} _{i}&\mathbf {I} _{n-i}\end{pmatrix}},}
1154:
284:
3909:
3255:
14758:
14683:
14420:
14149:
10263:
3080:
1036:
4225:
Which of the algorithms below is faster depends on the details of the implementation. Generally, the first algorithm will be slightly slower because it accesses the data in a less regular manner.
1253:
1109:
13807:
13202:
13081:
12957:
12792:
12581:
12104:
11816:
11541:
5638:
3346:
For linear systems that can be put into symmetric form, the
Cholesky decomposition (or its LDL variant) is the method of choice, for superior efficiency and numerical stability. Compared to the
1549:
1307:
15478:
12039:
13185:{\displaystyle {\begin{aligned}\mathbf {A} &={\begin{pmatrix}\mathbf {A} _{11}&\mathbf {A} _{13}\\\mathbf {A} _{13}^{\mathrm {T} }&\mathbf {A} _{33}\\\end{pmatrix}},\end{aligned}}}
691:
12004:
174:
14530:
14464:
14063:
14011:
1194:
13658:
3163:
453:{\displaystyle {\begin{bmatrix}0&0\\0&1\end{bmatrix}}=\mathbf {L} \mathbf {L} ^{*},\quad \quad \mathbf {L} ={\begin{bmatrix}0&0\\\cos \theta &\sin \theta \end{bmatrix}},}
14966:
3210:
2452:
2119:
12072:
4447:{\displaystyle \mathbf {A} ^{(i)}={\begin{pmatrix}\mathbf {I} _{i-1}&0&0\\0&a_{i,i}&\mathbf {b} _{i}^{*}\\0&\mathbf {b} _{i}&\mathbf {B} ^{(i)}\end{pmatrix}},}
3288:
3113:
3038:
2999:
2766:
11806:
11775:
11509:
11236:
10499:
10460:
10338:
15011:
13560:
4697:
4057:
13967:
2282:
3482:
2891:
14716:
14581:
13739:
2311:
5042:
2950:
2712:
2058:
542:
5114:
2653:
727:
in the decomposition. The main advantage is that the LDL decomposition can be computed and used with essentially the same algorithms, but avoids extracting square roots.
10206:{\displaystyle \mathbf {L} _{ij}=\left(\mathbf {A} _{ij}-\sum _{k=1}^{j-1}\mathbf {L} _{ik}\mathbf {D} _{k}\mathbf {L} _{jk}^{\mathrm {T} }\right)\mathbf {D} _{j}^{-1}.}
15293:
14802:
14780:
14633:
14603:
14552:
14486:
14171:
14107:
14085:
13526:
13500:
13071:
12094:
11531:
11277:
11134:
11053:
10521:
10429:
10307:
10285:
1064:
16543:
3833:
1901:
13552:
7847:
2491:
2124:
2006:
15616:
6225:
2604:
5925:
13929:
11241:
The code for the rank-one update shown above can easily be adapted to do a rank-one downdate: one merely needs to replace the two additions in the assignment to
4077:
4018:
3793:
3769:
3742:
3715:
3695:
3668:
3594:
3563:
3536:
3509:
3429:
2548:
2521:
1955:
1928:
13042:{\displaystyle {\begin{aligned}\mathbf {L} &={\begin{pmatrix}\mathbf {L} _{11}&\mathbf {L} _{13}\\0&\mathbf {L} _{33}\\\end{pmatrix}}\end{aligned}}}
10223:
A task that often arises in practice is that one needs to update a
Cholesky decomposition. In more details, one has already computed the Cholesky decomposition
2391:
3643:
The following simplified example shows the economy one gets from the
Cholesky decomposition: suppose the goal is to generate two correlated normal variables
15730:
15636:
14211:
14191:
6207:
For complex and real matrices, inconsequential arbitrary sign changes of diagonal and associated off-diagonal elements are allowed. The expression under the
2911:
2568:
2242:
11378:{\displaystyle \mathbf {A} ={\begin{pmatrix}\mathbf {A} _{11}&\mathbf {A} _{13}\\\mathbf {A} _{13}^{\mathrm {T} }&\mathbf {A} _{33}\\\end{pmatrix}}}
10343:
16424:
8891:
16531:
16694:
14924:{\textstyle A=\mathbf {B} \mathbf {B} ^{*}=(\mathbf {QR} )^{*}\mathbf {QR} =\mathbf {R} ^{*}\mathbf {Q} ^{*}\mathbf {QR} =\mathbf {R} ^{*}\mathbf {R} }
3922:
commonly use the
Cholesky decomposition to choose a set of so-called sigma points. The Kalman filter tracks the average state of a system as a vector
15235:
13411:
204:. Every Hermitian positive-definite matrix (and thus also every real-valued symmetric positive-definite matrix) has a unique Cholesky decomposition.
8594:
11151:
11058:
10526:
16496:
16341:
15377:
1114:
244:
103:
for real matrices, and posthumously published in 1924. When it is applicable, the
Cholesky decomposition is roughly twice as efficient as the
3384:
comes from an energy functional, which must be positive from physical considerations; this happens frequently in the numerical solution of
16815:
2771:
13744:
3299:
762:
10045:{\displaystyle \mathbf {D} _{j}=\mathbf {A} _{jj}-\sum _{k=1}^{j-1}\mathbf {L} _{jk}\mathbf {D} _{k}\mathbf {L} _{jk}^{\mathrm {T} },}
650:
16840:
7800:
Now, suppose that the
Cholesky decomposition is applicable. As mentioned above, the algorithm will be twice as fast. Furthermore, no
136:
1159:
4162:
There are various methods for calculating the Cholesky decomposition. The computational complexity of commonly used algorithms is
13655:. (This is an immediate consequence of, for example, the spectral mapping theorem for the polynomial functional calculus.) Also,
16835:
16687:
11471:{\displaystyle \mathbf {L} ={\begin{pmatrix}\mathbf {L} _{11}&\mathbf {L} _{13}\\0&\mathbf {L} _{33}\\\end{pmatrix}},}
6510:{\displaystyle L_{i,j}={\frac {1}{L_{j,j}}}\left(A_{i,j}-\sum _{k=1}^{j-1}L_{j,k}^{*}L_{i,k}\right)\quad {\text{for }}i>j.}
16450:
16395:
16362:
16139:
16010:
15981:
14021:(because the underlying vector space is finite-dimensional). Consequently, it has a convergent subsequence, also denoted by
9159:{\displaystyle L_{ij}={\frac {1}{D_{j}}}\left(A_{ij}-\sum _{k=1}^{j-1}L_{ik}L_{jk}^{*}D_{k}\right)\quad {\text{for }}i>j.}
6531:
entry if the entries to the left and above are known. The computation is usually arranged in either of the following orders:
3838:
13467:, which allows them to be efficiently calculated using the update and downdate procedures detailed in the previous section.
3215:
16630:
14721:
14646:
14422:. Because the underlying vector space is finite-dimensional, all topologies on the space of operators are equivalent. So
14383:
14112:
10226:
3043:
999:
6198:{\displaystyle L_{i,j}={\frac {1}{L_{j,j}}}\left(A_{i,j}-\sum _{k=1}^{j-1}L_{i,k}L_{j,k}\right)\quad {\text{for }}i>j.}
1219:
1075:
16599:
8844:{\displaystyle L_{ij}={\frac {1}{D_{j}}}\left(A_{ij}-\sum _{k=1}^{j-1}L_{ik}L_{jk}D_{k}\right)\quad {\text{for }}i>j.}
1267:. Some indefinite matrices for which no Cholesky decomposition exists have an LDL decomposition with negative entries in
9921:
where every element in the matrices above is a square submatrix. From this, these analogous recursive relations follow:
15440:
12009:
7956:
3375:
16666:
3993:
of a Hermitian matrix can be computed by Cholesky decomposition, in a manner similar to solving linear systems, using
16871:
16794:
16680:
16621:
16414:
16311:
16238:
15895:
15841:
15748:
66:
16340:
Dereniowski, Dariusz; Kubale, Marek (2004). "Cholesky Factorization of Matrices in Parallel and Ranking of Graphs".
13408:
Notice that the equations above that involve finding the Cholesky decomposition of a new matrix are all of the form
11979:
5117:
Access pattern (white) and writing pattern (yellow) for the in-place Cholesky—Banachiewicz algorithm on a 5×5 matrix
16825:
16572:
includes a partial port of the LAPACK to C++, C#, Delphi, Visual Basic, etc. (spdmatrixcholesky, hpdmatrixcholesky)
14491:
14425:
14024:
13972:
16434:", in Proc. AeroSense: 11th Int. Symp. Aerospace/Defence Sensing, Simulation and Controls, 1997, pp. 182–193.
15537:
numerical computations system provides several functions to calculate, update, and apply a Cholesky decomposition.
1280:
16752:
15773:
3601:
3128:
16866:
14976:
The Cholesky factorization can be generalized to (not necessarily finite) matrices with operator entries. Let
14934:
3181:
2396:
2063:
16524:
15768:
15556:
13902:{\displaystyle \|\mathbf {L} _{k}\|^{2}\leq \|\mathbf {L} _{k}\mathbf {L} _{k}^{*}\|=\|\mathbf {A} _{k}\|\,.}
12044:
3385:
16514:
13644:{\textstyle \left(\mathbf {A} _{k}\right)_{k}:=\left(\mathbf {A} +{\frac {1}{k}}\mathbf {I} _{n}\right)_{k}}
3260:
3085:
3010:
2971:
16784:
15653:
15544:
library provides a high performance implementation of the Cholesky decomposition that can be accessed from
2659:
306:
is only positive semidefinite, instead of positive definite, then it still has a decomposition of the form
16269:
Fang, Haw-ren (24 August 2007). "Analysis of Block LDLT Factorizations for Symmetric Indefinite Matrices".
2717:
16519:
16086:
A Semidefinite Programming Approach to the Graph Realization Problem: Theory, Applications and Extensions
11780:
11749:
11483:
11210:
10473:
10434:
10312:
3374:
symmetric and positive definite arise quite often in applications. For instance, the normal equations in
16474:
14979:
16738:
16368:
2966:
108:
4023:
611:{\textstyle \mathbf {L} ={\begin{bmatrix}\mathbf {L} _{1}&0\\\mathbf {L} _{2}&0\end{bmatrix}}}
16789:
15686:
13934:
16650:
16287:
16207:
16069:
13701:{\displaystyle \mathbf {A} _{k}\rightarrow \mathbf {A} \quad {\text{for}}\quad k\rightarrow \infty }
2247:
16861:
16703:
16559:
is a collection of FORTRAN subroutines for solving dense linear algebra problems (DPOTRF, DPOTRF2,
16461:
15796:
15758:
15642:
15549:
14636:
13652:
13554:
3772:
3717:. To accomplish that, it is necessary to first generate two uncorrelated Gaussian random variables
3434:
2849:
1854:
123:
84:
15919:
Schabauer, Hannes; Pacher, Christoph; Sunderland, Andrew G.; Gansterer, Wilfried N. (2010-05-01).
14692:
14557:
13715:
2287:
497:
Alternatively, the decomposition can be made unique when a pivoting choice is fixed. Formally, if
16029:
Krishnamoorthy, Aravindh; Menon, Deepak (2011). "Matrix Inversion Using Cholesky Decomposition".
15763:
14640:
3919:
3600:
formed by repeating rank-1 updates at each iteration. Two well-known update formulas are called
2916:
2665:
2313:
is the unit sphere in n dimensions. That is, the ellipsoid is a linear image of the unit sphere.
2011:
185:
88:
15833:
15827:
9168:
Again, the pattern of access allows the entire computation to be performed in-place if desired.
2609:
100:
16230:
15518:
15514:
11114:. This can be achieved by successively performing rank-one updates for each of the columns of
96:
16129:
14785:
14763:
14616:
14586:
14535:
14469:
14154:
14090:
14068:
13509:
13483:
13054:
12077:
11514:
11260:
11117:
11036:
10504:
10412:
10290:
10268:
10215:
This involves matrix products and explicit inversion, thus limiting the practical block size.
1047:
16748:
16274:
16194:
16056:
15697:
7457:
6799:
3798:
1866:
16477:" Universidade Federal Do Rio Grande Do Sul, Instituto De Informatica, 2016, pp. 29-30.
15825:
7784:
Either pattern of access allows the entire computation to be performed in-place if desired.
3396:
Non-linear multi-variate functions may be minimized over their parameters using variants of
645:
A closely related variant of the classical Cholesky decomposition is the LDL decomposition,
16743:
16179:
16044:
15753:
15719:
13531:
4778:{\displaystyle \mathbf {A} ^{(i)}=\mathbf {L} _{i}\mathbf {A} ^{(i+1)}\mathbf {L} _{i}^{*}}
4244:
3122:
2457:
1972:
1859:
702:
76:
15585:
2573:
8:
16722:
16654:
16625:
16590:
is a book explaining the implementation of the CF with TBB, threads and SSE (in Spanish).
16383:
1962:
195:
92:
16624:
is an open encyclopedia of algorithms’ properties and features of their implementations
16581:
16425:
A General Method for Approximating Nonlinear Transformations of ProbabilityDistributions
16349:. Lecture Notes on Computer Science. Vol. 3019. Springer-Verlag. pp. 985–992.
16084:
16048:
16438:
16223:
16154:
16034:
15826:
Press, William H.; Saul A. Teukolsky; William T. Vetterling; Brian P. Flannery (1992).
14018:
13914:
10402:{\textstyle {\tilde {\mathbf {A} }}={\tilde {\mathbf {L} }}{\tilde {\mathbf {L} }}^{*}}
3996:
3778:
3747:
3720:
3700:
3673:
3646:
3617:
3572:
3541:
3514:
3487:
3407:
3397:
2526:
2499:
1933:
1906:
241:
is a real matrix (hence symmetric positive-definite), the factorization may be written
16563:
11148:
is similar to a rank-one update, except that the addition is replaced by subtraction:
3981:. These sigma points completely capture the mean and covariance of the system state.
16560:
16446:
16410:
16391:
16358:
16307:
16234:
16135:
16006:
15977:
15942:
15837:
15638:
is upper triangular. A flag can be passed to use the lower triangular factor instead.
15524:
5099:{\displaystyle \mathbf {L} :=\mathbf {L} _{1}\mathbf {L} _{2}\dots \mathbf {L} _{n}.}
3621:
3172:
2319:
794:
The LDL decomposition is related to the classical Cholesky decomposition of the form
16587:
16758:
16350:
16115:
15932:
15805:
15621:
15359:
14804:
is upper triangular. Inserting the decomposition into the original equality yields
14686:
14196:
14176:
7793:
4211:
4059:
multiplications). The entire inversion can even be efficiently performed in-place.
3347:
2896:
2553:
2227:
788:
120:
104:
80:
34:
16463:
Bayesian Gaussian Processes for Sequential Prediction, Optimisation and Quadrature
16354:
15582:
uses the upper triangular factor of the input matrix by default, i.e. it computes
15349:{\displaystyle \mathbf {A} _{ij}:{\mathcal {H}}_{j}\rightarrow {\mathcal {H}}_{i}}
2493:. Different choices of the conjugate axes correspond to different decompositions.
16799:
16615:
13460:{\textstyle {\tilde {\mathbf {A} }}=\mathbf {A} \pm \mathbf {x} \mathbf {x} ^{*}}
7952:
4467:
712:
15937:
15921:"Toward a parallel solver for generalized complex symmetric eigenvalue problems"
15920:
16717:
16537:
15014:
7940:
7909:{\displaystyle \|\mathbf {E} \|_{2}\leq c_{n}\varepsilon \|\mathbf {A} \|_{2}.}
3990:
3597:
2217:{\displaystyle \left\{\sum _{i}x_{i}v_{i}:x^{T}x=1\right\}=f(\mathbb {S} ^{n})}
20:
16431:
16183:
15998:
11200:{\textstyle {\tilde {\mathbf {A} }}=\mathbf {A} -\mathbf {x} \mathbf {x} ^{*}}
11107:{\textstyle {\tilde {\mathbf {A} }}=\mathbf {A} +\mathbf {M} \mathbf {M} ^{*}}
10575:{\textstyle {\tilde {\mathbf {A} }}=\mathbf {A} +\mathbf {x} \mathbf {x} ^{*}}
6341:{\displaystyle L_{j,j}={\sqrt {A_{j,j}-\sum _{k=1}^{j-1}L_{j,k}^{*}L_{j,k}}},}
721:
are required to be 1 at the cost of introducing an additional diagonal matrix
16855:
16763:
16379:
15946:
13709:
10340:, and one wants to compute the Cholesky decomposition of the updated matrix:
7801:
6034:{\displaystyle L_{j,j}=(\pm ){\sqrt {A_{j,j}-\sum _{k=1}^{j-1}L_{j,k}^{2}}},}
5001:
16605:
16582:
Notes and video on high-performance implementation of Cholesky factorization
7959:, adding a diagonal matrix can improve stability when far from the optimum.
2550:
to be within the plane spanned by the first two axes, and so on. This makes
16672:
16343:
5th International Conference on Parallel Processing and Applied Mathematics
14014:
4148:{\displaystyle \mathbf {B} ^{-1}=\mathbf {B} ^{*}(\mathbf {BB} ^{*})^{-1}.}
1255:
decomposition exists where the number of non-zero elements on the diagonal
323:
are allowed to be zero. The decomposition need not be unique, for example:
5108:
3795:, the correlated normal variables can be obtained via the transformations
15668:
7921:
7453:
6795:
6208:
3605:
16575:
10409:. The question is now whether one can use the Cholesky decomposition of
2965:
The Cholesky decomposition is mainly used for the numerical solution of
16475:
Parallel Implementations of the Cholesky Decomposition on CPUs and GPUs
15809:
15743:
15534:
4190:
multiplications and the same number of additions) for real flavors and
16611:
16779:
3630:. Applying this to a vector of uncorrelated observations in a sample
1966:
15700:
supplies Cholesky factorizations for both sparse and dense matrices.
8998:{\displaystyle D_{j}=A_{jj}-\sum _{k=1}^{j-1}L_{jk}L_{jk}^{*}D_{k},}
7967:
An alternative form, eliminating the need to take square roots when
7804:
is necessary, and the error will always be small. Specifically, if
3566:
2960:
2839:{\textstyle \Sigma =\mathrm {diag} (\lambda _{1},...,\lambda _{n})}
1961:
The Cholesky decomposition is equivalent to a particular choice of
16039:
15999:"Analysis of the Cholesky Decomposition of a Semi-definite Matrix"
15918:
13795:{\textstyle \mathbf {A} _{k}=\mathbf {L} _{k}\mathbf {L} _{k}^{*}}
11746:
Now there is an interest in finding the Cholesky factorization of
10431:
that was computed before to compute the Cholesky decomposition of
3336:{\textstyle \mathbf {DL} ^{\mathrm {*} }\mathbf {x} =\mathbf {y} }
1863:
The ellipse is a linear image of the unit circle. The two vectors
292:
is a real lower triangular matrix with positive diagonal entries.
15545:
7461:
6803:
734:
decomposition. For real matrices, the factorization has the form
16445:. Philadelphia: Society for Industrial and Applied Mathematics.
15832:(second ed.). Cambridge University England EPress. p.
15682:, multiple linear algebra libraries support this decomposition:
15279:{\displaystyle {\mathcal {H}}=\bigoplus _{n}{\mathcal {H}}_{n},}
14173:
is lower triangular with non-negative diagonal entries: for all
16830:
16820:
16651:
Generating Correlated Random Variables and Stochastic Processes
15571:
15541:
10590:
7452:
The above algorithm can be succinctly expressed as combining a
6794:
The above algorithm can be succinctly expressed as combining a
3620:
for simulating systems with multiple correlated variables. The
1297:
Here is the Cholesky decomposition of a symmetric real matrix:
16493:
Sur la résolution numérique des systèmes d'équations linéaires
13480:
The above algorithms show that every positive definite matrix
8688:{\displaystyle D_{j}=A_{jj}-\sum _{k=1}^{j-1}L_{jk}^{2}D_{k},}
16306:. Philadelphia: Soc. for Industrial and Applied Mathematics.
16184:"Modified Cholesky Algorithms: A Catalog with New Approaches"
15679:
14639:
Hermitian matrix. Then it can be written as a product of its
6220:
For complex Hermitian matrix, the following formula applies:
4173:
in general. The algorithms described below all involve about
3040:
can be solved by first computing the Cholesky decomposition
16556:
15733:
which can be used in Java, Scala and any other JVM language.
15428:{\displaystyle h\in \bigoplus _{n=1}^{k}{\mathcal {H}}_{k},}
15368:
is positive (semidefinite) in the sense that for all finite
13470:
7951:. Unfortunately, the numbers can become negative because of
3954:
is always positive semi-definite and can be decomposed into
3640:
with the covariance properties of the system being modeled.
16593:
16028:
15704:
15521:
provides several implementations of Cholesky decomposition.
15508:
8579:
The following recursive relations apply for the entries of
4197:
4180:
3178:
An alternative way to eliminate taking square roots in the
730:
For this reason, the LDL decomposition is often called the
95:, which is useful for efficient numerical solutions, e.g.,
58:
49:
40:
16177:
3404:
methods. At iteration k, the search steps in a direction
3378:
problems are of this form. It may also happen that matrix
1149:{\displaystyle \mathbf {L} =\mathbf {C} \mathbf {S} ^{-1}}
37:
16569:
16432:
A new extension of the Kalman filter to nonlinear systems
16005:. Oxford, UK: Oxford University Press. pp. 161–185.
12041:, which can be found easily for triangular matrices, and
8853:
This works as long as the generated diagonal elements in
4068:
can also be inverted using the following identity, where
1156:(this rescales each column to make diagonal elements 1),
279:{\displaystyle \mathbf {A} =\mathbf {LL} ^{\mathsf {T}},}
14583:
is lower triangular with non-negative diagonal entries,
11808:, without directly computing the entire decomposition.
7973:
is symmetric, is the symmetric indefinite factorization
5914:
and therefore the following formulas for the entries of
3904:{\textstyle x_{2}=\rho z_{1}+{\sqrt {1-\rho ^{2}}}z_{2}}
1044:
is a diagonal matrix that contains the main diagonal of
15974:
Numerical Linear Algebra for Applications in Statistics
15829:
Numerical Recipes in C: The Art of Scientific Computing
15480:, then there exists a lower triangular operator matrix
11252:
7135:
and proceeds to calculate the matrix column by column.
5113:
5109:
The Cholesky–Banachiewicz and Cholesky–Crout algorithms
3250:{\textstyle \mathbf {A} =\mathbf {LDL} ^{\mathrm {*} }}
996:
Conversely, given the classical Cholesky decomposition
15624:
15588:
15443:
15039:
14982:
14937:
14810:
14788:
14766:
14753:{\textstyle \mathbf {B} ^{*}=\mathbf {Q} \mathbf {R} }
14724:
14695:
14678:{\textstyle \mathbf {A} =\mathbf {B} \mathbf {B} ^{*}}
14649:
14619:
14589:
14560:
14538:
14494:
14472:
14428:
14415:{\textstyle \mathbf {A} =\mathbf {L} \mathbf {L} ^{*}}
14386:
14199:
14179:
14157:
14144:{\textstyle \mathbf {A} =\mathbf {L} \mathbf {L} ^{*}}
14115:
14093:
14071:
14027:
13975:
13937:
13917:
13747:
13718:
13563:
13534:
13512:
13486:
13414:
13101:
13057:
12977:
12821:
12610:
12080:
12047:
12012:
11982:
11845:
11783:
11752:
11570:
11517:
11486:
11407:
11301:
11263:
11213:
11154:
11120:
11061:
11039:
10529:
10507:
10476:
10437:
10415:
10346:
10315:
10293:
10271:
10258:{\textstyle \mathbf {A} =\mathbf {L} \mathbf {L} ^{*}}
10229:
9546:
9419:
9329:
9233:
8286:
8192:
8108:
8024:
5654:
5387:
5272:
5167:
4824:
4518:
4317:
4028:
4026:
3999:
3841:
3801:
3781:
3750:
3723:
3703:
3676:
3649:
3575:
3544:
3517:
3490:
3437:
3410:
3302:
3263:
3218:
3184:
3131:
3088:
3075:{\textstyle \mathbf {A} =\mathbf {LL} ^{\mathrm {*} }}
3046:
3013:
2974:
2919:
2899:
2852:
2774:
2720:
2668:
2612:
2576:
2556:
2529:
2502:
2460:
2399:
2322:
2290:
2250:
2230:
2066:
2014:
1975:
1936:
1909:
1869:
1768:
1705:
1639:
1557:
1459:
1393:
1315:
1222:
1078:
1050:
1031:{\textstyle \mathbf {A} =\mathbf {C} \mathbf {C} ^{*}}
1002:
559:
545:
404:
335:
16633:
Intel-Optimized Math Library for Numerical Computing
16495:, Cholesky's 1910 manuscript, online and analyzed on
15896:"matrices - Diagonalizing a Complex Symmetric Matrix"
15578:
function gives the Cholesky decomposition. Note that
15380:
15296:
15238:
15025:
14221:
13810:
13661:
13200:
13079:
12955:
12790:
12579:
12102:
11814:
11539:
11393:
11287:
10057:
9929:
9188:
9010:
8894:
8700:
8597:
7979:
7850:
6353:
6228:
6046:
5928:
5636:
5127:
5045:
4791:
4700:
4497:
4290:
4080:
2127:
1957:
to be within the plane spanned by the first two axes.
1547:
1305:
1248:{\textstyle \mathbf {L} \mathbf {D} \mathbf {L} ^{*}}
1162:
1117:
1104:{\textstyle \mathbf {L} \mathbf {D} \mathbf {L} ^{*}}
809:
653:
329:
247:
139:
67:
55:
52:
16148:
13969:
equipped with the operator norm is a C* algebra. So
5033:
is obtained, and hence, the lower triangular matrix
46:
16669:
Performs Cholesky decomposition of matrices online.
3616:The Cholesky decomposition is commonly used in the
2496:The Cholesky decomposition corresponds to choosing
1903:are conjugate axes of the ellipse chosen such that
1204:is positive definite then the diagonal elements of
43:
16644:
16222:
16118:" Cornell University Report No. FDA (2008): 08-01.
15664:standard library gives the Cholesky decomposition.
15630:
15610:
15473:{\textstyle \langle h,\mathbf {A} h\rangle \geq 0}
15472:
15427:
15348:
15278:
15220:
15005:
14960:
14923:
14796:
14774:
14752:
14710:
14677:
14627:
14597:
14575:
14554:entrywise. This in turn implies that, since each
14546:
14524:
14480:
14458:
14414:
14370:
14205:
14185:
14165:
14143:
14101:
14079:
14057:
14005:
13961:
13923:
13901:
13794:
13733:
13700:
13643:
13546:
13520:
13494:
13459:
13398:
13184:
13065:
13041:
12939:
12776:
12560:
12088:
12066:
12034:{\textstyle \mathbf {A} \mathbf {x} =\mathbf {b} }
12033:
11998:
11966:
11800:
11769:
11736:
11525:
11503:
11470:
11377:
11271:
11230:
11199:
11128:
11106:
11047:
10574:
10515:
10493:
10454:
10423:
10401:
10332:
10301:
10279:
10257:
10205:
10044:
9911:
9158:
8997:
8843:
8687:
8569:
7908:
6509:
6340:
6197:
6033:
5904:
5618:
5098:
4972:
4777:
4662:
4446:
4147:
4051:
4012:
3903:
3827:
3787:
3763:
3736:
3709:
3689:
3662:
3588:
3557:
3530:
3503:
3476:
3423:
3335:
3282:
3249:
3212:decomposition is to compute the LDL decomposition
3204:
3157:
3107:
3074:
3032:
2993:
2944:
2905:
2885:
2838:
2760:
2706:
2647:
2598:
2562:
2542:
2515:
2485:
2446:
2385:
2305:
2276:
2236:
2216:
2113:
2052:
2000:
1949:
1922:
1895:
1838:
1529:
1247:
1188:
1148:
1103:
1058:
1030:
986:
685:
610:
452:
278:
168:
16339:
15731:Apache Commons Math library has an implementation
8859:stay non-zero. The decomposition is then unique.
295:
16853:
16473:Ruschel, João Paulo Tarasconi, Bachelor degree "
14285:
14250:
12949:and the desire to determine the Cholesky factor
11533:but with the insertion of new rows and columns,
7131:starts from the upper left corner of the matrix
6543:and proceeds to calculate the matrix row by row.
6539:starts from the upper left corner of the matrix
3775:). Given the required correlation coefficient
2961:Numerical solution of system of linear equations
686:{\displaystyle \mathbf {A} =\mathbf {LDL} ^{*},}
637:lower triangular matrix with positive diagonal.
517:, then there is at least one permutation matrix
11999:{\textstyle \mathbf {A} \setminus \mathbf {b} }
7787:
169:{\displaystyle \mathbf {A} =\mathbf {LL} ^{*},}
14525:{\textstyle \left(\mathbf {L} _{k}\right)_{k}}
14459:{\textstyle \left(\mathbf {L} _{k}\right)_{k}}
14058:{\textstyle \left(\mathbf {L} _{k}\right)_{k}}
14006:{\textstyle \left(\mathbf {L} _{k}\right)_{k}}
1189:{\displaystyle \mathbf {D} =\mathbf {S} ^{2}.}
16688:
16378:
16102:
15959:
15862:
13475:
10218:
7781:refers to complex conjugate of the elements.
7123:refers to complex conjugate of the elements.
4237:, used to calculate the decomposition matrix
1969:. In detail, let the ellipsoid be defined as
775:, but is quite different in practice because
763:eigendecomposition of real symmetric matrices
188:with real and positive diagonal entries, and
16702:
16437:
16405:Horn, Roger A.; Johnson, Charles R. (1985).
15870:
15499:. One can also take the diagonal entries of
15461:
15444:
15000:
14983:
14608:
14361:
14332:
14326:
14282:
14239:
14222:
13892:
13877:
13871:
13839:
13827:
13811:
11257:If a symmetric and positive definite matrix
10470:The specific case, where the updated matrix
7894:
7885:
7860:
7851:
7460:in vectorized programming languages such as
6802:in vectorized programming languages such as
2749:
2735:
1210:are all positive. For positive semidefinite
16404:
16001:. In Cox, M. G.; Hammarling, S. J. (eds.).
15882:
15866:
5628:is written out, the following is obtained:
3624:is decomposed to give the lower-triangular
3158:{\textstyle \mathbf {L^{*}x} =\mathbf {y} }
2570:an upper-triangular matrix. Then, there is
1848:
16695:
16681:
16390:(3rd ed.). Baltimore: Johns Hopkins.
15649:function gives the Cholesky decomposition.
15204:
15201:
15198:
15193:
15136:
15084:
14961:{\textstyle \mathbf {L} =\mathbf {R} ^{*}}
7792:Suppose that there is a desire to solve a
3964:can be added and subtracted from the mean
3391:
3007:is symmetric and positive definite, then
2913:is an orthogonal matrix. This then yields
715:matrix. That is, the diagonal elements of
474:, then there is a unique lower triangular
16578:is a C library with LAPACK functionality.
16038:
15936:
14364:
13952:
13895:
13471:Proof for positive semi-definite matrices
5004:, therefore this algorithm is called the
4228:
3611:
3205:{\textstyle \mathbf {LL} ^{\mathrm {*} }}
2447:{\textstyle v_{i}^{T}Av_{j}=\delta _{ij}}
2293:
2201:
2114:{\textstyle v_{i}^{T}Av_{j}=\delta _{ij}}
15509:Implementations in programming libraries
13712:. From the positive definite case, each
12096:, the following relations can be found:
12067:{\textstyle {\text{chol}}(\mathbf {M} )}
11055:one updates the decomposition such that
10593:syntax that realizes a rank-one update:
5112:
4210:. Hence, they have half the cost of the
2060:are conjugate axes of the ellipsoid iff
1858:
16826:Basic Linear Algebra Subprograms (BLAS)
16459:
16301:
16253:
16220:
15786:
15567:module performs Cholesky decomposition.
3353:
3283:{\textstyle \mathbf {Ly} =\mathbf {b} }
3108:{\textstyle \mathbf {Ly} =\mathbf {b} }
3033:{\textstyle \mathbf {Ax} =\mathbf {b} }
2994:{\textstyle \mathbf {Ax} =\mathbf {b} }
2008:, then by definition, a set of vectors
529:has a unique decomposition of the form
16854:
16024:
16022:
15996:
15971:
15792:
14087:. It can be easily checked that this
9176:When used on indefinite matrices, the
7962:
267:
226:, lower triangular or otherwise, then
16676:
16660:
16584:at The University of Texas at Austin.
16486:
16127:
10309:in some way into another matrix, say
6519:So it now is possible to compute the
2761:{\textstyle \lambda =1/\|v_{i}\|^{2}}
703:lower unit triangular (unitriangular)
513:positive semidefinite matrix of rank
16614:is a programming chrestomathy site.
16268:
15821:
15819:
13802:. By property of the operator norm,
11801:{\textstyle {\tilde {\mathbf {S} }}}
11770:{\textstyle {\tilde {\mathbf {A} }}}
11504:{\textstyle {\tilde {\mathbf {A} }}}
11253:Adding and removing rows and columns
11231:{\textstyle {\tilde {\mathbf {A} }}}
11207:. This only works if the new matrix
11139:
10494:{\textstyle {\tilde {\mathbf {A} }}}
10455:{\textstyle {\tilde {\mathbf {A} }}}
10333:{\textstyle {\tilde {\mathbf {A} }}}
7820:denotes the computed solution, then
4250:The recursive algorithm starts with
4222:FLOPs (see Trefethen and Bau 1997).
3350:, it is roughly twice as efficient.
640:
232:is Hermitian and positive definite.
16588:Cholesky : TBB + Threads + SSE
16225:Fundamentals of Matrix Computations
16019:
15006:{\textstyle \{{\mathcal {H}}_{n}\}}
12784:with known Cholesky decomposition
3984:
3697:with given correlation coefficient
1539:And here is its LDL decomposition:
13:
16544:Cholesky Decomposition Made Simple
16166:?potrf Intel® Math Kernel Library
16128:Arora, Jasbir Singh (2004-06-02).
16082:
15693:to perform Cholesky decomposition.
15527:computer algebra system: function
15411:
15335:
15318:
15262:
15241:
14989:
13695:
13366:
13332:
13310:
13307:
13304:
13301:
13147:
12742:
12721:
12670:
12528:
12494:
12472:
12469:
12466:
12463:
12413:
12371:
12309:
12272:
12269:
12266:
12263:
12177:
12074:for the Cholesky decomposition of
11702:
11681:
11630:
11347:
10465:
10169:
10033:
9873:
9827:
9755:
9666:
9592:
9589:
9586:
9583:
9580:
9577:
9574:
9571:
9568:
9498:
9463:
9442:
9215:
8330:
8327:
8324:
8321:
8318:
8315:
8312:
8309:
8306:
8006:
2929:
2863:
2791:
2788:
2785:
2782:
2775:
2523:to be parallel to the first axis,
2121:. Then, the ellipsoid is precisely
1930:to be parallel to the first axis,
1038:of a positive definite matrix, if
14:
16883:
16631:Intel® oneAPI Math Kernel Library
16481:
16430:S. J. Julier and J. K. Uhlmann, "
16423:S. J. Julier and J. K. Uhlmann. "
15816:
15749:Incomplete Cholesky factorization
15726:gives the Cholesky decomposition.
14971:
14109:has the desired properties, i.e.
12377:
12183:
11988:
8886:, the following formula applies:
7931:is a small constant depending on
4052:{\textstyle {\tfrac {1}{2}}n^{3}}
3914:
207:The converse holds trivially: if
131:, is a decomposition of the form
16550:
15531:computes Cholesky decomposition.
15454:
15299:
15181:
15162:
15143:
15124:
15110:
15091:
15072:
15058:
15044:
15027:
14948:
14939:
14917:
14906:
14897:
14894:
14883:
14871:
14862:
14859:
14844:
14841:
14824:
14818:
14790:
14768:
14746:
14741:
14727:
14698:
14665:
14659:
14651:
14621:
14591:
14563:
14540:
14502:
14474:
14436:
14402:
14396:
14388:
14342:
14336:
14302:
14290:
14255:
14226:
14159:
14131:
14125:
14117:
14095:
14073:
14035:
13983:
13962:{\textstyle M_{n}(\mathbb {C} )}
13882:
13856:
13844:
13816:
13777:
13765:
13750:
13721:
13678:
13664:
13620:
13601:
13571:
13514:
13488:
13447:
13441:
13433:
13419:
13374:
13355:
13340:
13321:
13283:
13264:
13245:
13226:
13207:
13157:
13136:
13120:
13106:
13085:
13059:
13017:
12996:
12982:
12961:
12915:
12889:
12875:
12854:
12840:
12826:
12799:
12752:
12731:
12710:
12694:
12680:
12659:
12643:
12629:
12615:
12588:
12536:
12517:
12502:
12483:
12445:
12421:
12402:
12387:
12360:
12341:
12317:
12298:
12283:
12245:
12226:
12207:
12188:
12166:
12147:
12128:
12109:
12082:
12057:
12027:
12019:
12014:
11992:
11984:
11939:
11913:
11899:
11878:
11864:
11850:
11823:
11788:
11757:
11712:
11691:
11670:
11654:
11640:
11619:
11603:
11589:
11575:
11548:
11519:
11491:
11447:
11426:
11412:
11395:
11357:
11336:
11320:
11306:
11289:
11279:is represented in block form as
11265:
11218:
11187:
11181:
11173:
11159:
11122:
11094:
11088:
11080:
11066:
11041:
10562:
10556:
10548:
10534:
10509:
10481:
10442:
10417:
10383:
10368:
10351:
10320:
10295:
10273:
10245:
10239:
10231:
10182:
10155:
10143:
10128:
10083:
10060:
10019:
10007:
9992:
9947:
9932:
9884:
9862:
9850:
9838:
9816:
9804:
9792:
9778:
9766:
9744:
9732:
9720:
9706:
9694:
9677:
9655:
9643:
9631:
9617:
9605:
9551:
9519:
9487:
9479:
9452:
9431:
9423:
9396:
9365:
9334:
9312:
9299:
9285:
9270:
9257:
9237:
9209:
9206:
9203:
9194:
9171:
8000:
7997:
7994:
7985:
7889:
7855:
6217:is real and positive-definite.
5642:
5145:
5142:
5133:
5083:
5068:
5056:
5047:
4944:
4932:
4888:
4829:
4794:
4760:
4736:
4724:
4703:
4633:
4619:
4523:
4500:
4417:
4403:
4377:
4322:
4293:
4119:
4116:
4101:
4083:
3606:Broyden–Fletcher–Goldfarb–Shanno
3329:
3321:
3308:
3305:
3276:
3268:
3265:
3235:
3232:
3229:
3220:
3190:
3187:
3151:
3143:
3138:
3134:
3101:
3093:
3090:
3060:
3057:
3048:
3026:
3018:
3015:
2987:
2979:
2976:
1235:
1229:
1224:
1173:
1164:
1133:
1127:
1119:
1091:
1085:
1080:
1052:
1018:
1012:
1004:
952:
946:
921:
915:
901:
871:
846:
840:
826:
823:
820:
811:
670:
667:
664:
655:
585:
564:
547:
392:
376:
370:
261:
258:
249:
153:
150:
141:
119:The Cholesky decomposition of a
33:
16645:Use of the matrix in simulation
16608:is a C++ linear algebra package
16469:(thesis). University of Oxford.
16326:Osborne, M. (2010), Appendix B.
16320:
16294:
16262:
16247:
16214:
16171:
16160:
16121:
16108:
16096:
16076:
15774:Symbolic Cholesky decomposition
15017:. Consider the operator matrix
13688:
13682:
13073:with rows and columns removed,
9138:
8823:
7957:Newton's method in optimization
6537:Cholesky–Banachiewicz algorithm
6489:
6177:
2955:
2714:to be perpendicular. Then, let
2277:{\textstyle e_{i}\mapsto v_{i}}
494:columns containing all zeroes.
484:positive diagonal elements and
390:
389:
16507:
16409:. Cambridge University Press.
16131:Introduction to Optimum Design
16003:Reliable Numerical Computation
15990:
15965:
15953:
15912:
15888:
15876:
15856:
15329:
14849:
14837:
13956:
13948:
13692:
13674:
13423:
12803:
12592:
12061:
12053:
11827:
11792:
11761:
11552:
11495:
11387:and its upper Cholesky factor
11222:
11163:
11070:
10589:Here is a function written in
10538:
10485:
10446:
10387:
10372:
10355:
10324:
10287:, then one changes the matrix
9596:
9564:
8334:
8302:
5954:
5948:
5416:
5408:
4899:
4893:
4811:
4799:
4753:
4741:
4714:
4708:
4428:
4422:
4304:
4298:
4157:
4130:
4111:
3477:{\textstyle B_{k}p_{k}=-g_{k}}
3386:partial differential equations
2886:{\textstyle V=U\Sigma ^{-1/2}}
2833:
2795:
2636:
2619:
2380:
2366:
2358:
2343:
2329:
2261:
2211:
2196:
317:where the diagonal entries of
296:Positive semidefinite matrices
1:
16534:, The Data Analysis BriefBook
16355:10.1007/978-3-540-24669-5_127
16333:
15675:" can be applied to a matrix.
14711:{\textstyle \mathbf {B} ^{*}}
14576:{\textstyle \mathbf {L} _{k}}
13734:{\textstyle \mathbf {A} _{k}}
13555:positive semi-definite matrix
8880:For complex Hermitian matrix
7826:solves the perturbed system (
2306:{\textstyle \mathbb {S} ^{n}}
1273:: it suffices that the first
761:). It is reminiscent of the
16083:So, Anthony Man-Cho (2007).
15997:Higham, Nicholas J. (1990).
13194:yields the following rules:
11238:is still positive definite.
7788:Stability of the computation
5037:sought for is calculated as
4688:is positive definite), then
4072:* will always be Hermitian:
2945:{\textstyle A=U\Sigma U^{T}}
2707:{\textstyle v_{1},...,v_{n}}
2660:principal component analysis
2053:{\textstyle v_{1},...,v_{n}}
744:and is often referred to as
114:
7:
16520:Encyclopedia of Mathematics
16229:. New York: Wiley. p.
15938:10.1016/j.procs.2010.04.047
15737:
13741:has Cholesky decomposition
5008:in (Golub & Van Loan).
4243:, is a modified version of
4200:for complex flavors, where
3596:is an approximation to the
2648:{\textstyle L=(V^{-1})^{T}}
109:systems of linear equations
16:Matrix decomposition method
10:
16888:
16739:System of linear equations
16155:Matlab randn documentation
16116:Algorithms for ellipsoids.
16103:Golub & Van Loan (1996
15960:Golub & Van Loan (1996
15863:Golub & Van Loan (1996
15769:Sylvester's law of inertia
13653:positive definite matrices
13476:Proof by limiting argument
11033:is one where for a matrix
10219:Updating the decomposition
4204:is the size of the matrix
1852:
1292:
464:. However, if the rank of
16808:
16790:Cache-oblivious algorithm
16772:
16731:
16710:
16460:Osborne, Michael (2010).
15972:Gentle, James E. (1998).
15925:Procedia Computer Science
15871:Trefethen & Bau (1997
15230:acting on the direct sum
14797:{\textstyle \mathbf {R} }
14775:{\textstyle \mathbf {Q} }
14628:{\textstyle \mathbf {A} }
14609:Proof by QR decomposition
14598:{\textstyle \mathbf {L} }
14547:{\textstyle \mathbf {L} }
14481:{\textstyle \mathbf {L} }
14166:{\textstyle \mathbf {L} }
14102:{\textstyle \mathbf {L} }
14080:{\textstyle \mathbf {L} }
13521:{\textstyle \mathbf {A} }
13495:{\textstyle \mathbf {A} }
13066:{\textstyle \mathbf {A} }
12089:{\textstyle \mathbf {M} }
11526:{\textstyle \mathbf {A} }
11272:{\textstyle \mathbf {A} }
11129:{\textstyle \mathbf {M} }
11048:{\textstyle \mathbf {M} }
10516:{\textstyle \mathbf {A} }
10501:is related to the matrix
10424:{\textstyle \mathbf {A} }
10302:{\textstyle \mathbf {A} }
10280:{\textstyle \mathbf {A} }
3636:produces a sample vector
1059:{\textstyle \mathbf {C} }
732:square-root-free Cholesky
16872:Numerical linear algebra
16841:General purpose software
16704:Numerical linear algebra
16667:Online Matrix Calculator
16515:"Cholesky factorization"
16443:Numerical linear algebra
15976:. Springer. p. 94.
15883:Horn & Johnson (1985
15867:Horn & Johnson (1985
15779:
15759:Minimum degree algorithm
14017:of operators, therefore
14013:is a bounded set in the
10595:
7466:
7136:
7129:Cholesky–Crout algorithm
6808:
6546:
4284:has the following form:
3920:Unscented Kalman filters
3828:{\textstyle x_{1}=z_{1}}
2662:corresponds to choosing
1896:{\textstyle v_{1},v_{2}}
1855:Whitening transformation
1849:Geometric interpretation
1281:leading principal minors
124:positive-definite matrix
85:positive-definite matrix
16594:library "Ceres Solver"
16500:(in French and English)
16302:Stewart, G. W. (1998).
16254:Nocedal, Jorge (2000).
15764:Square root of a matrix
15687:Armadillo (C++ library)
11511:, which is the same as
4062:A non-Hermitian matrix
3602:Davidon–Fletcher–Powell
3538:is the step direction,
3392:Non-linear optimization
1261:is exactly the rank of
186:lower triangular matrix
99:. It was discovered by
97:Monte Carlo simulations
89:lower triangular matrix
16546:on Science Meanderthal
16538:Cholesky Decomposition
16532:Cholesky Decomposition
16282:Cite journal requires
16256:Numerical Optimization
16202:Cite journal requires
16064:Cite journal requires
15632:
15612:
15519:GNU Scientific Library
15515:C programming language
15474:
15429:
15407:
15350:
15280:
15222:
15007:
14962:
14925:
14798:
14776:
14754:
14712:
14679:
14637:positive semi-definite
14629:
14599:
14577:
14548:
14526:
14482:
14460:
14416:
14372:
14207:
14187:
14167:
14145:
14103:
14081:
14059:
14007:
13963:
13925:
13903:
13796:
13735:
13702:
13645:
13548:
13547:{\textstyle n\times n}
13522:
13496:
13461:
13400:
13186:
13067:
13043:
12941:
12778:
12562:
12090:
12068:
12035:
12000:
11968:
11802:
11777:, which can be called
11771:
11738:
11527:
11505:
11480:then for a new matrix
11472:
11379:
11273:
11232:
11201:
11130:
11108:
11049:
10576:
10517:
10495:
10456:
10425:
10403:
10334:
10303:
10281:
10259:
10207:
10125:
10046:
9989:
9913:
9160:
9091:
8999:
8950:
8845:
8781:
8689:
8653:
8571:
7910:
6511:
6446:
6342:
6295:
6211:is always positive if
6199:
6139:
6035:
6004:
5906:
5620:
5118:
5100:
4974:
4779:
4664:
4448:
4229:The Cholesky algorithm
4149:
4053:
4014:
3905:
3829:
3789:
3765:
3738:
3711:
3691:
3664:
3612:Monte Carlo simulation
3590:
3559:
3532:
3505:
3478:
3425:
3337:
3296:, and finally solving
3284:
3251:
3206:
3159:
3125:, and finally solving
3109:
3076:
3034:
2995:
2946:
2907:
2887:
2840:
2762:
2708:
2649:
2600:
2564:
2544:
2517:
2487:
2486:{\textstyle V^{T}AV=I}
2448:
2387:
2307:
2278:
2244:maps the basis vector
2238:
2218:
2115:
2054:
2002:
2001:{\textstyle y^{T}Ay=1}
1958:
1951:
1924:
1897:
1840:
1531:
1249:
1190:
1150:
1105:
1060:
1032:
988:
687:
612:
454:
300:If a Hermitian matrix
280:
170:
87:into the product of a
29:Cholesky factorization
25:Cholesky decomposition
16867:Matrix decompositions
16836:Specialized libraries
16749:Matrix multiplication
16744:Matrix decompositions
16540:on www.math-linux.com
16441:; Bau, David (1997).
16092:(PhD). Theorem 2.2.6.
15689:supplies the command
15673:CholeskyDecomposition
15633:
15613:
15611:{\textstyle A=R^{*}R}
15475:
15430:
15387:
15351:
15281:
15223:
15008:
14968:completes the proof.
14963:
14926:
14799:
14777:
14755:
14713:
14680:
14630:
14600:
14578:
14549:
14527:
14483:
14461:
14417:
14373:
14208:
14188:
14168:
14146:
14104:
14082:
14060:
14008:
13964:
13926:
13904:
13797:
13736:
13703:
13646:
13549:
13523:
13497:
13462:
13401:
13187:
13068:
13044:
12942:
12779:
12563:
12091:
12069:
12036:
12001:
11969:
11803:
11772:
11739:
11528:
11506:
11473:
11380:
11274:
11233:
11202:
11131:
11109:
11050:
10577:
10518:
10496:
10457:
10426:
10404:
10335:
10304:
10282:
10260:
10208:
10099:
10047:
9963:
9914:
9161:
9065:
9000:
8924:
8846:
8755:
8690:
8627:
8572:
7911:
7458:matrix multiplication
6800:matrix multiplication
6512:
6420:
6343:
6269:
6200:
6113:
6036:
5978:
5907:
5621:
5116:
5101:
5011:This is repeated for
5006:outer-product version
4975:
4780:
4665:
4449:
4150:
4054:
4015:
3932:and covariance as an
3906:
3830:
3790:
3766:
3739:
3712:
3692:
3665:
3591:
3560:
3533:
3506:
3479:
3426:
3338:
3285:
3252:
3207:
3160:
3110:
3077:
3035:
2996:
2947:
2908:
2888:
2841:
2763:
2709:
2655:is lower-triangular.
2650:
2601:
2599:{\textstyle A=LL^{T}}
2565:
2545:
2518:
2488:
2449:
2388:
2308:
2279:
2239:
2219:
2116:
2055:
2003:
1952:
1925:
1898:
1862:
1841:
1532:
1250:
1191:
1151:
1106:
1072:can be decomposed as
1061:
1033:
989:
688:
613:
455:
281:
171:
16384:Van Loan, Charles F.
16304:Basic decompositions
16221:Watkins, D. (1991).
15754:Matrix decomposition
15622:
15586:
15441:
15378:
15294:
15236:
15023:
14980:
14935:
14808:
14786:
14764:
14722:
14693:
14647:
14617:
14587:
14558:
14536:
14492:
14470:
14426:
14384:
14219:
14197:
14177:
14155:
14113:
14091:
14069:
14025:
13973:
13935:
13915:
13808:
13745:
13716:
13659:
13561:
13557:, then the sequence
13532:
13510:
13484:
13412:
13198:
13077:
13055:
12953:
12788:
12577:
12100:
12078:
12045:
12010:
12006:for the solution of
11980:
11812:
11781:
11750:
11537:
11515:
11484:
11391:
11285:
11261:
11211:
11152:
11118:
11059:
11037:
10527:
10505:
10474:
10435:
10413:
10344:
10313:
10291:
10269:
10227:
10055:
9927:
9186:
9008:
8892:
8698:
8595:
7977:
7848:
6351:
6226:
6044:
5926:
5634:
5125:
5043:
4789:
4698:
4495:
4288:
4245:Gaussian elimination
4078:
4024:
3997:
3839:
3799:
3779:
3773:Box–Muller transform
3771:(for example, via a
3748:
3721:
3701:
3674:
3647:
3573:
3542:
3515:
3488:
3435:
3408:
3376:linear least squares
3358:Systems of the form
3354:Linear least squares
3300:
3261:
3216:
3182:
3129:
3123:forward substitution
3086:
3044:
3011:
2972:
2917:
2897:
2850:
2772:
2718:
2666:
2610:
2574:
2554:
2527:
2500:
2458:
2397:
2320:
2288:
2248:
2228:
2125:
2064:
2012:
1973:
1934:
1907:
1867:
1545:
1303:
1220:
1160:
1115:
1076:
1048:
1000:
807:
651:
543:
327:
245:
220:for some invertible
137:
101:André-Louis Cholesky
16723:Numerical stability
16655:Columbia University
16602:routines in Matlab.
16439:Trefethen, Lloyd N.
16388:Matrix Computations
16049:2011arXiv1111.4144K
15797:Bulletin Géodésique
15711:class is available.
15552:and most languages.
15176:
15157:
15105:
14316:
13870:
13791:
13371:
13337:
13152:
12747:
12726:
12675:
12533:
12499:
12418:
12376:
12314:
12182:
11707:
11686:
11635:
11352:
10199:
10174:
10038:
9878:
9832:
9760:
9671:
9503:
9468:
9447:
9122:
8981:
8671:
8525:
8497:
8377:
7949:in exact arithmetic
6467:
6316:
6025:
5887:
5869:
5741:
5600:
5582:
5564:
5477:
5459:
5404:
4958:
4774:
4391:
3431:defined by solving
2414:
2081:
196:conjugate transpose
93:conjugate transpose
16661:Online calculators
16487:History of science
16180:O'Leary, Dianne P.
16114:Pope, Stephen B. "
15810:10.1007/BF03031308
15660:function from the
15628:
15608:
15470:
15425:
15346:
15276:
15258:
15218:
15212:
15160:
15141:
15089:
15003:
14958:
14921:
14794:
14772:
14750:
14708:
14689:can be applied to
14675:
14641:square root matrix
14625:
14595:
14573:
14544:
14522:
14478:
14456:
14412:
14368:
14300:
14203:
14183:
14163:
14141:
14099:
14077:
14055:
14019:relatively compact
14003:
13959:
13924:{\textstyle \leq }
13921:
13899:
13854:
13792:
13775:
13731:
13698:
13641:
13544:
13518:
13492:
13457:
13396:
13394:
13353:
13319:
13182:
13180:
13169:
13134:
13063:
13039:
13037:
13029:
12937:
12935:
12927:
12774:
12772:
12764:
12729:
12708:
12657:
12558:
12556:
12515:
12481:
12400:
12358:
12296:
12164:
12086:
12064:
12031:
11996:
11964:
11962:
11951:
11798:
11767:
11734:
11732:
11724:
11689:
11668:
11617:
11523:
11501:
11468:
11459:
11375:
11369:
11334:
11269:
11228:
11197:
11126:
11104:
11045:
10572:
10513:
10491:
10452:
10421:
10399:
10330:
10299:
10277:
10255:
10203:
10180:
10153:
10042:
10017:
9909:
9907:
9896:
9860:
9814:
9742:
9653:
9525:
9485:
9450:
9429:
9408:
9318:
9156:
9105:
8995:
8964:
8841:
8685:
8654:
8567:
8565:
8554:
8511:
8483:
8363:
8265:
8181:
8097:
7906:
7464:as the following,
6806:as the following,
6507:
6447:
6338:
6296:
6195:
6031:
6005:
5902:
5900:
5892:
5873:
5855:
5727:
5616:
5614:
5603:
5586:
5568:
5550:
5463:
5445:
5390:
5366:
5261:
5119:
5096:
4970:
4961:
4942:
4775:
4758:
4694:can be written as
4660:
4651:
4444:
4435:
4375:
4235:Cholesky algorithm
4145:
4049:
4037:
4013:{\textstyle n^{3}}
4010:
3958:. The columns of
3901:
3825:
3788:{\textstyle \rho }
3785:
3764:{\textstyle z_{2}}
3761:
3737:{\textstyle z_{1}}
3734:
3710:{\textstyle \rho }
3707:
3690:{\textstyle x_{2}}
3687:
3663:{\textstyle x_{1}}
3660:
3618:Monte Carlo method
3589:{\textstyle B_{k}}
3586:
3558:{\textstyle g_{k}}
3555:
3531:{\textstyle p_{k}}
3528:
3504:{\textstyle p_{k}}
3501:
3474:
3424:{\textstyle p_{k}}
3421:
3333:
3280:
3247:
3202:
3155:
3105:
3072:
3030:
2991:
2942:
2903:
2883:
2836:
2758:
2704:
2645:
2596:
2560:
2543:{\textstyle v_{2}}
2540:
2516:{\textstyle v_{1}}
2513:
2483:
2444:
2400:
2383:
2316:Define the matrix
2303:
2274:
2234:
2214:
2142:
2111:
2067:
2050:
1998:
1959:
1950:{\textstyle v_{2}}
1947:
1923:{\textstyle v_{1}}
1920:
1893:
1836:
1834:
1823:
1757:
1694:
1621:
1527:
1525:
1514:
1448:
1379:
1289:are non-singular.
1245:
1186:
1146:
1101:
1056:
1028:
984:
757:decomposition, or
683:
608:
602:
450:
441:
360:
276:
213:can be written as
166:
16849:
16848:
16600:LDL decomposition
16503:
16452:978-0-89871-361-9
16397:978-0-8018-5414-9
16364:978-3-540-21946-0
16182:(8 August 2006).
16141:978-0-08-047025-2
16012:978-0-19-853564-5
15983:978-1-4612-0623-1
15249:
15013:be a sequence of
13686:
13616:
13426:
12806:
12595:
12051:
11830:
11795:
11764:
11555:
11498:
11249:by subtractions.
11225:
11166:
11146:rank-one downdate
11140:Rank-one downdate
11073:
10541:
10488:
10449:
10390:
10375:
10358:
10327:
9142:
9042:
8827:
8732:
7963:LDL decomposition
6493:
6394:
6333:
6181:
6087:
6026:
5888:
5742:
5669:
5414:
4928:
4615:
4614:
4576:
4036:
3970:to form a set of
3889:
3622:covariance matrix
3173:back substitution
2454:is equivalent to
2133:
641:LDL decomposition
16879:
16759:Matrix splitting
16697:
16690:
16683:
16674:
16673:
16653:, Martin Haugh,
16528:
16502:
16501:
16470:
16468:
16456:
16420:
16401:
16375:
16373:
16367:. Archived from
16348:
16327:
16324:
16318:
16317:
16298:
16292:
16291:
16285:
16280:
16278:
16270:
16266:
16260:
16259:
16251:
16245:
16244:
16228:
16218:
16212:
16211:
16205:
16200:
16198:
16190:
16188:
16175:
16169:
16164:
16158:
16157:. mathworks.com.
16152:
16146:
16145:
16125:
16119:
16112:
16106:
16105:, Theorem 4.1.3)
16100:
16094:
16093:
16091:
16080:
16074:
16073:
16067:
16062:
16060:
16052:
16042:
16026:
16017:
16016:
15994:
15988:
15987:
15969:
15963:
15957:
15951:
15950:
15940:
15916:
15910:
15909:
15907:
15906:
15892:
15886:
15880:
15874:
15869:, p. 407),
15865:, p. 143),
15860:
15854:
15853:
15851:
15850:
15823:
15814:
15813:
15790:
15725:
15710:
15692:
15674:
15671:, the function "
15663:
15659:
15648:
15637:
15635:
15634:
15629:
15617:
15615:
15614:
15609:
15604:
15603:
15581:
15577:
15566:
15562:
15530:
15505:to be positive.
15504:
15498:
15497:
15485:
15479:
15477:
15476:
15471:
15457:
15434:
15432:
15431:
15426:
15421:
15420:
15415:
15414:
15406:
15401:
15371:
15367:
15360:bounded operator
15355:
15353:
15352:
15347:
15345:
15344:
15339:
15338:
15328:
15327:
15322:
15321:
15311:
15310:
15302:
15285:
15283:
15282:
15277:
15272:
15271:
15266:
15265:
15257:
15245:
15244:
15227:
15225:
15224:
15219:
15217:
15216:
15190:
15189:
15184:
15175:
15170:
15165:
15156:
15151:
15146:
15133:
15132:
15127:
15119:
15118:
15113:
15104:
15099:
15094:
15081:
15080:
15075:
15067:
15066:
15061:
15053:
15052:
15047:
15030:
15012:
15010:
15009:
15004:
14999:
14998:
14993:
14992:
14967:
14965:
14964:
14959:
14957:
14956:
14951:
14942:
14930:
14928:
14927:
14922:
14920:
14915:
14914:
14909:
14900:
14892:
14891:
14886:
14880:
14879:
14874:
14865:
14857:
14856:
14847:
14833:
14832:
14827:
14821:
14803:
14801:
14800:
14795:
14793:
14781:
14779:
14778:
14773:
14771:
14759:
14757:
14756:
14751:
14749:
14744:
14736:
14735:
14730:
14717:
14715:
14714:
14709:
14707:
14706:
14701:
14687:QR decomposition
14684:
14682:
14681:
14676:
14674:
14673:
14668:
14662:
14654:
14634:
14632:
14631:
14626:
14624:
14604:
14602:
14601:
14596:
14594:
14582:
14580:
14579:
14574:
14572:
14571:
14566:
14553:
14551:
14550:
14545:
14543:
14531:
14529:
14528:
14523:
14521:
14520:
14515:
14511:
14510:
14505:
14487:
14485:
14484:
14479:
14477:
14465:
14463:
14462:
14457:
14455:
14454:
14449:
14445:
14444:
14439:
14421:
14419:
14418:
14413:
14411:
14410:
14405:
14399:
14391:
14377:
14375:
14374:
14369:
14351:
14350:
14345:
14339:
14315:
14310:
14305:
14299:
14298:
14293:
14278:
14274:
14264:
14263:
14258:
14229:
14212:
14210:
14209:
14204:
14192:
14190:
14189:
14184:
14172:
14170:
14169:
14164:
14162:
14150:
14148:
14147:
14142:
14140:
14139:
14134:
14128:
14120:
14108:
14106:
14105:
14100:
14098:
14086:
14084:
14083:
14078:
14076:
14064:
14062:
14061:
14056:
14054:
14053:
14048:
14044:
14043:
14038:
14012:
14010:
14009:
14004:
14002:
14001:
13996:
13992:
13991:
13986:
13968:
13966:
13965:
13960:
13955:
13947:
13946:
13930:
13928:
13927:
13922:
13908:
13906:
13905:
13900:
13891:
13890:
13885:
13869:
13864:
13859:
13853:
13852:
13847:
13835:
13834:
13825:
13824:
13819:
13801:
13799:
13798:
13793:
13790:
13785:
13780:
13774:
13773:
13768:
13759:
13758:
13753:
13740:
13738:
13737:
13732:
13730:
13729:
13724:
13707:
13705:
13704:
13699:
13687:
13684:
13681:
13673:
13672:
13667:
13650:
13648:
13647:
13642:
13640:
13639:
13634:
13630:
13629:
13628:
13623:
13617:
13609:
13604:
13590:
13589:
13584:
13580:
13579:
13574:
13553:
13551:
13550:
13545:
13527:
13525:
13524:
13519:
13517:
13501:
13499:
13498:
13493:
13491:
13466:
13464:
13463:
13458:
13456:
13455:
13450:
13444:
13436:
13428:
13427:
13422:
13417:
13405:
13403:
13402:
13397:
13395:
13388:
13384:
13383:
13382:
13377:
13370:
13369:
13363:
13358:
13349:
13348:
13343:
13336:
13335:
13329:
13324:
13313:
13292:
13291:
13286:
13273:
13272:
13267:
13254:
13253:
13248:
13235:
13234:
13229:
13216:
13215:
13210:
13191:
13189:
13188:
13183:
13181:
13174:
13173:
13166:
13165:
13160:
13151:
13150:
13144:
13139:
13129:
13128:
13123:
13115:
13114:
13109:
13088:
13072:
13070:
13069:
13064:
13062:
13048:
13046:
13045:
13040:
13038:
13034:
13033:
13026:
13025:
13020:
13005:
13004:
12999:
12991:
12990:
12985:
12964:
12946:
12944:
12943:
12938:
12936:
12932:
12931:
12924:
12923:
12918:
12898:
12897:
12892:
12884:
12883:
12878:
12863:
12862:
12857:
12849:
12848:
12843:
12835:
12834:
12829:
12808:
12807:
12802:
12797:
12783:
12781:
12780:
12775:
12773:
12769:
12768:
12761:
12760:
12755:
12746:
12745:
12739:
12734:
12725:
12724:
12718:
12713:
12703:
12702:
12697:
12689:
12688:
12683:
12674:
12673:
12667:
12662:
12652:
12651:
12646:
12638:
12637:
12632:
12624:
12623:
12618:
12597:
12596:
12591:
12586:
12567:
12565:
12564:
12559:
12557:
12550:
12546:
12545:
12544:
12539:
12532:
12531:
12525:
12520:
12511:
12510:
12505:
12498:
12497:
12491:
12486:
12475:
12454:
12453:
12448:
12435:
12431:
12430:
12429:
12424:
12417:
12416:
12410:
12405:
12396:
12395:
12390:
12375:
12374:
12368:
12363:
12350:
12349:
12344:
12331:
12327:
12326:
12325:
12320:
12313:
12312:
12306:
12301:
12292:
12291:
12286:
12275:
12254:
12253:
12248:
12235:
12234:
12229:
12216:
12215:
12210:
12197:
12196:
12191:
12181:
12180:
12174:
12169:
12156:
12155:
12150:
12137:
12136:
12131:
12118:
12117:
12112:
12095:
12093:
12092:
12087:
12085:
12073:
12071:
12070:
12065:
12060:
12052:
12049:
12040:
12038:
12037:
12032:
12030:
12022:
12017:
12005:
12003:
12002:
11997:
11995:
11987:
11973:
11971:
11970:
11965:
11963:
11956:
11955:
11948:
11947:
11942:
11922:
11921:
11916:
11908:
11907:
11902:
11887:
11886:
11881:
11873:
11872:
11867:
11859:
11858:
11853:
11832:
11831:
11826:
11821:
11807:
11805:
11804:
11799:
11797:
11796:
11791:
11786:
11776:
11774:
11773:
11768:
11766:
11765:
11760:
11755:
11743:
11741:
11740:
11735:
11733:
11729:
11728:
11721:
11720:
11715:
11706:
11705:
11699:
11694:
11685:
11684:
11678:
11673:
11663:
11662:
11657:
11649:
11648:
11643:
11634:
11633:
11627:
11622:
11612:
11611:
11606:
11598:
11597:
11592:
11584:
11583:
11578:
11557:
11556:
11551:
11546:
11532:
11530:
11529:
11524:
11522:
11510:
11508:
11507:
11502:
11500:
11499:
11494:
11489:
11477:
11475:
11474:
11469:
11464:
11463:
11456:
11455:
11450:
11435:
11434:
11429:
11421:
11420:
11415:
11398:
11384:
11382:
11381:
11376:
11374:
11373:
11366:
11365:
11360:
11351:
11350:
11344:
11339:
11329:
11328:
11323:
11315:
11314:
11309:
11292:
11278:
11276:
11275:
11270:
11268:
11248:
11244:
11237:
11235:
11234:
11229:
11227:
11226:
11221:
11216:
11206:
11204:
11203:
11198:
11196:
11195:
11190:
11184:
11176:
11168:
11167:
11162:
11157:
11135:
11133:
11132:
11127:
11125:
11113:
11111:
11110:
11105:
11103:
11102:
11097:
11091:
11083:
11075:
11074:
11069:
11064:
11054:
11052:
11051:
11046:
11044:
11025:
11022:
11019:
11016:
11013:
11010:
11007:
11004:
11001:
10998:
10995:
10992:
10989:
10986:
10983:
10980:
10977:
10974:
10971:
10968:
10965:
10962:
10959:
10956:
10953:
10950:
10947:
10944:
10941:
10938:
10935:
10932:
10929:
10926:
10923:
10920:
10917:
10914:
10911:
10908:
10905:
10902:
10899:
10896:
10893:
10890:
10887:
10884:
10881:
10878:
10875:
10872:
10869:
10866:
10863:
10860:
10857:
10854:
10851:
10848:
10845:
10842:
10839:
10836:
10833:
10830:
10827:
10824:
10821:
10818:
10815:
10812:
10809:
10806:
10803:
10800:
10797:
10794:
10791:
10788:
10785:
10782:
10779:
10776:
10773:
10770:
10767:
10764:
10761:
10758:
10755:
10752:
10749:
10746:
10743:
10740:
10737:
10734:
10731:
10728:
10725:
10722:
10719:
10716:
10713:
10710:
10707:
10704:
10701:
10697:
10694:
10691:
10688:
10685:
10682:
10678:
10675:
10672:
10669:
10666:
10663:
10660:
10657:
10654:
10651:
10648:
10645:
10642:
10639:
10636:
10633:
10630:
10627:
10624:
10621:
10618:
10615:
10612:
10608:
10605:
10602:
10599:
10582:, is known as a
10581:
10579:
10578:
10573:
10571:
10570:
10565:
10559:
10551:
10543:
10542:
10537:
10532:
10522:
10520:
10519:
10514:
10512:
10500:
10498:
10497:
10492:
10490:
10489:
10484:
10479:
10461:
10459:
10458:
10453:
10451:
10450:
10445:
10440:
10430:
10428:
10427:
10422:
10420:
10408:
10406:
10405:
10400:
10398:
10397:
10392:
10391:
10386:
10381:
10377:
10376:
10371:
10366:
10360:
10359:
10354:
10349:
10339:
10337:
10336:
10331:
10329:
10328:
10323:
10318:
10308:
10306:
10305:
10300:
10298:
10286:
10284:
10283:
10278:
10276:
10264:
10262:
10261:
10256:
10254:
10253:
10248:
10242:
10234:
10212:
10210:
10209:
10204:
10198:
10190:
10185:
10179:
10175:
10173:
10172:
10166:
10158:
10152:
10151:
10146:
10140:
10139:
10131:
10124:
10113:
10095:
10094:
10086:
10072:
10071:
10063:
10051:
10049:
10048:
10043:
10037:
10036:
10030:
10022:
10016:
10015:
10010:
10004:
10003:
9995:
9988:
9977:
9959:
9958:
9950:
9941:
9940:
9935:
9918:
9916:
9915:
9910:
9908:
9901:
9900:
9893:
9892:
9887:
9877:
9876:
9870:
9865:
9859:
9858:
9853:
9847:
9846:
9841:
9831:
9830:
9824:
9819:
9813:
9812:
9807:
9801:
9800:
9795:
9787:
9786:
9781:
9775:
9774:
9769:
9759:
9758:
9752:
9747:
9741:
9740:
9735:
9729:
9728:
9723:
9715:
9714:
9709:
9703:
9702:
9697:
9688:
9686:
9685:
9680:
9670:
9669:
9663:
9658:
9652:
9651:
9646:
9640:
9639:
9634:
9626:
9625:
9620:
9614:
9613:
9608:
9595:
9562:
9560:
9559:
9554:
9534:
9530:
9529:
9522:
9502:
9501:
9495:
9490:
9482:
9467:
9466:
9460:
9455:
9446:
9445:
9439:
9434:
9426:
9413:
9412:
9405:
9404:
9399:
9374:
9373:
9368:
9343:
9342:
9337:
9323:
9322:
9315:
9308:
9307:
9302:
9294:
9293:
9288:
9273:
9266:
9265:
9260:
9240:
9220:
9219:
9218:
9212:
9197:
9165:
9163:
9162:
9157:
9143:
9140:
9137:
9133:
9132:
9131:
9121:
9116:
9104:
9103:
9090:
9079:
9061:
9060:
9043:
9041:
9040:
9028:
9023:
9022:
9004:
9002:
9001:
8996:
8991:
8990:
8980:
8975:
8963:
8962:
8949:
8938:
8920:
8919:
8904:
8903:
8885:
8876:
8870:
8864:
8858:
8850:
8848:
8847:
8842:
8828:
8825:
8822:
8818:
8817:
8816:
8807:
8806:
8794:
8793:
8780:
8769:
8751:
8750:
8733:
8731:
8730:
8718:
8713:
8712:
8694:
8692:
8691:
8686:
8681:
8680:
8670:
8665:
8652:
8641:
8623:
8622:
8607:
8606:
8590:
8584:
8576:
8574:
8573:
8568:
8566:
8559:
8558:
8548:
8547:
8535:
8534:
8524:
8519:
8507:
8506:
8496:
8491:
8480:
8479:
8470:
8469:
8457:
8456:
8447:
8446:
8437:
8436:
8425:
8424:
8415:
8414:
8402:
8400:
8399:
8387:
8386:
8376:
8371:
8360:
8359:
8350:
8349:
8333:
8300:
8298:
8297:
8274:
8270:
8269:
8245:
8244:
8221:
8220:
8209:
8208:
8186:
8185:
8178:
8177:
8149:
8148:
8120:
8119:
8102:
8101:
8089:
8088:
8077:
8076:
8053:
8052:
8011:
8010:
8009:
8003:
7988:
7972:
7953:round-off errors
7938:
7934:
7915:
7913:
7912:
7907:
7902:
7901:
7892:
7881:
7880:
7868:
7867:
7858:
7843:
7825:
7819:
7813:
7794:well-conditioned
7780:
7773:
7770:
7767:
7764:
7761:
7758:
7755:
7752:
7749:
7746:
7743:
7740:
7737:
7734:
7731:
7728:
7725:
7722:
7719:
7716:
7713:
7710:
7707:
7704:
7701:
7698:
7695:
7692:
7689:
7686:
7683:
7680:
7677:
7674:
7671:
7668:
7665:
7662:
7659:
7656:
7653:
7650:
7647:
7644:
7641:
7638:
7635:
7632:
7629:
7626:
7623:
7620:
7617:
7614:
7611:
7608:
7605:
7602:
7599:
7596:
7593:
7590:
7587:
7584:
7581:
7578:
7575:
7572:
7569:
7566:
7563:
7560:
7557:
7554:
7551:
7548:
7545:
7542:
7539:
7536:
7533:
7530:
7527:
7524:
7521:
7518:
7515:
7512:
7509:
7506:
7503:
7500:
7497:
7494:
7491:
7488:
7485:
7482:
7479:
7476:
7473:
7470:
7446:
7443:
7440:
7437:
7434:
7431:
7428:
7425:
7422:
7419:
7416:
7413:
7410:
7407:
7404:
7401:
7398:
7395:
7392:
7389:
7386:
7383:
7380:
7377:
7374:
7371:
7368:
7365:
7362:
7359:
7356:
7353:
7350:
7347:
7344:
7341:
7338:
7335:
7332:
7329:
7326:
7323:
7320:
7317:
7314:
7311:
7308:
7305:
7302:
7299:
7296:
7293:
7290:
7287:
7284:
7281:
7278:
7275:
7272:
7269:
7266:
7263:
7260:
7257:
7254:
7251:
7248:
7245:
7242:
7239:
7236:
7233:
7230:
7227:
7224:
7221:
7218:
7215:
7212:
7209:
7206:
7203:
7200:
7197:
7194:
7191:
7188:
7185:
7182:
7179:
7176:
7173:
7170:
7167:
7164:
7161:
7158:
7155:
7152:
7149:
7146:
7143:
7140:
7134:
7122:
7115:
7112:
7109:
7106:
7103:
7100:
7097:
7094:
7091:
7088:
7085:
7082:
7079:
7076:
7073:
7070:
7067:
7064:
7061:
7058:
7055:
7052:
7049:
7046:
7043:
7040:
7037:
7034:
7031:
7028:
7025:
7022:
7019:
7016:
7013:
7010:
7007:
7004:
7001:
6998:
6995:
6992:
6989:
6986:
6983:
6980:
6977:
6974:
6971:
6968:
6965:
6962:
6959:
6956:
6953:
6950:
6947:
6944:
6941:
6938:
6935:
6932:
6929:
6926:
6923:
6920:
6917:
6914:
6911:
6908:
6905:
6902:
6899:
6896:
6893:
6890:
6887:
6884:
6881:
6878:
6875:
6872:
6869:
6866:
6863:
6860:
6857:
6854:
6851:
6848:
6845:
6842:
6839:
6836:
6833:
6830:
6827:
6824:
6821:
6818:
6815:
6812:
6790:
6787:
6784:
6781:
6778:
6775:
6772:
6769:
6766:
6763:
6760:
6757:
6754:
6751:
6748:
6745:
6742:
6739:
6736:
6733:
6730:
6727:
6724:
6721:
6718:
6715:
6712:
6709:
6706:
6703:
6700:
6697:
6694:
6691:
6688:
6685:
6682:
6679:
6676:
6673:
6670:
6667:
6664:
6661:
6658:
6655:
6652:
6649:
6646:
6643:
6640:
6637:
6634:
6631:
6628:
6625:
6622:
6619:
6616:
6613:
6610:
6607:
6604:
6601:
6598:
6595:
6592:
6589:
6586:
6583:
6580:
6577:
6574:
6571:
6568:
6565:
6562:
6559:
6556:
6553:
6550:
6542:
6530:
6516:
6514:
6513:
6508:
6494:
6491:
6488:
6484:
6483:
6482:
6466:
6461:
6445:
6434:
6416:
6415:
6395:
6393:
6392:
6374:
6369:
6368:
6347:
6345:
6344:
6339:
6334:
6332:
6331:
6315:
6310:
6294:
6283:
6265:
6264:
6249:
6244:
6243:
6216:
6204:
6202:
6201:
6196:
6182:
6179:
6176:
6172:
6171:
6170:
6155:
6154:
6138:
6127:
6109:
6108:
6088:
6086:
6085:
6067:
6062:
6061:
6040:
6038:
6037:
6032:
6027:
6024:
6019:
6003:
5992:
5974:
5973:
5958:
5944:
5943:
5919:
5911:
5909:
5908:
5903:
5901:
5897:
5896:
5889:
5886:
5881:
5868:
5863:
5851:
5850:
5841:
5837:
5836:
5827:
5822:
5818:
5817:
5816:
5807:
5806:
5794:
5793:
5777:
5776:
5767:
5762:
5761:
5743:
5740:
5735:
5723:
5722:
5713:
5709:
5708:
5699:
5694:
5693:
5670:
5668:
5667:
5658:
5645:
5625:
5623:
5622:
5617:
5615:
5608:
5607:
5599:
5594:
5581:
5576:
5563:
5558:
5547:
5546:
5537:
5536:
5524:
5523:
5514:
5513:
5502:
5501:
5492:
5491:
5479:
5476:
5471:
5458:
5453:
5442:
5441:
5432:
5431:
5415:
5412:
5406:
5403:
5398:
5375:
5371:
5370:
5363:
5362:
5339:
5338:
5327:
5326:
5308:
5307:
5296:
5295:
5284:
5283:
5266:
5265:
5258:
5257:
5246:
5245:
5234:
5233:
5215:
5214:
5203:
5202:
5179:
5178:
5154:
5153:
5148:
5136:
5121:If the equation
5105:
5103:
5102:
5097:
5092:
5091:
5086:
5077:
5076:
5071:
5065:
5064:
5059:
5050:
5036:
5032:
5022:
5018:
5014:
4999:
4979:
4977:
4976:
4971:
4966:
4965:
4957:
4952:
4947:
4941:
4940:
4935:
4929:
4927:
4926:
4908:
4903:
4902:
4891:
4844:
4843:
4832:
4815:
4814:
4797:
4784:
4782:
4781:
4776:
4773:
4768:
4763:
4757:
4756:
4739:
4733:
4732:
4727:
4718:
4717:
4706:
4693:
4687:
4681:
4669:
4667:
4666:
4661:
4656:
4655:
4648:
4647:
4636:
4628:
4627:
4622:
4616:
4613:
4612:
4597:
4593:
4577:
4575:
4574:
4559:
4538:
4537:
4526:
4509:
4508:
4503:
4490:
4476:
4465:
4453:
4451:
4450:
4445:
4440:
4439:
4432:
4431:
4420:
4412:
4411:
4406:
4390:
4385:
4380:
4372:
4371:
4337:
4336:
4325:
4308:
4307:
4296:
4283:
4277:
4269:
4256:
4242:
4221:
4212:LU decomposition
4209:
4203:
4196:
4189:
4179:
4172:
4154:
4152:
4151:
4146:
4141:
4140:
4128:
4127:
4122:
4110:
4109:
4104:
4095:
4094:
4086:
4067:
4058:
4056:
4055:
4050:
4048:
4047:
4038:
4029:
4019:
4017:
4016:
4011:
4009:
4008:
3985:Matrix inversion
3976:
3969:
3963:
3953:
3947:
3941:
3931:
3927:
3910:
3908:
3907:
3902:
3900:
3899:
3890:
3888:
3887:
3872:
3867:
3866:
3851:
3850:
3834:
3832:
3831:
3826:
3824:
3823:
3811:
3810:
3794:
3792:
3791:
3786:
3770:
3768:
3767:
3762:
3760:
3759:
3743:
3741:
3740:
3735:
3733:
3732:
3716:
3714:
3713:
3708:
3696:
3694:
3693:
3688:
3686:
3685:
3669:
3667:
3666:
3661:
3659:
3658:
3635:
3629:
3595:
3593:
3592:
3587:
3585:
3584:
3564:
3562:
3561:
3556:
3554:
3553:
3537:
3535:
3534:
3529:
3527:
3526:
3510:
3508:
3507:
3502:
3500:
3499:
3483:
3481:
3480:
3475:
3473:
3472:
3457:
3456:
3447:
3446:
3430:
3428:
3427:
3422:
3420:
3419:
3383:
3373:
3367:
3348:LU decomposition
3342:
3340:
3339:
3334:
3332:
3324:
3319:
3318:
3317:
3311:
3295:
3289:
3287:
3286:
3281:
3279:
3271:
3256:
3254:
3253:
3248:
3246:
3245:
3244:
3238:
3223:
3211:
3209:
3208:
3203:
3201:
3200:
3199:
3193:
3170:
3164:
3162:
3161:
3156:
3154:
3146:
3142:
3141:
3120:
3114:
3112:
3111:
3106:
3104:
3096:
3081:
3079:
3078:
3073:
3071:
3070:
3069:
3063:
3051:
3039:
3037:
3036:
3031:
3029:
3021:
3006:
3000:
2998:
2997:
2992:
2990:
2982:
2967:linear equations
2951:
2949:
2948:
2943:
2941:
2940:
2912:
2910:
2909:
2904:
2892:
2890:
2889:
2884:
2882:
2881:
2877:
2846:, and there is
2845:
2843:
2842:
2837:
2832:
2831:
2807:
2806:
2794:
2767:
2765:
2764:
2759:
2757:
2756:
2747:
2746:
2734:
2713:
2711:
2710:
2705:
2703:
2702:
2678:
2677:
2654:
2652:
2651:
2646:
2644:
2643:
2634:
2633:
2605:
2603:
2602:
2597:
2595:
2594:
2569:
2567:
2566:
2561:
2549:
2547:
2546:
2541:
2539:
2538:
2522:
2520:
2519:
2514:
2512:
2511:
2492:
2490:
2489:
2484:
2470:
2469:
2453:
2451:
2450:
2445:
2443:
2442:
2427:
2426:
2413:
2408:
2392:
2390:
2389:
2386:{\textstyle V:=}
2384:
2379:
2378:
2369:
2361:
2356:
2355:
2346:
2341:
2340:
2312:
2310:
2309:
2304:
2302:
2301:
2296:
2283:
2281:
2280:
2275:
2273:
2272:
2260:
2259:
2243:
2241:
2240:
2235:
2223:
2221:
2220:
2215:
2210:
2209:
2204:
2189:
2185:
2175:
2174:
2162:
2161:
2152:
2151:
2141:
2120:
2118:
2117:
2112:
2110:
2109:
2094:
2093:
2080:
2075:
2059:
2057:
2056:
2051:
2049:
2048:
2024:
2023:
2007:
2005:
2004:
1999:
1985:
1984:
1956:
1954:
1953:
1948:
1946:
1945:
1929:
1927:
1926:
1921:
1919:
1918:
1902:
1900:
1899:
1894:
1892:
1891:
1879:
1878:
1845:
1843:
1842:
1837:
1835:
1828:
1827:
1762:
1761:
1699:
1698:
1626:
1625:
1536:
1534:
1533:
1528:
1526:
1519:
1518:
1453:
1452:
1384:
1383:
1288:
1279:
1272:
1266:
1260:
1254:
1252:
1251:
1246:
1244:
1243:
1238:
1232:
1227:
1215:
1209:
1203:
1195:
1193:
1192:
1187:
1182:
1181:
1176:
1167:
1155:
1153:
1152:
1147:
1145:
1144:
1136:
1130:
1122:
1110:
1108:
1107:
1102:
1100:
1099:
1094:
1088:
1083:
1071:
1065:
1063:
1062:
1057:
1055:
1043:
1037:
1035:
1034:
1029:
1027:
1026:
1021:
1015:
1007:
993:
991:
990:
985:
980:
979:
974:
970:
969:
968:
964:
955:
949:
938:
937:
933:
924:
918:
910:
909:
904:
898:
897:
892:
888:
887:
883:
874:
863:
862:
858:
849:
843:
835:
834:
829:
814:
800:
789:similar matrices
786:
780:
774:
756:
748:
743:
726:
720:
710:
700:
692:
690:
689:
684:
679:
678:
673:
658:
636:
626:
617:
615:
614:
609:
607:
606:
594:
593:
588:
573:
572:
567:
550:
538:
528:
522:
516:
512:
502:
493:
483:
479:
473:
469:
463:
459:
457:
456:
451:
446:
445:
395:
385:
384:
379:
373:
365:
364:
322:
316:
305:
291:
285:
283:
282:
277:
272:
271:
270:
264:
252:
240:
231:
225:
219:
212:
203:
193:
183:
175:
173:
172:
167:
162:
161:
156:
144:
130:
105:LU decomposition
71:
65:
64:
61:
60:
57:
54:
51:
48:
45:
42:
39:
16887:
16886:
16882:
16881:
16880:
16878:
16877:
16876:
16862:Operator theory
16852:
16851:
16850:
16845:
16804:
16800:Multiprocessing
16768:
16764:Sparse problems
16727:
16706:
16701:
16663:
16647:
16553:
16513:
16510:
16499:
16489:
16484:
16466:
16453:
16417:
16407:Matrix Analysis
16398:
16371:
16365:
16346:
16336:
16331:
16330:
16325:
16321:
16314:
16299:
16295:
16283:
16281:
16272:
16271:
16267:
16263:
16252:
16248:
16241:
16219:
16215:
16203:
16201:
16192:
16191:
16186:
16178:Fang, Haw-ren;
16176:
16172:
16165:
16161:
16153:
16149:
16142:
16126:
16122:
16113:
16109:
16101:
16097:
16089:
16081:
16077:
16065:
16063:
16054:
16053:
16027:
16020:
16013:
15995:
15991:
15984:
15970:
15966:
15962:, p. 147).
15958:
15954:
15917:
15913:
15904:
15902:
15894:
15893:
15889:
15885:, p. 407).
15881:
15877:
15873:, p. 174).
15861:
15857:
15848:
15846:
15844:
15824:
15817:
15791:
15787:
15782:
15740:
15723:
15722:, the function
15708:
15690:
15672:
15661:
15657:
15646:
15623:
15620:
15619:
15599:
15595:
15587:
15584:
15583:
15579:
15575:
15564:
15560:
15559:, the function
15528:
15511:
15500:
15492:
15487:
15481:
15453:
15442:
15439:
15438:
15416:
15410:
15409:
15408:
15402:
15391:
15379:
15376:
15375:
15369:
15363:
15340:
15334:
15333:
15332:
15323:
15317:
15316:
15315:
15303:
15298:
15297:
15295:
15292:
15291:
15267:
15261:
15260:
15259:
15253:
15240:
15239:
15237:
15234:
15233:
15211:
15210:
15205:
15202:
15199:
15195:
15194:
15191:
15185:
15180:
15179:
15177:
15171:
15166:
15161:
15158:
15152:
15147:
15142:
15138:
15137:
15134:
15128:
15123:
15122:
15120:
15114:
15109:
15108:
15106:
15100:
15095:
15090:
15086:
15085:
15082:
15076:
15071:
15070:
15068:
15062:
15057:
15056:
15054:
15048:
15043:
15042:
15035:
15034:
15026:
15024:
15021:
15020:
14994:
14988:
14987:
14986:
14981:
14978:
14977:
14974:
14952:
14947:
14946:
14938:
14936:
14933:
14932:
14916:
14910:
14905:
14904:
14893:
14887:
14882:
14881:
14875:
14870:
14869:
14858:
14852:
14848:
14840:
14828:
14823:
14822:
14817:
14809:
14806:
14805:
14789:
14787:
14784:
14783:
14782:is unitary and
14767:
14765:
14762:
14761:
14745:
14740:
14731:
14726:
14725:
14723:
14720:
14719:
14718:, resulting in
14702:
14697:
14696:
14694:
14691:
14690:
14669:
14664:
14663:
14658:
14650:
14648:
14645:
14644:
14620:
14618:
14615:
14614:
14611:
14590:
14588:
14585:
14584:
14567:
14562:
14561:
14559:
14556:
14555:
14539:
14537:
14534:
14533:
14516:
14506:
14501:
14500:
14496:
14495:
14493:
14490:
14489:
14473:
14471:
14468:
14467:
14450:
14440:
14435:
14434:
14430:
14429:
14427:
14424:
14423:
14406:
14401:
14400:
14395:
14387:
14385:
14382:
14381:
14346:
14341:
14340:
14335:
14311:
14306:
14301:
14294:
14289:
14288:
14259:
14254:
14253:
14249:
14245:
14225:
14220:
14217:
14216:
14198:
14195:
14194:
14178:
14175:
14174:
14158:
14156:
14153:
14152:
14135:
14130:
14129:
14124:
14116:
14114:
14111:
14110:
14094:
14092:
14089:
14088:
14072:
14070:
14067:
14066:
14049:
14039:
14034:
14033:
14029:
14028:
14026:
14023:
14022:
13997:
13987:
13982:
13981:
13977:
13976:
13974:
13971:
13970:
13951:
13942:
13938:
13936:
13933:
13932:
13916:
13913:
13912:
13886:
13881:
13880:
13865:
13860:
13855:
13848:
13843:
13842:
13830:
13826:
13820:
13815:
13814:
13809:
13806:
13805:
13786:
13781:
13776:
13769:
13764:
13763:
13754:
13749:
13748:
13746:
13743:
13742:
13725:
13720:
13719:
13717:
13714:
13713:
13683:
13677:
13668:
13663:
13662:
13660:
13657:
13656:
13635:
13624:
13619:
13618:
13608:
13600:
13599:
13595:
13594:
13585:
13575:
13570:
13569:
13565:
13564:
13562:
13559:
13558:
13533:
13530:
13529:
13513:
13511:
13508:
13507:
13487:
13485:
13482:
13481:
13478:
13473:
13451:
13446:
13445:
13440:
13432:
13418:
13416:
13415:
13413:
13410:
13409:
13393:
13392:
13378:
13373:
13372:
13365:
13364:
13359:
13354:
13344:
13339:
13338:
13331:
13330:
13325:
13320:
13318:
13314:
13300:
13293:
13287:
13282:
13281:
13278:
13277:
13268:
13263:
13262:
13255:
13249:
13244:
13243:
13240:
13239:
13230:
13225:
13224:
13217:
13211:
13206:
13205:
13201:
13199:
13196:
13195:
13179:
13178:
13168:
13167:
13161:
13156:
13155:
13153:
13146:
13145:
13140:
13135:
13131:
13130:
13124:
13119:
13118:
13116:
13110:
13105:
13104:
13097:
13096:
13089:
13084:
13080:
13078:
13075:
13074:
13058:
13056:
13053:
13052:
13036:
13035:
13028:
13027:
13021:
13016:
13015:
13013:
13007:
13006:
13000:
12995:
12994:
12992:
12986:
12981:
12980:
12973:
12972:
12965:
12960:
12956:
12954:
12951:
12950:
12934:
12933:
12926:
12925:
12919:
12914:
12913:
12911:
12906:
12900:
12899:
12893:
12888:
12887:
12885:
12879:
12874:
12873:
12871:
12865:
12864:
12858:
12853:
12852:
12850:
12844:
12839:
12838:
12836:
12830:
12825:
12824:
12817:
12816:
12809:
12798:
12796:
12795:
12791:
12789:
12786:
12785:
12771:
12770:
12763:
12762:
12756:
12751:
12750:
12748:
12741:
12740:
12735:
12730:
12727:
12720:
12719:
12714:
12709:
12705:
12704:
12698:
12693:
12692:
12690:
12684:
12679:
12678:
12676:
12669:
12668:
12663:
12658:
12654:
12653:
12647:
12642:
12641:
12639:
12633:
12628:
12627:
12625:
12619:
12614:
12613:
12606:
12605:
12598:
12587:
12585:
12584:
12580:
12578:
12575:
12574:
12555:
12554:
12540:
12535:
12534:
12527:
12526:
12521:
12516:
12506:
12501:
12500:
12493:
12492:
12487:
12482:
12480:
12476:
12462:
12455:
12449:
12444:
12443:
12440:
12439:
12425:
12420:
12419:
12412:
12411:
12406:
12401:
12391:
12386:
12385:
12384:
12380:
12370:
12369:
12364:
12359:
12351:
12345:
12340:
12339:
12336:
12335:
12321:
12316:
12315:
12308:
12307:
12302:
12297:
12287:
12282:
12281:
12280:
12276:
12262:
12255:
12249:
12244:
12243:
12240:
12239:
12230:
12225:
12224:
12217:
12211:
12206:
12205:
12202:
12201:
12192:
12187:
12186:
12176:
12175:
12170:
12165:
12157:
12151:
12146:
12145:
12142:
12141:
12132:
12127:
12126:
12119:
12113:
12108:
12107:
12103:
12101:
12098:
12097:
12081:
12079:
12076:
12075:
12056:
12048:
12046:
12043:
12042:
12026:
12018:
12013:
12011:
12008:
12007:
11991:
11983:
11981:
11978:
11977:
11961:
11960:
11950:
11949:
11943:
11938:
11937:
11935:
11930:
11924:
11923:
11917:
11912:
11911:
11909:
11903:
11898:
11897:
11895:
11889:
11888:
11882:
11877:
11876:
11874:
11868:
11863:
11862:
11860:
11854:
11849:
11848:
11841:
11840:
11833:
11822:
11820:
11819:
11815:
11813:
11810:
11809:
11787:
11785:
11784:
11782:
11779:
11778:
11756:
11754:
11753:
11751:
11748:
11747:
11731:
11730:
11723:
11722:
11716:
11711:
11710:
11708:
11701:
11700:
11695:
11690:
11687:
11680:
11679:
11674:
11669:
11665:
11664:
11658:
11653:
11652:
11650:
11644:
11639:
11638:
11636:
11629:
11628:
11623:
11618:
11614:
11613:
11607:
11602:
11601:
11599:
11593:
11588:
11587:
11585:
11579:
11574:
11573:
11566:
11565:
11558:
11547:
11545:
11544:
11540:
11538:
11535:
11534:
11518:
11516:
11513:
11512:
11490:
11488:
11487:
11485:
11482:
11481:
11458:
11457:
11451:
11446:
11445:
11443:
11437:
11436:
11430:
11425:
11424:
11422:
11416:
11411:
11410:
11403:
11402:
11394:
11392:
11389:
11388:
11368:
11367:
11361:
11356:
11355:
11353:
11346:
11345:
11340:
11335:
11331:
11330:
11324:
11319:
11318:
11316:
11310:
11305:
11304:
11297:
11296:
11288:
11286:
11283:
11282:
11264:
11262:
11259:
11258:
11255:
11246:
11242:
11217:
11215:
11214:
11212:
11209:
11208:
11191:
11186:
11185:
11180:
11172:
11158:
11156:
11155:
11153:
11150:
11149:
11142:
11121:
11119:
11116:
11115:
11098:
11093:
11092:
11087:
11079:
11065:
11063:
11062:
11060:
11057:
11056:
11040:
11038:
11035:
11034:
11027:
11026:
11023:
11020:
11017:
11014:
11011:
11008:
11005:
11002:
10999:
10996:
10993:
10990:
10987:
10984:
10981:
10978:
10975:
10972:
10969:
10966:
10963:
10960:
10957:
10954:
10951:
10948:
10945:
10942:
10939:
10936:
10933:
10930:
10927:
10924:
10921:
10918:
10915:
10912:
10909:
10906:
10903:
10900:
10897:
10894:
10891:
10888:
10885:
10882:
10879:
10876:
10873:
10870:
10867:
10864:
10861:
10858:
10855:
10852:
10849:
10846:
10843:
10840:
10837:
10834:
10831:
10828:
10825:
10822:
10819:
10816:
10813:
10810:
10807:
10804:
10801:
10798:
10795:
10792:
10789:
10786:
10783:
10780:
10777:
10774:
10771:
10768:
10765:
10762:
10759:
10756:
10753:
10750:
10747:
10744:
10741:
10738:
10735:
10732:
10729:
10726:
10723:
10720:
10717:
10714:
10711:
10708:
10705:
10702:
10699:
10695:
10692:
10689:
10686:
10683:
10680:
10676:
10673:
10670:
10667:
10664:
10661:
10658:
10655:
10652:
10649:
10646:
10643:
10640:
10637:
10634:
10631:
10628:
10625:
10622:
10619:
10616:
10613:
10610:
10606:
10603:
10600:
10597:
10584:rank-one update
10566:
10561:
10560:
10555:
10547:
10533:
10531:
10530:
10528:
10525:
10524:
10508:
10506:
10503:
10502:
10480:
10478:
10477:
10475:
10472:
10471:
10468:
10466:Rank-one update
10441:
10439:
10438:
10436:
10433:
10432:
10416:
10414:
10411:
10410:
10393:
10382:
10380:
10379:
10378:
10367:
10365:
10364:
10350:
10348:
10347:
10345:
10342:
10341:
10319:
10317:
10316:
10314:
10311:
10310:
10294:
10292:
10289:
10288:
10272:
10270:
10267:
10266:
10265:of some matrix
10249:
10244:
10243:
10238:
10230:
10228:
10225:
10224:
10221:
10191:
10186:
10181:
10168:
10167:
10159:
10154:
10147:
10142:
10141:
10132:
10127:
10126:
10114:
10103:
10087:
10082:
10081:
10080:
10076:
10064:
10059:
10058:
10056:
10053:
10052:
10032:
10031:
10023:
10018:
10011:
10006:
10005:
9996:
9991:
9990:
9978:
9967:
9951:
9946:
9945:
9936:
9931:
9930:
9928:
9925:
9924:
9906:
9905:
9895:
9894:
9888:
9883:
9882:
9872:
9871:
9866:
9861:
9854:
9849:
9848:
9842:
9837:
9836:
9826:
9825:
9820:
9815:
9808:
9803:
9802:
9796:
9791:
9790:
9788:
9782:
9777:
9776:
9770:
9765:
9764:
9754:
9753:
9748:
9743:
9736:
9731:
9730:
9724:
9719:
9718:
9716:
9710:
9705:
9704:
9698:
9693:
9692:
9689:
9687:
9681:
9676:
9675:
9665:
9664:
9659:
9654:
9647:
9642:
9641:
9635:
9630:
9629:
9627:
9621:
9616:
9615:
9609:
9604:
9603:
9600:
9599:
9567:
9561:
9555:
9550:
9549:
9542:
9541:
9532:
9531:
9524:
9523:
9518:
9516:
9511:
9505:
9504:
9497:
9496:
9491:
9486:
9483:
9478:
9476:
9470:
9469:
9462:
9461:
9456:
9451:
9448:
9441:
9440:
9435:
9430:
9427:
9422:
9415:
9414:
9407:
9406:
9400:
9395:
9394:
9392:
9387:
9381:
9380:
9375:
9369:
9364:
9363:
9361:
9355:
9354:
9349:
9344:
9338:
9333:
9332:
9325:
9324:
9317:
9316:
9311:
9309:
9303:
9298:
9297:
9295:
9289:
9284:
9283:
9280:
9279:
9274:
9269:
9267:
9261:
9256:
9255:
9252:
9251:
9246:
9241:
9236:
9229:
9228:
9221:
9214:
9213:
9202:
9201:
9193:
9189:
9187:
9184:
9183:
9174:
9139:
9127:
9123:
9117:
9109:
9096:
9092:
9080:
9069:
9053:
9049:
9048:
9044:
9036:
9032:
9027:
9015:
9011:
9009:
9006:
9005:
8986:
8982:
8976:
8968:
8955:
8951:
8939:
8928:
8912:
8908:
8899:
8895:
8893:
8890:
8889:
8881:
8872:
8866:
8860:
8854:
8824:
8812:
8808:
8799:
8795:
8786:
8782:
8770:
8759:
8743:
8739:
8738:
8734:
8726:
8722:
8717:
8705:
8701:
8699:
8696:
8695:
8676:
8672:
8666:
8658:
8642:
8631:
8615:
8611:
8602:
8598:
8596:
8593:
8592:
8586:
8580:
8564:
8563:
8553:
8552:
8543:
8539:
8530:
8526:
8520:
8515:
8502:
8498:
8492:
8487:
8481:
8475:
8471:
8465:
8461:
8452:
8448:
8442:
8438:
8432:
8428:
8426:
8420:
8416:
8410:
8406:
8403:
8401:
8395:
8391:
8382:
8378:
8372:
8367:
8361:
8355:
8351:
8345:
8341:
8338:
8337:
8305:
8299:
8293:
8289:
8282:
8281:
8272:
8271:
8264:
8263:
8258:
8253:
8247:
8246:
8240:
8236:
8234:
8229:
8223:
8222:
8216:
8212:
8210:
8204:
8200:
8198:
8188:
8187:
8180:
8179:
8173:
8169:
8167:
8162:
8156:
8155:
8150:
8144:
8140:
8138:
8132:
8131:
8126:
8121:
8115:
8111:
8104:
8103:
8096:
8095:
8090:
8084:
8080:
8078:
8072:
8068:
8065:
8064:
8059:
8054:
8048:
8044:
8041:
8040:
8035:
8030:
8020:
8019:
8012:
8005:
8004:
7993:
7992:
7984:
7980:
7978:
7975:
7974:
7968:
7965:
7936:
7932:
7929:
7919:
7897:
7893:
7888:
7876:
7872:
7863:
7859:
7854:
7849:
7846:
7845:
7827:
7821:
7815:
7805:
7790:
7778:
7775:
7774:
7771:
7768:
7765:
7762:
7759:
7756:
7753:
7750:
7747:
7744:
7741:
7738:
7735:
7732:
7729:
7726:
7723:
7720:
7717:
7714:
7711:
7708:
7705:
7702:
7699:
7696:
7693:
7690:
7687:
7684:
7681:
7678:
7675:
7672:
7669:
7666:
7663:
7660:
7657:
7654:
7651:
7648:
7645:
7642:
7639:
7636:
7633:
7630:
7627:
7624:
7621:
7618:
7615:
7612:
7609:
7606:
7603:
7600:
7597:
7594:
7591:
7588:
7585:
7582:
7579:
7576:
7573:
7570:
7567:
7564:
7561:
7558:
7555:
7552:
7549:
7546:
7543:
7540:
7537:
7534:
7531:
7528:
7525:
7522:
7519:
7516:
7513:
7510:
7507:
7504:
7501:
7498:
7495:
7492:
7489:
7486:
7483:
7480:
7477:
7474:
7471:
7468:
7448:
7447:
7444:
7441:
7438:
7435:
7432:
7429:
7426:
7423:
7420:
7417:
7414:
7411:
7408:
7405:
7402:
7399:
7396:
7393:
7390:
7387:
7384:
7381:
7378:
7375:
7372:
7369:
7366:
7363:
7360:
7357:
7354:
7351:
7348:
7345:
7342:
7339:
7336:
7333:
7330:
7327:
7324:
7321:
7318:
7315:
7312:
7309:
7306:
7303:
7300:
7297:
7294:
7291:
7288:
7285:
7282:
7279:
7276:
7273:
7270:
7267:
7264:
7261:
7258:
7255:
7252:
7249:
7246:
7243:
7240:
7237:
7234:
7231:
7228:
7225:
7222:
7219:
7216:
7213:
7210:
7207:
7204:
7201:
7198:
7195:
7192:
7189:
7186:
7183:
7180:
7177:
7174:
7171:
7168:
7165:
7162:
7159:
7156:
7153:
7150:
7147:
7144:
7141:
7138:
7132:
7120:
7117:
7116:
7113:
7110:
7107:
7104:
7101:
7098:
7095:
7092:
7089:
7086:
7083:
7080:
7077:
7074:
7071:
7068:
7065:
7062:
7059:
7056:
7053:
7050:
7047:
7044:
7041:
7038:
7035:
7032:
7029:
7026:
7023:
7020:
7017:
7014:
7011:
7008:
7005:
7002:
6999:
6996:
6993:
6990:
6987:
6984:
6981:
6978:
6975:
6972:
6969:
6966:
6963:
6960:
6957:
6954:
6951:
6948:
6945:
6942:
6939:
6936:
6933:
6930:
6927:
6924:
6921:
6918:
6915:
6912:
6909:
6906:
6903:
6900:
6897:
6894:
6891:
6888:
6885:
6882:
6879:
6876:
6873:
6870:
6867:
6864:
6861:
6858:
6855:
6852:
6849:
6846:
6843:
6840:
6837:
6834:
6831:
6828:
6825:
6822:
6819:
6816:
6813:
6810:
6792:
6791:
6788:
6785:
6782:
6779:
6776:
6773:
6770:
6767:
6764:
6761:
6758:
6755:
6752:
6749:
6746:
6743:
6740:
6737:
6734:
6731:
6728:
6725:
6722:
6719:
6716:
6713:
6710:
6707:
6704:
6701:
6698:
6695:
6692:
6689:
6686:
6683:
6680:
6677:
6674:
6671:
6668:
6665:
6662:
6659:
6656:
6653:
6650:
6647:
6644:
6641:
6638:
6635:
6632:
6629:
6626:
6623:
6620:
6617:
6614:
6611:
6608:
6605:
6602:
6599:
6596:
6593:
6590:
6587:
6584:
6581:
6578:
6575:
6572:
6569:
6566:
6563:
6560:
6557:
6554:
6551:
6548:
6540:
6520:
6490:
6472:
6468:
6462:
6451:
6435:
6424:
6405:
6401:
6400:
6396:
6382:
6378:
6373:
6358:
6354:
6352:
6349:
6348:
6321:
6317:
6311:
6300:
6284:
6273:
6254:
6250:
6248:
6233:
6229:
6227:
6224:
6223:
6212:
6178:
6160:
6156:
6144:
6140:
6128:
6117:
6098:
6094:
6093:
6089:
6075:
6071:
6066:
6051:
6047:
6045:
6042:
6041:
6020:
6009:
5993:
5982:
5963:
5959:
5957:
5933:
5929:
5927:
5924:
5923:
5915:
5899:
5898:
5891:
5890:
5882:
5877:
5864:
5859:
5846:
5842:
5840:
5838:
5832:
5828:
5823:
5812:
5808:
5802:
5798:
5789:
5785:
5784:
5780:
5778:
5772:
5768:
5763:
5757:
5753:
5750:
5749:
5744:
5736:
5731:
5718:
5714:
5712:
5710:
5704:
5700:
5695:
5689:
5685:
5682:
5681:
5676:
5671:
5663:
5659:
5657:
5650:
5649:
5641:
5637:
5635:
5632:
5631:
5613:
5612:
5602:
5601:
5595:
5590:
5577:
5572:
5559:
5554:
5548:
5542:
5538:
5532:
5528:
5519:
5515:
5509:
5505:
5503:
5497:
5493:
5487:
5483:
5480:
5478:
5472:
5467:
5454:
5449:
5443:
5437:
5433:
5427:
5423:
5420:
5419:
5411:
5405:
5399:
5394:
5383:
5382:
5373:
5372:
5365:
5364:
5358:
5354:
5352:
5347:
5341:
5340:
5334:
5330:
5328:
5322:
5318:
5316:
5310:
5309:
5303:
5299:
5297:
5291:
5287:
5285:
5279:
5275:
5268:
5267:
5260:
5259:
5253:
5249:
5247:
5241:
5237:
5235:
5229:
5225:
5222:
5221:
5216:
5210:
5206:
5204:
5198:
5194:
5191:
5190:
5185:
5180:
5174:
5170:
5163:
5162:
5155:
5149:
5141:
5140:
5132:
5128:
5126:
5123:
5122:
5111:
5087:
5082:
5081:
5072:
5067:
5066:
5060:
5055:
5054:
5046:
5044:
5041:
5040:
5034:
5024:
5020:
5016:
5012:
4998:
4989:
4981:
4960:
4959:
4953:
4948:
4943:
4936:
4931:
4930:
4916:
4912:
4907:
4892:
4887:
4886:
4884:
4879:
4873:
4872:
4867:
4862:
4856:
4855:
4850:
4845:
4833:
4828:
4827:
4820:
4819:
4798:
4793:
4792:
4790:
4787:
4786:
4769:
4764:
4759:
4740:
4735:
4734:
4728:
4723:
4722:
4707:
4702:
4701:
4699:
4696:
4695:
4689:
4683:
4679:
4671:
4650:
4649:
4637:
4632:
4631:
4629:
4623:
4618:
4617:
4602:
4598:
4592:
4590:
4584:
4583:
4578:
4564:
4560:
4558:
4556:
4550:
4549:
4544:
4539:
4527:
4522:
4521:
4514:
4513:
4504:
4499:
4498:
4496:
4493:
4492:
4489:
4481:
4471:
4468:identity matrix
4464:
4455:
4434:
4433:
4421:
4416:
4415:
4413:
4407:
4402:
4401:
4399:
4393:
4392:
4386:
4381:
4376:
4373:
4361:
4357:
4355:
4349:
4348:
4343:
4338:
4326:
4321:
4320:
4313:
4312:
4297:
4292:
4291:
4289:
4286:
4285:
4279:
4275:
4261:
4251:
4238:
4231:
4215:
4205:
4201:
4191:
4184:
4174:
4163:
4160:
4133:
4129:
4123:
4115:
4114:
4105:
4100:
4099:
4087:
4082:
4081:
4079:
4076:
4075:
4063:
4043:
4039:
4027:
4025:
4022:
4021:
4004:
4000:
3998:
3995:
3994:
3987:
3977:vectors called
3971:
3965:
3959:
3949:
3943:
3933:
3929:
3923:
3917:
3895:
3891:
3883:
3879:
3871:
3862:
3858:
3846:
3842:
3840:
3837:
3836:
3819:
3815:
3806:
3802:
3800:
3797:
3796:
3780:
3777:
3776:
3755:
3751:
3749:
3746:
3745:
3728:
3724:
3722:
3719:
3718:
3702:
3699:
3698:
3681:
3677:
3675:
3672:
3671:
3654:
3650:
3648:
3645:
3644:
3631:
3625:
3614:
3580:
3576:
3574:
3571:
3570:
3549:
3545:
3543:
3540:
3539:
3522:
3518:
3516:
3513:
3512:
3495:
3491:
3489:
3486:
3485:
3468:
3464:
3452:
3448:
3442:
3438:
3436:
3433:
3432:
3415:
3411:
3409:
3406:
3405:
3398:Newton's method
3394:
3379:
3369:
3359:
3356:
3328:
3320:
3313:
3312:
3304:
3303:
3301:
3298:
3297:
3291:
3275:
3264:
3262:
3259:
3258:
3257:, then solving
3240:
3239:
3228:
3227:
3219:
3217:
3214:
3213:
3195:
3194:
3186:
3185:
3183:
3180:
3179:
3166:
3150:
3137:
3133:
3132:
3130:
3127:
3126:
3116:
3100:
3089:
3087:
3084:
3083:
3082:, then solving
3065:
3064:
3056:
3055:
3047:
3045:
3042:
3041:
3025:
3014:
3012:
3009:
3008:
3002:
2986:
2975:
2973:
2970:
2969:
2963:
2958:
2936:
2932:
2918:
2915:
2914:
2898:
2895:
2894:
2873:
2866:
2862:
2851:
2848:
2847:
2827:
2823:
2802:
2798:
2781:
2773:
2770:
2769:
2752:
2748:
2742:
2738:
2730:
2719:
2716:
2715:
2698:
2694:
2673:
2669:
2667:
2664:
2663:
2639:
2635:
2626:
2622:
2611:
2608:
2607:
2590:
2586:
2575:
2572:
2571:
2555:
2552:
2551:
2534:
2530:
2528:
2525:
2524:
2507:
2503:
2501:
2498:
2497:
2465:
2461:
2459:
2456:
2455:
2435:
2431:
2422:
2418:
2409:
2404:
2398:
2395:
2394:
2374:
2370:
2365:
2357:
2351:
2347:
2342:
2336:
2332:
2321:
2318:
2317:
2297:
2292:
2291:
2289:
2286:
2285:
2268:
2264:
2255:
2251:
2249:
2246:
2245:
2229:
2226:
2225:
2205:
2200:
2199:
2170:
2166:
2157:
2153:
2147:
2143:
2137:
2132:
2128:
2126:
2123:
2122:
2102:
2098:
2089:
2085:
2076:
2071:
2065:
2062:
2061:
2044:
2040:
2019:
2015:
2013:
2010:
2009:
1980:
1976:
1974:
1971:
1970:
1941:
1937:
1935:
1932:
1931:
1914:
1910:
1908:
1905:
1904:
1887:
1883:
1874:
1870:
1868:
1865:
1864:
1857:
1851:
1833:
1832:
1822:
1821:
1816:
1811:
1805:
1804:
1799:
1794:
1788:
1787:
1779:
1774:
1764:
1763:
1756:
1755:
1750:
1745:
1739:
1738:
1733:
1728:
1722:
1721:
1716:
1711:
1701:
1700:
1693:
1692:
1687:
1682:
1673:
1672:
1667:
1662:
1656:
1655:
1650:
1645:
1635:
1634:
1627:
1620:
1619:
1614:
1606:
1597:
1596:
1588:
1583:
1577:
1576:
1568:
1563:
1553:
1552:
1548:
1546:
1543:
1542:
1524:
1523:
1513:
1512:
1507:
1502:
1496:
1495:
1490:
1485:
1479:
1478:
1470:
1465:
1455:
1454:
1447:
1446:
1441:
1436:
1427:
1426:
1421:
1416:
1410:
1409:
1404:
1399:
1389:
1388:
1378:
1377:
1372:
1364:
1355:
1354:
1346:
1341:
1335:
1334:
1326:
1321:
1311:
1310:
1306:
1304:
1301:
1300:
1295:
1284:
1274:
1268:
1262:
1256:
1239:
1234:
1233:
1228:
1223:
1221:
1218:
1217:
1211:
1205:
1199:
1177:
1172:
1171:
1163:
1161:
1158:
1157:
1137:
1132:
1131:
1126:
1118:
1116:
1113:
1112:
1095:
1090:
1089:
1084:
1079:
1077:
1074:
1073:
1067:
1051:
1049:
1046:
1045:
1039:
1022:
1017:
1016:
1011:
1003:
1001:
998:
997:
975:
960:
956:
951:
950:
945:
944:
940:
939:
929:
925:
920:
919:
914:
905:
900:
899:
893:
879:
875:
870:
869:
865:
864:
854:
850:
845:
844:
839:
830:
819:
818:
810:
808:
805:
804:
795:
782:
776:
766:
752:
746:
735:
722:
716:
706:
696:
674:
663:
662:
654:
652:
649:
648:
643:
628:
625:
619:
601:
600:
595:
589:
584:
583:
580:
579:
574:
568:
563:
562:
555:
554:
546:
544:
541:
540:
530:
524:
518:
514:
504:
498:
485:
481:
475:
471:
465:
461:
440:
439:
428:
416:
415:
410:
400:
399:
391:
380:
375:
374:
369:
359:
358:
353:
347:
346:
341:
331:
330:
328:
325:
324:
318:
307:
301:
298:
287:
266:
265:
257:
256:
248:
246:
243:
242:
236:
227:
221:
214:
208:
199:
189:
179:
157:
149:
148:
140:
138:
135:
134:
126:
117:
69:
36:
32:
17:
12:
11:
5:
16885:
16875:
16874:
16869:
16864:
16847:
16846:
16844:
16843:
16838:
16833:
16828:
16823:
16818:
16812:
16810:
16806:
16805:
16803:
16802:
16797:
16792:
16787:
16782:
16776:
16774:
16770:
16769:
16767:
16766:
16761:
16756:
16746:
16741:
16735:
16733:
16729:
16728:
16726:
16725:
16720:
16718:Floating point
16714:
16712:
16708:
16707:
16700:
16699:
16692:
16685:
16677:
16671:
16670:
16662:
16659:
16658:
16657:
16646:
16643:
16642:
16641:
16628:
16619:
16609:
16603:
16597:
16591:
16585:
16579:
16573:
16567:
16552:
16549:
16548:
16547:
16541:
16535:
16529:
16509:
16506:
16505:
16504:
16488:
16485:
16483:
16482:External links
16480:
16479:
16478:
16471:
16457:
16451:
16435:
16428:
16421:
16415:
16402:
16396:
16380:Golub, Gene H.
16376:
16374:on 2011-07-16.
16363:
16335:
16332:
16329:
16328:
16319:
16312:
16293:
16284:|journal=
16261:
16246:
16239:
16213:
16204:|journal=
16170:
16159:
16147:
16140:
16120:
16107:
16095:
16075:
16066:|journal=
16018:
16011:
15989:
15982:
15964:
15952:
15931:(1): 437–445.
15911:
15887:
15875:
15855:
15842:
15815:
15784:
15783:
15781:
15778:
15777:
15776:
15771:
15766:
15761:
15756:
15751:
15746:
15739:
15736:
15735:
15734:
15727:
15715:
15714:
15713:
15712:
15701:
15694:
15676:
15665:
15650:
15639:
15631:{\textstyle R}
15627:
15607:
15602:
15598:
15594:
15591:
15568:
15553:
15538:
15532:
15522:
15510:
15507:
15469:
15466:
15463:
15460:
15456:
15452:
15449:
15446:
15424:
15419:
15413:
15405:
15400:
15397:
15394:
15390:
15386:
15383:
15343:
15337:
15331:
15326:
15320:
15314:
15309:
15306:
15301:
15275:
15270:
15264:
15256:
15252:
15248:
15243:
15215:
15209:
15206:
15203:
15200:
15197:
15196:
15192:
15188:
15183:
15178:
15174:
15169:
15164:
15159:
15155:
15150:
15145:
15140:
15139:
15135:
15131:
15126:
15121:
15117:
15112:
15107:
15103:
15098:
15093:
15088:
15087:
15083:
15079:
15074:
15069:
15065:
15060:
15055:
15051:
15046:
15041:
15040:
15038:
15033:
15029:
15015:Hilbert spaces
15002:
14997:
14991:
14985:
14973:
14972:Generalization
14970:
14955:
14950:
14945:
14941:
14919:
14913:
14908:
14903:
14899:
14896:
14890:
14885:
14878:
14873:
14868:
14864:
14861:
14855:
14851:
14846:
14843:
14839:
14836:
14831:
14826:
14820:
14816:
14813:
14792:
14770:
14748:
14743:
14739:
14734:
14729:
14705:
14700:
14672:
14667:
14661:
14657:
14653:
14623:
14610:
14607:
14593:
14570:
14565:
14542:
14519:
14514:
14509:
14504:
14499:
14488:in norm means
14476:
14453:
14448:
14443:
14438:
14433:
14409:
14404:
14398:
14394:
14390:
14367:
14363:
14360:
14357:
14354:
14349:
14344:
14338:
14334:
14331:
14328:
14325:
14322:
14319:
14314:
14309:
14304:
14297:
14292:
14287:
14284:
14281:
14277:
14273:
14270:
14267:
14262:
14257:
14252:
14248:
14244:
14241:
14238:
14235:
14232:
14228:
14224:
14206:{\textstyle y}
14202:
14186:{\textstyle x}
14182:
14161:
14138:
14133:
14127:
14123:
14119:
14097:
14075:
14052:
14047:
14042:
14037:
14032:
14000:
13995:
13990:
13985:
13980:
13958:
13954:
13950:
13945:
13941:
13931:holds because
13920:
13898:
13894:
13889:
13884:
13879:
13876:
13873:
13868:
13863:
13858:
13851:
13846:
13841:
13838:
13833:
13829:
13823:
13818:
13813:
13789:
13784:
13779:
13772:
13767:
13762:
13757:
13752:
13728:
13723:
13697:
13694:
13691:
13680:
13676:
13671:
13666:
13638:
13633:
13627:
13622:
13615:
13612:
13607:
13603:
13598:
13593:
13588:
13583:
13578:
13573:
13568:
13543:
13540:
13537:
13516:
13490:
13477:
13474:
13472:
13469:
13454:
13449:
13443:
13439:
13435:
13431:
13425:
13421:
13391:
13387:
13381:
13376:
13368:
13362:
13357:
13352:
13347:
13342:
13334:
13328:
13323:
13317:
13312:
13309:
13306:
13303:
13299:
13296:
13294:
13290:
13285:
13280:
13279:
13276:
13271:
13266:
13261:
13258:
13256:
13252:
13247:
13242:
13241:
13238:
13233:
13228:
13223:
13220:
13218:
13214:
13209:
13204:
13203:
13177:
13172:
13164:
13159:
13154:
13149:
13143:
13138:
13133:
13132:
13127:
13122:
13117:
13113:
13108:
13103:
13102:
13100:
13095:
13092:
13090:
13087:
13083:
13082:
13061:
13051:of the matrix
13032:
13024:
13019:
13014:
13012:
13009:
13008:
13003:
12998:
12993:
12989:
12984:
12979:
12978:
12976:
12971:
12968:
12966:
12963:
12959:
12958:
12930:
12922:
12917:
12912:
12910:
12907:
12905:
12902:
12901:
12896:
12891:
12886:
12882:
12877:
12872:
12870:
12867:
12866:
12861:
12856:
12851:
12847:
12842:
12837:
12833:
12828:
12823:
12822:
12820:
12815:
12812:
12810:
12805:
12801:
12794:
12793:
12767:
12759:
12754:
12749:
12744:
12738:
12733:
12728:
12723:
12717:
12712:
12707:
12706:
12701:
12696:
12691:
12687:
12682:
12677:
12672:
12666:
12661:
12656:
12655:
12650:
12645:
12640:
12636:
12631:
12626:
12622:
12617:
12612:
12611:
12609:
12604:
12601:
12599:
12594:
12590:
12583:
12582:
12553:
12549:
12543:
12538:
12530:
12524:
12519:
12514:
12509:
12504:
12496:
12490:
12485:
12479:
12474:
12471:
12468:
12465:
12461:
12458:
12456:
12452:
12447:
12442:
12441:
12438:
12434:
12428:
12423:
12415:
12409:
12404:
12399:
12394:
12389:
12383:
12379:
12373:
12367:
12362:
12357:
12354:
12352:
12348:
12343:
12338:
12337:
12334:
12330:
12324:
12319:
12311:
12305:
12300:
12295:
12290:
12285:
12279:
12274:
12271:
12268:
12265:
12261:
12258:
12256:
12252:
12247:
12242:
12241:
12238:
12233:
12228:
12223:
12220:
12218:
12214:
12209:
12204:
12203:
12200:
12195:
12190:
12185:
12179:
12173:
12168:
12163:
12160:
12158:
12154:
12149:
12144:
12143:
12140:
12135:
12130:
12125:
12122:
12120:
12116:
12111:
12106:
12105:
12084:
12063:
12059:
12055:
12029:
12025:
12021:
12016:
11994:
11990:
11986:
11959:
11954:
11946:
11941:
11936:
11934:
11931:
11929:
11926:
11925:
11920:
11915:
11910:
11906:
11901:
11896:
11894:
11891:
11890:
11885:
11880:
11875:
11871:
11866:
11861:
11857:
11852:
11847:
11846:
11844:
11839:
11836:
11834:
11829:
11825:
11818:
11817:
11794:
11790:
11763:
11759:
11727:
11719:
11714:
11709:
11704:
11698:
11693:
11688:
11683:
11677:
11672:
11667:
11666:
11661:
11656:
11651:
11647:
11642:
11637:
11632:
11626:
11621:
11616:
11615:
11610:
11605:
11600:
11596:
11591:
11586:
11582:
11577:
11572:
11571:
11569:
11564:
11561:
11559:
11554:
11550:
11543:
11542:
11521:
11497:
11493:
11467:
11462:
11454:
11449:
11444:
11442:
11439:
11438:
11433:
11428:
11423:
11419:
11414:
11409:
11408:
11406:
11401:
11397:
11372:
11364:
11359:
11354:
11349:
11343:
11338:
11333:
11332:
11327:
11322:
11317:
11313:
11308:
11303:
11302:
11300:
11295:
11291:
11267:
11254:
11251:
11224:
11220:
11194:
11189:
11183:
11179:
11175:
11171:
11165:
11161:
11141:
11138:
11124:
11101:
11096:
11090:
11086:
11082:
11078:
11072:
11068:
11043:
10596:
10569:
10564:
10558:
10554:
10550:
10546:
10540:
10536:
10511:
10487:
10483:
10467:
10464:
10448:
10444:
10419:
10396:
10389:
10385:
10374:
10370:
10363:
10357:
10353:
10326:
10322:
10297:
10275:
10252:
10247:
10241:
10237:
10233:
10220:
10217:
10202:
10197:
10194:
10189:
10184:
10178:
10171:
10165:
10162:
10157:
10150:
10145:
10138:
10135:
10130:
10123:
10120:
10117:
10112:
10109:
10106:
10102:
10098:
10093:
10090:
10085:
10079:
10075:
10070:
10067:
10062:
10041:
10035:
10029:
10026:
10021:
10014:
10009:
10002:
9999:
9994:
9987:
9984:
9981:
9976:
9973:
9970:
9966:
9962:
9957:
9954:
9949:
9944:
9939:
9934:
9904:
9899:
9891:
9886:
9881:
9875:
9869:
9864:
9857:
9852:
9845:
9840:
9835:
9829:
9823:
9818:
9811:
9806:
9799:
9794:
9789:
9785:
9780:
9773:
9768:
9763:
9757:
9751:
9746:
9739:
9734:
9727:
9722:
9717:
9713:
9708:
9701:
9696:
9691:
9690:
9684:
9679:
9674:
9668:
9662:
9657:
9650:
9645:
9638:
9633:
9628:
9624:
9619:
9612:
9607:
9602:
9601:
9598:
9594:
9591:
9588:
9585:
9582:
9579:
9576:
9573:
9570:
9566:
9563:
9558:
9553:
9548:
9547:
9545:
9540:
9537:
9535:
9533:
9528:
9521:
9517:
9515:
9512:
9510:
9507:
9506:
9500:
9494:
9489:
9484:
9481:
9477:
9475:
9472:
9471:
9465:
9459:
9454:
9449:
9444:
9438:
9433:
9428:
9425:
9421:
9420:
9418:
9411:
9403:
9398:
9393:
9391:
9388:
9386:
9383:
9382:
9379:
9376:
9372:
9367:
9362:
9360:
9357:
9356:
9353:
9350:
9348:
9345:
9341:
9336:
9331:
9330:
9328:
9321:
9314:
9310:
9306:
9301:
9296:
9292:
9287:
9282:
9281:
9278:
9275:
9272:
9268:
9264:
9259:
9254:
9253:
9250:
9247:
9245:
9242:
9239:
9235:
9234:
9232:
9227:
9224:
9222:
9217:
9211:
9208:
9205:
9200:
9196:
9192:
9191:
9173:
9170:
9155:
9152:
9149:
9146:
9136:
9130:
9126:
9120:
9115:
9112:
9108:
9102:
9099:
9095:
9089:
9086:
9083:
9078:
9075:
9072:
9068:
9064:
9059:
9056:
9052:
9047:
9039:
9035:
9031:
9026:
9021:
9018:
9014:
8994:
8989:
8985:
8979:
8974:
8971:
8967:
8961:
8958:
8954:
8948:
8945:
8942:
8937:
8934:
8931:
8927:
8923:
8918:
8915:
8911:
8907:
8902:
8898:
8840:
8837:
8834:
8831:
8821:
8815:
8811:
8805:
8802:
8798:
8792:
8789:
8785:
8779:
8776:
8773:
8768:
8765:
8762:
8758:
8754:
8749:
8746:
8742:
8737:
8729:
8725:
8721:
8716:
8711:
8708:
8704:
8684:
8679:
8675:
8669:
8664:
8661:
8657:
8651:
8648:
8645:
8640:
8637:
8634:
8630:
8626:
8621:
8618:
8614:
8610:
8605:
8601:
8562:
8557:
8551:
8546:
8542:
8538:
8533:
8529:
8523:
8518:
8514:
8510:
8505:
8501:
8495:
8490:
8486:
8482:
8478:
8474:
8468:
8464:
8460:
8455:
8451:
8445:
8441:
8435:
8431:
8427:
8423:
8419:
8413:
8409:
8405:
8404:
8398:
8394:
8390:
8385:
8381:
8375:
8370:
8366:
8362:
8358:
8354:
8348:
8344:
8340:
8339:
8336:
8332:
8329:
8326:
8323:
8320:
8317:
8314:
8311:
8308:
8304:
8301:
8296:
8292:
8288:
8287:
8285:
8280:
8277:
8275:
8273:
8268:
8262:
8259:
8257:
8254:
8252:
8249:
8248:
8243:
8239:
8235:
8233:
8230:
8228:
8225:
8224:
8219:
8215:
8211:
8207:
8203:
8199:
8197:
8194:
8193:
8191:
8184:
8176:
8172:
8168:
8166:
8163:
8161:
8158:
8157:
8154:
8151:
8147:
8143:
8139:
8137:
8134:
8133:
8130:
8127:
8125:
8122:
8118:
8114:
8110:
8109:
8107:
8100:
8094:
8091:
8087:
8083:
8079:
8075:
8071:
8067:
8066:
8063:
8060:
8058:
8055:
8051:
8047:
8043:
8042:
8039:
8036:
8034:
8031:
8029:
8026:
8025:
8023:
8018:
8015:
8013:
8008:
8002:
7999:
7996:
7991:
7987:
7983:
7982:
7964:
7961:
7941:unit round-off
7927:
7917:
7905:
7900:
7896:
7891:
7887:
7884:
7879:
7875:
7871:
7866:
7862:
7857:
7853:
7789:
7786:
7467:
7450:
7449:
7137:
6809:
6547:
6545:
6544:
6506:
6503:
6500:
6497:
6487:
6481:
6478:
6475:
6471:
6465:
6460:
6457:
6454:
6450:
6444:
6441:
6438:
6433:
6430:
6427:
6423:
6419:
6414:
6411:
6408:
6404:
6399:
6391:
6388:
6385:
6381:
6377:
6372:
6367:
6364:
6361:
6357:
6337:
6330:
6327:
6324:
6320:
6314:
6309:
6306:
6303:
6299:
6293:
6290:
6287:
6282:
6279:
6276:
6272:
6268:
6263:
6260:
6257:
6253:
6247:
6242:
6239:
6236:
6232:
6194:
6191:
6188:
6185:
6175:
6169:
6166:
6163:
6159:
6153:
6150:
6147:
6143:
6137:
6134:
6131:
6126:
6123:
6120:
6116:
6112:
6107:
6104:
6101:
6097:
6092:
6084:
6081:
6078:
6074:
6070:
6065:
6060:
6057:
6054:
6050:
6030:
6023:
6018:
6015:
6012:
6008:
6002:
5999:
5996:
5991:
5988:
5985:
5981:
5977:
5972:
5969:
5966:
5962:
5956:
5953:
5950:
5947:
5942:
5939:
5936:
5932:
5895:
5885:
5880:
5876:
5872:
5867:
5862:
5858:
5854:
5849:
5845:
5839:
5835:
5831:
5826:
5821:
5815:
5811:
5805:
5801:
5797:
5792:
5788:
5783:
5779:
5775:
5771:
5766:
5760:
5756:
5752:
5751:
5748:
5745:
5739:
5734:
5730:
5726:
5721:
5717:
5711:
5707:
5703:
5698:
5692:
5688:
5684:
5683:
5680:
5677:
5675:
5672:
5666:
5662:
5656:
5655:
5653:
5648:
5644:
5640:
5639:
5611:
5606:
5598:
5593:
5589:
5585:
5580:
5575:
5571:
5567:
5562:
5557:
5553:
5549:
5545:
5541:
5535:
5531:
5527:
5522:
5518:
5512:
5508:
5504:
5500:
5496:
5490:
5486:
5482:
5481:
5475:
5470:
5466:
5462:
5457:
5452:
5448:
5444:
5440:
5436:
5430:
5426:
5422:
5421:
5418:
5410:
5407:
5402:
5397:
5393:
5389:
5388:
5386:
5381:
5378:
5376:
5374:
5369:
5361:
5357:
5353:
5351:
5348:
5346:
5343:
5342:
5337:
5333:
5329:
5325:
5321:
5317:
5315:
5312:
5311:
5306:
5302:
5298:
5294:
5290:
5286:
5282:
5278:
5274:
5273:
5271:
5264:
5256:
5252:
5248:
5244:
5240:
5236:
5232:
5228:
5224:
5223:
5220:
5217:
5213:
5209:
5205:
5201:
5197:
5193:
5192:
5189:
5186:
5184:
5181:
5177:
5173:
5169:
5168:
5166:
5161:
5158:
5156:
5152:
5147:
5144:
5139:
5135:
5131:
5130:
5110:
5107:
5095:
5090:
5085:
5080:
5075:
5070:
5063:
5058:
5053:
5049:
4994:
4985:
4969:
4964:
4956:
4951:
4946:
4939:
4934:
4925:
4922:
4919:
4915:
4911:
4906:
4901:
4898:
4895:
4890:
4885:
4883:
4880:
4878:
4875:
4874:
4871:
4868:
4866:
4863:
4861:
4858:
4857:
4854:
4851:
4849:
4846:
4842:
4839:
4836:
4831:
4826:
4825:
4823:
4818:
4813:
4810:
4807:
4804:
4801:
4796:
4772:
4767:
4762:
4755:
4752:
4749:
4746:
4743:
4738:
4731:
4726:
4721:
4716:
4713:
4710:
4705:
4675:
4659:
4654:
4646:
4643:
4640:
4635:
4630:
4626:
4621:
4611:
4608:
4605:
4601:
4596:
4591:
4589:
4586:
4585:
4582:
4579:
4573:
4570:
4567:
4563:
4557:
4555:
4552:
4551:
4548:
4545:
4543:
4540:
4536:
4533:
4530:
4525:
4520:
4519:
4517:
4512:
4507:
4502:
4491:is defined by
4485:
4480:If the matrix
4459:
4443:
4438:
4430:
4427:
4424:
4419:
4414:
4410:
4405:
4400:
4398:
4395:
4394:
4389:
4384:
4379:
4374:
4370:
4367:
4364:
4360:
4356:
4354:
4351:
4350:
4347:
4344:
4342:
4339:
4335:
4332:
4329:
4324:
4319:
4318:
4316:
4311:
4306:
4303:
4300:
4295:
4272:
4271:
4230:
4227:
4159:
4156:
4144:
4139:
4136:
4132:
4126:
4121:
4118:
4113:
4108:
4103:
4098:
4093:
4090:
4085:
4046:
4042:
4035:
4032:
4007:
4003:
3986:
3983:
3948:. The matrix
3916:
3915:Kalman filters
3913:
3898:
3894:
3886:
3882:
3878:
3875:
3870:
3865:
3861:
3857:
3854:
3849:
3845:
3822:
3818:
3814:
3809:
3805:
3784:
3758:
3754:
3731:
3727:
3706:
3684:
3680:
3657:
3653:
3613:
3610:
3598:Hessian matrix
3583:
3579:
3552:
3548:
3525:
3521:
3498:
3494:
3471:
3467:
3463:
3460:
3455:
3451:
3445:
3441:
3418:
3414:
3393:
3390:
3355:
3352:
3331:
3327:
3323:
3316:
3310:
3307:
3278:
3274:
3270:
3267:
3243:
3237:
3234:
3231:
3226:
3222:
3198:
3192:
3189:
3153:
3149:
3145:
3140:
3136:
3103:
3099:
3095:
3092:
3068:
3062:
3059:
3054:
3050:
3028:
3024:
3020:
3017:
2989:
2985:
2981:
2978:
2962:
2959:
2957:
2954:
2939:
2935:
2931:
2928:
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2906:{\textstyle U}
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2563:{\textstyle V}
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2237:{\textstyle f}
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1963:conjugate axes
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194:* denotes the
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21:linear algebra
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16626:on page topic
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16551:Computer code
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15927:. ICCS 2010.
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15843:0-521-43108-5
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15800:(in French).
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15707:package, the
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15698:Eigen library
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15662:LinearAlgebra
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11031:rank-n update
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7922:matrix 2-norm
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7313:dimensionSize
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6110:
6105:
6102:
6099:
6095:
6090:
6082:
6079:
6076:
6072:
6068:
6063:
6058:
6055:
6052:
6048:
6028:
6021:
6016:
6013:
6010:
6006:
6000:
5997:
5994:
5989:
5986:
5983:
5979:
5975:
5970:
5967:
5964:
5960:
5951:
5945:
5940:
5937:
5934:
5930:
5921:
5918:
5912:
5893:
5883:
5878:
5874:
5870:
5865:
5860:
5856:
5852:
5847:
5843:
5833:
5829:
5824:
5819:
5813:
5809:
5803:
5799:
5795:
5790:
5786:
5781:
5773:
5769:
5764:
5758:
5754:
5746:
5737:
5732:
5728:
5724:
5719:
5715:
5705:
5701:
5696:
5690:
5686:
5678:
5673:
5664:
5660:
5651:
5646:
5629:
5626:
5609:
5604:
5596:
5591:
5587:
5583:
5578:
5573:
5569:
5565:
5560:
5555:
5551:
5543:
5539:
5533:
5529:
5525:
5520:
5516:
5510:
5506:
5498:
5494:
5488:
5484:
5473:
5468:
5464:
5460:
5455:
5450:
5446:
5438:
5434:
5428:
5424:
5400:
5395:
5391:
5384:
5379:
5377:
5367:
5359:
5355:
5349:
5344:
5335:
5331:
5323:
5319:
5313:
5304:
5300:
5292:
5288:
5280:
5276:
5269:
5262:
5254:
5250:
5242:
5238:
5230:
5226:
5218:
5211:
5207:
5199:
5195:
5187:
5182:
5175:
5171:
5164:
5159:
5157:
5150:
5137:
5115:
5106:
5093:
5088:
5078:
5073:
5061:
5051:
5038:
5031:
5027:
5009:
5007:
5003:
5002:outer product
4997:
4992:
4988:
4984:
4967:
4962:
4954:
4949:
4937:
4923:
4920:
4917:
4913:
4909:
4904:
4896:
4881:
4876:
4869:
4864:
4859:
4852:
4847:
4840:
4837:
4834:
4821:
4816:
4808:
4805:
4802:
4770:
4765:
4750:
4747:
4744:
4729:
4719:
4711:
4692:
4686:
4678:
4674:
4657:
4652:
4644:
4641:
4638:
4624:
4609:
4606:
4603:
4599:
4594:
4587:
4580:
4571:
4568:
4565:
4561:
4553:
4546:
4541:
4534:
4531:
4528:
4515:
4510:
4505:
4488:
4484:
4478:
4474:
4470:of dimension
4469:
4462:
4458:
4441:
4436:
4425:
4408:
4396:
4387:
4382:
4368:
4365:
4362:
4358:
4352:
4345:
4340:
4333:
4330:
4327:
4314:
4309:
4301:
4282:
4278:, the matrix
4268:
4264:
4260:
4259:
4258:
4254:
4248:
4246:
4241:
4236:
4226:
4223:
4219:
4214:, which uses
4213:
4208:
4199:
4195:
4187:
4182:
4178:
4170:
4166:
4155:
4142:
4137:
4134:
4124:
4106:
4096:
4091:
4088:
4073:
4071:
4066:
4060:
4044:
4040:
4033:
4030:
4005:
4001:
3992:
3989:The explicit
3982:
3980:
3975:
3968:
3962:
3957:
3952:
3946:
3940:
3936:
3926:
3921:
3912:
3896:
3892:
3884:
3880:
3876:
3873:
3868:
3863:
3859:
3855:
3852:
3847:
3843:
3820:
3816:
3812:
3807:
3803:
3782:
3774:
3756:
3752:
3729:
3725:
3704:
3682:
3678:
3655:
3651:
3641:
3639:
3634:
3628:
3623:
3619:
3609:
3607:
3603:
3599:
3581:
3577:
3568:
3550:
3546:
3523:
3519:
3496:
3492:
3469:
3465:
3461:
3458:
3453:
3449:
3443:
3439:
3416:
3412:
3403:
3399:
3389:
3387:
3382:
3377:
3372:
3366:
3362:
3351:
3349:
3344:
3325:
3314:
3294:
3272:
3241:
3224:
3196:
3176:
3174:
3169:
3147:
3124:
3119:
3097:
3066:
3052:
3022:
3005:
2983:
2968:
2953:
2937:
2933:
2926:
2923:
2920:
2900:
2878:
2874:
2870:
2867:
2859:
2856:
2853:
2828:
2824:
2820:
2817:
2814:
2811:
2808:
2803:
2799:
2778:
2753:
2743:
2739:
2731:
2727:
2724:
2721:
2699:
2695:
2691:
2688:
2685:
2682:
2679:
2674:
2670:
2661:
2656:
2640:
2630:
2627:
2623:
2616:
2613:
2591:
2587:
2583:
2580:
2577:
2557:
2535:
2531:
2508:
2504:
2494:
2480:
2477:
2474:
2471:
2466:
2462:
2439:
2436:
2432:
2428:
2423:
2419:
2415:
2410:
2405:
2401:
2375:
2371:
2362:
2352:
2348:
2337:
2333:
2326:
2323:
2314:
2298:
2269:
2265:
2256:
2252:
2231:
2206:
2193:
2190:
2186:
2182:
2179:
2176:
2171:
2167:
2163:
2158:
2154:
2148:
2144:
2138:
2134:
2129:
2106:
2103:
2099:
2095:
2090:
2086:
2082:
2077:
2072:
2068:
2045:
2041:
2037:
2034:
2031:
2028:
2025:
2020:
2016:
1995:
1992:
1989:
1986:
1981:
1977:
1968:
1964:
1942:
1938:
1915:
1911:
1888:
1884:
1880:
1875:
1871:
1861:
1856:
1846:
1829:
1824:
1818:
1813:
1808:
1801:
1796:
1791:
1784:
1781:
1776:
1771:
1765:
1758:
1752:
1747:
1742:
1735:
1730:
1725:
1718:
1713:
1708:
1702:
1695:
1689:
1684:
1679:
1676:
1669:
1664:
1659:
1652:
1647:
1642:
1636:
1631:
1629:
1622:
1616:
1611:
1608:
1603:
1600:
1593:
1590:
1585:
1580:
1573:
1570:
1565:
1560:
1554:
1540:
1537:
1520:
1515:
1509:
1504:
1499:
1492:
1487:
1482:
1475:
1472:
1467:
1462:
1456:
1449:
1443:
1438:
1433:
1430:
1423:
1418:
1413:
1406:
1401:
1396:
1390:
1385:
1380:
1374:
1369:
1366:
1361:
1358:
1351:
1348:
1343:
1338:
1331:
1328:
1323:
1318:
1312:
1298:
1290:
1287:
1282:
1277:
1271:
1265:
1259:
1240:
1214:
1208:
1202:
1196:
1183:
1178:
1168:
1141:
1138:
1123:
1096:
1070:
1042:
1023:
1008:
994:
981:
976:
971:
965:
961:
957:
941:
934:
930:
926:
911:
906:
894:
889:
884:
880:
876:
866:
859:
855:
851:
836:
831:
815:
802:
798:
792:
790:
785:
779:
773:
769:
764:
760:
755:
750:
749:decomposition
742:
738:
733:
728:
725:
719:
714:
709:
704:
699:
693:
680:
675:
659:
646:
638:
635:
631:
622:
603:
597:
590:
576:
569:
556:
551:
537:
533:
527:
521:
511:
507:
501:
495:
492:
488:
480:with exactly
478:
468:
447:
442:
436:
433:
430:
425:
422:
419:
412:
407:
401:
396:
386:
381:
366:
361:
355:
350:
343:
338:
332:
321:
314:
310:
304:
293:
290:
273:
253:
239:
233:
230:
224:
217:
211:
205:
202:
197:
192:
187:
182:
176:
163:
158:
145:
132:
129:
125:
122:
112:
110:
106:
102:
98:
94:
90:
86:
82:
78:
77:decomposition
74:
73:
63:
30:
26:
22:
16711:Key concepts
16612:Rosetta Code
16518:
16492:
16462:
16442:
16406:
16387:
16369:the original
16342:
16322:
16303:
16296:
16275:cite journal
16264:
16255:
16249:
16224:
16216:
16195:cite journal
16173:
16162:
16150:
16134:. Elsevier.
16130:
16123:
16110:
16098:
16085:
16078:
16057:cite journal
16030:
16002:
15992:
15973:
15967:
15955:
15928:
15924:
15914:
15903:. Retrieved
15900:MathOverflow
15899:
15890:
15878:
15858:
15847:. Retrieved
15828:
15801:
15795:
15788:
15565:numpy.linalg
15501:
15493:
15488:
15482:
15436:
15374:
15372:and for any
15364:
15357:
15290:
15287:
15232:
15229:
15019:
14975:
14612:
14379:
14215:
14015:Banach space
13910:
13804:
13651:consists of
13505:
13479:
13407:
13193:
13050:
12948:
12573:
12569:
11975:
11745:
11479:
11386:
11281:
11256:
11240:
11145:
11143:
11030:
11028:
10588:
10583:
10469:
10222:
10214:
9923:
9920:
9182:
9177:
9175:
9167:
8888:
8882:
8879:
8873:
8871:are real if
8867:
8861:
8855:
8852:
8587:
8581:
8578:
7969:
7966:
7948:
7945:
7939:denotes the
7925:
7840:
7836:
7832:
7828:
7822:
7816:
7810:
7806:
7799:
7791:
7783:
7776:
7451:
7128:
7118:
6793:
6536:
6526:
6522:
6518:
6222:
6219:
6213:
6206:
5922:
5916:
5913:
5630:
5627:
5120:
5039:
5029:
5025:
5010:
5005:
4995:
4990:
4986:
4982:
4690:
4684:
4676:
4672:
4486:
4482:
4479:
4472:
4466:denotes the
4460:
4456:
4280:
4273:
4266:
4262:
4252:
4249:
4239:
4234:
4232:
4224:
4217:
4206:
4193:
4185:
4176:
4168:
4164:
4161:
4074:
4069:
4064:
4061:
4020:operations (
3988:
3979:sigma points
3978:
3973:
3966:
3960:
3955:
3950:
3944:
3938:
3934:
3924:
3918:
3642:
3637:
3632:
3626:
3615:
3402:quasi-Newton
3401:
3395:
3380:
3370:
3364:
3360:
3357:
3345:
3292:
3177:
3167:
3117:
3003:
2964:
2956:Applications
2657:
2495:
2315:
1960:
1541:
1538:
1299:
1296:
1285:
1275:
1269:
1263:
1257:
1212:
1206:
1200:
1197:
1068:
1040:
995:
803:
801:as follows:
796:
793:
783:
777:
771:
767:
758:
753:
745:
740:
736:
731:
729:
723:
717:
707:
705:matrix, and
697:
694:
647:
644:
633:
629:
620:
535:
531:
525:
519:
509:
505:
499:
496:
490:
486:
476:
466:
319:
312:
308:
302:
299:
288:
237:
234:
228:
222:
215:
209:
206:
200:
190:
180:
177:
133:
127:
118:
107:for solving
31:(pronounced
28:
24:
18:
16564:performance
16508:Information
16258:. Springer.
15709:TDecompChol
15669:Mathematica
15288:where each
14380:Therefore,
7550:dot_product
7454:dot product
6892:dot_product
6796:dot product
6209:square root
4670:(note that
4158:Computation
2658:Similarly,
16856:Categories
16753:algorithms
16596:by Google.
16334:References
16300:Based on:
15905:2020-01-25
15849:2009-01-28
15744:Cycle rank
15535:GNU Octave
15486:such that
14931:. Setting
10604:cholupdate
7916:Here ||·||
5015:from 1 to
4980:Note that
4255: := 1
3928:of length
3604:(DFP) and
1853:See also:
523:such that
16780:CPU cache
16606:Armadillo
16525:EMS Press
16040:1111.4144
15947:1877-0509
15804:: 66–67.
15724:Decompose
15720:Analytica
15601:∗
15563:from the
15465:≥
15462:⟩
15445:⟨
15437:there is
15389:⨁
15385:∈
15330:→
15251:⨁
15208:⋱
15173:∗
15154:∗
15102:∗
14954:∗
14912:∗
14889:∗
14877:∗
14854:∗
14830:∗
14733:∗
14704:∗
14671:∗
14605:is also.
14532:tends to
14466:tends to
14408:∗
14362:⟩
14348:∗
14333:⟨
14327:⟩
14313:∗
14283:⟨
14240:⟩
14223:⟨
14137:∗
13919:≤
13893:‖
13878:‖
13872:‖
13867:∗
13840:‖
13837:≤
13828:‖
13812:‖
13788:∗
13696:∞
13693:→
13675:→
13539:×
13453:∗
13438:±
13424:~
12804:~
12593:~
12513:−
12398:−
12378:∖
12294:−
12184:∖
11989:∖
11828:~
11793:~
11762:~
11553:~
11496:~
11223:~
11193:∗
11178:−
11164:~
11100:∗
11071:~
10568:∗
10539:~
10486:~
10447:~
10395:∗
10388:~
10373:~
10356:~
10325:~
10251:∗
10193:−
10119:−
10101:∑
10097:−
9983:−
9965:∑
9961:−
9141:for
9119:∗
9085:−
9067:∑
9063:−
8978:∗
8944:−
8926:∑
8922:−
8877:is real.
8826:for
8775:−
8757:∑
8753:−
8647:−
8629:∑
8625:−
7895:‖
7886:‖
7883:ε
7870:≤
7861:‖
7852:‖
6492:for
6464:∗
6440:−
6422:∑
6418:−
6313:∗
6289:−
6271:∑
6267:−
6180:for
6133:−
6115:∑
6111:−
5998:−
5980:∑
5976:−
5952:±
5871:−
5853:−
5796:−
5725:−
5413:symmetric
5079:…
4955:∗
4905:−
4838:−
4771:∗
4642:−
4532:−
4388:∗
4331:−
4265: :=
4135:−
4125:∗
4107:∗
4089:−
3881:ρ
3877:−
3856:ρ
3783:ρ
3705:ρ
3462:−
3315:∗
3242:∗
3197:∗
3139:∗
3067:∗
2930:Σ
2868:−
2864:Σ
2825:λ
2800:λ
2776:Σ
2750:‖
2736:‖
2722:λ
2628:−
2433:δ
2363:⋯
2262:↦
2135:∑
2100:δ
1967:ellipsoid
1782:−
1677:−
1609:−
1601:−
1591:−
1571:−
1473:−
1431:−
1367:−
1359:−
1349:−
1329:−
1241:∗
1139:−
1097:∗
1024:∗
977:∗
907:∗
895:∗
832:∗
676:∗
437:θ
434:
426:θ
423:
382:∗
159:∗
121:Hermitian
115:Statement
81:Hermitian
16809:Software
16773:Hardware
16732:Problems
16622:AlgoWiki
16576:libflame
16386:(1996).
16033:: 4144.
15738:See also
15658:cholesky
15561:cholesky
15529:cholesky
14760:, where
14276:⟩
14247:⟨
11976:Writing
10598:function
7844:, where
7802:pivoting
5019:. After
4274:At step
3567:gradient
3511:, where
2606:, where
787:are not
713:diagonal
618:, where
460:for any
91:and its
16561:details
16527:, 2001
16045:Bibcode
15703:In the
15546:Fortran
7920:is the
7462:Fortran
6804:Fortran
5023:steps,
4682:since
3991:inverse
3942:matrix
3565:is the
3400:called
2393:, then
1293:Example
1066:, then
75:) is a
16831:LAPACK
16821:MATLAB
16639:?potrs
16635:?potrf
16570:ALGLIB
16557:LAPACK
16497:BibNum
16449:
16413:
16394:
16361:
16310:
16237:
16138:
16009:
15980:
15945:
15840:
15656:, the
15645:, the
15618:where
15574:, the
15572:Matlab
15557:Python
15542:LAPACK
15525:Maxima
15517:: the
14685:. Now
14151:, and
13528:is an
10620:length
10591:Matlab
7935:, and
7814:, and
7777:where
7772:end do
7673:matmul
7119:where
7114:end do
7015:matmul
5000:is an
4785:where
4680:> 0
4454:where
3569:, and
2893:where
2284:, and
2224:where
1965:of an
1216:, an
1111:where
695:where
627:is an
503:is an
286:where
178:where
23:, the
16816:ATLAS
16467:(PDF)
16372:(PDF)
16347:(PDF)
16187:(PDF)
16090:(PDF)
16035:arXiv
15780:Notes
15654:Julia
15362:. If
15358:is a
14635:be a
7779:conjg
7679:conjg
7181:float
7121:conjg
7021:conjg
6633:float
6612:<=
4198:FLOPs
4192:(4/3)
4181:FLOPs
4175:(1/3)
3368:with
3001:. If
711:is a
701:is a
539:with
532:P A P
526:P A P
235:When
184:is a
79:of a
16795:SIMD
16447:ISBN
16411:ISBN
16392:ISBN
16359:ISBN
16308:ISBN
16288:help
16235:ISBN
16208:help
16136:ISBN
16070:help
16031:1111
16007:ISBN
15978:ISBN
15943:ISSN
15838:ISBN
15729:The
15705:ROOT
15696:The
15691:chol
15685:The
15647:chol
15580:chol
15576:chol
15540:The
14613:Let
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13911:The
12050:chol
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6747:else
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4257:and
4233:The
3835:and
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3670:and
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3290:for
3165:for
3115:for
2768:and
781:and
759:LDL′
751:(or
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72:-kee
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16785:TLB
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15933:doi
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11021:end
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15214:]
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15073:A
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15032:=
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15001:}
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14984:{
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14838:(
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13761:=
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13727:k
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13430:=
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9074:=
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9025:=
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8764:=
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8715:=
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8636:=
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8561:.
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8017:=
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7990:=
7986:A
7970:A
7937:ε
7933:n
7928:n
7926:c
7918:2
7904:.
7899:2
7890:A
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7661:+
7658:i
7655:,
7652:i
7649:(
7646:A
7643:(
7640:=
7634:1
7631:+
7628:i
7625:,
7622:i
7619:(
7616:L
7610:i
7607:,
7604:1
7601:-
7598:i
7595::
7592:1
7589:(
7586:L
7580:i
7577:,
7574:1
7571:-
7568:i
7565::
7562:1
7559:(
7556:L
7553:(
7547:-
7544:)
7541:i
7538:,
7535:i
7532:(
7529:A
7526:(
7520:=
7517:)
7514:i
7511:,
7508:i
7505:(
7502:L
7499:)
7496:1
7493:,
7490:A
7487:(
7481:,
7478:1
7475:=
7472:i
7445:}
7442:}
7433:-
7430:A
7427:(
7424:*
7421:L
7418:/
7412:(
7409:=
7406:L
7403:}
7400:;
7397:L
7394:*
7391:L
7382:{
7379:)
7373:k
7370:;
7367:j
7361:k
7358:;
7355:0
7352:=
7349:k
7346:(
7340:;
7337:0
7334:=
7328:{
7325:)
7319:i
7316:;
7307:i
7304:;
7301:1
7298:+
7295:j
7292:=
7289:i
7286:(
7274:-
7271:A
7268:(
7262:=
7259:L
7256:}
7253:;
7250:L
7247:*
7244:L
7235:{
7232:)
7226:k
7223:;
7220:j
7214:k
7211:;
7208:0
7205:=
7202:k
7199:(
7193:;
7190:0
7187:=
7178:{
7175:)
7169:j
7166:;
7157:j
7154:;
7151:0
7148:=
7145:j
7142:(
7133:L
7111:)
7108:i
7105:,
7102:i
7099:(
7096:L
7093:/
7087:1
7084:-
7081:i
7078::
7075:1
7069:1
7066:+
7063:i
7060:(
7057:L
7051:1
7048:-
7045:i
7042::
7039:1
7036:,
7033:i
7030:(
7027:L
7024:(
7018:(
7012:-
7009:)
7006:i
7000:1
6997:+
6994:i
6991:(
6988:A
6985:(
6982:=
6979:)
6976:i
6970:1
6967:+
6964:i
6961:(
6958:L
6952:1
6949:-
6946:i
6943::
6940:1
6937:,
6934:i
6931:(
6928:L
6922:1
6919:-
6916:i
6913::
6910:1
6907:,
6904:i
6901:(
6898:L
6895:(
6889:-
6886:)
6883:i
6880:,
6877:i
6874:(
6871:A
6868:(
6862:=
6859:)
6856:i
6853:,
6850:i
6847:(
6844:L
6841:)
6838:1
6835:,
6832:A
6829:(
6823:,
6820:1
6817:=
6814:i
6789:}
6786:}
6777:-
6774:A
6771:(
6768:*
6765:L
6762:/
6756:(
6753:=
6750:L
6738:-
6735:A
6732:(
6726:=
6723:L
6720:)
6717:j
6711:i
6708:(
6702:;
6699:L
6696:*
6693:L
6684:)
6678:k
6675:;
6672:j
6666:k
6663:;
6660:0
6657:=
6654:k
6651:(
6645:;
6642:0
6639:=
6630:{
6627:)
6621:j
6618:;
6615:i
6609:j
6606:;
6603:0
6600:=
6597:j
6594:(
6588:{
6585:)
6579:i
6576:;
6567:i
6564:;
6561:0
6558:=
6555:i
6552:(
6541:L
6529:)
6527:j
6523:i
6521:(
6505:.
6502:j
6496:i
6486:)
6480:k
6477:,
6474:i
6470:L
6459:k
6456:,
6453:j
6449:L
6443:1
6437:j
6432:1
6429:=
6426:k
6413:j
6410:,
6407:i
6403:A
6398:(
6390:j
6387:,
6384:j
6380:L
6376:1
6371:=
6366:j
6363:,
6360:i
6356:L
6336:,
6329:k
6326:,
6323:j
6319:L
6308:k
6305:,
6302:j
6298:L
6292:1
6286:j
6281:1
6278:=
6275:k
6262:j
6259:,
6256:j
6252:A
6246:=
6241:j
6238:,
6235:j
6231:L
6214:A
6193:.
6190:j
6184:i
6174:)
6168:k
6165:,
6162:j
6158:L
6152:k
6149:,
6146:i
6142:L
6136:1
6130:j
6125:1
6122:=
6119:k
6106:j
6103:,
6100:i
6096:A
6091:(
6083:j
6080:,
6077:j
6073:L
6069:1
6064:=
6059:j
6056:,
6053:i
6049:L
6029:,
6022:2
6017:k
6014:,
6011:j
6007:L
6001:1
5995:j
5990:1
5987:=
5984:k
5971:j
5968:,
5965:j
5961:A
5955:)
5949:(
5946:=
5941:j
5938:,
5935:j
5931:L
5917:L
5894:)
5884:2
5875:L
5866:2
5857:L
5844:A
5830:L
5825:/
5820:)
5810:L
5800:L
5787:A
5782:(
5770:L
5765:/
5755:A
5747:0
5738:2
5729:L
5716:A
5702:L
5697:/
5687:A
5679:0
5674:0
5661:A
5652:(
5647:=
5643:L
5610:,
5605:)
5597:2
5588:L
5584:+
5579:2
5570:L
5566:+
5561:2
5552:L
5540:L
5530:L
5526:+
5517:L
5507:L
5495:L
5485:L
5474:2
5465:L
5461:+
5456:2
5447:L
5435:L
5425:L
5417:)
5409:(
5401:2
5392:L
5385:(
5380:=
5368:)
5356:L
5350:0
5345:0
5332:L
5320:L
5314:0
5301:L
5289:L
5277:L
5270:(
5263:)
5251:L
5239:L
5227:L
5219:0
5208:L
5196:L
5188:0
5183:0
5172:L
5165:(
5160:=
5151:T
5146:L
5143:L
5138:=
5134:A
5094:.
5089:n
5084:L
5074:2
5069:L
5062:1
5057:L
5048:L
5035:L
5030:I
5026:A
5021:n
5017:n
5013:i
4996:i
4993:*
4991:b
4987:i
4983:b
4968:.
4963:)
4950:i
4945:b
4938:i
4933:b
4924:i
4921:,
4918:i
4914:a
4910:1
4900:)
4897:i
4894:(
4889:B
4882:0
4877:0
4870:0
4865:1
4860:0
4853:0
4848:0
4841:1
4835:i
4830:I
4822:(
4817:=
4812:)
4809:1
4806:+
4803:i
4800:(
4795:A
4766:i
4761:L
4754:)
4751:1
4748:+
4745:i
4742:(
4737:A
4730:i
4725:L
4720:=
4715:)
4712:i
4709:(
4704:A
4691:A
4685:A
4673:a
4658:,
4653:)
4645:i
4639:n
4634:I
4625:i
4620:b
4610:i
4607:,
4604:i
4600:a
4595:1
4588:0
4581:0
4572:i
4569:,
4566:i
4562:a
4554:0
4547:0
4542:0
4535:1
4529:i
4524:I
4516:(
4506:i
4501:L
4487:i
4483:L
4473:i
4461:i
4457:I
4442:,
4437:)
4429:)
4426:i
4423:(
4418:B
4409:i
4404:b
4397:0
4383:i
4378:b
4369:i
4366:,
4363:i
4359:a
4353:0
4346:0
4341:0
4334:1
4328:i
4323:I
4315:(
4310:=
4305:)
4302:i
4299:(
4294:A
4281:A
4276:i
4270:.
4267:A
4263:A
4253:i
4240:L
4218:n
4216:2
4207:A
4202:n
4194:n
4186:n
4183:(
4177:n
4171:)
4169:n
4167:(
4165:O
4143:.
4138:1
4131:)
4120:B
4117:B
4112:(
4102:B
4097:=
4092:1
4084:B
4065:B
4045:3
4041:n
4034:2
4031:1
4006:3
4002:n
3974:N
3972:2
3967:x
3961:L
3951:P
3945:P
3939:N
3935:N
3930:N
3925:x
3897:2
3893:z
3885:2
3874:1
3869:+
3864:1
3860:z
3853:=
3848:2
3844:x
3821:1
3817:z
3813:=
3808:1
3804:x
3757:2
3753:z
3730:1
3726:z
3683:2
3679:x
3656:1
3652:x
3633:u
3627:L
3582:k
3578:B
3551:k
3547:g
3524:k
3520:p
3497:k
3493:p
3470:k
3466:g
3459:=
3454:k
3450:p
3444:k
3440:B
3417:k
3413:p
3381:A
3371:A
3365:b
3330:y
3326:=
3322:x
3309:L
3306:D
3293:y
3277:b
3273:=
3269:y
3266:L
3236:L
3233:D
3230:L
3225:=
3221:A
3191:L
3188:L
3168:x
3152:y
3148:=
3144:x
3135:L
3118:y
3102:b
3098:=
3094:y
3091:L
3061:L
3058:L
3053:=
3049:A
3027:b
3023:=
3019:x
3016:A
3004:A
2988:b
2984:=
2980:x
2977:A
2938:T
2934:U
2927:U
2924:=
2921:A
2901:U
2879:2
2875:/
2871:1
2860:U
2857:=
2854:V
2834:)
2829:n
2821:,
2818:.
2815:.
2812:.
2809:,
2804:1
2796:(
2792:g
2789:a
2786:i
2783:d
2779:=
2754:2
2744:i
2740:v
2732:/
2728:1
2725:=
2700:n
2696:v
2692:,
2689:.
2686:.
2683:.
2680:,
2675:1
2671:v
2641:T
2637:)
2631:1
2624:V
2620:(
2617:=
2614:L
2592:T
2588:L
2584:L
2581:=
2578:A
2558:V
2536:2
2532:v
2509:1
2505:v
2481:I
2478:=
2475:V
2472:A
2467:T
2463:V
2440:j
2437:i
2429:=
2424:j
2420:v
2416:A
2411:T
2406:i
2402:v
2381:]
2376:n
2372:v
2367:|
2359:|
2353:2
2349:v
2344:|
2338:1
2334:v
2330:[
2324:V
2299:n
2294:S
2270:i
2266:v
2257:i
2253:e
2232:f
2212:)
2207:n
2202:S
2197:(
2194:f
2191:=
2187:}
2183:1
2180:=
2177:x
2172:T
2168:x
2164::
2159:i
2155:v
2149:i
2145:x
2139:i
2130:{
2107:j
2104:i
2096:=
2091:j
2087:v
2083:A
2078:T
2073:i
2069:v
2046:n
2042:v
2038:,
2035:.
2032:.
2029:.
2026:,
2021:1
2017:v
1996:1
1993:=
1990:y
1987:A
1982:T
1978:y
1943:2
1939:v
1916:1
1912:v
1889:2
1885:v
1881:,
1876:1
1872:v
1830:.
1825:)
1819:1
1814:0
1809:0
1802:5
1797:1
1792:0
1785:4
1777:3
1772:1
1766:(
1759:)
1753:9
1748:0
1743:0
1736:0
1731:1
1726:0
1719:0
1714:0
1709:4
1703:(
1696:)
1690:1
1685:5
1680:4
1670:0
1665:1
1660:3
1653:0
1648:0
1643:1
1637:(
1632:=
1623:)
1561:4
1555:(
1521:.
1516:)
1510:3
1505:0
1500:0
1493:5
1488:1
1483:0
1476:8
1468:6
1463:2
1457:(
1450:)
1444:3
1439:5
1434:8
1424:0
1419:1
1414:6
1407:0
1402:0
1397:2
1391:(
1386:=
1381:)
1319:4
1313:(
1286:A
1276:n
1270:D
1264:A
1258:D
1236:L
1230:D
1225:L
1213:A
1207:D
1201:A
1184:.
1179:2
1174:S
1169:=
1165:D
1142:1
1134:S
1128:C
1124:=
1120:L
1092:L
1086:D
1081:L
1069:A
1053:C
1041:S
1019:C
1013:C
1009:=
1005:A
982:.
972:)
966:2
962:/
958:1
953:D
947:L
942:(
935:2
931:/
927:1
922:D
916:L
912:=
902:L
890:)
885:2
881:/
877:1
872:D
867:(
860:2
856:/
852:1
847:D
841:L
837:=
827:L
824:D
821:L
816:=
812:A
799:*
784:D
778:Λ
768:A
737:A
724:D
718:L
708:D
698:L
681:,
671:L
668:D
665:L
660:=
656:A
634:r
630:r
624:1
621:L
604:]
598:0
591:2
586:L
577:0
570:1
565:L
557:[
552:=
548:L
520:P
515:r
510:n
506:n
500:A
491:r
487:n
482:r
477:L
472:r
467:A
462:θ
448:,
443:]
413:0
408:0
402:[
397:=
393:L
387:,
377:L
371:L
367:=
362:]
356:1
351:0
344:0
339:0
333:[
320:L
315:*
309:A
303:A
289:L
274:,
268:T
262:L
259:L
254:=
250:A
238:A
229:A
223:L
218:*
210:A
201:L
191:L
181:L
164:,
154:L
151:L
146:=
142:A
128:A
62:/
59:i
56:k
53:s
50:ɛ
47:l
44:ˈ
41:ə
38:ʃ
35:/
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