7951:
7519:
5055:
7946:{\displaystyle \varepsilon ={\frac {\left\langle \displaystyle \sum _{i=1}^{N}c_{i}\Psi _{i}\right|{\hat {H}}\left|\displaystyle \sum _{i=1}^{N}c_{i}\Psi _{i}\right\rangle }{\left\langle \left.\displaystyle \sum _{i=1}^{N}c_{i}\Psi _{i}\right|\displaystyle \sum _{i=1}^{N}c_{i}\Psi _{i}\right\rangle }}={\frac {\displaystyle \sum _{i=1}^{N}\displaystyle \sum _{j=1}^{N}c_{i}^{*}c_{j}H_{ij}}{\displaystyle \sum _{i=1}^{N}\displaystyle \sum _{j=1}^{N}c_{i}^{*}c_{j}S_{ij}}}\equiv {\frac {A}{B}}.}
6504:
4712:
25:
5948:
6223:
10306:
5304:
7166:
6892:
5050:{\displaystyle A={\begin{bmatrix}0&0&0&1\\0&0&1&0\\0&1&0&0\\1&0&0&0\\0&0&0&0\end{bmatrix}}{\begin{bmatrix}4&0&0&0\\0&3&0&0\\0&0&2&0\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}0&0&0&1\\0&0&1&0\\0&1&0&0\\1&0&0&0\end{bmatrix}},}
5701:
2549:
10159:
5080:
3035:
10152:
6935:
6662:
6499:{\displaystyle \mathbf {V} _{h}={\begin{bmatrix}1/{\sqrt {2}}&-1/{\sqrt {2}}\\1/{\sqrt {2}}&1/{\sqrt {2}}\end{bmatrix}}\,{\begin{bmatrix}1/{\sqrt {2}}&1/{\sqrt {2}}&0&0\\1/{\sqrt {2}}&-1/{\sqrt {2}}&0&0\end{bmatrix}}={\begin{bmatrix}0&1&0&0\\1&0&0&0\end{bmatrix}}}
2361:
7246:
is the set of discrete energy levels allowed by a quantum mechanical system, the
Rayleigh–Ritz method is used to approximate the energy states and wavefunctions of a complicated atomic or nuclear system. In fact, for any system more complicated than a single hydrogen atom, there is no known exact
4524:
10401:
8244:
244:
on the other hand for numerical calculation of the solutions, by substituting for the variational problems simpler approximating extremum problems in which a finite number of parameters need to be determined. Ironically for the debate, the modern justification of the algorithm drops the
211:
wrote a paper congratulating Ritz on his work in 1911, but stating that he himself had used Ritz's method in many places in his book and in another publication. This statement, although later disputed, and the fact that the method in the trivial case of a single vector results in the
4666:
10876:
5943:{\displaystyle \mathbf {U} ={\begin{bmatrix}0&1\\1&0\\0&0\\0&0\\0&0\end{bmatrix}},\quad \Sigma ={\begin{bmatrix}2&0\\0&1\end{bmatrix}},\quad \mathbf {V} _{h}={\begin{bmatrix}1/{\sqrt {2}}&-1/{\sqrt {2}}\\1/{\sqrt {2}}&1/{\sqrt {2}}\end{bmatrix}}.}
2842:
7274:, is tested on the system. This trial function is selected to meet boundary conditions (and any other physical constraints). The exact function is not known; the trial function contains one or more adjustable parameters, which are varied to find a lowest energy configuration.
2873:
10023:
5441:
5645:
10301:{\displaystyle \mathbf {c} ^{\mathsf {T}}M\mathbf {c} {\frac {\partial \mathbf {c} ^{\mathsf {T}}K\mathbf {c} }{\partial \mathbf {c} }}-\mathbf {c} ^{\mathsf {T}}K\mathbf {c} {\frac {\partial \mathbf {c} ^{\mathsf {T}}M\mathbf {c} }{\partial \mathbf {c} }}=0}
8384:
5299:{\displaystyle {\begin{bmatrix}0&0&0&1\\0&0&1&0\\0&1&0&0\\1&0&0&0\end{bmatrix}}^{*}\quad =\quad {\begin{bmatrix}0&0&0&1\\0&0&1&0\\0&1&0&0\\1&0&0&0\end{bmatrix}}}
9606:
will be close to the lowest possible actual eigenmode of the system. Thus, this finds the lowest eigenfrequency. If we find eigenmodes orthogonal to this approximated lowest eigenmode, we can approximately find the next few eigenfrequencies as well.
7394:
11026:
may fit most of the easy problems of simply linked beams even if the order of the deformed solution may be lower. The springs and masses do not have to be discrete, they can be continuous (or a mixture), and this method can be easily used in a
7161:{\displaystyle {\begin{bmatrix}1&0&0&0&0\\0&2&0&0&0\\0&0&3&0&0\\0&0&0&4&0\end{bmatrix}}\,{\begin{bmatrix}0\\1\\0\\0\\0\end{bmatrix}}=\,2\,{\begin{bmatrix}0\\1\\0\\0\end{bmatrix}}.}
1655:
6887:{\displaystyle {\begin{bmatrix}1&0&0&0\\0&2&0&0\\0&0&3&0\\0&0&0&4\\0&0&0&0\end{bmatrix}}\,{\begin{bmatrix}0\\1\\0\\0\end{bmatrix}}=\,2\,{\begin{bmatrix}0\\1\\0\\0\\0\end{bmatrix}}}
875:
It is possible for the
Rayleigh–Ritz method to produce values which do not converge to actual values in the spectrum of the operator as the truncation gets large. These values are known as spectral pollution. In some cases (such as for the
4384:
9948:
9830:
2324:
10310:
8110:
7188:
with its column-space that is spanned by two exact right singular vectors, we determine these right singular vectors, as well as the corresponding left singular vectors and the singular values, all exactly. For an arbitrary matrix
4529:
10719:
5518:
2726:
2544:{\displaystyle \mathbf {x} _{\lambda =1}={\begin{bmatrix}0\\1\\-1\end{bmatrix}},\quad \mathbf {x} _{\lambda =2}={\begin{bmatrix}1\\0\\0\end{bmatrix}},\quad \mathbf {x} _{\lambda =3}={\begin{bmatrix}0\\1\\1\end{bmatrix}}.}
8980:
5312:
2623:
5526:
2695:
599:
2151:
1426:
10454:
1251:
425:
1309:
11034:
This method could be used iteratively, adding additional mode shapes to the previous best solution, or you can build up a long expression with many Bs and many mode shapes, and then differentiate them
8256:
5696:
3989:
10651:
8502:
8097:
8037:
7512:
7445:
3619:
3478:
3427:
937:
11031:
to find the natural frequencies of quite complex distributed systems, if you can describe the distributed KE and PE terms easily, or else break the continuous elements up into discrete parts.
9341:
8664:
8441:
7997:
3030:{\displaystyle \mathbf {\tilde {x}} _{{\tilde {\lambda }}=1}={\begin{bmatrix}0\\1\\-1\end{bmatrix}},\quad \mathbf {\tilde {x}} _{{\tilde {\lambda }}=3}={\begin{bmatrix}0\\1\\1\end{bmatrix}}.}
1358:
10147:{\displaystyle {\frac {\partial \omega ^{2}}{\partial c_{i}}}={\frac {\partial }{\partial c_{i}}}{\frac {\mathbf {c} ^{\mathsf {T}}K\mathbf {c} }{\mathbf {c} ^{\mathsf {T}}M\mathbf {c} }}=0}
7307:
3832:
3712:
3245:
1495:
1065:
978:
804:
9086:
6219:
4221:
642:
10711:
9604:
9518:
9459:
9400:
3189:
3154:
8761:
1878:
10507:
8792:. Next, find the total energy of the system, consisting of a kinetic energy term and a potential energy term. The kinetic energy term involves the square of the time derivative of
2085:
9834:
9716:
2243:
1979:
8621:
8594:
8567:
4138:
3559:
836:
671:
511:
6931:
4263:
3370:
762:
357:
4169:
719:
322:
10949:
8526:
8406:
8057:
5990:
4037:
3515:
10018:
9545:
8852:
6658:
3331:
1570:
888:
11152:
4109:
4063:
4015:
3911:
3858:
3742:
3671:
3645:
3271:
1776:
1710:
1099:
9260:
4371:
2178:
11015:, but we have found the lowest value of that upper bound, given our assumptions, because B is used to find the optimal 'mix' of the two assumed mode shape functions.
11009:
10989:
10969:
10905:
10585:
5968:
5446:
4704:
4083:
3299:
1937:
1809:
1521:
1132:
11128:
11094:
8825:
5077:, the diagonal entries of the middle term are the singular values, and the columns of the last multiplier transposed (although the transposition does not change it)
1024:
9711:
7302:
7272:
2356:
1159:
9224:
9119:
8790:
7398:
That is, the ground-state energy is less than this value. The trial wave-function will always give an expectation value larger than or equal to the ground-energy.
6626:
6575:
6124:
6067:
8857:
2868:
2721:
2029:
1904:
1684:
3934:
3881:
11172:
9988:
9968:
9684:
9646:
9195:
9157:
7227:
7207:
7186:
6524:
6164:
6144:
6010:
5075:
4331:
4307:
4283:
3119:
3095:
3075:
3055:
2554:
2230:
2202:
1999:
1750:
1730:
1565:
1541:
1450:
1179:
998:
860:
695:
482:
291:
10880:
The overall amplitude of the mode shape cancels out from each side, always. That is, the actual size of the assumed deflection does not matter, just the mode
10523:
The following discussion uses the simplest case, where the system has two lumped springs and two lumped masses, and only two mode shapes are assumed. Hence
2628:
518:
10405:
4519:{\displaystyle M={\begin{bmatrix}1&0&0&0\\0&2&0&0\\0&0&3&0\\0&0&0&4\\0&0&0&0\end{bmatrix}}}
3480:, whichever one is smaller size, one could determine the other set of left of right singular vectors simply by dividing by the singular values, i.e.,
11019:
10396:{\displaystyle K\mathbf {c} -{\frac {\mathbf {c} ^{\mathsf {T}}K\mathbf {c} }{\mathbf {c} ^{\mathsf {T}}M\mathbf {c} }}M\mathbf {c} =\mathbf {0} }
8239:{\displaystyle {\frac {\partial \varepsilon }{\partial c_{k}^{*}}}={\frac {\displaystyle \sum _{j=1}^{N}c_{j}(H_{kj}-\varepsilon S_{kj})}{B}}=0,}
11058:. In general, both of these problems are difficult to solve, but for the latter we can use the Ritz-Galerkin method to approximate a solution.
3372:. Having found one set (left of right) of approximate singular vectors and singular values by applying naively the Rayleigh–Ritz method to the
4265:, called the Ritz singular triplets, to selected singular values and the corresponding left and right singular vectors of the original matrix
4661:{\displaystyle A=M^{*}M={\begin{bmatrix}1&0&0&0\\0&4&0&0\\0&0&9&0\\0&0&0&16\\\end{bmatrix}},}
10871:{\displaystyle \sum _{i=1}^{2}\left({\frac {1}{2}}\omega ^{2}Y_{i}^{2}M_{i}\right)=\sum _{i=1}^{2}\left({\frac {1}{2}}K_{i}Y_{i}^{2}\right)}
11018:
There are many tricks with this method, the most important is to try and choose realistic assumed mode shapes. For example, in the case of
2090:
1365:
89:
8456:
7450:
2837:{\displaystyle \mathbf {y} _{\mu =1}={\begin{bmatrix}1\\-1\end{bmatrix}},\quad \mathbf {y} _{\mu =3}={\begin{bmatrix}1\\1\end{bmatrix}},}
61:
906:
9264:
8701:
Consider the case whereby we want to find the resonant frequency of oscillation of a system. First, write the oscillation in the form,
8683:
1186:
362:
11650:
3747:
42:
68:
8698:. It is an extension of Rayleigh's method. It can also be used for finding buckling loads and post-buckling behaviour for columns.
5436:{\displaystyle W={\begin{bmatrix}1/{\sqrt {2}}&1/{\sqrt {2}}\\1/{\sqrt {2}}&-1/{\sqrt {2}}\\0&0\\0&0\end{bmatrix}}}
1258:
8987:
8039:(the conjugation of the first) can be used to minimize the expectation value. For instance, by making the partial derivatives of
5640:{\displaystyle MW={\begin{bmatrix}1/{\sqrt {2}}&1/{\sqrt {2}}\\{\sqrt {2}}&-{\sqrt {2}}\\0&0\\0&0\end{bmatrix}},}
2212:(and thus its every Ritz value) is real and takes values within the closed interval of the smallest and largest eigenvalues of
8704:
5653:
75:
11564:
3945:
3883:. This interpretation allows simple simultaneous calculation of both left and right approximate singular vectors as follows.
183:. The Rayleigh–Ritz method or Ritz method terminology is typical in mechanical and structural engineering to approximate the
10461:
9197:, and in turn get the eigenfrequency. However, we do not yet know the mode shape. In order to find this, we can approximate
11598:
8062:
8002:
7415:
3567:
10594:
4286:
3199:
57:
3432:
3381:
11519:
Pokrzywa, Andrzej (1979). "Method of orthogonal projections and approximation of the spectrum of a bounded operator".
8634:
8411:
7967:
1314:
108:
5648:
4707:
3940:
8379:{\displaystyle \sum _{j=1}^{N}c_{j}\left(H_{kj}-\varepsilon S_{kj}\right)=0\quad {\text{for}}\quad k=1,2,\dots ,N.}
3676:
3209:
1459:
1029:
942:
770:
199:
The name of the method and its origin story have been debated by historians. It has been called Ritz method after
7243:
6172:
4174:
606:
168:
11442:
11392:
9550:
9464:
9405:
9346:
3159:
3124:
237:
46:
10664:
11482:
1817:
880:), there is no approximation which both includes all eigenvalues of the equation, and contains no pollution.
8984:
By conservation of energy, the average kinetic energy must be equal to the average potential energy. Thus,
5057:
where the columns of the first multiplier from the complete set of the left singular vectors of the matrix
2041:
1429:
160:
126:
11267:
457:
One could use the orthonormal basis generated from the eigenfunctions of the operator, which will produce
82:
11609:
7389:{\displaystyle E_{0}\leq {\frac {\langle \Psi |{\hat {H}}|\Psi \rangle }{\langle \Psi |\Psi \rangle }}.}
11655:
11501:
11384:
11220:
11193:
1942:
674:
149:
11022:
problems it is wise to use a deformed shape that is analytically similar to the expected solution. A
8599:
8572:
8545:
4114:
3520:
812:
647:
487:
6897:
4229:
3336:
735:
330:
4146:
700:
303:
10914:
10515:
eigenfrequencies and eigenmodes of the system, with N being the number of approximating functions.
10458:
For a non-trivial solution of c, we require determinant of the matrix coefficient of c to be zero.
8695:
1650:{\displaystyle \|A{\tilde {\mathbf {x} }}_{i}-{\tilde {\lambda }}_{i}{\tilde {\mathbf {x} }}_{i}\|}
842:
of orthonormal basis functions, and as a result they will be approximations of the eigenvectors of
8627:
energies are estimates of excited state energies. An approximation for the wave function of state
8511:
8391:
8042:
5973:
4020:
3483:
11035:
9996:
9523:
8830:
8623:, will be the best approximation to the ground state for the basis functions used. The remaining
6631:
3304:
729:
35:
877:
11137:
10557:
9461:, it will describe a superposition of the actual eigenmodes of the system. However, if we seek
8675:
4088:
4042:
3994:
3890:
3837:
3721:
3650:
3624:
3250:
1755:
1689:
1078:
725:
246:
241:
233:
130:
9229:
8854:. Thus, we can calculate the total energy of the system and express it in the following form:
4340:
2156:
461:
approximating matrices, but in this case we would have already had to calculate the spectrum.
11465:
11309:
Ilanko, Sinniah (2009). "Comments on the historical bases of the
Rayleigh and Ritz methods".
11237:
10994:
10974:
10954:
10890:
10570:
5953:
4671:
4068:
3284:
1909:
1781:
1500:
1104:
887:
of the operator; in many cases it is bounded by a subset of the numerical range known as the
250:
176:
11103:
11069:
9943:{\displaystyle A=\sum _{i}\sum _{j}c_{i}c_{j}M_{ij}=\mathbf {c} ^{\mathsf {T}}M\mathbf {c} }
9825:{\displaystyle B=\sum _{i}\sum _{j}c_{i}c_{j}K_{ij}=\mathbf {c} ^{\mathsf {T}}K\mathbf {c} }
8795:
1003:
11318:
11280:
11011:
is hoped to be the predicted fundamental frequency of the system because the mode shape is
9689:
7405:
to the ground state, then it will provide a boundary for the energy of some excited state.
7280:
7257:
2329:
2319:{\displaystyle A={\begin{bmatrix}2&0&0\\0&2&1\\0&1&2\end{bmatrix}}}
1137:
439:
262:
180:
153:
9200:
9095:
8766:
6580:
6529:
6072:
6015:
8:
11221:"Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik"
2847:
2700:
2008:
1883:
1663:
839:
11322:
11284:
3916:
3863:
11157:
9973:
9953:
9651:
9613:
9162:
9124:
8687:
8679:
8529:
7212:
7192:
7171:
6509:
6149:
6129:
5995:
5060:
4316:
4292:
4268:
3104:
3097:. A mathematical explanation for the exact approximation is based on the fact that the
3080:
3060:
3040:
2215:
2187:
1984:
1735:
1715:
1550:
1526:
1435:
1164:
983:
845:
680:
467:
276:
1567:
Ritz vectors that well approximate these eigenvectors. The easily computable quantity
809:
If a subset of the orthonormal basis was used to find the matrix, the eigenvectors of
11560:
11438:
11388:
11188:
11183:
10565:
10544:
is assumed for the system, with two terms, one of which is weighted by a factor
9089:
3273:
in given subspaces by turning the singular value problem into an eigenvalue problem.
2209:
2181:
2035:
2002:
1812:
432:
213:
164:
11629:
11461:
11326:
11288:
11051:
11047:
11023:
10654:
8539:
8250:
3373:
3037:
We observe that each one of the Ritz vectors is exactly one of the eigenvectors of
2205:
765:
11554:
11413:"Spectral pollution and second order relative spectra for self-adjoint operators"
11351:
11203:
10658:
884:
458:
443:
294:
254:
221:
145:
11617:
5513:{\displaystyle {\begin{bmatrix}0&1\\1&0\\0&0\\0&0\end{bmatrix}}}
3561:. However, the division is unstable or fails for small or zero singular values.
10716:
We also know that without damping, the maximal KE equals the maximal PE. Thus,
10588:
8449:, this is a homogeneous set of linear equations, which has a solution when the
7956:
5443:
with the column-space that is spanned by the two exact right singular vectors
3121:
happens to be exactly the same as the subspace spanned by the two eigenvectors
450:
258:
11330:
11292:
1456:
If the subspace with the orthonormal basis given by the columns of the matrix
11644:
11604:
11198:
11055:
7402:
325:
298:
225:
208:
175:
to approximate the ground state eigenfunction with the lowest energy. In the
134:
11376:
10587:
times the deflection (y) at time of maximum deflection. In this example the
9990:
are the stiffness matrix and mass matrix of a discrete system respectively.
8975:{\displaystyle E=T+V\equiv A\omega ^{2}\sin ^{2}\omega t+B\cos ^{2}\omega t}
11216:
3098:
229:
204:
200:
138:
2618:{\displaystyle V={\begin{bmatrix}0&0\\1&0\\0&1\end{bmatrix}},}
11100:
of the
Hamiltonian in the piecewise linear element space, and the matrix
11028:
9402:
are constants to be determined. In general, if we choose a random set of
8450:
3715:
1068:
184:
4289:
with left singular vectors restricted to the column-space of the matrix
3206:
to find approximations to left and right singular vectors of the matrix
3077:
as well as the Ritz values give exactly two of the three eigenvalues of
11599:
Course on
Calculus of Variations, has a section on Rayleigh–Ritz method
10908:
10541:
4337:, approximating numerically selected eigenvectors of the normal matrix
883:
The spectrum of the compression (and thus pollution) is bounded by the
11633:
2690:{\displaystyle V^{*}AV={\begin{bmatrix}2&1\\1&2\end{bmatrix}}}
1657:
determines the accuracy of such an approximation for every Ritz pair.
594:{\displaystyle (T_{\mathcal {L}})_{i,j}=(T\varphi _{i},\varphi _{j}).}
232:
independently conceived the idea of utilizing the equivalence between
11412:
11355:
4313:
The algorithm can be used as a post-processing step where the matrix
2146:{\displaystyle ({\tilde {\lambda }}_{i},{\tilde {\mathbf {x} }}_{i})}
1421:{\displaystyle ({\tilde {\lambda }}_{i},{\tilde {\mathbf {x} }}_{i})}
188:
24:
11581:
11535:
10561:
10449:{\displaystyle K\mathbf {c} -\omega ^{2}M\mathbf {c} =\mathbf {0} }
8691:
3718:
columns the eigenvalue problem of the
Rayleigh–Ritz method for the
2001:
itself. Thus, the
Rayleigh–Ritz method turns into computing of the
11061:
7209:, we obtain approximate singular triplets which are optimal given
179:
context, mathematically the same algorithm is commonly called the
11238:"Successive Approximations by the Rayleigh-Ritz Variation Method"
1071:
columns. The matrix version of the algorithm is the most simple:
11154:
is an eigenvalue of the discretized
Hamiltonian, and the vector
1246:{\displaystyle V^{*}AV\mathbf {y} _{i}=\mu _{i}\mathbf {y} _{i}}
420:{\displaystyle {\mathcal {L}}=\{\varphi _{1},...,\varphi _{n}\}}
7251:
4334:
172:
11536:"The essential numerical range for unbounded linear operators"
7516:
With a known
Hamiltonian, we can write its expected value as
3564:
An alternative approach, e.g., defining the normal matrix as
1304:{\displaystyle {\tilde {\mathbf {x} }}_{i}=V\mathbf {y} _{i}}
11534:
Bögli, Sabine; Marletta, Marco; Tretter, Christiane (2020).
7955:
The basis functions are usually not orthogonal, so that the
6146:, explaining why the approximations are exact for the given
10518:
7653:
903:
is commonly applied to approximate an eigenvalue problem
159:
It is used in all applications that involve approximating
11582:"A Koopman Operator Tutorial with Orthogonal Polynomials"
11500:
Colbrook, Matthew; Roman, Bogdan; Hansen, Anders (2019).
11066:
In the language of the finite element method, the matrix
10887:
Mathematical manipulations then obtain an expression for
3301:
and the corresponding left and right singular vectors is
7229:
in the sense of optimality of the
Rayleigh–Ritz method.
6506:
recovering from its rows the two right singular vectors
5950:
Thus we already obtain the singular values 2 and 1 from
5691:{\displaystyle MW=\mathbf {U} {\Sigma }\mathbf {V} _{h}}
11268:"The historical bases of the Rayleigh and Ritz methods"
9686:
as a collection of terms quadratic in the coefficients
3984:{\displaystyle MW=\mathbf {U} \Sigma \mathbf {V} _{h},}
3834:
can be interpreted as a singular value problem for the
427:. Depending on the application these functions may be:
10667:
10597:
7120:
7065:
6944:
6842:
6794:
6671:
6445:
6335:
6247:
5855:
5800:
5718:
5544:
5455:
5327:
5201:
5090:
4949:
4849:
4727:
4560:
4399:
2996:
2919:
2810:
2756:
2656:
2569:
2510:
2452:
2391:
2258:
203:, since the numerical procedure has been published by
11406:
11404:
11160:
11140:
11106:
11072:
10997:
10977:
10957:
10917:
10893:
10722:
10573:
10464:
10408:
10313:
10162:
10026:
9999:
9976:
9956:
9837:
9719:
9692:
9654:
9616:
9553:
9547:
is minimised, then the mode described by this set of
9526:
9467:
9408:
9349:
9267:
9232:
9203:
9165:
9127:
9098:
8990:
8860:
8833:
8798:
8769:
8707:
8637:
8602:
8575:
8548:
8514:
8459:
8414:
8394:
8259:
8151:
8113:
8065:
8045:
8005:
7970:
7863:
7841:
7779:
7757:
7702:
7655:
7599:
7536:
7522:
7453:
7418:
7310:
7283:
7260:
7215:
7195:
7174:
6938:
6900:
6665:
6634:
6583:
6532:
6512:
6226:
6175:
6152:
6132:
6075:
6018:
5998:
5976:
5956:
5704:
5656:
5529:
5449:
5315:
5083:
5063:
4715:
4674:
4532:
4387:
4343:
4319:
4295:
4271:
4232:
4177:
4149:
4117:
4091:
4071:
4045:
4023:
3997:
3948:
3919:
3893:
3866:
3840:
3750:
3724:
3679:
3653:
3627:
3570:
3523:
3486:
3435:
3384:
3339:
3307:
3287:
3253:
3212:
3194:
3162:
3127:
3107:
3083:
3063:
3043:
2876:
2850:
2729:
2703:
2631:
2557:
2364:
2332:
2246:
2218:
2190:
2159:
2093:
2044:
2011:
1987:
1945:
1912:
1886:
1820:
1784:
1758:
1738:
1718:
1692:
1666:
1573:
1553:
1529:
1523:
vectors that are close to eigenvectors of the matrix
1503:
1462:
1438:
1368:
1317:
1261:
1189:
1167:
1140:
1107:
1081:
1032:
1006:
986:
945:
909:
848:
815:
773:
738:
703:
683:
650:
609:
521:
490:
470:
365:
333:
306:
279:
249:
in favor of the simpler and more general approach of
11618:"From Euler, Ritz, and Galerkin to Modern Computing"
11533:
11483:"Unscrambling the Infinite: Can we Compute Spectra?"
10646:{\textstyle {\frac {1}{2}}\omega ^{2}Y_{1}^{2}m_{1}}
8497:{\displaystyle \det \left(H-\varepsilon S\right)=0,}
8092:{\displaystyle \left\lbrace c_{i}^{*}\right\rbrace }
8032:{\displaystyle \left\lbrace c_{i}^{*}\right\rbrace }
7507:{\displaystyle \Psi =\sum _{i=1}^{N}c_{i}\Psi _{i}.}
7440:{\displaystyle \left\lbrace \Psi _{i}\right\rbrace }
7408:
The Ritz ansatz function is a linear combination of
4333:
is an output of an eigenvalue solver, e.g., such as
3614:{\displaystyle A=M^{*}M\in \mathbb {C} ^{N\times N}}
2153:, allowing to derive some properties of Ritz values
11499:
11410:
8099:zero, the following equality is obtained for every
167:, where a system of particles is described using a
49:. Unsourced material may be challenged and removed.
11580:Servadio, Simone; Arnas, David; Linares, Richard.
11401:
11266:
11166:
11146:
11122:
11088:
11003:
10983:
10963:
10943:
10911:with respect to B, to find the minimum, i.e. when
10899:
10870:
10705:
10645:
10579:
10501:
10448:
10395:
10300:
10146:
10012:
9982:
9962:
9942:
9824:
9705:
9678:
9640:
9598:
9539:
9512:
9453:
9394:
9335:
9254:
9226:as a combination of a few approximating functions
9218:
9189:
9151:
9113:
9080:
8974:
8846:
8819:
8784:
8755:
8658:
8615:
8588:
8561:
8520:
8496:
8435:
8400:
8378:
8238:
8091:
8051:
8031:
7991:
7945:
7506:
7439:
7388:
7296:
7266:
7221:
7201:
7180:
7160:
6925:
6886:
6652:
6620:
6569:
6518:
6498:
6213:
6158:
6138:
6118:
6061:
6004:
5984:
5962:
5942:
5690:
5639:
5512:
5435:
5298:
5069:
5049:
4698:
4660:
4518:
4365:
4325:
4301:
4277:
4257:
4215:
4163:
4132:
4103:
4077:
4057:
4031:
4009:
3983:
3928:
3905:
3875:
3852:
3826:
3736:
3706:
3665:
3639:
3613:
3553:
3509:
3473:{\displaystyle MM^{*}\in \mathbb {C} ^{M\times M}}
3472:
3422:{\displaystyle M^{*}M\in \mathbb {C} ^{N\times N}}
3421:
3364:
3325:
3293:
3265:
3239:
3183:
3148:
3113:
3089:
3069:
3049:
3029:
2862:
2836:
2715:
2689:
2617:
2543:
2350:
2318:
2224:
2196:
2172:
2145:
2079:
2023:
1993:
1973:
1931:
1898:
1872:
1803:
1770:
1744:
1724:
1704:
1678:
1649:
1559:
1535:
1515:
1489:
1444:
1420:
1352:
1303:
1245:
1173:
1153:
1126:
1093:
1059:
1018:
992:
972:
932:{\displaystyle A\mathbf {x} =\lambda \mathbf {x} }
931:
854:
830:
798:
756:
713:
689:
665:
636:
593:
505:
476:
419:
351:
316:
285:
11552:
9336:{\displaystyle Y(x)=\sum _{i=1}^{N}c_{i}Y_{i}(x)}
2961:
2884:
894:
11642:
11579:
11546:
11489:. Institute of Mathematics and its Applications.
10465:
8659:{\displaystyle \left\lbrace c_{j}\right\rbrace }
8460:
8436:{\displaystyle \left\lbrace c_{j}\right\rbrace }
7992:{\displaystyle \left\lbrace c_{i}\right\rbrace }
1353:{\displaystyle {\tilde {\lambda }}_{i}=\mu _{i}}
129:, originated in the context of solving physical
11411:Levitin, Michael; Shargorodsky, Eugene (2004).
11225:Journal für die Reine und Angewandte Mathematik
11134:. In the language of linear algebra, the value
11062:The relationship with the finite element method
10971:is lowest. This is an upper bound solution for
8453:of the coefficients to these unknowns is zero:
3827:{\displaystyle W^{*}AW=W^{*}M^{*}MW=(MW)^{*}MW}
3647:, takes advantage of the fact that for a given
11435:Numerical Solution of Sturm-Liouville Problems
7277:It can be shown that the ground state energy,
7247:solution for the spectrum of the Hamiltonian.
7242:In quantum physics, where the spectrum of the
5520:corresponding to the singular values 1 and 2.
5306:are the corresponding right singular vectors.
513:, which is defined as the matrix with entries
125:is a direct numerical method of approximating
11553:Trefethen, Lloyd N.; Bau, III, David (1997).
11456:
11454:
3707:{\displaystyle W\in \mathbb {C} ^{N\times m}}
3240:{\displaystyle M\in \mathbb {C} ^{M\times N}}
3202:in numerical linear algebra can also use the
1490:{\displaystyle V\in \mathbb {C} ^{N\times m}}
1060:{\displaystyle V\in \mathbb {C} ^{N\times m}}
973:{\displaystyle A\in \mathbb {C} ^{N\times N}}
799:{\displaystyle {\mathcal {A}}(\cdot ,\cdot )}
11615:
11371:
11369:
11346:
11344:
11342:
11340:
10564:at the time when deflection is zero, is the
9081:{\displaystyle \omega ^{2}={\frac {B}{A}}=R}
8669:
8631:can be obtained by finding the coefficients
7377:
7363:
7358:
7327:
7232:
6126:, which span the column-space of the matrix
5992:the corresponding two left singular vectors
4287:Truncated singular value decomposition (SVD)
3200:Truncated singular value decomposition (SVD)
1644:
1574:
414:
376:
11616:Gander, Martin J.; Wanner, Gerhard (2012).
11502:"How to Compute Spectra with Error Control"
11460:
6214:{\displaystyle V_{h}=\mathbf {V} _{h}W^{*}}
5523:Following the algorithm step 1, we compute
4216:{\displaystyle V_{h}=\mathbf {V} _{h}W^{*}}
1161:denotes the complex-conjugate transpose of
1000:using a projected matrix of a smaller size
637:{\displaystyle T_{\mathcal {L}}u=\lambda u}
11451:
11375:
11350:
8674:The Rayleigh–Ritz method is often used in
8528:. Furthermore, since the Hamiltonian is a
7401:If the trial wave function is known to be
3276:
11366:
11337:
11235:
11054:to be encoded as an infinite-dimensional
10706:{\textstyle {\frac {1}{2}}k_{1}Y_{1}^{2}}
9599:{\displaystyle c_{1},c_{2},\cdots ,c_{N}}
9513:{\displaystyle c_{1},c_{2},\cdots ,c_{N}}
9454:{\displaystyle c_{1},c_{2},\cdots ,c_{N}}
9395:{\displaystyle c_{1},c_{2},\cdots ,c_{N}}
8666:from the corresponding secular equation.
7964:has nonzero nondiagonal elements. Either
7114:
7110:
7059:
6836:
6832:
6788:
6329:
3688:
3595:
3454:
3403:
3221:
3184:{\displaystyle \mathbf {x} _{\lambda =3}}
3149:{\displaystyle \mathbf {x} _{\lambda =1}}
1471:
1041:
954:
449:A set of functions which approximate the
359:. Now consider a finite set of functions
207:in 1908-1909. According to A. W. Leissa,
194:
109:Learn how and when to remove this message
11518:
11304:
11302:
11260:
11258:
10519:Simple case of double spring-mass system
8756:{\displaystyle y(x,t)=Y(x)\cos \omega t}
7447:, parametrized by unknown coefficients:
148:is approximated by a finite-dimensional
144:In this method, an infinite-dimensional
11428:
11426:
11209:
11041:
1873:{\displaystyle \rho (v)=v^{*}Av/v^{*}v}
220:method. According to S. Ilanko, citing
11643:
11308:
11264:
10951:. This gives the value of B for which
10502:{\displaystyle \det(K-\omega ^{2}M)=0}
10360:
10336:
10265:
10237:
10199:
10171:
10121:
10097:
9926:
9808:
4143:Compute the matrices of the Ritz left
2180:from the corresponding theory for the
1906:solution to the eigenvalue problem is
11432:
11381:An Introduction to Numerical Analysis
11299:
11255:
8569:will be real. The lowest value among
7237:
3281:The definition of the singular value
2080:{\displaystyle \mu _{i}=\rho (v_{i})}
870:
11480:
11423:
11215:
10511:This gives a solution for the first
6169:Finally, step 3 computes the matrix
240:on the one hand and problems of the
47:adding citations to reliable sources
18:
16:Method for approximating eigenvalues
11467:Mathematical Methods For Physicists
2723:and the corresponding eigenvectors
2358:and the corresponding eigenvectors
13:
10281:
10254:
10215:
10188:
10070:
10066:
10045:
10030:
9092:. Thus, if we knew the mode shape
8125:
8117:
7735:
7688:
7632:
7569:
7492:
7454:
7424:
7374:
7366:
7355:
7330:
7261:
5957:
5789:
5672:
4239:
4072:
3963:
3195:For matrix singular value problems
1981:, and the only one Ritz vector is
822:
776:
706:
657:
644:. It can be shown that the matrix
616:
531:
497:
368:
309:
163:, often under different names. In
14:
11667:
11592:
11417:IMA Journal of Numerical Analysis
11360:IMA Journal of Numerical Analysis
8678:for finding the approximate real
2034:Another useful connection to the
1974:{\displaystyle \mu _{i}=\rho (v)}
1811:is a scalar that is equal to the
899:In numerical linear algebra, the
603:and solve the eigenvalue problem
11651:Numerical differential equations
10442:
10434:
10413:
10389:
10381:
10370:
10354:
10346:
10330:
10318:
10285:
10275:
10259:
10247:
10231:
10219:
10209:
10193:
10181:
10165:
10131:
10115:
10107:
10091:
9936:
9920:
9818:
9802:
9121:, we would be able to calculate
8616:{\displaystyle \varepsilon _{0}}
8589:{\displaystyle \varepsilon _{i}}
8562:{\displaystyle \varepsilon _{i}}
6628:. We validate the first vector:
6229:
6191:
5978:
5837:
5706:
5678:
5667:
4193:
4157:
4133:{\displaystyle \mathbf {V} _{h}}
4120:
4025:
3968:
3959:
3554:{\displaystyle v=M^{*}u/\sigma }
3165:
3130:
2958:
2881:
2786:
2732:
2486:
2428:
2367:
2124:
1732:turns into a unit column-vector
1628:
1585:
1399:
1291:
1267:
1233:
1208:
1026:, generated from a given matrix
925:
914:
831:{\displaystyle T_{\mathcal {L}}}
666:{\displaystyle T_{\mathcal {L}}}
506:{\displaystyle T_{\mathcal {L}}}
23:
11573:
11527:
11512:
11470:(6th ed.). Academic Press.
8504:which in turn is true only for
8388:In the above equations, energy
8345:
8339:
6926:{\displaystyle M^{*}u=\sigma v}
5834:
5788:
5195:
5191:
4258:{\displaystyle U,\Sigma ,V_{h}}
3365:{\displaystyle M^{*}u=\sigma v}
2953:
2783:
2483:
2425:
757:{\displaystyle (\cdot ,\cdot )}
352:{\displaystyle (\cdot ,\cdot )}
34:needs additional citations for
11540:Journal of Functional Analysis
11493:
11474:
11311:Journal of Sound and Vibration
11273:Journal of Sound and Vibration
11174:is a discretized eigenvector.
10907:, in terms of B, which can be
10490:
10468:
9856:
9853:
9847:
9841:
9738:
9735:
9729:
9723:
9673:
9670:
9664:
9658:
9635:
9632:
9626:
9620:
9330:
9324:
9277:
9271:
9249:
9243:
9213:
9207:
9184:
9181:
9175:
9169:
9146:
9143:
9137:
9131:
9108:
9102:
9075:
9072:
9066:
9060:
9048:
9045:
9039:
9033:
9025:
9022:
9016:
9010:
8950:
8947:
8941:
8935:
8897:
8894:
8888:
8882:
8814:
8802:
8779:
8773:
8738:
8732:
8723:
8711:
8218:
8183:
7589:
7370:
7351:
7344:
7334:
6609:
6584:
6558:
6533:
6107:
6076:
6050:
6019:
4164:{\displaystyle U=\mathbf {U} }
3809:
3799:
2974:
2897:
2140:
2128:
2104:
2094:
2074:
2061:
1968:
1962:
1830:
1824:
1632:
1611:
1589:
1415:
1403:
1379:
1369:
1325:
1271:
895:For matrix eigenvalue problems
793:
781:
751:
739:
714:{\displaystyle {\mathcal {L}}}
585:
556:
538:
522:
346:
334:
317:{\displaystyle {\mathcal {H}}}
238:partial differential equations
1:
10944:{\displaystyle d\omega /dB=0}
9520:such that the eigenfrequency
8443:are unknown. With respect to
1183:Solve the eigenvalue problem
865:
11194:Sturm–Liouville theory
11050:allows a finite-dimensional
8521:{\displaystyle \varepsilon }
8401:{\displaystyle \varepsilon }
8052:{\displaystyle \varepsilon }
5985:{\displaystyle \mathbf {U} }
4376:
4285:representing an approximate
4032:{\displaystyle \mathbf {U} }
3510:{\displaystyle u=Mv/\sigma }
2844:so that the Ritz values are
1430:eigenvalues and eigenvectors
1428:, called the Ritz pairs, to
161:eigenvalues and eigenvectors
7:
11610:Encyclopedia of Mathematics
11437:. Oxford University Press.
11177:
10013:{\displaystyle \omega ^{2}}
9610:In general, we can express
9540:{\displaystyle \omega ^{2}}
9088:which is also known as the
8847:{\displaystyle \omega ^{2}}
8827:and thus gains a factor of
8763:with an unknown mode shape
7304:, satisfies an inequality:
7168:Thus, for the given matrix
6653:{\displaystyle Mv=\sigma u}
3941:thin, or economy-sized, SVD
3326:{\displaystyle Mv=\sigma u}
261:, thus leading also to the
216:make the case for the name
10:
11672:
11385:Cambridge University Press
2235:
11464:; Weber, Hans J. (2005).
11331:10.1016/j.jsv.2008.06.001
11293:10.1016/j.jsv.2004.12.021
11236:MacDonald, J. K. (1933).
11147:{\displaystyle \epsilon }
8670:In mechanical engineering
7233:Applications and examples
4104:{\displaystyle m\times m}
4058:{\displaystyle m\times m}
4010:{\displaystyle N\times m}
3906:{\displaystyle N\times m}
3853:{\displaystyle N\times m}
3737:{\displaystyle m\times m}
3666:{\displaystyle N\times m}
3640:{\displaystyle N\times N}
3266:{\displaystyle M\times N}
2870:and the Ritz vectors are
1771:{\displaystyle m\times m}
1705:{\displaystyle N\times m}
1255:Compute the Ritz vectors
1094:{\displaystyle m\times m}
889:essential numerical range
730:Sturm-Liouville operators
435:of the original operator;
268:
152:, on which we can use an
11556:Numerical Linear Algebra
11379:; Mayers, David (2003).
9255:{\displaystyle Y_{i}(x)}
8694:on a shaft with varying
8246:which leads to a set of
4366:{\displaystyle A=M^{*}M}
2173:{\displaystyle \mu _{i}}
11506:Physical Review Letters
11433:Pryce, John D. (1994).
11004:{\displaystyle \omega }
10984:{\displaystyle \omega }
10964:{\displaystyle \omega }
10900:{\displaystyle \omega }
10580:{\displaystyle \omega }
5963:{\displaystyle \Sigma }
4699:{\displaystyle 1,2,3,4}
4526:has its normal matrix
4078:{\displaystyle \Sigma }
3294:{\displaystyle \sigma }
3277:Using the normal matrix
1932:{\displaystyle y_{i}=1}
1804:{\displaystyle V^{*}AV}
1516:{\displaystyle k\leq m}
1432:of the original matrix
1127:{\displaystyle V^{*}AV}
764:can be replaced by the
234:boundary value problems
171:, the Ritz method uses
131:boundary value problems
11168:
11148:
11124:
11123:{\displaystyle S_{kj}}
11090:
11089:{\displaystyle H_{kj}}
11005:
10985:
10965:
10945:
10901:
10872:
10822:
10743:
10707:
10647:
10591:(KE) for each mass is
10581:
10558:Simple harmonic motion
10503:
10450:
10397:
10302:
10148:
10014:
9984:
9964:
9944:
9826:
9707:
9680:
9642:
9600:
9541:
9514:
9455:
9396:
9337:
9303:
9256:
9220:
9191:
9153:
9115:
9082:
8976:
8848:
8821:
8820:{\displaystyle y(x,t)}
8786:
8757:
8676:mechanical engineering
8660:
8617:
8590:
8563:
8522:
8498:
8437:
8402:
8380:
8280:
8240:
8172:
8093:
8053:
8033:
7993:
7947:
7884:
7862:
7800:
7778:
7723:
7676:
7620:
7557:
7508:
7480:
7441:
7412:known basis functions
7390:
7298:
7268:
7223:
7203:
7182:
7162:
6927:
6888:
6654:
6622:
6571:
6520:
6500:
6215:
6160:
6140:
6120:
6063:
6006:
5986:
5964:
5944:
5692:
5641:
5514:
5437:
5300:
5071:
5051:
4706:and the corresponding
4700:
4662:
4520:
4367:
4327:
4303:
4279:
4259:
4226:Output approximations
4217:
4165:
4134:
4105:
4079:
4059:
4033:
4011:
3985:
3930:
3907:
3877:
3854:
3828:
3738:
3708:
3667:
3641:
3615:
3555:
3511:
3474:
3423:
3366:
3327:
3295:
3267:
3241:
3185:
3150:
3115:
3091:
3071:
3051:
3031:
2864:
2838:
2717:
2691:
2619:
2545:
2352:
2320:
2226:
2198:
2174:
2147:
2081:
2025:
1995:
1975:
1933:
1900:
1874:
1805:
1772:
1746:
1726:
1706:
1680:
1651:
1561:
1537:
1517:
1491:
1446:
1422:
1362:Output approximations
1354:
1305:
1247:
1175:
1155:
1128:
1095:
1061:
1020:
1019:{\displaystyle m<N}
994:
974:
933:
856:
832:
800:
758:
726:differential operators
715:
691:
667:
638:
595:
507:
478:
421:
353:
318:
287:
247:calculus of variations
242:calculus of variations
195:Naming and attribution
58:"Rayleigh–Ritz method"
11559:. SIAM. p. 254.
11265:Leissa, A.W. (2005).
11169:
11149:
11125:
11091:
11006:
10986:
10966:
10946:
10902:
10873:
10802:
10723:
10708:
10648:
10582:
10560:theory says that the
10504:
10451:
10398:
10303:
10149:
10015:
9985:
9965:
9945:
9827:
9708:
9706:{\displaystyle c_{i}}
9681:
9643:
9601:
9542:
9515:
9456:
9397:
9338:
9283:
9257:
9221:
9192:
9154:
9116:
9083:
8977:
8849:
8822:
8787:
8758:
8661:
8618:
8591:
8564:
8523:
8499:
8438:
8408:and the coefficients
8403:
8381:
8260:
8241:
8152:
8094:
8054:
8034:
7994:
7948:
7864:
7842:
7780:
7758:
7703:
7656:
7600:
7537:
7509:
7460:
7442:
7391:
7299:
7297:{\displaystyle E_{0}}
7269:
7267:{\displaystyle \Psi }
7224:
7204:
7183:
7163:
6928:
6889:
6655:
6623:
6572:
6521:
6501:
6216:
6161:
6141:
6121:
6064:
6007:
5987:
5965:
5945:
5693:
5642:
5515:
5438:
5301:
5072:
5052:
4701:
4663:
4521:
4368:
4328:
4304:
4280:
4260:
4218:
4166:
4135:
4106:
4080:
4060:
4034:
4012:
3986:
3931:
3908:
3878:
3855:
3829:
3739:
3709:
3668:
3642:
3616:
3556:
3512:
3475:
3424:
3367:
3328:
3296:
3268:
3242:
3186:
3151:
3116:
3092:
3072:
3052:
3032:
2865:
2839:
2718:
2692:
2620:
2546:
2353:
2351:{\displaystyle 1,2,3}
2321:
2227:
2199:
2175:
2148:
2082:
2026:
1996:
1976:
1934:
1901:
1875:
1806:
1773:
1747:
1727:
1707:
1681:
1652:
1562:
1538:
1518:
1492:
1447:
1423:
1355:
1306:
1248:
1176:
1156:
1154:{\displaystyle V^{*}}
1129:
1096:
1062:
1021:
995:
975:
934:
857:
833:
801:
759:
732:), the inner product
716:
692:
668:
639:
596:
508:
479:
422:
354:
319:
288:
251:orthogonal projection
177:finite element method
11356:"Spectral Pollution"
11210:Notes and references
11158:
11138:
11104:
11070:
11042:In dynamical systems
10995:
10975:
10955:
10915:
10891:
10720:
10665:
10595:
10571:
10552:= +
10462:
10406:
10311:
10160:
10024:
9997:
9993:The minimization of
9974:
9954:
9835:
9717:
9690:
9652:
9614:
9551:
9524:
9465:
9406:
9347:
9265:
9230:
9219:{\displaystyle Y(x)}
9201:
9163:
9125:
9114:{\displaystyle Y(x)}
9096:
8988:
8858:
8831:
8796:
8785:{\displaystyle Y(x)}
8767:
8705:
8680:resonant frequencies
8635:
8600:
8573:
8546:
8512:
8457:
8412:
8392:
8257:
8111:
8063:
8043:
8003:
7968:
7520:
7451:
7416:
7308:
7281:
7258:
7213:
7193:
7172:
6936:
6898:
6663:
6632:
6621:{\displaystyle ^{*}}
6581:
6570:{\displaystyle ^{*}}
6530:
6510:
6224:
6173:
6150:
6130:
6119:{\displaystyle ^{*}}
6073:
6062:{\displaystyle ^{*}}
6016:
5996:
5974:
5954:
5702:
5654:
5527:
5447:
5313:
5081:
5061:
4713:
4672:
4530:
4385:
4341:
4317:
4293:
4269:
4230:
4175:
4147:
4115:
4089:
4069:
4043:
4021:
3995:
3946:
3917:
3891:
3864:
3838:
3748:
3722:
3677:
3651:
3625:
3568:
3521:
3484:
3433:
3382:
3337:
3305:
3285:
3251:
3210:
3204:Rayleigh–Ritz method
3160:
3125:
3105:
3081:
3061:
3041:
2874:
2848:
2727:
2701:
2629:
2555:
2362:
2330:
2244:
2216:
2188:
2157:
2091:
2087:for every Ritz pair
2042:
2009:
1985:
1943:
1910:
1884:
1818:
1782:
1756:
1736:
1716:
1690:
1664:
1660:In the easiest case
1571:
1551:
1545:Rayleigh–Ritz method
1527:
1501:
1460:
1436:
1366:
1315:
1259:
1187:
1165:
1138:
1105:
1079:
1030:
1004:
984:
943:
907:
901:Rayleigh–Ritz method
878:Schrödinger equation
846:
813:
771:
736:
701:
681:
648:
607:
519:
488:
468:
363:
331:
304:
277:
263:Ritz-Galerkin method
189:resonant frequencies
181:Ritz-Galerkin method
173:trial wave functions
154:eigenvalue algorithm
123:Rayleigh–Ritz method
43:improve this article
11481:Colbrook, Matthew.
11354:; Plum, M. (2003).
11323:2009JSV...319..731I
11285:2005JSV...287..961L
10862:
10783:
10702:
10632:
8688:spring mass systems
8142:
8084:
8024:
7899:
7815:
7252:trial wave function
2863:{\displaystyle 1,3}
2716:{\displaystyle 1,3}
2024:{\displaystyle m=1}
1899:{\displaystyle i=1}
1679:{\displaystyle m=1}
1311:and the Ritz value
840:linear combinations
464:We now approximate
11521:Studia Mathematica
11164:
11144:
11120:
11086:
11001:
10981:
10961:
10941:
10897:
10868:
10848:
10769:
10703:
10688:
10643:
10618:
10577:
10499:
10446:
10393:
10298:
10144:
10010:
9980:
9960:
9940:
9881:
9871:
9822:
9763:
9753:
9703:
9676:
9638:
9596:
9537:
9510:
9451:
9392:
9333:
9252:
9216:
9187:
9149:
9111:
9078:
8972:
8844:
8817:
8782:
8753:
8656:
8613:
8586:
8559:
8542:and the values of
8530:hermitian operator
8518:
8494:
8433:
8398:
8376:
8236:
8221:
8128:
8089:
8070:
8049:
8029:
8010:
7989:
7943:
7924:
7923:
7885:
7840:
7839:
7801:
7744:
7697:
7641:
7578:
7504:
7437:
7386:
7294:
7264:
7238:In quantum physics
7219:
7199:
7178:
7158:
7149:
7101:
7053:
6923:
6884:
6878:
6823:
6782:
6650:
6618:
6567:
6516:
6496:
6490:
6431:
6323:
6211:
6156:
6136:
6116:
6059:
6002:
5982:
5960:
5940:
5931:
5825:
5779:
5688:
5647:and on step 2 its
5637:
5628:
5510:
5504:
5433:
5427:
5296:
5290:
5179:
5067:
5047:
5038:
4938:
4838:
4696:
4658:
4649:
4516:
4510:
4363:
4323:
4299:
4275:
4255:
4213:
4161:
4130:
4101:
4075:
4055:
4029:
4007:
3981:
3929:{\displaystyle MW}
3926:
3903:
3876:{\displaystyle MW}
3873:
3850:
3824:
3734:
3704:
3663:
3637:
3611:
3551:
3507:
3470:
3419:
3362:
3323:
3291:
3263:
3237:
3181:
3146:
3111:
3087:
3067:
3047:
3027:
3018:
2944:
2860:
2834:
2825:
2774:
2713:
2687:
2681:
2615:
2606:
2541:
2532:
2474:
2416:
2348:
2316:
2310:
2222:
2194:
2184:. For example, if
2170:
2143:
2077:
2021:
1991:
1971:
1929:
1896:
1870:
1801:
1768:
1742:
1722:
1702:
1676:
1647:
1557:
1533:
1513:
1487:
1442:
1418:
1350:
1301:
1243:
1171:
1151:
1124:
1091:
1057:
1016:
990:
970:
929:
871:Spectral pollution
852:
828:
796:
754:
711:
687:
663:
634:
591:
503:
474:
417:
349:
314:
283:
11656:Dynamical systems
11634:10.1137/100804036
11566:978-0-89871-957-4
11487:Mathematics Today
11462:Arfken, George B.
11189:Arnoldi iteration
11184:Rayleigh quotient
11167:{\displaystyle c}
11096:is precisely the
10836:
10757:
10676:
10606:
10566:angular frequency
10375:
10290:
10224:
10136:
10084:
10059:
9983:{\displaystyle M}
9963:{\displaystyle K}
9872:
9862:
9754:
9744:
9679:{\displaystyle B}
9641:{\displaystyle A}
9190:{\displaystyle B}
9152:{\displaystyle A}
9090:Rayleigh quotient
9052:
8686:systems, such as
8684:degree of freedom
8343:
8251:secular equations
8225:
8144:
7938:
7925:
7750:
7592:
7381:
7347:
7222:{\displaystyle W}
7202:{\displaystyle W}
7181:{\displaystyle W}
6519:{\displaystyle v}
6417:
6397:
6368:
6351:
6319:
6302:
6283:
6263:
6159:{\displaystyle W}
6139:{\displaystyle W}
6005:{\displaystyle u}
5927:
5910:
5891:
5871:
5600:
5588:
5577:
5560:
5399:
5379:
5360:
5343:
5070:{\displaystyle A}
4326:{\displaystyle W}
4302:{\displaystyle W}
4278:{\displaystyle M}
4223:singular vectors.
3191:in this example.
3114:{\displaystyle V}
3090:{\displaystyle A}
3070:{\displaystyle V}
3050:{\displaystyle A}
2977:
2964:
2900:
2887:
2697:with eigenvalues
2225:{\displaystyle A}
2210:Rayleigh quotient
2197:{\displaystyle A}
2182:Rayleigh quotient
2131:
2107:
2036:Rayleigh quotient
2003:Rayleigh quotient
1994:{\displaystyle v}
1813:Rayleigh quotient
1745:{\displaystyle v}
1725:{\displaystyle V}
1635:
1614:
1592:
1560:{\displaystyle k}
1536:{\displaystyle A}
1445:{\displaystyle A}
1406:
1382:
1328:
1274:
1174:{\displaystyle V}
993:{\displaystyle N}
855:{\displaystyle T}
690:{\displaystyle T}
477:{\displaystyle T}
433:orthonormal basis
286:{\displaystyle T}
214:Rayleigh quotient
165:quantum mechanics
119:
118:
111:
93:
11663:
11637:
11586:
11585:
11577:
11571:
11570:
11550:
11544:
11543:
11531:
11525:
11524:
11516:
11510:
11509:
11497:
11491:
11490:
11478:
11472:
11471:
11458:
11449:
11448:
11430:
11421:
11420:
11408:
11399:
11398:
11373:
11364:
11363:
11348:
11335:
11334:
11317:(1–2): 731–733.
11306:
11297:
11296:
11279:(4–5): 961–978.
11270:
11262:
11249:
11232:
11173:
11171:
11170:
11165:
11153:
11151:
11150:
11145:
11129:
11127:
11126:
11121:
11119:
11118:
11098:stiffness matrix
11095:
11093:
11092:
11087:
11085:
11084:
11052:nonlinear system
11048:Koopman operator
11010:
11008:
11007:
11002:
10990:
10988:
10987:
10982:
10970:
10968:
10967:
10962:
10950:
10948:
10947:
10942:
10928:
10906:
10904:
10903:
10898:
10877:
10875:
10874:
10869:
10867:
10863:
10861:
10856:
10847:
10846:
10837:
10829:
10821:
10816:
10798:
10794:
10793:
10792:
10782:
10777:
10768:
10767:
10758:
10750:
10742:
10737:
10712:
10710:
10709:
10704:
10701:
10696:
10687:
10686:
10677:
10669:
10655:potential energy
10652:
10650:
10649:
10644:
10642:
10641:
10631:
10626:
10617:
10616:
10607:
10599:
10586:
10584:
10583:
10578:
10536:
10529:
10508:
10506:
10505:
10500:
10486:
10485:
10455:
10453:
10452:
10447:
10445:
10437:
10429:
10428:
10416:
10402:
10400:
10399:
10394:
10392:
10384:
10376:
10374:
10373:
10365:
10364:
10363:
10357:
10350:
10349:
10341:
10340:
10339:
10333:
10326:
10321:
10307:
10305:
10304:
10299:
10291:
10289:
10288:
10279:
10278:
10270:
10269:
10268:
10262:
10252:
10250:
10242:
10241:
10240:
10234:
10225:
10223:
10222:
10213:
10212:
10204:
10203:
10202:
10196:
10186:
10184:
10176:
10175:
10174:
10168:
10153:
10151:
10150:
10145:
10137:
10135:
10134:
10126:
10125:
10124:
10118:
10111:
10110:
10102:
10101:
10100:
10094:
10087:
10085:
10083:
10082:
10081:
10065:
10060:
10058:
10057:
10056:
10043:
10042:
10041:
10028:
10019:
10017:
10016:
10011:
10009:
10008:
9989:
9987:
9986:
9981:
9969:
9967:
9966:
9961:
9949:
9947:
9946:
9941:
9939:
9931:
9930:
9929:
9923:
9914:
9913:
9901:
9900:
9891:
9890:
9880:
9870:
9831:
9829:
9828:
9823:
9821:
9813:
9812:
9811:
9805:
9796:
9795:
9783:
9782:
9773:
9772:
9762:
9752:
9712:
9710:
9709:
9704:
9702:
9701:
9685:
9683:
9682:
9677:
9647:
9645:
9644:
9639:
9605:
9603:
9602:
9597:
9595:
9594:
9576:
9575:
9563:
9562:
9546:
9544:
9543:
9538:
9536:
9535:
9519:
9517:
9516:
9511:
9509:
9508:
9490:
9489:
9477:
9476:
9460:
9458:
9457:
9452:
9450:
9449:
9431:
9430:
9418:
9417:
9401:
9399:
9398:
9393:
9391:
9390:
9372:
9371:
9359:
9358:
9342:
9340:
9339:
9334:
9323:
9322:
9313:
9312:
9302:
9297:
9261:
9259:
9258:
9253:
9242:
9241:
9225:
9223:
9222:
9217:
9196:
9194:
9193:
9188:
9158:
9156:
9155:
9150:
9120:
9118:
9117:
9112:
9087:
9085:
9084:
9079:
9053:
9051:
9028:
9005:
9000:
8999:
8981:
8979:
8978:
8973:
8962:
8961:
8919:
8918:
8909:
8908:
8853:
8851:
8850:
8845:
8843:
8842:
8826:
8824:
8823:
8818:
8791:
8789:
8788:
8783:
8762:
8760:
8759:
8754:
8665:
8663:
8662:
8657:
8655:
8651:
8650:
8622:
8620:
8619:
8614:
8612:
8611:
8595:
8593:
8592:
8587:
8585:
8584:
8568:
8566:
8565:
8560:
8558:
8557:
8527:
8525:
8524:
8519:
8503:
8501:
8500:
8495:
8484:
8480:
8442:
8440:
8439:
8434:
8432:
8428:
8427:
8407:
8405:
8404:
8399:
8385:
8383:
8382:
8377:
8344:
8341:
8332:
8328:
8327:
8326:
8308:
8307:
8290:
8289:
8279:
8274:
8245:
8243:
8242:
8237:
8226:
8217:
8216:
8198:
8197:
8182:
8181:
8171:
8166:
8150:
8145:
8143:
8141:
8136:
8123:
8115:
8098:
8096:
8095:
8090:
8088:
8083:
8078:
8058:
8056:
8055:
8050:
8038:
8036:
8035:
8030:
8028:
8023:
8018:
7998:
7996:
7995:
7990:
7988:
7984:
7983:
7952:
7950:
7949:
7944:
7939:
7931:
7926:
7922:
7921:
7909:
7908:
7898:
7893:
7883:
7878:
7861:
7856:
7838:
7837:
7825:
7824:
7814:
7809:
7799:
7794:
7777:
7772:
7756:
7751:
7749:
7745:
7743:
7742:
7733:
7732:
7722:
7717:
7701:
7696:
7695:
7686:
7685:
7675:
7670:
7646:
7645:
7640:
7639:
7630:
7629:
7619:
7614:
7594:
7593:
7585:
7582:
7577:
7576:
7567:
7566:
7556:
7551:
7530:
7513:
7511:
7510:
7505:
7500:
7499:
7490:
7489:
7479:
7474:
7446:
7444:
7443:
7438:
7436:
7432:
7431:
7395:
7393:
7392:
7387:
7382:
7380:
7373:
7361:
7354:
7349:
7348:
7340:
7337:
7325:
7320:
7319:
7303:
7301:
7300:
7295:
7293:
7292:
7273:
7271:
7270:
7265:
7250:In this case, a
7228:
7226:
7225:
7220:
7208:
7206:
7205:
7200:
7187:
7185:
7184:
7179:
7167:
7165:
7164:
7159:
7154:
7153:
7106:
7105:
7058:
7057:
6932:
6930:
6929:
6924:
6910:
6909:
6893:
6891:
6890:
6885:
6883:
6882:
6828:
6827:
6787:
6786:
6659:
6657:
6656:
6651:
6627:
6625:
6624:
6619:
6617:
6616:
6576:
6574:
6573:
6568:
6566:
6565:
6525:
6523:
6522:
6517:
6505:
6503:
6502:
6497:
6495:
6494:
6436:
6435:
6418:
6413:
6411:
6398:
6393:
6391:
6369:
6364:
6362:
6352:
6347:
6345:
6328:
6327:
6320:
6315:
6313:
6303:
6298:
6296:
6284:
6279:
6277:
6264:
6259:
6257:
6238:
6237:
6232:
6220:
6218:
6217:
6212:
6210:
6209:
6200:
6199:
6194:
6185:
6184:
6165:
6163:
6162:
6157:
6145:
6143:
6142:
6137:
6125:
6123:
6122:
6117:
6115:
6114:
6068:
6066:
6065:
6060:
6058:
6057:
6011:
6009:
6008:
6003:
5991:
5989:
5988:
5983:
5981:
5969:
5967:
5966:
5961:
5949:
5947:
5946:
5941:
5936:
5935:
5928:
5923:
5921:
5911:
5906:
5904:
5892:
5887:
5885:
5872:
5867:
5865:
5846:
5845:
5840:
5830:
5829:
5784:
5783:
5709:
5697:
5695:
5694:
5689:
5687:
5686:
5681:
5675:
5670:
5646:
5644:
5643:
5638:
5633:
5632:
5601:
5596:
5589:
5584:
5578:
5573:
5571:
5561:
5556:
5554:
5519:
5517:
5516:
5511:
5509:
5508:
5442:
5440:
5439:
5434:
5432:
5431:
5400:
5395:
5393:
5380:
5375:
5373:
5361:
5356:
5354:
5344:
5339:
5337:
5305:
5303:
5302:
5297:
5295:
5294:
5190:
5189:
5184:
5183:
5076:
5074:
5073:
5068:
5056:
5054:
5053:
5048:
5043:
5042:
4943:
4942:
4843:
4842:
4705:
4703:
4702:
4697:
4668:singular values
4667:
4665:
4664:
4659:
4654:
4653:
4548:
4547:
4525:
4523:
4522:
4517:
4515:
4514:
4372:
4370:
4369:
4364:
4359:
4358:
4332:
4330:
4329:
4324:
4308:
4306:
4305:
4300:
4284:
4282:
4281:
4276:
4264:
4262:
4261:
4256:
4254:
4253:
4222:
4220:
4219:
4214:
4212:
4211:
4202:
4201:
4196:
4187:
4186:
4170:
4168:
4167:
4162:
4160:
4139:
4137:
4136:
4131:
4129:
4128:
4123:
4110:
4108:
4107:
4102:
4084:
4082:
4081:
4076:
4065:diagonal matrix
4064:
4062:
4061:
4056:
4038:
4036:
4035:
4030:
4028:
4016:
4014:
4013:
4008:
3990:
3988:
3987:
3982:
3977:
3976:
3971:
3962:
3935:
3933:
3932:
3927:
3912:
3910:
3909:
3904:
3882:
3880:
3879:
3874:
3859:
3857:
3856:
3851:
3833:
3831:
3830:
3825:
3817:
3816:
3789:
3788:
3779:
3778:
3760:
3759:
3743:
3741:
3740:
3735:
3713:
3711:
3710:
3705:
3703:
3702:
3691:
3672:
3670:
3669:
3664:
3646:
3644:
3643:
3638:
3620:
3618:
3617:
3612:
3610:
3609:
3598:
3586:
3585:
3560:
3558:
3557:
3552:
3547:
3539:
3538:
3516:
3514:
3513:
3508:
3503:
3479:
3477:
3476:
3471:
3469:
3468:
3457:
3448:
3447:
3428:
3426:
3425:
3420:
3418:
3417:
3406:
3394:
3393:
3371:
3369:
3368:
3363:
3349:
3348:
3332:
3330:
3329:
3324:
3300:
3298:
3297:
3292:
3272:
3270:
3269:
3264:
3246:
3244:
3243:
3238:
3236:
3235:
3224:
3190:
3188:
3187:
3182:
3180:
3179:
3168:
3155:
3153:
3152:
3147:
3145:
3144:
3133:
3120:
3118:
3117:
3112:
3096:
3094:
3093:
3088:
3076:
3074:
3073:
3068:
3056:
3054:
3053:
3048:
3036:
3034:
3033:
3028:
3023:
3022:
2987:
2986:
2979:
2978:
2970:
2966:
2965:
2957:
2949:
2948:
2910:
2909:
2902:
2901:
2893:
2889:
2888:
2880:
2869:
2867:
2866:
2861:
2843:
2841:
2840:
2835:
2830:
2829:
2801:
2800:
2789:
2779:
2778:
2747:
2746:
2735:
2722:
2720:
2719:
2714:
2696:
2694:
2693:
2688:
2686:
2685:
2641:
2640:
2624:
2622:
2621:
2616:
2611:
2610:
2550:
2548:
2547:
2542:
2537:
2536:
2501:
2500:
2489:
2479:
2478:
2443:
2442:
2431:
2421:
2420:
2382:
2381:
2370:
2357:
2355:
2354:
2349:
2326:has eigenvalues
2325:
2323:
2322:
2317:
2315:
2314:
2231:
2229:
2228:
2223:
2206:Hermitian matrix
2203:
2201:
2200:
2195:
2179:
2177:
2176:
2171:
2169:
2168:
2152:
2150:
2149:
2144:
2139:
2138:
2133:
2132:
2127:
2122:
2115:
2114:
2109:
2108:
2100:
2086:
2084:
2083:
2078:
2073:
2072:
2054:
2053:
2030:
2028:
2027:
2022:
2000:
1998:
1997:
1992:
1980:
1978:
1977:
1972:
1955:
1954:
1938:
1936:
1935:
1930:
1922:
1921:
1905:
1903:
1902:
1897:
1879:
1877:
1876:
1871:
1866:
1865:
1856:
1845:
1844:
1810:
1808:
1807:
1802:
1794:
1793:
1777:
1775:
1774:
1769:
1751:
1749:
1748:
1743:
1731:
1729:
1728:
1723:
1711:
1709:
1708:
1703:
1685:
1683:
1682:
1677:
1656:
1654:
1653:
1648:
1643:
1642:
1637:
1636:
1631:
1626:
1622:
1621:
1616:
1615:
1607:
1600:
1599:
1594:
1593:
1588:
1583:
1566:
1564:
1563:
1558:
1542:
1540:
1539:
1534:
1522:
1520:
1519:
1514:
1496:
1494:
1493:
1488:
1486:
1485:
1474:
1451:
1449:
1448:
1443:
1427:
1425:
1424:
1419:
1414:
1413:
1408:
1407:
1402:
1397:
1390:
1389:
1384:
1383:
1375:
1359:
1357:
1356:
1351:
1349:
1348:
1336:
1335:
1330:
1329:
1321:
1310:
1308:
1307:
1302:
1300:
1299:
1294:
1282:
1281:
1276:
1275:
1270:
1265:
1252:
1250:
1249:
1244:
1242:
1241:
1236:
1230:
1229:
1217:
1216:
1211:
1199:
1198:
1180:
1178:
1177:
1172:
1160:
1158:
1157:
1152:
1150:
1149:
1133:
1131:
1130:
1125:
1117:
1116:
1100:
1098:
1097:
1092:
1066:
1064:
1063:
1058:
1056:
1055:
1044:
1025:
1023:
1022:
1017:
999:
997:
996:
991:
979:
977:
976:
971:
969:
968:
957:
938:
936:
935:
930:
928:
917:
861:
859:
858:
853:
837:
835:
834:
829:
827:
826:
825:
805:
803:
802:
797:
780:
779:
766:weak formulation
763:
761:
760:
755:
720:
718:
717:
712:
710:
709:
696:
694:
693:
688:
672:
670:
669:
664:
662:
661:
660:
643:
641:
640:
635:
621:
620:
619:
600:
598:
597:
592:
584:
583:
571:
570:
552:
551:
536:
535:
534:
512:
510:
509:
504:
502:
501:
500:
483:
481:
480:
475:
453:of the operator.
431:A subset of the
426:
424:
423:
418:
413:
412:
388:
387:
372:
371:
358:
356:
355:
350:
323:
321:
320:
315:
313:
312:
292:
290:
289:
284:
191:of a structure.
133:and named after
114:
107:
103:
100:
94:
92:
51:
27:
19:
11671:
11670:
11666:
11665:
11664:
11662:
11661:
11660:
11641:
11640:
11595:
11590:
11589:
11578:
11574:
11567:
11551:
11547:
11532:
11528:
11517:
11513:
11498:
11494:
11479:
11475:
11459:
11452:
11445:
11431:
11424:
11409:
11402:
11395:
11374:
11367:
11349:
11338:
11307:
11300:
11263:
11256:
11212:
11204:Galerkin method
11180:
11159:
11156:
11155:
11139:
11136:
11135:
11111:
11107:
11105:
11102:
11101:
11077:
11073:
11071:
11068:
11067:
11064:
11044:
11020:beam deflection
10996:
10993:
10992:
10976:
10973:
10972:
10956:
10953:
10952:
10924:
10916:
10913:
10912:
10892:
10889:
10888:
10857:
10852:
10842:
10838:
10828:
10827:
10823:
10817:
10806:
10788:
10784:
10778:
10773:
10763:
10759:
10749:
10748:
10744:
10738:
10727:
10721:
10718:
10717:
10697:
10692:
10682:
10678:
10668:
10666:
10663:
10662:
10637:
10633:
10627:
10622:
10612:
10608:
10598:
10596:
10593:
10592:
10572:
10569:
10568:
10531:
10524:
10521:
10481:
10477:
10463:
10460:
10459:
10441:
10433:
10424:
10420:
10412:
10407:
10404:
10403:
10388:
10380:
10369:
10359:
10358:
10353:
10352:
10351:
10345:
10335:
10334:
10329:
10328:
10327:
10325:
10317:
10312:
10309:
10308:
10284:
10280:
10274:
10264:
10263:
10258:
10257:
10253:
10251:
10246:
10236:
10235:
10230:
10229:
10218:
10214:
10208:
10198:
10197:
10192:
10191:
10187:
10185:
10180:
10170:
10169:
10164:
10163:
10161:
10158:
10157:
10130:
10120:
10119:
10114:
10113:
10112:
10106:
10096:
10095:
10090:
10089:
10088:
10086:
10077:
10073:
10069:
10064:
10052:
10048:
10044:
10037:
10033:
10029:
10027:
10025:
10022:
10021:
10004:
10000:
9998:
9995:
9994:
9975:
9972:
9971:
9955:
9952:
9951:
9935:
9925:
9924:
9919:
9918:
9906:
9902:
9896:
9892:
9886:
9882:
9876:
9866:
9836:
9833:
9832:
9817:
9807:
9806:
9801:
9800:
9788:
9784:
9778:
9774:
9768:
9764:
9758:
9748:
9718:
9715:
9714:
9697:
9693:
9691:
9688:
9687:
9653:
9650:
9649:
9615:
9612:
9611:
9590:
9586:
9571:
9567:
9558:
9554:
9552:
9549:
9548:
9531:
9527:
9525:
9522:
9521:
9504:
9500:
9485:
9481:
9472:
9468:
9466:
9463:
9462:
9445:
9441:
9426:
9422:
9413:
9409:
9407:
9404:
9403:
9386:
9382:
9367:
9363:
9354:
9350:
9348:
9345:
9344:
9318:
9314:
9308:
9304:
9298:
9287:
9266:
9263:
9262:
9237:
9233:
9231:
9228:
9227:
9202:
9199:
9198:
9164:
9161:
9160:
9126:
9123:
9122:
9097:
9094:
9093:
9029:
9006:
9004:
8995:
8991:
8989:
8986:
8985:
8957:
8953:
8914:
8910:
8904:
8900:
8859:
8856:
8855:
8838:
8834:
8832:
8829:
8828:
8797:
8794:
8793:
8768:
8765:
8764:
8706:
8703:
8702:
8672:
8646:
8642:
8638:
8636:
8633:
8632:
8607:
8603:
8601:
8598:
8597:
8580:
8576:
8574:
8571:
8570:
8553:
8549:
8547:
8544:
8543:
8538:matrix is also
8513:
8510:
8509:
8467:
8463:
8458:
8455:
8454:
8423:
8419:
8415:
8413:
8410:
8409:
8393:
8390:
8389:
8340:
8319:
8315:
8300:
8296:
8295:
8291:
8285:
8281:
8275:
8264:
8258:
8255:
8254:
8209:
8205:
8190:
8186:
8177:
8173:
8167:
8156:
8149:
8137:
8132:
8124:
8116:
8114:
8112:
8109:
8108:
8079:
8074:
8066:
8064:
8061:
8060:
8044:
8041:
8040:
8019:
8014:
8006:
8004:
8001:
8000:
7979:
7975:
7971:
7969:
7966:
7965:
7930:
7914:
7910:
7904:
7900:
7894:
7889:
7879:
7868:
7857:
7846:
7830:
7826:
7820:
7816:
7810:
7805:
7795:
7784:
7773:
7762:
7755:
7738:
7734:
7728:
7724:
7718:
7707:
7691:
7687:
7681:
7677:
7671:
7660:
7652:
7651:
7647:
7635:
7631:
7625:
7621:
7615:
7604:
7595:
7584:
7583:
7572:
7568:
7562:
7558:
7552:
7541:
7532:
7531:
7529:
7521:
7518:
7517:
7495:
7491:
7485:
7481:
7475:
7464:
7452:
7449:
7448:
7427:
7423:
7419:
7417:
7414:
7413:
7369:
7362:
7350:
7339:
7338:
7333:
7326:
7324:
7315:
7311:
7309:
7306:
7305:
7288:
7284:
7282:
7279:
7278:
7259:
7256:
7255:
7240:
7235:
7214:
7211:
7210:
7194:
7191:
7190:
7173:
7170:
7169:
7148:
7147:
7141:
7140:
7134:
7133:
7127:
7126:
7116:
7115:
7100:
7099:
7093:
7092:
7086:
7085:
7079:
7078:
7072:
7071:
7061:
7060:
7052:
7051:
7046:
7041:
7036:
7031:
7025:
7024:
7019:
7014:
7009:
7004:
6998:
6997:
6992:
6987:
6982:
6977:
6971:
6970:
6965:
6960:
6955:
6950:
6940:
6939:
6937:
6934:
6933:
6905:
6901:
6899:
6896:
6895:
6877:
6876:
6870:
6869:
6863:
6862:
6856:
6855:
6849:
6848:
6838:
6837:
6822:
6821:
6815:
6814:
6808:
6807:
6801:
6800:
6790:
6789:
6781:
6780:
6775:
6770:
6765:
6759:
6758:
6753:
6748:
6743:
6737:
6736:
6731:
6726:
6721:
6715:
6714:
6709:
6704:
6699:
6693:
6692:
6687:
6682:
6677:
6667:
6666:
6664:
6661:
6660:
6633:
6630:
6629:
6612:
6608:
6582:
6579:
6578:
6561:
6557:
6531:
6528:
6527:
6511:
6508:
6507:
6489:
6488:
6483:
6478:
6473:
6467:
6466:
6461:
6456:
6451:
6441:
6440:
6430:
6429:
6424:
6419:
6412:
6407:
6399:
6392:
6387:
6381:
6380:
6375:
6370:
6363:
6358:
6353:
6346:
6341:
6331:
6330:
6322:
6321:
6314:
6309:
6304:
6297:
6292:
6286:
6285:
6278:
6273:
6265:
6258:
6253:
6243:
6242:
6233:
6228:
6227:
6225:
6222:
6221:
6205:
6201:
6195:
6190:
6189:
6180:
6176:
6174:
6171:
6170:
6151:
6148:
6147:
6131:
6128:
6127:
6110:
6106:
6074:
6071:
6070:
6053:
6049:
6017:
6014:
6013:
5997:
5994:
5993:
5977:
5975:
5972:
5971:
5955:
5952:
5951:
5930:
5929:
5922:
5917:
5912:
5905:
5900:
5894:
5893:
5886:
5881:
5873:
5866:
5861:
5851:
5850:
5841:
5836:
5835:
5824:
5823:
5818:
5812:
5811:
5806:
5796:
5795:
5778:
5777:
5772:
5766:
5765:
5760:
5754:
5753:
5748:
5742:
5741:
5736:
5730:
5729:
5724:
5714:
5713:
5705:
5703:
5700:
5699:
5682:
5677:
5676:
5671:
5666:
5655:
5652:
5651:
5627:
5626:
5621:
5615:
5614:
5609:
5603:
5602:
5595:
5590:
5583:
5580:
5579:
5572:
5567:
5562:
5555:
5550:
5540:
5539:
5528:
5525:
5524:
5503:
5502:
5497:
5491:
5490:
5485:
5479:
5478:
5473:
5467:
5466:
5461:
5451:
5450:
5448:
5445:
5444:
5426:
5425:
5420:
5414:
5413:
5408:
5402:
5401:
5394:
5389:
5381:
5374:
5369:
5363:
5362:
5355:
5350:
5345:
5338:
5333:
5323:
5322:
5314:
5311:
5310:
5289:
5288:
5283:
5278:
5273:
5267:
5266:
5261:
5256:
5251:
5245:
5244:
5239:
5234:
5229:
5223:
5222:
5217:
5212:
5207:
5197:
5196:
5185:
5178:
5177:
5172:
5167:
5162:
5156:
5155:
5150:
5145:
5140:
5134:
5133:
5128:
5123:
5118:
5112:
5111:
5106:
5101:
5096:
5086:
5085:
5084:
5082:
5079:
5078:
5062:
5059:
5058:
5037:
5036:
5031:
5026:
5021:
5015:
5014:
5009:
5004:
4999:
4993:
4992:
4987:
4982:
4977:
4971:
4970:
4965:
4960:
4955:
4945:
4944:
4937:
4936:
4931:
4926:
4921:
4915:
4914:
4909:
4904:
4899:
4893:
4892:
4887:
4882:
4877:
4871:
4870:
4865:
4860:
4855:
4845:
4844:
4837:
4836:
4831:
4826:
4821:
4815:
4814:
4809:
4804:
4799:
4793:
4792:
4787:
4782:
4777:
4771:
4770:
4765:
4760:
4755:
4749:
4748:
4743:
4738:
4733:
4723:
4722:
4714:
4711:
4710:
4673:
4670:
4669:
4648:
4647:
4642:
4637:
4632:
4626:
4625:
4620:
4615:
4610:
4604:
4603:
4598:
4593:
4588:
4582:
4581:
4576:
4571:
4566:
4556:
4555:
4543:
4539:
4531:
4528:
4527:
4509:
4508:
4503:
4498:
4493:
4487:
4486:
4481:
4476:
4471:
4465:
4464:
4459:
4454:
4449:
4443:
4442:
4437:
4432:
4427:
4421:
4420:
4415:
4410:
4405:
4395:
4394:
4386:
4383:
4382:
4379:
4354:
4350:
4342:
4339:
4338:
4318:
4315:
4314:
4294:
4291:
4290:
4270:
4267:
4266:
4249:
4245:
4231:
4228:
4227:
4207:
4203:
4197:
4192:
4191:
4182:
4178:
4176:
4173:
4172:
4156:
4148:
4145:
4144:
4124:
4119:
4118:
4116:
4113:
4112:
4090:
4087:
4086:
4070:
4067:
4066:
4044:
4041:
4040:
4024:
4022:
4019:
4018:
3996:
3993:
3992:
3972:
3967:
3966:
3958:
3947:
3944:
3943:
3918:
3915:
3914:
3892:
3889:
3888:
3865:
3862:
3861:
3839:
3836:
3835:
3812:
3808:
3784:
3780:
3774:
3770:
3755:
3751:
3749:
3746:
3745:
3723:
3720:
3719:
3692:
3687:
3686:
3678:
3675:
3674:
3652:
3649:
3648:
3626:
3623:
3622:
3599:
3594:
3593:
3581:
3577:
3569:
3566:
3565:
3543:
3534:
3530:
3522:
3519:
3518:
3499:
3485:
3482:
3481:
3458:
3453:
3452:
3443:
3439:
3434:
3431:
3430:
3407:
3402:
3401:
3389:
3385:
3383:
3380:
3379:
3344:
3340:
3338:
3335:
3334:
3306:
3303:
3302:
3286:
3283:
3282:
3279:
3252:
3249:
3248:
3225:
3220:
3219:
3211:
3208:
3207:
3197:
3169:
3164:
3163:
3161:
3158:
3157:
3134:
3129:
3128:
3126:
3123:
3122:
3106:
3103:
3102:
3082:
3079:
3078:
3062:
3059:
3058:
3042:
3039:
3038:
3017:
3016:
3010:
3009:
3003:
3002:
2992:
2991:
2969:
2968:
2967:
2956:
2955:
2954:
2943:
2942:
2933:
2932:
2926:
2925:
2915:
2914:
2892:
2891:
2890:
2879:
2878:
2877:
2875:
2872:
2871:
2849:
2846:
2845:
2824:
2823:
2817:
2816:
2806:
2805:
2790:
2785:
2784:
2773:
2772:
2763:
2762:
2752:
2751:
2736:
2731:
2730:
2728:
2725:
2724:
2702:
2699:
2698:
2680:
2679:
2674:
2668:
2667:
2662:
2652:
2651:
2636:
2632:
2630:
2627:
2626:
2605:
2604:
2599:
2593:
2592:
2587:
2581:
2580:
2575:
2565:
2564:
2556:
2553:
2552:
2531:
2530:
2524:
2523:
2517:
2516:
2506:
2505:
2490:
2485:
2484:
2473:
2472:
2466:
2465:
2459:
2458:
2448:
2447:
2432:
2427:
2426:
2415:
2414:
2405:
2404:
2398:
2397:
2387:
2386:
2371:
2366:
2365:
2363:
2360:
2359:
2331:
2328:
2327:
2309:
2308:
2303:
2298:
2292:
2291:
2286:
2281:
2275:
2274:
2269:
2264:
2254:
2253:
2245:
2242:
2241:
2238:
2217:
2214:
2213:
2189:
2186:
2185:
2164:
2160:
2158:
2155:
2154:
2134:
2123:
2121:
2120:
2119:
2110:
2099:
2098:
2097:
2092:
2089:
2088:
2068:
2064:
2049:
2045:
2043:
2040:
2039:
2010:
2007:
2006:
1986:
1983:
1982:
1950:
1946:
1944:
1941:
1940:
1917:
1913:
1911:
1908:
1907:
1885:
1882:
1881:
1861:
1857:
1852:
1840:
1836:
1819:
1816:
1815:
1789:
1785:
1783:
1780:
1779:
1757:
1754:
1753:
1737:
1734:
1733:
1717:
1714:
1713:
1691:
1688:
1687:
1665:
1662:
1661:
1638:
1627:
1625:
1624:
1623:
1617:
1606:
1605:
1604:
1595:
1584:
1582:
1581:
1580:
1572:
1569:
1568:
1552:
1549:
1548:
1528:
1525:
1524:
1502:
1499:
1498:
1475:
1470:
1469:
1461:
1458:
1457:
1437:
1434:
1433:
1409:
1398:
1396:
1395:
1394:
1385:
1374:
1373:
1372:
1367:
1364:
1363:
1344:
1340:
1331:
1320:
1319:
1318:
1316:
1313:
1312:
1295:
1290:
1289:
1277:
1266:
1264:
1263:
1262:
1260:
1257:
1256:
1237:
1232:
1231:
1225:
1221:
1212:
1207:
1206:
1194:
1190:
1188:
1185:
1184:
1166:
1163:
1162:
1145:
1141:
1139:
1136:
1135:
1112:
1108:
1106:
1103:
1102:
1080:
1077:
1076:
1045:
1040:
1039:
1031:
1028:
1027:
1005:
1002:
1001:
985:
982:
981:
958:
953:
952:
944:
941:
940:
939:for the matrix
924:
913:
908:
905:
904:
897:
885:numerical range
873:
868:
847:
844:
843:
821:
820:
816:
814:
811:
810:
775:
774:
772:
769:
768:
737:
734:
733:
705:
704:
702:
699:
698:
682:
679:
678:
656:
655:
651:
649:
646:
645:
615:
614:
610:
608:
605:
604:
579:
575:
566:
562:
541:
537:
530:
529:
525:
520:
517:
516:
496:
495:
491:
489:
486:
485:
469:
466:
465:
444:Galerkin method
408:
404:
383:
379:
367:
366:
364:
361:
360:
332:
329:
328:
308:
307:
305:
302:
301:
295:linear operator
278:
275:
274:
271:
255:Galerkin method
222:Richard Courant
197:
146:linear operator
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
11669:
11659:
11658:
11653:
11639:
11638:
11628:(4): 627–666.
11613:
11602:
11594:
11593:External links
11591:
11588:
11587:
11572:
11565:
11545:
11526:
11511:
11492:
11473:
11450:
11443:
11422:
11400:
11393:
11365:
11336:
11298:
11253:
11252:
11251:
11250:
11233:
11211:
11208:
11207:
11206:
11201:
11196:
11191:
11186:
11179:
11176:
11163:
11143:
11117:
11114:
11110:
11083:
11080:
11076:
11063:
11060:
11043:
11040:
11000:
10980:
10960:
10940:
10937:
10934:
10931:
10927:
10923:
10920:
10909:differentiated
10896:
10866:
10860:
10855:
10851:
10845:
10841:
10835:
10832:
10826:
10820:
10815:
10812:
10809:
10805:
10801:
10797:
10791:
10787:
10781:
10776:
10772:
10766:
10762:
10756:
10753:
10747:
10741:
10736:
10733:
10730:
10726:
10700:
10695:
10691:
10685:
10681:
10675:
10672:
10657:(PE) for each
10653:etc., and the
10640:
10636:
10630:
10625:
10621:
10615:
10611:
10605:
10602:
10589:kinetic energy
10576:
10520:
10517:
10498:
10495:
10492:
10489:
10484:
10480:
10476:
10473:
10470:
10467:
10444:
10440:
10436:
10432:
10427:
10423:
10419:
10415:
10411:
10391:
10387:
10383:
10379:
10372:
10368:
10362:
10356:
10348:
10344:
10338:
10332:
10324:
10320:
10316:
10297:
10294:
10287:
10283:
10277:
10273:
10267:
10261:
10256:
10249:
10245:
10239:
10233:
10228:
10221:
10217:
10211:
10207:
10201:
10195:
10190:
10183:
10179:
10173:
10167:
10156:Solving this,
10143:
10140:
10133:
10129:
10123:
10117:
10109:
10105:
10099:
10093:
10080:
10076:
10072:
10068:
10063:
10055:
10051:
10047:
10040:
10036:
10032:
10007:
10003:
9979:
9959:
9938:
9934:
9928:
9922:
9917:
9912:
9909:
9905:
9899:
9895:
9889:
9885:
9879:
9875:
9869:
9865:
9861:
9858:
9855:
9852:
9849:
9846:
9843:
9840:
9820:
9816:
9810:
9804:
9799:
9794:
9791:
9787:
9781:
9777:
9771:
9767:
9761:
9757:
9751:
9747:
9743:
9740:
9737:
9734:
9731:
9728:
9725:
9722:
9700:
9696:
9675:
9672:
9669:
9666:
9663:
9660:
9657:
9637:
9634:
9631:
9628:
9625:
9622:
9619:
9593:
9589:
9585:
9582:
9579:
9574:
9570:
9566:
9561:
9557:
9534:
9530:
9507:
9503:
9499:
9496:
9493:
9488:
9484:
9480:
9475:
9471:
9448:
9444:
9440:
9437:
9434:
9429:
9425:
9421:
9416:
9412:
9389:
9385:
9381:
9378:
9375:
9370:
9366:
9362:
9357:
9353:
9332:
9329:
9326:
9321:
9317:
9311:
9307:
9301:
9296:
9293:
9290:
9286:
9282:
9279:
9276:
9273:
9270:
9251:
9248:
9245:
9240:
9236:
9215:
9212:
9209:
9206:
9186:
9183:
9180:
9177:
9174:
9171:
9168:
9148:
9145:
9142:
9139:
9136:
9133:
9130:
9110:
9107:
9104:
9101:
9077:
9074:
9071:
9068:
9065:
9062:
9059:
9056:
9050:
9047:
9044:
9041:
9038:
9035:
9032:
9027:
9024:
9021:
9018:
9015:
9012:
9009:
9003:
8998:
8994:
8971:
8968:
8965:
8960:
8956:
8952:
8949:
8946:
8943:
8940:
8937:
8934:
8931:
8928:
8925:
8922:
8917:
8913:
8907:
8903:
8899:
8896:
8893:
8890:
8887:
8884:
8881:
8878:
8875:
8872:
8869:
8866:
8863:
8841:
8837:
8816:
8813:
8810:
8807:
8804:
8801:
8781:
8778:
8775:
8772:
8752:
8749:
8746:
8743:
8740:
8737:
8734:
8731:
8728:
8725:
8722:
8719:
8716:
8713:
8710:
8671:
8668:
8654:
8649:
8645:
8641:
8610:
8606:
8596:(i=1,2,..,N),
8583:
8579:
8556:
8552:
8517:
8493:
8490:
8487:
8483:
8479:
8476:
8473:
8470:
8466:
8462:
8431:
8426:
8422:
8418:
8397:
8375:
8372:
8369:
8366:
8363:
8360:
8357:
8354:
8351:
8348:
8338:
8335:
8331:
8325:
8322:
8318:
8314:
8311:
8306:
8303:
8299:
8294:
8288:
8284:
8278:
8273:
8270:
8267:
8263:
8235:
8232:
8229:
8224:
8220:
8215:
8212:
8208:
8204:
8201:
8196:
8193:
8189:
8185:
8180:
8176:
8170:
8165:
8162:
8159:
8155:
8148:
8140:
8135:
8131:
8127:
8122:
8119:
8087:
8082:
8077:
8073:
8069:
8048:
8027:
8022:
8017:
8013:
8009:
7987:
7982:
7978:
7974:
7957:overlap matrix
7942:
7937:
7934:
7929:
7920:
7917:
7913:
7907:
7903:
7897:
7892:
7888:
7882:
7877:
7874:
7871:
7867:
7860:
7855:
7852:
7849:
7845:
7836:
7833:
7829:
7823:
7819:
7813:
7808:
7804:
7798:
7793:
7790:
7787:
7783:
7776:
7771:
7768:
7765:
7761:
7754:
7748:
7741:
7737:
7731:
7727:
7721:
7716:
7713:
7710:
7706:
7700:
7694:
7690:
7684:
7680:
7674:
7669:
7666:
7663:
7659:
7654:
7650:
7644:
7638:
7634:
7628:
7624:
7618:
7613:
7610:
7607:
7603:
7598:
7591:
7588:
7581:
7575:
7571:
7565:
7561:
7555:
7550:
7547:
7544:
7540:
7535:
7528:
7525:
7503:
7498:
7494:
7488:
7484:
7478:
7473:
7470:
7467:
7463:
7459:
7456:
7435:
7430:
7426:
7422:
7385:
7379:
7376:
7372:
7368:
7365:
7360:
7357:
7353:
7346:
7343:
7336:
7332:
7329:
7323:
7318:
7314:
7291:
7287:
7263:
7239:
7236:
7234:
7231:
7218:
7198:
7177:
7157:
7152:
7146:
7143:
7142:
7139:
7136:
7135:
7132:
7129:
7128:
7125:
7122:
7121:
7119:
7113:
7109:
7104:
7098:
7095:
7094:
7091:
7088:
7087:
7084:
7081:
7080:
7077:
7074:
7073:
7070:
7067:
7066:
7064:
7056:
7050:
7047:
7045:
7042:
7040:
7037:
7035:
7032:
7030:
7027:
7026:
7023:
7020:
7018:
7015:
7013:
7010:
7008:
7005:
7003:
7000:
6999:
6996:
6993:
6991:
6988:
6986:
6983:
6981:
6978:
6976:
6973:
6972:
6969:
6966:
6964:
6961:
6959:
6956:
6954:
6951:
6949:
6946:
6945:
6943:
6922:
6919:
6916:
6913:
6908:
6904:
6881:
6875:
6872:
6871:
6868:
6865:
6864:
6861:
6858:
6857:
6854:
6851:
6850:
6847:
6844:
6843:
6841:
6835:
6831:
6826:
6820:
6817:
6816:
6813:
6810:
6809:
6806:
6803:
6802:
6799:
6796:
6795:
6793:
6785:
6779:
6776:
6774:
6771:
6769:
6766:
6764:
6761:
6760:
6757:
6754:
6752:
6749:
6747:
6744:
6742:
6739:
6738:
6735:
6732:
6730:
6727:
6725:
6722:
6720:
6717:
6716:
6713:
6710:
6708:
6705:
6703:
6700:
6698:
6695:
6694:
6691:
6688:
6686:
6683:
6681:
6678:
6676:
6673:
6672:
6670:
6649:
6646:
6643:
6640:
6637:
6615:
6611:
6607:
6604:
6601:
6598:
6595:
6592:
6589:
6586:
6564:
6560:
6556:
6553:
6550:
6547:
6544:
6541:
6538:
6535:
6515:
6493:
6487:
6484:
6482:
6479:
6477:
6474:
6472:
6469:
6468:
6465:
6462:
6460:
6457:
6455:
6452:
6450:
6447:
6446:
6444:
6439:
6434:
6428:
6425:
6423:
6420:
6416:
6410:
6406:
6403:
6400:
6396:
6390:
6386:
6383:
6382:
6379:
6376:
6374:
6371:
6367:
6361:
6357:
6354:
6350:
6344:
6340:
6337:
6336:
6334:
6326:
6318:
6312:
6308:
6305:
6301:
6295:
6291:
6288:
6287:
6282:
6276:
6272:
6269:
6266:
6262:
6256:
6252:
6249:
6248:
6246:
6241:
6236:
6231:
6208:
6204:
6198:
6193:
6188:
6183:
6179:
6155:
6135:
6113:
6109:
6105:
6102:
6099:
6096:
6093:
6090:
6087:
6084:
6081:
6078:
6056:
6052:
6048:
6045:
6042:
6039:
6036:
6033:
6030:
6027:
6024:
6021:
6001:
5980:
5959:
5939:
5934:
5926:
5920:
5916:
5913:
5909:
5903:
5899:
5896:
5895:
5890:
5884:
5880:
5877:
5874:
5870:
5864:
5860:
5857:
5856:
5854:
5849:
5844:
5839:
5833:
5828:
5822:
5819:
5817:
5814:
5813:
5810:
5807:
5805:
5802:
5801:
5799:
5794:
5791:
5787:
5782:
5776:
5773:
5771:
5768:
5767:
5764:
5761:
5759:
5756:
5755:
5752:
5749:
5747:
5744:
5743:
5740:
5737:
5735:
5732:
5731:
5728:
5725:
5723:
5720:
5719:
5717:
5712:
5708:
5685:
5680:
5674:
5669:
5665:
5662:
5659:
5636:
5631:
5625:
5622:
5620:
5617:
5616:
5613:
5610:
5608:
5605:
5604:
5599:
5594:
5591:
5587:
5582:
5581:
5576:
5570:
5566:
5563:
5559:
5553:
5549:
5546:
5545:
5543:
5538:
5535:
5532:
5507:
5501:
5498:
5496:
5493:
5492:
5489:
5486:
5484:
5481:
5480:
5477:
5474:
5472:
5469:
5468:
5465:
5462:
5460:
5457:
5456:
5454:
5430:
5424:
5421:
5419:
5416:
5415:
5412:
5409:
5407:
5404:
5403:
5398:
5392:
5388:
5385:
5382:
5378:
5372:
5368:
5365:
5364:
5359:
5353:
5349:
5346:
5342:
5336:
5332:
5329:
5328:
5326:
5321:
5318:
5293:
5287:
5284:
5282:
5279:
5277:
5274:
5272:
5269:
5268:
5265:
5262:
5260:
5257:
5255:
5252:
5250:
5247:
5246:
5243:
5240:
5238:
5235:
5233:
5230:
5228:
5225:
5224:
5221:
5218:
5216:
5213:
5211:
5208:
5206:
5203:
5202:
5200:
5194:
5188:
5182:
5176:
5173:
5171:
5168:
5166:
5163:
5161:
5158:
5157:
5154:
5151:
5149:
5146:
5144:
5141:
5139:
5136:
5135:
5132:
5129:
5127:
5124:
5122:
5119:
5117:
5114:
5113:
5110:
5107:
5105:
5102:
5100:
5097:
5095:
5092:
5091:
5089:
5066:
5046:
5041:
5035:
5032:
5030:
5027:
5025:
5022:
5020:
5017:
5016:
5013:
5010:
5008:
5005:
5003:
5000:
4998:
4995:
4994:
4991:
4988:
4986:
4983:
4981:
4978:
4976:
4973:
4972:
4969:
4966:
4964:
4961:
4959:
4956:
4954:
4951:
4950:
4948:
4941:
4935:
4932:
4930:
4927:
4925:
4922:
4920:
4917:
4916:
4913:
4910:
4908:
4905:
4903:
4900:
4898:
4895:
4894:
4891:
4888:
4886:
4883:
4881:
4878:
4876:
4873:
4872:
4869:
4866:
4864:
4861:
4859:
4856:
4854:
4851:
4850:
4848:
4841:
4835:
4832:
4830:
4827:
4825:
4822:
4820:
4817:
4816:
4813:
4810:
4808:
4805:
4803:
4800:
4798:
4795:
4794:
4791:
4788:
4786:
4783:
4781:
4778:
4776:
4773:
4772:
4769:
4766:
4764:
4761:
4759:
4756:
4754:
4751:
4750:
4747:
4744:
4742:
4739:
4737:
4734:
4732:
4729:
4728:
4726:
4721:
4718:
4695:
4692:
4689:
4686:
4683:
4680:
4677:
4657:
4652:
4646:
4643:
4641:
4638:
4636:
4633:
4631:
4628:
4627:
4624:
4621:
4619:
4616:
4614:
4611:
4609:
4606:
4605:
4602:
4599:
4597:
4594:
4592:
4589:
4587:
4584:
4583:
4580:
4577:
4575:
4572:
4570:
4567:
4565:
4562:
4561:
4559:
4554:
4551:
4546:
4542:
4538:
4535:
4513:
4507:
4504:
4502:
4499:
4497:
4494:
4492:
4489:
4488:
4485:
4482:
4480:
4477:
4475:
4472:
4470:
4467:
4466:
4463:
4460:
4458:
4455:
4453:
4450:
4448:
4445:
4444:
4441:
4438:
4436:
4433:
4431:
4428:
4426:
4423:
4422:
4419:
4416:
4414:
4411:
4409:
4406:
4404:
4401:
4400:
4398:
4393:
4390:
4378:
4375:
4362:
4357:
4353:
4349:
4346:
4322:
4311:
4310:
4298:
4274:
4252:
4248:
4244:
4241:
4238:
4235:
4224:
4210:
4206:
4200:
4195:
4190:
4185:
4181:
4159:
4155:
4152:
4141:
4127:
4122:
4100:
4097:
4094:
4074:
4054:
4051:
4048:
4027:
4006:
4003:
4000:
3980:
3975:
3970:
3965:
3961:
3957:
3954:
3951:
3937:
3925:
3922:
3902:
3899:
3896:
3872:
3869:
3849:
3846:
3843:
3823:
3820:
3815:
3811:
3807:
3804:
3801:
3798:
3795:
3792:
3787:
3783:
3777:
3773:
3769:
3766:
3763:
3758:
3754:
3733:
3730:
3727:
3701:
3698:
3695:
3690:
3685:
3682:
3662:
3659:
3656:
3636:
3633:
3630:
3608:
3605:
3602:
3597:
3592:
3589:
3584:
3580:
3576:
3573:
3550:
3546:
3542:
3537:
3533:
3529:
3526:
3506:
3502:
3498:
3495:
3492:
3489:
3467:
3464:
3461:
3456:
3451:
3446:
3442:
3438:
3416:
3413:
3410:
3405:
3400:
3397:
3392:
3388:
3361:
3358:
3355:
3352:
3347:
3343:
3322:
3319:
3316:
3313:
3310:
3290:
3278:
3275:
3262:
3259:
3256:
3234:
3231:
3228:
3223:
3218:
3215:
3196:
3193:
3178:
3175:
3172:
3167:
3143:
3140:
3137:
3132:
3110:
3101:of the matrix
3086:
3066:
3057:for the given
3046:
3026:
3021:
3015:
3012:
3011:
3008:
3005:
3004:
3001:
2998:
2997:
2995:
2990:
2985:
2982:
2976:
2973:
2963:
2960:
2952:
2947:
2941:
2938:
2935:
2934:
2931:
2928:
2927:
2924:
2921:
2920:
2918:
2913:
2908:
2905:
2899:
2896:
2886:
2883:
2859:
2856:
2853:
2833:
2828:
2822:
2819:
2818:
2815:
2812:
2811:
2809:
2804:
2799:
2796:
2793:
2788:
2782:
2777:
2771:
2768:
2765:
2764:
2761:
2758:
2757:
2755:
2750:
2745:
2742:
2739:
2734:
2712:
2709:
2706:
2684:
2678:
2675:
2673:
2670:
2669:
2666:
2663:
2661:
2658:
2657:
2655:
2650:
2647:
2644:
2639:
2635:
2614:
2609:
2603:
2600:
2598:
2595:
2594:
2591:
2588:
2586:
2583:
2582:
2579:
2576:
2574:
2571:
2570:
2568:
2563:
2560:
2540:
2535:
2529:
2526:
2525:
2522:
2519:
2518:
2515:
2512:
2511:
2509:
2504:
2499:
2496:
2493:
2488:
2482:
2477:
2471:
2468:
2467:
2464:
2461:
2460:
2457:
2454:
2453:
2451:
2446:
2441:
2438:
2435:
2430:
2424:
2419:
2413:
2410:
2407:
2406:
2403:
2400:
2399:
2396:
2393:
2392:
2390:
2385:
2380:
2377:
2374:
2369:
2347:
2344:
2341:
2338:
2335:
2313:
2307:
2304:
2302:
2299:
2297:
2294:
2293:
2290:
2287:
2285:
2282:
2280:
2277:
2276:
2273:
2270:
2268:
2265:
2263:
2260:
2259:
2257:
2252:
2249:
2237:
2234:
2221:
2193:
2167:
2163:
2142:
2137:
2130:
2126:
2118:
2113:
2106:
2103:
2096:
2076:
2071:
2067:
2063:
2060:
2057:
2052:
2048:
2020:
2017:
2014:
1990:
1970:
1967:
1964:
1961:
1958:
1953:
1949:
1928:
1925:
1920:
1916:
1895:
1892:
1889:
1869:
1864:
1860:
1855:
1851:
1848:
1843:
1839:
1835:
1832:
1829:
1826:
1823:
1800:
1797:
1792:
1788:
1767:
1764:
1761:
1741:
1721:
1701:
1698:
1695:
1675:
1672:
1669:
1646:
1641:
1634:
1630:
1620:
1613:
1610:
1603:
1598:
1591:
1587:
1579:
1576:
1556:
1532:
1512:
1509:
1506:
1484:
1481:
1478:
1473:
1468:
1465:
1454:
1453:
1441:
1417:
1412:
1405:
1401:
1393:
1388:
1381:
1378:
1371:
1360:
1347:
1343:
1339:
1334:
1327:
1324:
1298:
1293:
1288:
1285:
1280:
1273:
1269:
1253:
1240:
1235:
1228:
1224:
1220:
1215:
1210:
1205:
1202:
1197:
1193:
1181:
1170:
1148:
1144:
1123:
1120:
1115:
1111:
1090:
1087:
1084:
1054:
1051:
1048:
1043:
1038:
1035:
1015:
1012:
1009:
989:
967:
964:
961:
956:
951:
948:
927:
923:
920:
916:
912:
896:
893:
872:
869:
867:
864:
851:
824:
819:
795:
792:
789:
786:
783:
778:
753:
750:
747:
744:
741:
708:
686:
659:
654:
633:
630:
627:
624:
618:
613:
590:
587:
582:
578:
574:
569:
565:
561:
558:
555:
550:
547:
544:
540:
533:
528:
524:
499:
494:
473:
455:
454:
451:eigenfunctions
447:
436:
416:
411:
407:
403:
400:
397:
394:
391:
386:
382:
378:
375:
370:
348:
345:
342:
339:
336:
311:
282:
270:
267:
259:Boris Galerkin
196:
193:
117:
116:
31:
29:
22:
15:
9:
6:
4:
3:
2:
11668:
11657:
11654:
11652:
11649:
11648:
11646:
11635:
11631:
11627:
11623:
11619:
11614:
11612:
11611:
11606:
11603:
11600:
11597:
11596:
11583:
11576:
11568:
11562:
11558:
11557:
11549:
11541:
11537:
11530:
11522:
11515:
11507:
11503:
11496:
11488:
11484:
11477:
11469:
11468:
11463:
11457:
11455:
11446:
11440:
11436:
11429:
11427:
11418:
11414:
11407:
11405:
11396:
11390:
11386:
11382:
11378:
11372:
11370:
11361:
11357:
11353:
11352:Davies, E. B.
11347:
11345:
11343:
11341:
11332:
11328:
11324:
11320:
11316:
11312:
11305:
11303:
11294:
11290:
11286:
11282:
11278:
11274:
11269:
11261:
11259:
11254:
11247:
11243:
11239:
11234:
11230:
11226:
11222:
11218:
11217:Ritz, Walther
11214:
11213:
11205:
11202:
11200:
11199:Hilbert space
11197:
11195:
11192:
11190:
11187:
11185:
11182:
11181:
11175:
11161:
11141:
11133:
11115:
11112:
11108:
11099:
11081:
11078:
11074:
11059:
11057:
11056:linear system
11053:
11049:
11039:
11037:
11032:
11030:
11025:
11021:
11016:
11014:
10998:
10978:
10958:
10938:
10935:
10932:
10929:
10925:
10921:
10918:
10910:
10894:
10885:
10883:
10878:
10864:
10858:
10853:
10849:
10843:
10839:
10833:
10830:
10824:
10818:
10813:
10810:
10807:
10803:
10799:
10795:
10789:
10785:
10779:
10774:
10770:
10764:
10760:
10754:
10751:
10745:
10739:
10734:
10731:
10728:
10724:
10714:
10698:
10693:
10689:
10683:
10679:
10673:
10670:
10660:
10656:
10638:
10634:
10628:
10623:
10619:
10613:
10609:
10603:
10600:
10590:
10574:
10567:
10563:
10559:
10555:
10551:
10547:
10543:
10538:
10534:
10527:
10516:
10514:
10509:
10496:
10493:
10487:
10482:
10478:
10474:
10471:
10456:
10438:
10430:
10425:
10421:
10417:
10409:
10385:
10377:
10366:
10342:
10322:
10314:
10295:
10292:
10271:
10243:
10226:
10205:
10177:
10154:
10141:
10138:
10127:
10103:
10078:
10074:
10061:
10053:
10049:
10038:
10034:
10005:
10001:
9991:
9977:
9957:
9932:
9915:
9910:
9907:
9903:
9897:
9893:
9887:
9883:
9877:
9873:
9867:
9863:
9859:
9850:
9844:
9838:
9814:
9797:
9792:
9789:
9785:
9779:
9775:
9769:
9765:
9759:
9755:
9749:
9745:
9741:
9732:
9726:
9720:
9698:
9694:
9667:
9661:
9655:
9629:
9623:
9617:
9608:
9591:
9587:
9583:
9580:
9577:
9572:
9568:
9564:
9559:
9555:
9532:
9528:
9505:
9501:
9497:
9494:
9491:
9486:
9482:
9478:
9473:
9469:
9446:
9442:
9438:
9435:
9432:
9427:
9423:
9419:
9414:
9410:
9387:
9383:
9379:
9376:
9373:
9368:
9364:
9360:
9355:
9351:
9327:
9319:
9315:
9309:
9305:
9299:
9294:
9291:
9288:
9284:
9280:
9274:
9268:
9246:
9238:
9234:
9210:
9204:
9178:
9172:
9166:
9140:
9134:
9128:
9105:
9099:
9091:
9069:
9063:
9057:
9054:
9042:
9036:
9030:
9019:
9013:
9007:
9001:
8996:
8992:
8982:
8969:
8966:
8963:
8958:
8954:
8944:
8938:
8932:
8929:
8926:
8923:
8920:
8915:
8911:
8905:
8901:
8891:
8885:
8879:
8876:
8873:
8870:
8867:
8864:
8861:
8839:
8835:
8811:
8808:
8805:
8799:
8776:
8770:
8750:
8747:
8744:
8741:
8735:
8729:
8726:
8720:
8717:
8714:
8708:
8699:
8697:
8696:cross section
8693:
8689:
8685:
8681:
8677:
8667:
8652:
8647:
8643:
8639:
8630:
8626:
8608:
8604:
8581:
8577:
8554:
8550:
8541:
8537:
8536:
8531:
8515:
8507:
8491:
8488:
8485:
8481:
8477:
8474:
8471:
8468:
8464:
8452:
8448:
8447:
8429:
8424:
8420:
8416:
8395:
8386:
8373:
8370:
8367:
8364:
8361:
8358:
8355:
8352:
8349:
8346:
8336:
8333:
8329:
8323:
8320:
8316:
8312:
8309:
8304:
8301:
8297:
8292:
8286:
8282:
8276:
8271:
8268:
8265:
8261:
8252:
8249:
8233:
8230:
8227:
8222:
8213:
8210:
8206:
8202:
8199:
8194:
8191:
8187:
8178:
8174:
8168:
8163:
8160:
8157:
8153:
8146:
8138:
8133:
8129:
8120:
8106:
8103:= 1, 2, ...,
8102:
8085:
8080:
8075:
8071:
8067:
8046:
8025:
8020:
8015:
8011:
8007:
7985:
7980:
7976:
7972:
7963:
7962:
7958:
7953:
7940:
7935:
7932:
7927:
7918:
7915:
7911:
7905:
7901:
7895:
7890:
7886:
7880:
7875:
7872:
7869:
7865:
7858:
7853:
7850:
7847:
7843:
7834:
7831:
7827:
7821:
7817:
7811:
7806:
7802:
7796:
7791:
7788:
7785:
7781:
7774:
7769:
7766:
7763:
7759:
7752:
7746:
7739:
7729:
7725:
7719:
7714:
7711:
7708:
7704:
7698:
7692:
7682:
7678:
7672:
7667:
7664:
7661:
7657:
7648:
7642:
7636:
7626:
7622:
7616:
7611:
7608:
7605:
7601:
7596:
7586:
7579:
7573:
7563:
7559:
7553:
7548:
7545:
7542:
7538:
7533:
7526:
7523:
7514:
7501:
7496:
7486:
7482:
7476:
7471:
7468:
7465:
7461:
7457:
7433:
7428:
7420:
7411:
7406:
7404:
7399:
7396:
7383:
7341:
7321:
7316:
7312:
7289:
7285:
7275:
7253:
7248:
7245:
7230:
7216:
7196:
7175:
7155:
7150:
7144:
7137:
7130:
7123:
7117:
7111:
7107:
7102:
7096:
7089:
7082:
7075:
7068:
7062:
7054:
7048:
7043:
7038:
7033:
7028:
7021:
7016:
7011:
7006:
7001:
6994:
6989:
6984:
6979:
6974:
6967:
6962:
6957:
6952:
6947:
6941:
6920:
6917:
6914:
6911:
6906:
6902:
6879:
6873:
6866:
6859:
6852:
6845:
6839:
6833:
6829:
6824:
6818:
6811:
6804:
6797:
6791:
6783:
6777:
6772:
6767:
6762:
6755:
6750:
6745:
6740:
6733:
6728:
6723:
6718:
6711:
6706:
6701:
6696:
6689:
6684:
6679:
6674:
6668:
6647:
6644:
6641:
6638:
6635:
6613:
6605:
6602:
6599:
6596:
6593:
6590:
6587:
6562:
6554:
6551:
6548:
6545:
6542:
6539:
6536:
6513:
6491:
6485:
6480:
6475:
6470:
6463:
6458:
6453:
6448:
6442:
6437:
6432:
6426:
6421:
6414:
6408:
6404:
6401:
6394:
6388:
6384:
6377:
6372:
6365:
6359:
6355:
6348:
6342:
6338:
6332:
6324:
6316:
6310:
6306:
6299:
6293:
6289:
6280:
6274:
6270:
6267:
6260:
6254:
6250:
6244:
6239:
6234:
6206:
6202:
6196:
6186:
6181:
6177:
6167:
6153:
6133:
6111:
6103:
6100:
6097:
6094:
6091:
6088:
6085:
6082:
6079:
6054:
6046:
6043:
6040:
6037:
6034:
6031:
6028:
6025:
6022:
5999:
5937:
5932:
5924:
5918:
5914:
5907:
5901:
5897:
5888:
5882:
5878:
5875:
5868:
5862:
5858:
5852:
5847:
5842:
5831:
5826:
5820:
5815:
5808:
5803:
5797:
5792:
5785:
5780:
5774:
5769:
5762:
5757:
5750:
5745:
5738:
5733:
5726:
5721:
5715:
5710:
5683:
5663:
5660:
5657:
5650:
5634:
5629:
5623:
5618:
5611:
5606:
5597:
5592:
5585:
5574:
5568:
5564:
5557:
5551:
5547:
5541:
5536:
5533:
5530:
5521:
5505:
5499:
5494:
5487:
5482:
5475:
5470:
5463:
5458:
5452:
5428:
5422:
5417:
5410:
5405:
5396:
5390:
5386:
5383:
5376:
5370:
5366:
5357:
5351:
5347:
5340:
5334:
5330:
5324:
5319:
5316:
5307:
5291:
5285:
5280:
5275:
5270:
5263:
5258:
5253:
5248:
5241:
5236:
5231:
5226:
5219:
5214:
5209:
5204:
5198:
5192:
5186:
5180:
5174:
5169:
5164:
5159:
5152:
5147:
5142:
5137:
5130:
5125:
5120:
5115:
5108:
5103:
5098:
5093:
5087:
5064:
5044:
5039:
5033:
5028:
5023:
5018:
5011:
5006:
5001:
4996:
4989:
4984:
4979:
4974:
4967:
4962:
4957:
4952:
4946:
4939:
4933:
4928:
4923:
4918:
4911:
4906:
4901:
4896:
4889:
4884:
4879:
4874:
4867:
4862:
4857:
4852:
4846:
4839:
4833:
4828:
4823:
4818:
4811:
4806:
4801:
4796:
4789:
4784:
4779:
4774:
4767:
4762:
4757:
4752:
4745:
4740:
4735:
4730:
4724:
4719:
4716:
4709:
4693:
4690:
4687:
4684:
4681:
4678:
4675:
4655:
4650:
4644:
4639:
4634:
4629:
4622:
4617:
4612:
4607:
4600:
4595:
4590:
4585:
4578:
4573:
4568:
4563:
4557:
4552:
4549:
4544:
4540:
4536:
4533:
4511:
4505:
4500:
4495:
4490:
4483:
4478:
4473:
4468:
4461:
4456:
4451:
4446:
4439:
4434:
4429:
4424:
4417:
4412:
4407:
4402:
4396:
4391:
4388:
4374:
4360:
4355:
4351:
4347:
4344:
4336:
4320:
4296:
4288:
4272:
4250:
4246:
4242:
4236:
4233:
4225:
4208:
4204:
4198:
4188:
4183:
4179:
4153:
4150:
4142:
4125:
4098:
4095:
4092:
4052:
4049:
4046:
4004:
4001:
3998:
3978:
3973:
3955:
3952:
3949:
3942:
3938:
3923:
3920:
3900:
3897:
3894:
3886:
3885:
3884:
3870:
3867:
3847:
3844:
3841:
3821:
3818:
3813:
3805:
3802:
3796:
3793:
3790:
3785:
3781:
3775:
3771:
3767:
3764:
3761:
3756:
3752:
3731:
3728:
3725:
3717:
3699:
3696:
3693:
3683:
3680:
3660:
3657:
3654:
3634:
3631:
3628:
3606:
3603:
3600:
3590:
3587:
3582:
3578:
3574:
3571:
3562:
3548:
3544:
3540:
3535:
3531:
3527:
3524:
3504:
3500:
3496:
3493:
3490:
3487:
3465:
3462:
3459:
3449:
3444:
3440:
3436:
3414:
3411:
3408:
3398:
3395:
3390:
3386:
3378:
3377:normal matrix
3375:
3359:
3356:
3353:
3350:
3345:
3341:
3320:
3317:
3314:
3311:
3308:
3288:
3274:
3260:
3257:
3254:
3232:
3229:
3226:
3216:
3213:
3205:
3201:
3192:
3176:
3173:
3170:
3141:
3138:
3135:
3108:
3100:
3084:
3064:
3044:
3024:
3019:
3013:
3006:
2999:
2993:
2988:
2983:
2980:
2971:
2950:
2945:
2939:
2936:
2929:
2922:
2916:
2911:
2906:
2903:
2894:
2857:
2854:
2851:
2831:
2826:
2820:
2813:
2807:
2802:
2797:
2794:
2791:
2780:
2775:
2769:
2766:
2759:
2753:
2748:
2743:
2740:
2737:
2710:
2707:
2704:
2682:
2676:
2671:
2664:
2659:
2653:
2648:
2645:
2642:
2637:
2633:
2612:
2607:
2601:
2596:
2589:
2584:
2577:
2572:
2566:
2561:
2558:
2538:
2533:
2527:
2520:
2513:
2507:
2502:
2497:
2494:
2491:
2480:
2475:
2469:
2462:
2455:
2449:
2444:
2439:
2436:
2433:
2422:
2417:
2411:
2408:
2401:
2394:
2388:
2383:
2378:
2375:
2372:
2345:
2342:
2339:
2336:
2333:
2311:
2305:
2300:
2295:
2288:
2283:
2278:
2271:
2266:
2261:
2255:
2250:
2247:
2233:
2219:
2211:
2207:
2191:
2183:
2165:
2161:
2135:
2116:
2111:
2101:
2069:
2065:
2058:
2055:
2050:
2046:
2037:
2032:
2018:
2015:
2012:
2004:
1988:
1965:
1959:
1956:
1951:
1947:
1926:
1923:
1918:
1914:
1893:
1890:
1887:
1867:
1862:
1858:
1853:
1849:
1846:
1841:
1837:
1833:
1827:
1821:
1814:
1798:
1795:
1790:
1786:
1765:
1762:
1759:
1739:
1719:
1699:
1696:
1693:
1673:
1670:
1667:
1658:
1639:
1618:
1608:
1601:
1596:
1577:
1554:
1546:
1530:
1510:
1507:
1504:
1482:
1479:
1476:
1466:
1463:
1439:
1431:
1410:
1391:
1386:
1376:
1361:
1345:
1341:
1337:
1332:
1322:
1296:
1286:
1283:
1278:
1254:
1238:
1226:
1222:
1218:
1213:
1203:
1200:
1195:
1191:
1182:
1168:
1146:
1142:
1121:
1118:
1113:
1109:
1088:
1085:
1082:
1074:
1073:
1072:
1070:
1052:
1049:
1046:
1036:
1033:
1013:
1010:
1007:
987:
965:
962:
959:
949:
946:
921:
918:
910:
902:
892:
890:
886:
881:
879:
863:
849:
841:
817:
807:
790:
787:
784:
767:
748:
745:
742:
731:
727:
722:
684:
676:
652:
631:
628:
625:
622:
611:
601:
588:
580:
576:
572:
567:
563:
559:
553:
548:
545:
542:
526:
514:
492:
471:
462:
460:
452:
448:
445:
441:
437:
434:
430:
429:
428:
409:
405:
401:
398:
395:
392:
389:
384:
380:
373:
343:
340:
337:
327:
326:inner product
300:
299:Hilbert space
296:
280:
266:
264:
260:
256:
252:
248:
243:
239:
235:
231:
227:
226:Lord Rayleigh
223:
219:
218:Rayleigh–Ritz
215:
210:
209:Lord Rayleigh
206:
202:
192:
190:
186:
182:
178:
174:
170:
166:
162:
157:
155:
151:
147:
142:
140:
136:
135:Lord Rayleigh
132:
128:
124:
113:
110:
102:
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
11625:
11621:
11608:
11575:
11555:
11548:
11539:
11529:
11520:
11514:
11505:
11495:
11486:
11476:
11466:
11434:
11416:
11380:
11359:
11314:
11310:
11276:
11272:
11245:
11241:
11228:
11224:
11131:
11097:
11065:
11045:
11033:
11017:
11012:
10886:
10881:
10879:
10715:
10553:
10549:
10545:
10539:
10532:
10525:
10522:
10512:
10510:
10457:
10155:
9992:
9609:
8983:
8700:
8673:
8628:
8624:
8534:
8533:
8505:
8445:
8444:
8387:
8247:
8104:
8100:
7960:
7959:
7954:
7515:
7409:
7407:
7400:
7397:
7276:
7249:
7241:
6168:
5522:
5309:Let us take
5308:
4381:The matrix
4380:
4312:
3939:Compute the
3887:Compute the
3563:
3376:
3280:
3203:
3198:
3099:column space
2551:Let us take
2239:
2033:
1659:
1547:above finds
1544:
1455:
1075:Compute the
900:
898:
882:
874:
808:
723:
602:
515:
463:
456:
272:
257:named after
230:Walther Ritz
217:
205:Walther Ritz
201:Walther Ritz
198:
158:
143:
139:Walther Ritz
122:
120:
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
11622:SIAM Review
11605:Ritz method
11377:Süli, Endre
11132:mass matrix
11029:spreadsheet
8451:determinant
7244:Hamiltonian
3716:orthonormal
2240:The matrix
1880:, the only
1069:orthonormal
675:compression
442:(as in the
438:A space of
169:Hamiltonian
150:compression
127:eigenvalues
11645:Categories
11444:0198534159
11394:0521007941
10542:mode shape
8508:values of
7403:orthogonal
4171:and right
866:Properties
185:eigenmodes
69:newspapers
11242:Phys. Rev
11142:ϵ
11036:partially
10999:ω
10979:ω
10959:ω
10922:ω
10895:ω
10804:∑
10761:ω
10725:∑
10610:ω
10575:ω
10479:ω
10475:−
10422:ω
10418:−
10323:−
10282:∂
10255:∂
10227:−
10216:∂
10189:∂
10071:∂
10067:∂
10046:∂
10035:ω
10031:∂
10020:becomes:
10002:ω
9874:∑
9864:∑
9756:∑
9746:∑
9581:⋯
9529:ω
9495:⋯
9436:⋯
9377:⋯
9285:∑
8993:ω
8967:ω
8964:
8924:ω
8921:
8902:ω
8877:≡
8836:ω
8748:ω
8745:
8692:flywheels
8682:of multi
8605:ε
8578:ε
8551:ε
8540:hermitian
8516:ε
8475:ε
8472:−
8396:ε
8365:…
8313:ε
8310:−
8262:∑
8203:ε
8200:−
8154:∑
8139:∗
8126:∂
8121:ε
8118:∂
8081:∗
8047:ε
8021:∗
7928:≡
7896:∗
7866:∑
7844:∑
7812:∗
7782:∑
7760:∑
7736:Ψ
7705:∑
7689:Ψ
7658:∑
7633:Ψ
7602:∑
7590:^
7570:Ψ
7539:∑
7524:ε
7493:Ψ
7462:∑
7455:Ψ
7425:Ψ
7378:⟩
7375:Ψ
7367:Ψ
7364:⟨
7359:⟩
7356:Ψ
7345:^
7331:Ψ
7328:⟨
7322:≤
7262:Ψ
6918:σ
6907:∗
6645:σ
6614:∗
6563:∗
6402:−
6268:−
6207:∗
6112:∗
6055:∗
5970:and from
5958:Σ
5876:−
5790:Σ
5673:Σ
5593:−
5384:−
5187:∗
4545:∗
4356:∗
4240:Σ
4209:∗
4096:×
4073:Σ
4050:×
4002:×
3964:Σ
3898:×
3845:×
3814:∗
3786:∗
3776:∗
3757:∗
3729:×
3697:×
3684:∈
3658:×
3632:×
3604:×
3591:∈
3583:∗
3549:σ
3536:∗
3505:σ
3463:×
3450:∈
3445:∗
3412:×
3399:∈
3391:∗
3374:Hermitian
3357:σ
3346:∗
3318:σ
3289:σ
3258:×
3230:×
3217:∈
3171:λ
3136:λ
2975:~
2972:λ
2962:~
2937:−
2898:~
2895:λ
2885:~
2792:μ
2767:−
2738:μ
2638:∗
2492:λ
2434:λ
2409:−
2373:λ
2162:μ
2129:~
2105:~
2102:λ
2059:ρ
2047:μ
1960:ρ
1948:μ
1863:∗
1842:∗
1822:ρ
1791:∗
1763:×
1697:×
1645:‖
1633:~
1612:~
1609:λ
1602:−
1590:~
1575:‖
1508:≤
1497:contains
1480:×
1467:∈
1404:~
1380:~
1377:λ
1342:μ
1326:~
1323:λ
1272:~
1223:μ
1196:∗
1147:∗
1114:∗
1086:×
1050:×
1037:∈
963:×
950:∈
922:λ
791:⋅
785:⋅
749:⋅
743:⋅
728:(such as
629:λ
577:φ
564:φ
406:φ
381:φ
344:⋅
338:⋅
99:June 2024
11584:. arXiv.
11219:(1909).
11178:See also
10562:velocity
7747:⟩
7649:⟨
7643:⟩
7534:⟨
5649:thin SVD
4708:thin SVD
3744:matrix
3621:of size
3247:of size
2038:is that
1134:, where
980:of size
838:will be
459:diagonal
265:naming.
11607:in the
11319:Bibcode
11281:Bibcode
11231:: 1–61.
11130:is the
11024:quartic
11013:assumed
10548:, e.g.
4377:Example
4111:matrix
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3913:matrix
3860:matrix
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2236:Example
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253:as in
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78:
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1694:N
1674:1
1671:=
1668:m
1640:i
1629:x
1619:i
1597:i
1586:x
1578:A
1555:k
1531:A
1511:m
1505:k
1483:m
1477:N
1472:C
1464:V
1452:.
1440:A
1416:)
1411:i
1400:x
1392:,
1387:i
1370:(
1346:i
1338:=
1333:i
1297:i
1292:y
1287:V
1284:=
1279:i
1268:x
1239:i
1234:y
1227:i
1219:=
1214:i
1209:y
1204:V
1201:A
1192:V
1169:V
1143:V
1122:V
1119:A
1110:V
1089:m
1083:m
1053:m
1047:N
1042:C
1034:V
1014:N
1008:m
988:N
966:N
960:N
955:C
947:A
926:x
919:=
915:x
911:A
850:T
823:L
818:T
794:)
788:,
782:(
777:A
752:)
746:,
740:(
707:L
685:T
658:L
653:T
632:u
626:=
623:u
617:L
612:T
589:.
586:)
581:j
573:,
568:i
560:T
557:(
554:=
549:j
546:,
543:i
539:)
532:L
527:T
523:(
498:L
493:T
472:T
415:}
410:n
402:,
399:.
396:.
393:.
390:,
385:1
377:{
374:=
369:L
347:)
341:,
335:(
310:H
281:T
112:)
106:(
101:)
97:(
87:·
80:·
73:·
66:·
39:.
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