5006:
3953:
5001:{\displaystyle \mathbf {H} (\Lambda )={\begin{bmatrix}{\dfrac {\partial ^{2}\Lambda }{\partial \lambda ^{2}}}&{\dfrac {\partial ^{2}\Lambda }{\partial \lambda \partial \mathbf {x} }}\\\left({\dfrac {\partial ^{2}\Lambda }{\partial \lambda \partial \mathbf {x} }}\right)^{\mathsf {T}}&{\dfrac {\partial ^{2}\Lambda }{\partial \mathbf {x} ^{2}}}\end{bmatrix}}={\begin{bmatrix}0&{\dfrac {\partial g}{\partial x_{1}}}&{\dfrac {\partial g}{\partial x_{2}}}&\cdots &{\dfrac {\partial g}{\partial x_{n}}}\\{\dfrac {\partial g}{\partial x_{1}}}&{\dfrac {\partial ^{2}\Lambda }{\partial x_{1}^{2}}}&{\dfrac {\partial ^{2}\Lambda }{\partial x_{1}\,\partial x_{2}}}&\cdots &{\dfrac {\partial ^{2}\Lambda }{\partial x_{1}\,\partial x_{n}}}\\{\dfrac {\partial g}{\partial x_{2}}}&{\dfrac {\partial ^{2}\Lambda }{\partial x_{2}\,\partial x_{1}}}&{\dfrac {\partial ^{2}\Lambda }{\partial x_{2}^{2}}}&\cdots &{\dfrac {\partial ^{2}\Lambda }{\partial x_{2}\,\partial x_{n}}}\\\vdots &\vdots &\vdots &\ddots &\vdots \\{\dfrac {\partial g}{\partial x_{n}}}&{\dfrac {\partial ^{2}\Lambda }{\partial x_{n}\,\partial x_{1}}}&{\dfrac {\partial ^{2}\Lambda }{\partial x_{n}\,\partial x_{2}}}&\cdots &{\dfrac {\partial ^{2}\Lambda }{\partial x_{n}^{2}}}\end{bmatrix}}={\begin{bmatrix}0&{\dfrac {\partial g}{\partial \mathbf {x} }}\\\left({\dfrac {\partial g}{\partial \mathbf {x} }}\right)^{\mathsf {T}}&{\dfrac {\partial ^{2}\Lambda }{\partial \mathbf {x} ^{2}}}\end{bmatrix}}}
1759:
1233:
1754:{\displaystyle \mathbf {H} _{f}={\begin{bmatrix}{\dfrac {\partial ^{2}f}{\partial x_{1}^{2}}}&{\dfrac {\partial ^{2}f}{\partial x_{1}\,\partial x_{2}}}&\cdots &{\dfrac {\partial ^{2}f}{\partial x_{1}\,\partial x_{n}}}\\{\dfrac {\partial ^{2}f}{\partial x_{2}\,\partial x_{1}}}&{\dfrac {\partial ^{2}f}{\partial x_{2}^{2}}}&\cdots &{\dfrac {\partial ^{2}f}{\partial x_{2}\,\partial x_{n}}}\\\vdots &\vdots &\ddots &\vdots \\{\dfrac {\partial ^{2}f}{\partial x_{n}\,\partial x_{1}}}&{\dfrac {\partial ^{2}f}{\partial x_{n}\,\partial x_{2}}}&\cdots &{\dfrac {\partial ^{2}f}{\partial x_{n}^{2}}}\end{bmatrix}}.}
8575:
7103:
2453:(determinants of sub-matrices) of the Hessian; these conditions are a special case of those given in the next section for bordered Hessians for constrained optimization—the case in which the number of constraints is zero. Specifically, the sufficient condition for a minimum is that all of these principal minors be positive, while the sufficient condition for a maximum is that the minors alternate in sign, with the
6884:
2932:
3528:
6659:
7090:
2779:
3291:
3357:
6619:
2445:
can be used, because the determinant is the product of the eigenvalues. If it is positive, then the eigenvalues are both positive, or both negative. If it is negative, then the two eigenvalues have different signs. If it is zero, then the second-derivative test is inconclusive.
6148:
5952:
6928:
5099:
The above rules stating that extrema are characterized (among critical points with a non-singular
Hessian) by a positive-definite or negative-definite Hessian cannot apply here since a bordered Hessian can neither be negative-definite nor positive-definite, as
3166:
3051:
2389:
For positive-semidefinite and negative-semidefinite
Hessians the test is inconclusive (a critical point where the Hessian is semidefinite but not definite may be a local extremum or a saddle point). However, more can be said from the point of view of
6498:
1869:
6879:{\displaystyle \operatorname {Hess} (f)=\nabla _{i}\,\partial _{j}f\ dx^{i}\!\otimes \!dx^{j}=\left({\frac {\partial ^{2}f}{\partial x^{i}\partial x^{j}}}-\Gamma _{ij}^{k}{\frac {\partial f}{\partial x^{k}}}\right)dx^{i}\otimes dx^{j}}
3658:
so if the gradient is already computed, the approximate
Hessian can be computed by a linear (in the size of the gradient) number of scalar operations. (While simple to program, this approximation scheme is not numerically stable since
3656:
2927:{\displaystyle y=f(\mathbf {x} +\Delta \mathbf {x} )\approx f(\mathbf {x} )+\nabla f(\mathbf {x} )^{\mathrm {T} }\Delta \mathbf {x} +{\frac {1}{2}}\,\Delta \mathbf {x} ^{\mathrm {T} }\mathbf {H} (\mathbf {x} )\,\Delta \mathbf {x} }
2024:
6367:
3523:{\displaystyle \mathbf {H} (\mathbf {x} )\,\Delta \mathbf {x} =\mathbf {H} (\mathbf {x} )r\mathbf {v} =r\mathbf {H} (\mathbf {x} )\mathbf {v} =\nabla f(\mathbf {x} +r\mathbf {v} )-\nabla f(\mathbf {x} )+{\mathcal {O}}(r^{2}),}
3949:
5145:
6029:
5833:
5828:
3723:
a scalar factor and small random fluctuations. This result has been formally proven for a single-parent strategy and a static model, as the population size increases, relying on the quadratic approximation.
2751:
of the function considered as a manifold. The eigenvalues of the
Hessian at that point are the principal curvatures of the function, and the eigenvectors are the principal directions of curvature. (See
6231:
5452:
118:
6293:
1057:
3329:
1136:
1094:
5308:
3161:
3090:
1772:
5457:
Specifically, sign conditions are imposed on the sequence of leading principal minors (determinants of upper-left-justified sub-matrices) of the bordered
Hessian, for which the first
2963:
6654:
6493:
5367:
3533:
3857:
7085:{\displaystyle \operatorname {Hess} (f)(X,Y)=\langle \nabla _{X}\operatorname {grad} f,Y\rangle \quad {\text{ and }}\quad \operatorname {Hess} (f)(X,Y)=X(Yf)-df(\nabla _{X}Y).}
3710:
6919:
3286:{\displaystyle \nabla f(\mathbf {x} +\Delta \mathbf {x} )=\nabla f(\mathbf {x} )+\mathbf {H} (\mathbf {x} )\,\Delta \mathbf {x} +{\mathcal {O}}(\|\Delta \mathbf {x} \|^{2})}
2591:
2534:
2386:
Otherwise the test is inconclusive. This implies that at a local minimum the
Hessian is positive-semidefinite, and at a local maximum the Hessian is negative-semidefinite.
6024:
5167:
2643:
2617:
1182:
5673:
5978:
5716:
5054:
2477:
1228:
2955:
6455:
2401:
for functions of one and two variables is simpler than the general case. In one variable, the
Hessian contains exactly one second derivative; if it is positive, then
2449:
Equivalently, the second-order conditions that are sufficient for a local minimum or maximum can be expressed in terms of the sequence of principal (upper-leftmost)
6431:
6174:
5594:
5565:
5536:
5507:
5742:
5620:
5242:
5196:
2086:
6614:{\displaystyle \operatorname {Hess} (f)\in \Gamma \left(T^{*}M\otimes T^{*}M\right)\quad {\text{ by }}\quad \operatorname {Hess} (f):=\nabla \nabla f=\nabla df,}
5478:
3352:
2724:
2697:
2670:
2384:
2333:
2290:
2263:
2220:
2146:
6387:
6002:
5770:
5216:
5094:
5074:
5028:
3817:
3797:
3677:
2554:
2509:
2439:
2419:
2357:
2310:
2240:
2190:
2126:
2056:
1941:
1913:
1202:
1160:
3118:
6621:
where this takes advantage of the fact that the first covariant derivative of a function is the same as its ordinary differential. Choosing local coordinates
1946:
6298:
2266:
3117:
algorithms have been developed. The latter family of algorithms use approximations to the
Hessian; one of the most popular quasi-Newton algorithms is
8233:
3865:
5103:
214:
5778:
5744:
these conditions coincide with the conditions for the unbordered
Hessian to be negative definite or positive definite respectively).
8447:
7666:
8538:
6190:
1004:
and later named after him. Hesse originally used the term "functional determinants". The
Hessian is sometimes denoted by H or,
7572:
7539:
7508:
7284:
7256:
7204:
7169:
5372:
6143:{\displaystyle \mathbf {H} (\mathbf {f} )=\left(\mathbf {H} (f_{1}),\mathbf {H} (f_{2}),\ldots ,\mathbf {H} (f_{m})\right).}
5947:{\displaystyle \mathbf {f} (\mathbf {x} )=\left(f_{1}(\mathbf {x} ),f_{2}(\mathbf {x} ),\ldots ,f_{m}(\mathbf {x} )\right),}
8457:
8223:
475:
455:
6236:
2753:
1019:
3298:
2769:
951:
514:
1099:
1062:
37:
7591:
7474:
7231:
5247:
470:
193:
460:
1000:
of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician
8626:
8611:
8258:
3130:
791:
465:
445:
127:
7805:
3057:
3046:{\displaystyle \left({\frac {\partial f}{\partial x_{1}}},\ldots ,{\frac {\partial f}{\partial x_{n}}}\right).}
2156:
1878:
3745:
7618:
2166:
573:
520:
406:
8022:
7659:
7608:
6390:
2558:
2170:
1005:
232:
204:
6624:
315:
8631:
8097:
7613:
6464:
5313:
2747:
The determinant of the Hessian matrix, when evaluated at a critical point of a function, is equal to the
824:
437:
275:
247:
3822:
3779:
is used for the second-derivative test in certain constrained optimization problems. Given the function
8253:
7775:
7117:
3682:
695:
659:
441:
320:
209:
199:
6889:
5172:
The second derivative test consists here of sign restrictions of the determinants of a certain set of
2100:
of the curve are exactly the non-singular points where the Hessian determinant is zero. It follows by
8357:
8228:
8142:
7531:
7525:
7442:"Matrix differential calculus with applications in the multivariate linear model and its diagnostics"
6184:
3098:
2765:
3163:
and proceed by first noticing that the Hessian also appears in the local expansion of the gradient:
300:
8462:
8352:
8060:
7740:
3102:
2571:
2514:
2196:
594:
159:
6007:
5150:
2626:
2600:
1864:{\displaystyle (\mathbf {H} _{f})_{i,j}={\frac {\partial ^{2}f}{\partial x_{i}\,\partial x_{j}}}.}
1165:
8497:
8426:
8308:
8168:
7765:
7652:
7440:
Liu, Shuangzhe; Leiva, Victor; Zhuang, Dan; Ma, Tiefeng; Figueroa-Zúñiga, Jorge I. (March 2022).
5633:
3110:
908:
700:
589:
5623:
8367:
7950:
7755:
6458:
5957:
5682:
5033:
3124:
Such approximations may use the fact that an optimization algorithm uses the Hessian only as a
2456:
2450:
2398:
2093:
2059:
1207:
989:
944:
873:
834:
718:
654:
578:
7221:
7194:
2937:
2101:
8621:
8313:
8050:
7900:
7895:
7730:
7705:
7700:
7122:
6440:
5480:
leading principal minors are neglected, the smallest minor consisting of the truncated first
3757:
3741:
918:
584:
360:
305:
266:
172:
3651:{\displaystyle \mathbf {H} (\mathbf {x} )\mathbf {v} ={\frac {1}{r}}\left+{\mathcal {O}}(r)}
8507:
7865:
7695:
7675:
7500:
6404:
923:
903:
829:
498:
422:
396:
310:
7301:
6153:
5570:
5541:
5512:
5483:
1873:
If furthermore the second partial derivatives are all continuous, the Hessian matrix is a
8:
8528:
8502:
8080:
7885:
7875:
6922:
6434:
5721:
5599:
5221:
5175:
3860:
3114:
2065:
898:
868:
858:
745:
599:
401:
257:
140:
135:
5460:
4171:
3334:
2706:
2679:
2652:
2441:
is a local maximum; if it is zero, then the test is inconclusive. In two variables, the
2366:
2315:
2272:
2245:
2202:
2131:
8579:
8533:
8523:
8477:
8472:
8401:
8337:
8203:
7940:
7935:
7870:
7860:
7725:
7422:
7355:
7324:
7108:
6372:
5987:
5755:
5201:
5079:
5059:
5013:
3802:
3782:
3716:
3662:
2748:
2737:
2733:
2539:
2494:
2424:
2404:
2342:
2295:
2225:
2175:
2111:
2041:
2019:{\displaystyle \mathbf {H} (f(\mathbf {x} ))=\mathbf {J} (\nabla f(\mathbf {x} ))^{T}.}
1926:
1898:
1187:
1145:
1139:
985:
863:
766:
750:
690:
685:
680:
644:
525:
449:
355:
350:
154:
149:
8616:
8590:
8574:
8377:
8372:
8362:
8342:
8303:
8298:
8127:
8122:
8107:
8102:
8093:
8088:
8035:
7930:
7880:
7825:
7795:
7790:
7770:
7760:
7720:
7627:
7587:
7568:
7535:
7504:
7414:
7406:
7386:
7280:
7252:
7227:
7200:
7175:
7165:
7102:
6362:{\displaystyle {\frac {\partial ^{2}f}{\partial z_{i}\partial {\overline {z_{j}}}}}.}
3106:
2105:
2089:
937:
771:
549:
432:
385:
242:
237:
7584:
Matrix Differential Calculus : With Applications in Statistics and Econometrics
8585:
8553:
8482:
8421:
8416:
8396:
8332:
8238:
8208:
8193:
8173:
8112:
8065:
8040:
8030:
8001:
7920:
7915:
7890:
7820:
7800:
7710:
7690:
7561:
7453:
7426:
7398:
7365:
7328:
7316:
7137:
3733:
2773:
2564:
2097:
1874:
781:
675:
649:
510:
427:
391:
8178:
8283:
8218:
8198:
8183:
8163:
8147:
8045:
7976:
7966:
7925:
7810:
7780:
7157:
7131: – Matrix of all first-order partial derivatives of a vector-valued function
7128:
5538:
rows and columns, and so on, with the last being the entire bordered Hessian; if
3737:
3732:
The Hessian matrix is commonly used for expressing image processing operators in
3125:
2162:
1916:
913:
786:
740:
735:
622:
535:
480:
5630:
is that these minors alternate in sign with the smallest one having the sign of
5596:
then the smallest leading principal minor is the Hessian itself. There are thus
2772:-type methods because they are the coefficient of the quadratic term of a local
1257:
8543:
8487:
8467:
8452:
8411:
8288:
8213:
8137:
8076:
8055:
7996:
7986:
7971:
7905:
7850:
7840:
7835:
7745:
7458:
7441:
5622:
minors to consider, each evaluated at the specific point being considered as a
604:
376:
7402:
7370:
7343:
7320:
8605:
8548:
8406:
8347:
8278:
8268:
8263:
8188:
8117:
7991:
7981:
7910:
7830:
7815:
7750:
7630:
7497:
Matrix Differential Calculus with Applications in Statistics and Econometrics
7410:
7272:
7179:
3094:
981:
776:
540:
295:
252:
8431:
8388:
8293:
8006:
7945:
7855:
7735:
7418:
5773:
2729:
2620:
2391:
2360:
2128:
inflection points, since the Hessian determinant is a polynomial of degree
993:
530:
280:
7220:
Casciaro, B.; Fortunato, D.; Francaviglia, M.; Masiello, A., eds. (2011).
8273:
8243:
8011:
7845:
7715:
7246:
3944:{\displaystyle \Lambda (\mathbf {x} ,\lambda )=f(\mathbf {x} )+\lambda :}
3753:
3749:
2594:
2442:
1885:
893:
3760:. It can also be used in local sensitivity and statistical diagnostics.
8324:
7785:
6925:
of the connection. Other equivalent forms for the Hessian are given by
3093:
memory, which is infeasible for high-dimensional functions such as the
2741:
2336:
1001:
965:
639:
563:
290:
285:
189:
8558:
8132:
7635:
7219:
3054:
997:
568:
558:
7582:
Magnus, Jan R.; Neudecker, Heinz (1999). "The Second Differential".
5140:{\displaystyle \mathbf {z} ^{\mathsf {T}}\mathbf {H} \mathbf {z} =0}
8492:
7360:
2958:
2488:
1920:
634:
381:
338:
27:
5823:{\displaystyle \mathbf {f} :\mathbb {R} ^{n}\to \mathbb {R} ^{m},}
5218:
constraints can be thought of as reducing the problem to one with
3719:'s covariance matrix adapts to the inverse of the Hessian matrix,
2192:
is a local maximum, local minimum, or a saddle point, as follows:
7644:
5981:
5509:
rows and columns, the next consisting of the truncated first
3756:
analysis to calculate the different molecular frequencies in
3720:
7344:"On the covariance-Hessian relation in evolution strategies"
3712:
term, but decreasing it loses precision in the first term.)
5954:
then the collection of second partial derivatives is not a
7116:
The determinant of the Hessian matrix is a covariant; see
6396:
2754:
Gaussian curvature § Relation to principal curvatures
6226:{\displaystyle f\colon \mathbb {C} ^{n}\to \mathbb {C} ,}
6150:
This tensor degenerates to the usual Hessian matrix when
5030:
constraints then the zero in the upper-left corner is an
3799:
considered previously, but adding a constraint function
7475:"Econ 500: Quantitative Methods in Economic Analysis I"
7133:
Pages displaying short descriptions of redirect targets
6393:, then the complex Hessian matrix is identically zero.
5447:{\displaystyle f\left(x_{1},x_{2},1-x_{1}-x_{2}\right)}
3109:
with large numbers of parameters. For such situations,
7586:(Revised ed.). New York: Wiley. pp. 99–115.
7226:. Springer Science & Business Media. p. 178.
7199:. Springer Science & Business Media. p. 248.
5198:
submatrices of the bordered Hessian. Intuitively, the
5169:
is any vector whose sole non-zero entry is its first.
4873:
3979:
2491:(the vector of the partial derivatives) of a function
6931:
6892:
6662:
6627:
6501:
6467:
6443:
6407:
6375:
6301:
6239:
6193:
6178:
6156:
6032:
6010:
5990:
5960:
5836:
5781:
5758:
5724:
5685:
5636:
5602:
5573:
5544:
5515:
5486:
5463:
5375:
5316:
5250:
5224:
5204:
5178:
5153:
5106:
5082:
5062:
5036:
5016:
4953:
4915:
4882:
4817:
4760:
4708:
4677:
4596:
4548:
4496:
4465:
4411:
4354:
4311:
4280:
4247:
4211:
4180:
4118:
4067:
4021:
3983:
3956:
3868:
3825:
3805:
3785:
3685:
3665:
3536:
3360:
3337:
3301:
3169:
3133:
3060:
2966:
2940:
2782:
2709:
2682:
2655:
2629:
2603:
2574:
2542:
2517:
2497:
2459:
2427:
2407:
2369:
2345:
2318:
2298:
2275:
2248:
2228:
2205:
2178:
2169:. Refining this property allows us to test whether a
2134:
2114:
2068:
2044:
1949:
1929:
1901:
1775:
1700:
1643:
1591:
1515:
1467:
1415:
1361:
1304:
1261:
1236:
1210:
1190:
1168:
1148:
1102:
1065:
1022:
40:
7247:
Domenico P. L. Castrigiano; Sandra A. Hayes (2004).
7098:
3715:
Notably regarding Randomized Search Heuristics, the
3053:
Computing and storing the full Hessian matrix takes
7125:, useful for rapid calculations involving Hessians.
6495:be a smooth function. Define the Hessian tensor by
7560:
7084:
6913:
6878:
6648:
6613:
6487:
6449:
6425:
6381:
6361:
6288:{\displaystyle f\left(z_{1},\ldots ,z_{n}\right).}
6287:
6225:
6168:
6142:
6018:
5996:
5972:
5946:
5822:
5764:
5736:
5710:
5667:
5614:
5588:
5559:
5530:
5501:
5472:
5446:
5361:
5302:
5244:free variables. (For example, the maximization of
5236:
5210:
5190:
5161:
5139:
5088:
5068:
5048:
5022:
5000:
3943:
3851:
3811:
3791:
3704:
3671:
3650:
3522:
3346:
3323:
3285:
3155:
3084:
3045:
2949:
2926:
2718:
2691:
2664:
2637:
2611:
2585:
2548:
2528:
2503:
2471:
2433:
2413:
2378:
2351:
2327:
2304:
2284:
2257:
2234:
2214:
2184:
2140:
2120:
2080:
2050:
2018:
1935:
1907:
1863:
1753:
1222:
1196:
1176:
1154:
1130:
1088:
1052:{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }
1051:
112:
7387:"Calculation of the infrared spectra of proteins"
7341:
6725:
6721:
3679:has to be made small to prevent error due to the
3324:{\displaystyle \Delta \mathbf {x} =r\mathbf {v} }
8603:
7625:
7439:
2421:is a local minimum, and if it is negative, then
1131:{\displaystyle f(\mathbf {x} )\in \mathbb {R} .}
1089:{\displaystyle \mathbf {x} \in \mathbb {R} ^{n}}
113:{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}
7581:
7494:
7385:Mott, Adam J.; Rez, Peter (December 24, 2014).
5303:{\displaystyle f\left(x_{1},x_{2},x_{3}\right)}
3746:the determinant of Hessian (DoH) blob detector
2728:The Hessian matrix plays an important role in
2335:If the Hessian has both positive and negative
7660:
7527:Fundamental Methods of Mathematical Economics
7271:
7156:
5679:is that all of these minors have the sign of
2744:allow classification of the critical points.
2699:Otherwise it is non-degenerate, and called a
1011:
945:
6993:
6965:
6656:gives a local expression for the Hessian as
6004:Hessian matrices, one for each component of
3271:
3259:
7299:
7164:. Cambridge University Press. p. 190.
6187:, the Hessian may be generalized. Suppose
3859:the bordered Hessian is the Hessian of the
3156:{\displaystyle \mathbf {H} (\mathbf {v} ),}
16:(Mathematical) matrix of second derivatives
8234:Fundamental (linear differential equation)
7667:
7653:
7302:"Fast exact multiplication by the Hessian"
5747:
952:
938:
7495:Neudecker, Heinz; Magnus, Jan R. (1988).
7457:
7369:
7359:
7223:Recent Developments in General Relativity
6691:
6481:
6216:
6202:
5807:
5792:
4791:
4739:
4627:
4527:
4442:
4385:
3377:
3237:
3085:{\displaystyle \Theta \left(n^{2}\right)}
2915:
2881:
2764:Hessian matrices are used in large-scale
2150:
1841:
1674:
1622:
1546:
1446:
1392:
1335:
1230:matrix, usually defined and arranged as
1121:
1076:
1045:
1031:
73:
7192:
5984:. This can be thought of as an array of
8539:Matrix representation of conic sections
7530:(Third ed.). McGraw-Hill. p.
7384:
6397:Generalizations to Riemannian manifolds
1059:is a function taking as input a vector
476:Differentiating under the integral sign
8604:
7523:
7472:
5369:can be reduced to the maximization of
5115:
4944:
4109:
2759:
7648:
7626:
7558:
5626:. A sufficient condition for a local
3727:
2312:attains an isolated local maximum at
2242:attains an isolated local minimum at
6649:{\displaystyle \left\{x^{i}\right\}}
2033:
1888:of the Hessian matrix is called the
7196:Advanced Calculus: A Geometric View
6488:{\displaystyle f:M\to \mathbb {R} }
5675:A sufficient condition for a local
5362:{\displaystyle x_{1}+x_{2}+x_{3}=1}
3768:
2623:. If this determinant is zero then
13:
7674:
7552:
7342:Shir, O.M.; A. Yehudayoff (2020).
7064:
6969:
6894:
6826:
6818:
6798:
6778:
6765:
6751:
6693:
6682:
6599:
6590:
6587:
6520:
6444:
6333:
6320:
6306:
6179:Generalization to the complex case
4971:
4966:
4957:
4926:
4918:
4893:
4885:
4835:
4830:
4821:
4792:
4778:
4773:
4764:
4740:
4726:
4721:
4712:
4688:
4680:
4628:
4614:
4609:
4600:
4566:
4561:
4552:
4528:
4514:
4509:
4500:
4476:
4468:
4443:
4429:
4424:
4415:
4386:
4372:
4367:
4358:
4329:
4324:
4315:
4291:
4283:
4258:
4250:
4222:
4214:
4191:
4183:
4136:
4131:
4122:
4091:
4085:
4080:
4071:
4045:
4039:
4034:
4025:
4001:
3996:
3987:
3965:
3869:
3852:{\displaystyle g(\mathbf {x} )=c,}
3763:
3688:
3634:
3607:
3576:
3496:
3474:
3443:
3378:
3302:
3262:
3251:
3238:
3201:
3187:
3170:
3061:
3019:
3011:
2983:
2975:
2941:
2916:
2893:
2882:
2860:
2854:
2834:
2803:
2482:
1986:
1842:
1828:
1814:
1718:
1704:
1675:
1661:
1647:
1623:
1609:
1595:
1547:
1533:
1519:
1485:
1471:
1447:
1433:
1419:
1393:
1379:
1365:
1336:
1322:
1308:
1279:
1265:
22:Part of a series of articles about
14:
8643:
7601:
5980:matrix, but rather a third-order
3705:{\displaystyle {\mathcal {O}}(r)}
2062:in three variables, the equation
1895:The Hessian matrix of a function
8573:
7473:Hallam, Arne (October 7, 2004).
7446:Journal of Multivariate Analysis
7101:
6914:{\displaystyle \Gamma _{ij}^{k}}
6295:Then the generalized Hessian is
6112:
6082:
6058:
6042:
6034:
6012:
5929:
5899:
5875:
5846:
5838:
5783:
5155:
5127:
5122:
5109:
4976:
4930:
4897:
4141:
4095:
4049:
3958:
3922:
3899:
3876:
3833:
3617:
3597:
3586:
3554:
3546:
3538:
3484:
3464:
3453:
3436:
3428:
3420:
3409:
3398:
3390:
3382:
3370:
3362:
3317:
3306:
3266:
3242:
3230:
3222:
3211:
3191:
3180:
3143:
3135:
2920:
2908:
2900:
2887:
2864:
2844:
2824:
2807:
2796:
2631:
2605:
2576:
2519:
1996:
1979:
1965:
1951:
1781:
1239:
1170:
1110:
1067:
8441:Used in science and engineering
7567:. Singapore: World Scientific.
7517:
7488:
7466:
7002:
6996:
6568:
6562:
2619:is called, in some contexts, a
2028:
1162:exist, then the Hessian matrix
7684:Explicitly constrained entries
7433:
7378:
7335:
7300:Pearlmutter, Barak A. (1994).
7293:
7265:
7251:. Westview Press. p. 18.
7240:
7213:
7186:
7150:
7076:
7060:
7048:
7039:
7030:
7018:
7015:
7009:
6959:
6947:
6944:
6938:
6675:
6669:
6581:
6575:
6514:
6508:
6477:
6420:
6408:
6212:
6129:
6116:
6099:
6086:
6075:
6062:
6046:
6038:
5933:
5925:
5903:
5895:
5879:
5871:
5850:
5842:
5802:
5718:(In the unconstrained case of
5696:
5686:
5647:
5637:
5056:block of zeros, and there are
3968:
3962:
3935:
3926:
3918:
3912:
3903:
3895:
3886:
3872:
3837:
3829:
3699:
3693:
3645:
3639:
3621:
3613:
3601:
3582:
3550:
3542:
3514:
3501:
3488:
3480:
3468:
3449:
3432:
3424:
3402:
3394:
3374:
3366:
3280:
3256:
3234:
3226:
3215:
3207:
3195:
3176:
3147:
3139:
2912:
2904:
2849:
2840:
2828:
2820:
2811:
2792:
2157:Second partial derivative test
2004:
2000:
1992:
1983:
1972:
1969:
1961:
1955:
1879:symmetry of second derivatives
1792:
1776:
1114:
1106:
1041:
107:
101:
92:
86:
70:
64:
1:
8458:Fundamental (computer vision)
7162:Calculus Concepts and Methods
2586:{\displaystyle \mathbf {x} .}
2529:{\displaystyle \mathbf {x} ,}
407:Integral of inverse functions
7348:Theoretical Computer Science
6389:satisfies the n-dimensional
6348:
6019:{\displaystyle \mathbf {f} }
5624:candidate maximum or minimum
5162:{\displaystyle \mathbf {z} }
5096:border columns at the left.
2638:{\displaystyle \mathbf {x} }
2612:{\displaystyle \mathbf {x} }
1177:{\displaystyle \mathbf {H} }
7:
8224:Duplication and elimination
8023:eigenvalues or eigenvectors
7614:Encyclopedia of Mathematics
7391:European Biophysics Journal
7193:Callahan, James J. (2010).
7094:
5668:{\displaystyle (-1)^{m+1}.}
5076:border rows at the top and
825:Calculus on Euclidean space
248:Logarithmic differentiation
10:
8648:
8157:With specific applications
7786:Discrete Fourier Transform
7459:10.1016/j.jmva.2021.104849
7275:; Wright, Stephen (2000).
7118:Invariant of a binary form
5310:subject to the constraint
2154:
1761:That is, the entry of the
1012:Definitions and properties
8567:
8516:
8448:Cabibbo–Kobayashi–Maskawa
8440:
8386:
8322:
8156:
8075:Satisfying conditions on
8074:
8020:
7959:
7683:
7524:Chiang, Alpha C. (1984).
7403:10.1007/s00249-014-1005-6
7371:10.1016/j.tcs.2019.09.002
7321:10.1162/neco.1994.6.1.147
6391:Cauchy–Riemann conditions
6185:several complex variables
5973:{\displaystyle n\times n}
5711:{\displaystyle (-1)^{m}.}
5049:{\displaystyle m\times m}
3103:conditional random fields
2647:degenerate critical point
2472:{\displaystyle 1\times 1}
1223:{\displaystyle n\times n}
996:. It describes the local
559:Summand limit (term test)
7559:Lewis, David W. (1991).
7143:
2950:{\displaystyle \nabla f}
2776:of a function. That is,
2674:non-Morse critical point
2161:The Hessian matrix of a
1915:is the transpose of the
1096:and outputting a scalar
243:Implicit differentiation
233:Differentiation notation
160:Inverse function theorem
7806:Generalized permutation
7609:"Hessian of a function"
7160:; Davies, Joan (2007).
6450:{\displaystyle \nabla }
5748:Vector-valued functions
701:Helmholtz decomposition
8627:Multivariable calculus
8612:Differential operators
8580:Mathematics portal
7277:Numerical Optimization
7086:
6915:
6880:
6650:
6615:
6489:
6459:Levi-Civita connection
6451:
6427:
6383:
6363:
6289:
6227:
6170:
6144:
6020:
5998:
5974:
5948:
5824:
5766:
5738:
5712:
5669:
5616:
5590:
5561:
5532:
5503:
5474:
5448:
5363:
5304:
5238:
5212:
5192:
5163:
5141:
5090:
5070:
5050:
5024:
5002:
3945:
3853:
3813:
3793:
3706:
3673:
3652:
3524:
3348:
3325:
3287:
3157:
3086:
3047:
2951:
2928:
2720:
2693:
2666:
2639:
2613:
2587:
2550:
2530:
2511:is zero at some point
2505:
2479:minor being negative.
2473:
2435:
2415:
2399:second-derivative test
2380:
2353:
2329:
2306:
2286:
2259:
2236:
2216:
2186:
2167:positive semi-definite
2151:Second-derivative test
2142:
2122:
2094:plane projective curve
2082:
2060:homogeneous polynomial
2052:
2020:
1937:
1909:
1865:
1755:
1224:
1198:
1178:
1156:
1132:
1090:
1053:
835:Limit of distributions
655:Directional derivative
316:Faà di Bruno's formula
114:
7501:John Wiley & Sons
7354:. Elsevier: 157–174.
7123:Polarization identity
7087:
6916:
6881:
6651:
6616:
6490:
6452:
6428:
6426:{\displaystyle (M,g)}
6384:
6364:
6290:
6228:
6171:
6145:
6021:
5999:
5975:
5949:
5825:
5767:
5739:
5713:
5670:
5617:
5591:
5562:
5533:
5504:
5475:
5454:without constraint.)
5449:
5364:
5305:
5239:
5213:
5193:
5164:
5142:
5091:
5071:
5051:
5025:
5003:
3946:
3854:
3814:
3794:
3758:infrared spectroscopy
3752:). It can be used in
3744:(LoG) blob detector,
3742:Laplacian of Gaussian
3707:
3674:
3653:
3525:
3349:
3326:
3288:
3158:
3087:
3048:
2952:
2929:
2721:
2694:
2667:
2640:
2614:
2588:
2551:
2531:
2506:
2474:
2436:
2416:
2381:
2354:
2330:
2307:
2287:
2260:
2237:
2217:
2187:
2143:
2123:
2083:
2053:
2021:
1938:
1910:
1866:
1756:
1225:
1199:
1179:
1157:
1133:
1091:
1054:
919:Mathematical analysis
830:Generalized functions
515:arithmetico-geometric
361:Leibniz integral rule
115:
6929:
6890:
6660:
6625:
6499:
6465:
6441:
6405:
6373:
6299:
6237:
6191:
6169:{\displaystyle m=1.}
6154:
6030:
6008:
5988:
5958:
5834:
5779:
5756:
5722:
5683:
5634:
5600:
5589:{\displaystyle n+m,}
5571:
5560:{\displaystyle 2m+1}
5542:
5531:{\displaystyle 2m+2}
5513:
5502:{\displaystyle 2m+1}
5484:
5461:
5373:
5314:
5248:
5222:
5202:
5176:
5151:
5104:
5080:
5060:
5034:
5014:
3954:
3866:
3823:
3803:
3783:
3683:
3663:
3534:
3358:
3335:
3299:
3167:
3131:
3058:
2964:
2938:
2780:
2707:
2701:Morse critical point
2680:
2653:
2627:
2601:
2572:
2540:
2515:
2495:
2457:
2425:
2405:
2367:
2343:
2316:
2296:
2273:
2246:
2226:
2203:
2176:
2132:
2112:
2066:
2042:
1947:
1927:
1899:
1773:
1234:
1208:
1188:
1166:
1146:
1138:If all second-order
1100:
1063:
1020:
924:Nonstandard analysis
397:Lebesgue integration
267:Rules and identities
38:
8529:Linear independence
7776:Diagonally dominant
7279:. Springer Verlag.
6923:Christoffel symbols
6910:
6814:
6435:Riemannian manifold
5737:{\displaystyle m=0}
5615:{\displaystyle n-m}
5237:{\displaystyle n-m}
5191:{\displaystyle n-m}
5010:If there are, say,
4852:
4583:
4346:
2760:Use in optimization
2081:{\displaystyle f=0}
1890:Hessian determinant
1735:
1502:
1296:
1140:partial derivatives
988:of a scalar-valued
986:partial derivatives
976:or (less commonly)
595:Cauchy condensation
402:Contour integration
128:Fundamental theorem
55:
8632:Singularity theory
8534:Matrix exponential
8524:Jordan normal form
8358:Fisher information
8229:Euclidean distance
8143:Totally unimodular
7628:Weisstein, Eric W.
7309:Neural Computation
7249:Catastrophe theory
7109:Mathematics portal
7082:
6911:
6893:
6876:
6797:
6646:
6611:
6485:
6447:
6423:
6379:
6359:
6285:
6223:
6183:In the context of
6166:
6140:
6016:
5994:
5970:
5944:
5820:
5762:
5734:
5708:
5665:
5612:
5586:
5557:
5528:
5499:
5473:{\displaystyle 2m}
5470:
5444:
5359:
5300:
5234:
5208:
5188:
5159:
5137:
5086:
5066:
5046:
5020:
4998:
4992:
4988:
4936:
4903:
4859:
4855:
4838:
4807:
4755:
4703:
4643:
4586:
4569:
4543:
4491:
4458:
4401:
4349:
4332:
4306:
4273:
4237:
4206:
4157:
4153:
4101:
4055:
4016:
3941:
3849:
3809:
3789:
3728:Other applications
3717:evolution strategy
3702:
3669:
3648:
3520:
3347:{\displaystyle r,}
3344:
3321:
3283:
3153:
3107:statistical models
3082:
3043:
2947:
2924:
2749:Gaussian curvature
2734:catastrophe theory
2719:{\displaystyle f.}
2716:
2692:{\displaystyle f.}
2689:
2665:{\displaystyle f,}
2662:
2635:
2609:
2597:of the Hessian at
2583:
2546:
2526:
2501:
2469:
2431:
2411:
2379:{\displaystyle f.}
2376:
2349:
2328:{\displaystyle x.}
2325:
2302:
2285:{\displaystyle x,}
2282:
2265:If the Hessian is
2258:{\displaystyle x.}
2255:
2232:
2215:{\displaystyle x,}
2212:
2195:If the Hessian is
2182:
2141:{\displaystyle 3.}
2138:
2118:
2078:
2048:
2016:
1933:
1905:
1861:
1751:
1742:
1738:
1721:
1690:
1638:
1562:
1505:
1488:
1462:
1408:
1351:
1299:
1282:
1220:
1194:
1174:
1152:
1128:
1086:
1049:
767:Partial derivative
696:generalized Stokes
590:Alternating series
471:Reduction formulae
446:tangent half-angle
433:Cylindrical shells
356:Integral transform
351:Lists of integrals
155:Mean value theorem
110:
41:
8599:
8598:
8591:Category:Matrices
8463:Fuzzy associative
8353:Doubly stochastic
8061:Positive-definite
7741:Block tridiagonal
7574:978-981-02-0689-5
7541:978-0-07-010813-4
7510:978-0-471-91516-4
7286:978-0-387-98793-4
7258:978-0-8133-4126-2
7206:978-1-4419-7332-0
7171:978-0-521-77541-0
7000:
6840:
6792:
6707:
6566:
6382:{\displaystyle f}
6354:
6351:
5997:{\displaystyle m}
5765:{\displaystyle f}
5211:{\displaystyle m}
5089:{\displaystyle m}
5069:{\displaystyle m}
5023:{\displaystyle m}
4987:
4935:
4902:
4854:
4806:
4754:
4702:
4642:
4585:
4542:
4490:
4457:
4400:
4348:
4305:
4272:
4236:
4205:
4152:
4100:
4054:
4015:
3861:Lagrange function
3812:{\displaystyle g}
3792:{\displaystyle f}
3672:{\displaystyle r}
3569:
3033:
2997:
2879:
2549:{\displaystyle f}
2504:{\displaystyle f}
2434:{\displaystyle x}
2414:{\displaystyle x}
2352:{\displaystyle x}
2305:{\displaystyle f}
2267:negative-definite
2235:{\displaystyle f}
2197:positive-definite
2185:{\displaystyle x}
2121:{\displaystyle 9}
2106:cubic plane curve
2098:inflection points
2090:implicit equation
2051:{\displaystyle f}
2034:Inflection points
1936:{\displaystyle f}
1908:{\displaystyle f}
1856:
1737:
1689:
1637:
1561:
1504:
1461:
1407:
1350:
1298:
1197:{\displaystyle f}
1155:{\displaystyle f}
1006:ambiguously, by ∇
1002:Ludwig Otto Hesse
962:
961:
842:
841:
804:
803:
772:Multiple integral
708:
707:
612:
611:
579:Direct comparison
550:Convergence tests
488:
487:
461:Partial fractions
328:
327:
238:Second derivative
8639:
8586:List of matrices
8578:
8577:
8554:Row echelon form
8498:State transition
8427:Seidel adjacency
8309:Totally positive
8169:Alternating sign
7766:Complex Hadamard
7669:
7662:
7655:
7646:
7645:
7641:
7640:
7622:
7597:
7578:
7566:
7546:
7545:
7521:
7515:
7514:
7492:
7486:
7485:
7479:
7470:
7464:
7463:
7461:
7437:
7431:
7430:
7382:
7376:
7375:
7373:
7363:
7339:
7333:
7332:
7306:
7297:
7291:
7290:
7269:
7263:
7262:
7244:
7238:
7237:
7217:
7211:
7210:
7190:
7184:
7183:
7154:
7138:Hessian equation
7134:
7111:
7106:
7105:
7091:
7089:
7088:
7083:
7072:
7071:
7001:
6998:
6977:
6976:
6920:
6918:
6917:
6912:
6909:
6904:
6885:
6883:
6882:
6877:
6875:
6874:
6859:
6858:
6846:
6842:
6841:
6839:
6838:
6837:
6824:
6816:
6813:
6808:
6793:
6791:
6790:
6789:
6777:
6776:
6763:
6759:
6758:
6748:
6738:
6737:
6720:
6719:
6705:
6701:
6700:
6690:
6689:
6655:
6653:
6652:
6647:
6645:
6641:
6640:
6620:
6618:
6617:
6612:
6567:
6564:
6561:
6557:
6553:
6552:
6537:
6536:
6494:
6492:
6491:
6486:
6484:
6456:
6454:
6453:
6448:
6432:
6430:
6429:
6424:
6388:
6386:
6385:
6380:
6368:
6366:
6365:
6360:
6355:
6353:
6352:
6347:
6346:
6337:
6332:
6331:
6318:
6314:
6313:
6303:
6294:
6292:
6291:
6286:
6281:
6277:
6276:
6275:
6257:
6256:
6232:
6230:
6229:
6224:
6219:
6211:
6210:
6205:
6175:
6173:
6172:
6167:
6149:
6147:
6146:
6141:
6136:
6132:
6128:
6127:
6115:
6098:
6097:
6085:
6074:
6073:
6061:
6045:
6037:
6025:
6023:
6022:
6017:
6015:
6003:
6001:
6000:
5995:
5979:
5977:
5976:
5971:
5953:
5951:
5950:
5945:
5940:
5936:
5932:
5924:
5923:
5902:
5894:
5893:
5878:
5870:
5869:
5849:
5841:
5829:
5827:
5826:
5821:
5816:
5815:
5810:
5801:
5800:
5795:
5786:
5771:
5769:
5768:
5763:
5743:
5741:
5740:
5735:
5717:
5715:
5714:
5709:
5704:
5703:
5674:
5672:
5671:
5666:
5661:
5660:
5621:
5619:
5618:
5613:
5595:
5593:
5592:
5587:
5566:
5564:
5563:
5558:
5537:
5535:
5534:
5529:
5508:
5506:
5505:
5500:
5479:
5477:
5476:
5471:
5453:
5451:
5450:
5445:
5443:
5439:
5438:
5437:
5425:
5424:
5406:
5405:
5393:
5392:
5368:
5366:
5365:
5360:
5352:
5351:
5339:
5338:
5326:
5325:
5309:
5307:
5306:
5301:
5299:
5295:
5294:
5293:
5281:
5280:
5268:
5267:
5243:
5241:
5240:
5235:
5217:
5215:
5214:
5209:
5197:
5195:
5194:
5189:
5168:
5166:
5165:
5160:
5158:
5146:
5144:
5143:
5138:
5130:
5125:
5120:
5119:
5118:
5112:
5095:
5093:
5092:
5087:
5075:
5073:
5072:
5067:
5055:
5053:
5052:
5047:
5029:
5027:
5026:
5021:
5007:
5005:
5004:
4999:
4997:
4996:
4989:
4986:
4985:
4984:
4979:
4969:
4965:
4964:
4954:
4949:
4948:
4947:
4941:
4937:
4934:
4933:
4924:
4916:
4904:
4901:
4900:
4891:
4883:
4864:
4863:
4856:
4853:
4851:
4846:
4833:
4829:
4828:
4818:
4808:
4805:
4804:
4803:
4790:
4789:
4776:
4772:
4771:
4761:
4756:
4753:
4752:
4751:
4738:
4737:
4724:
4720:
4719:
4709:
4704:
4701:
4700:
4699:
4686:
4678:
4644:
4641:
4640:
4639:
4626:
4625:
4612:
4608:
4607:
4597:
4587:
4584:
4582:
4577:
4564:
4560:
4559:
4549:
4544:
4541:
4540:
4539:
4526:
4525:
4512:
4508:
4507:
4497:
4492:
4489:
4488:
4487:
4474:
4466:
4459:
4456:
4455:
4454:
4441:
4440:
4427:
4423:
4422:
4412:
4402:
4399:
4398:
4397:
4384:
4383:
4370:
4366:
4365:
4355:
4350:
4347:
4345:
4340:
4327:
4323:
4322:
4312:
4307:
4304:
4303:
4302:
4289:
4281:
4274:
4271:
4270:
4269:
4256:
4248:
4238:
4235:
4234:
4233:
4220:
4212:
4207:
4204:
4203:
4202:
4189:
4181:
4162:
4161:
4154:
4151:
4150:
4149:
4144:
4134:
4130:
4129:
4119:
4114:
4113:
4112:
4106:
4102:
4099:
4098:
4083:
4079:
4078:
4068:
4056:
4053:
4052:
4037:
4033:
4032:
4022:
4017:
4014:
4013:
4012:
3999:
3995:
3994:
3984:
3961:
3950:
3948:
3947:
3942:
3925:
3902:
3879:
3858:
3856:
3855:
3850:
3836:
3818:
3816:
3815:
3810:
3798:
3796:
3795:
3790:
3776:bordered Hessian
3769:Bordered Hessian
3734:image processing
3711:
3709:
3708:
3703:
3692:
3691:
3678:
3676:
3675:
3670:
3657:
3655:
3654:
3649:
3638:
3637:
3628:
3624:
3620:
3600:
3589:
3570:
3562:
3557:
3549:
3541:
3529:
3527:
3526:
3521:
3513:
3512:
3500:
3499:
3487:
3467:
3456:
3439:
3431:
3423:
3412:
3401:
3393:
3385:
3373:
3365:
3353:
3351:
3350:
3345:
3331:for some scalar
3330:
3328:
3327:
3322:
3320:
3309:
3292:
3290:
3289:
3284:
3279:
3278:
3269:
3255:
3254:
3245:
3233:
3225:
3214:
3194:
3183:
3162:
3160:
3159:
3154:
3146:
3138:
3111:truncated-Newton
3091:
3089:
3088:
3083:
3081:
3077:
3076:
3052:
3050:
3049:
3044:
3039:
3035:
3034:
3032:
3031:
3030:
3017:
3009:
2998:
2996:
2995:
2994:
2981:
2973:
2956:
2954:
2953:
2948:
2933:
2931:
2930:
2925:
2923:
2911:
2903:
2898:
2897:
2896:
2890:
2880:
2872:
2867:
2859:
2858:
2857:
2847:
2827:
2810:
2799:
2774:Taylor expansion
2768:problems within
2725:
2723:
2722:
2717:
2698:
2696:
2695:
2690:
2671:
2669:
2668:
2663:
2644:
2642:
2641:
2636:
2634:
2618:
2616:
2615:
2610:
2608:
2592:
2590:
2589:
2584:
2579:
2565:stationary point
2555:
2553:
2552:
2547:
2535:
2533:
2532:
2527:
2522:
2510:
2508:
2507:
2502:
2478:
2476:
2475:
2470:
2440:
2438:
2437:
2432:
2420:
2418:
2417:
2412:
2385:
2383:
2382:
2377:
2358:
2356:
2355:
2350:
2334:
2332:
2331:
2326:
2311:
2309:
2308:
2303:
2291:
2289:
2288:
2283:
2264:
2262:
2261:
2256:
2241:
2239:
2238:
2233:
2221:
2219:
2218:
2213:
2191:
2189:
2188:
2183:
2147:
2145:
2144:
2139:
2127:
2125:
2124:
2119:
2102:Bézout's theorem
2087:
2085:
2084:
2079:
2057:
2055:
2054:
2049:
2025:
2023:
2022:
2017:
2012:
2011:
1999:
1982:
1968:
1954:
1942:
1940:
1939:
1934:
1923:of the function
1914:
1912:
1911:
1906:
1875:symmetric matrix
1870:
1868:
1867:
1862:
1857:
1855:
1854:
1853:
1840:
1839:
1826:
1822:
1821:
1811:
1806:
1805:
1790:
1789:
1784:
1768:
1764:
1760:
1758:
1757:
1752:
1747:
1746:
1739:
1736:
1734:
1729:
1716:
1712:
1711:
1701:
1691:
1688:
1687:
1686:
1673:
1672:
1659:
1655:
1654:
1644:
1639:
1636:
1635:
1634:
1621:
1620:
1607:
1603:
1602:
1592:
1563:
1560:
1559:
1558:
1545:
1544:
1531:
1527:
1526:
1516:
1506:
1503:
1501:
1496:
1483:
1479:
1478:
1468:
1463:
1460:
1459:
1458:
1445:
1444:
1431:
1427:
1426:
1416:
1409:
1406:
1405:
1404:
1391:
1390:
1377:
1373:
1372:
1362:
1352:
1349:
1348:
1347:
1334:
1333:
1320:
1316:
1315:
1305:
1300:
1297:
1295:
1290:
1277:
1273:
1272:
1262:
1248:
1247:
1242:
1229:
1227:
1226:
1221:
1203:
1201:
1200:
1195:
1183:
1181:
1180:
1175:
1173:
1161:
1159:
1158:
1153:
1137:
1135:
1134:
1129:
1124:
1113:
1095:
1093:
1092:
1087:
1085:
1084:
1079:
1070:
1058:
1056:
1055:
1050:
1048:
1040:
1039:
1034:
984:of second-order
954:
947:
940:
888:
853:
819:
818:
815:
782:Surface integral
725:
724:
721:
629:
628:
625:
585:Limit comparison
505:
504:
501:
392:Riemann integral
345:
344:
341:
301:L'Hôpital's rule
258:Taylor's theorem
179:
178:
175:
119:
117:
116:
111:
63:
54:
49:
19:
18:
8647:
8646:
8642:
8641:
8640:
8638:
8637:
8636:
8602:
8601:
8600:
8595:
8572:
8563:
8512:
8436:
8382:
8318:
8152:
8070:
8016:
7955:
7756:Centrosymmetric
7679:
7673:
7607:
7604:
7594:
7575:
7555:
7553:Further reading
7550:
7549:
7542:
7522:
7518:
7511:
7503:. p. 136.
7493:
7489:
7477:
7471:
7467:
7438:
7434:
7383:
7379:
7340:
7336:
7304:
7298:
7294:
7287:
7270:
7266:
7259:
7245:
7241:
7234:
7218:
7214:
7207:
7191:
7187:
7172:
7155:
7151:
7146:
7132:
7129:Jacobian matrix
7107:
7100:
7097:
7067:
7063:
6999: and
6997:
6972:
6968:
6930:
6927:
6926:
6905:
6897:
6891:
6888:
6887:
6870:
6866:
6854:
6850:
6833:
6829:
6825:
6817:
6815:
6809:
6801:
6785:
6781:
6772:
6768:
6764:
6754:
6750:
6749:
6747:
6746:
6742:
6733:
6729:
6715:
6711:
6696:
6692:
6685:
6681:
6661:
6658:
6657:
6636:
6632:
6628:
6626:
6623:
6622:
6563:
6548:
6544:
6532:
6528:
6527:
6523:
6500:
6497:
6496:
6480:
6466:
6463:
6462:
6442:
6439:
6438:
6406:
6403:
6402:
6399:
6374:
6371:
6370:
6342:
6338:
6336:
6327:
6323:
6319:
6309:
6305:
6304:
6302:
6300:
6297:
6296:
6271:
6267:
6252:
6248:
6247:
6243:
6238:
6235:
6234:
6215:
6206:
6201:
6200:
6192:
6189:
6188:
6181:
6155:
6152:
6151:
6123:
6119:
6111:
6093:
6089:
6081:
6069:
6065:
6057:
6056:
6052:
6041:
6033:
6031:
6028:
6027:
6011:
6009:
6006:
6005:
5989:
5986:
5985:
5959:
5956:
5955:
5928:
5919:
5915:
5898:
5889:
5885:
5874:
5865:
5861:
5860:
5856:
5845:
5837:
5835:
5832:
5831:
5811:
5806:
5805:
5796:
5791:
5790:
5782:
5780:
5777:
5776:
5757:
5754:
5753:
5750:
5723:
5720:
5719:
5699:
5695:
5684:
5681:
5680:
5650:
5646:
5635:
5632:
5631:
5601:
5598:
5597:
5572:
5569:
5568:
5567:is larger than
5543:
5540:
5539:
5514:
5511:
5510:
5485:
5482:
5481:
5462:
5459:
5458:
5433:
5429:
5420:
5416:
5401:
5397:
5388:
5384:
5383:
5379:
5374:
5371:
5370:
5347:
5343:
5334:
5330:
5321:
5317:
5315:
5312:
5311:
5289:
5285:
5276:
5272:
5263:
5259:
5258:
5254:
5249:
5246:
5245:
5223:
5220:
5219:
5203:
5200:
5199:
5177:
5174:
5173:
5154:
5152:
5149:
5148:
5126:
5121:
5114:
5113:
5108:
5107:
5105:
5102:
5101:
5081:
5078:
5077:
5061:
5058:
5057:
5035:
5032:
5031:
5015:
5012:
5011:
4991:
4990:
4980:
4975:
4974:
4970:
4960:
4956:
4955:
4952:
4950:
4943:
4942:
4929:
4925:
4917:
4914:
4910:
4909:
4906:
4905:
4896:
4892:
4884:
4881:
4879:
4869:
4868:
4858:
4857:
4847:
4842:
4834:
4824:
4820:
4819:
4816:
4814:
4809:
4799:
4795:
4785:
4781:
4777:
4767:
4763:
4762:
4759:
4757:
4747:
4743:
4733:
4729:
4725:
4715:
4711:
4710:
4707:
4705:
4695:
4691:
4687:
4679:
4676:
4673:
4672:
4667:
4662:
4657:
4652:
4646:
4645:
4635:
4631:
4621:
4617:
4613:
4603:
4599:
4598:
4595:
4593:
4588:
4578:
4573:
4565:
4555:
4551:
4550:
4547:
4545:
4535:
4531:
4521:
4517:
4513:
4503:
4499:
4498:
4495:
4493:
4483:
4479:
4475:
4467:
4464:
4461:
4460:
4450:
4446:
4436:
4432:
4428:
4418:
4414:
4413:
4410:
4408:
4403:
4393:
4389:
4379:
4375:
4371:
4361:
4357:
4356:
4353:
4351:
4341:
4336:
4328:
4318:
4314:
4313:
4310:
4308:
4298:
4294:
4290:
4282:
4279:
4276:
4275:
4265:
4261:
4257:
4249:
4246:
4244:
4239:
4229:
4225:
4221:
4213:
4210:
4208:
4198:
4194:
4190:
4182:
4179:
4177:
4167:
4166:
4156:
4155:
4145:
4140:
4139:
4135:
4125:
4121:
4120:
4117:
4115:
4108:
4107:
4094:
4084:
4074:
4070:
4069:
4066:
4062:
4061:
4058:
4057:
4048:
4038:
4028:
4024:
4023:
4020:
4018:
4008:
4004:
4000:
3990:
3986:
3985:
3982:
3975:
3974:
3957:
3955:
3952:
3951:
3921:
3898:
3875:
3867:
3864:
3863:
3832:
3824:
3821:
3820:
3804:
3801:
3800:
3784:
3781:
3780:
3771:
3766:
3764:Generalizations
3738:computer vision
3730:
3687:
3686:
3684:
3681:
3680:
3664:
3661:
3660:
3633:
3632:
3616:
3596:
3585:
3575:
3571:
3561:
3553:
3545:
3537:
3535:
3532:
3531:
3508:
3504:
3495:
3494:
3483:
3463:
3452:
3435:
3427:
3419:
3408:
3397:
3389:
3381:
3369:
3361:
3359:
3356:
3355:
3336:
3333:
3332:
3316:
3305:
3300:
3297:
3296:
3274:
3270:
3265:
3250:
3249:
3241:
3229:
3221:
3210:
3190:
3179:
3168:
3165:
3164:
3142:
3134:
3132:
3129:
3128:
3126:linear operator
3072:
3068:
3064:
3059:
3056:
3055:
3026:
3022:
3018:
3010:
3008:
2990:
2986:
2982:
2974:
2972:
2971:
2967:
2965:
2962:
2961:
2939:
2936:
2935:
2919:
2907:
2899:
2892:
2891:
2886:
2885:
2871:
2863:
2853:
2852:
2848:
2843:
2823:
2806:
2795:
2781:
2778:
2777:
2762:
2708:
2705:
2704:
2681:
2678:
2677:
2654:
2651:
2650:
2630:
2628:
2625:
2624:
2604:
2602:
2599:
2598:
2575:
2573:
2570:
2569:
2541:
2538:
2537:
2518:
2516:
2513:
2512:
2496:
2493:
2492:
2485:
2483:Critical points
2458:
2455:
2454:
2426:
2423:
2422:
2406:
2403:
2402:
2368:
2365:
2364:
2344:
2341:
2340:
2317:
2314:
2313:
2297:
2294:
2293:
2274:
2271:
2270:
2247:
2244:
2243:
2227:
2224:
2223:
2204:
2201:
2200:
2177:
2174:
2173:
2163:convex function
2159:
2153:
2133:
2130:
2129:
2113:
2110:
2109:
2067:
2064:
2063:
2043:
2040:
2039:
2036:
2031:
2007:
2003:
1995:
1978:
1964:
1950:
1948:
1945:
1944:
1928:
1925:
1924:
1917:Jacobian matrix
1900:
1897:
1896:
1849:
1845:
1835:
1831:
1827:
1817:
1813:
1812:
1810:
1795:
1791:
1785:
1780:
1779:
1774:
1771:
1770:
1766:
1765:th row and the
1762:
1741:
1740:
1730:
1725:
1717:
1707:
1703:
1702:
1699:
1697:
1692:
1682:
1678:
1668:
1664:
1660:
1650:
1646:
1645:
1642:
1640:
1630:
1626:
1616:
1612:
1608:
1598:
1594:
1593:
1590:
1587:
1586:
1581:
1576:
1571:
1565:
1564:
1554:
1550:
1540:
1536:
1532:
1522:
1518:
1517:
1514:
1512:
1507:
1497:
1492:
1484:
1474:
1470:
1469:
1466:
1464:
1454:
1450:
1440:
1436:
1432:
1422:
1418:
1417:
1414:
1411:
1410:
1400:
1396:
1386:
1382:
1378:
1368:
1364:
1363:
1360:
1358:
1353:
1343:
1339:
1329:
1325:
1321:
1311:
1307:
1306:
1303:
1301:
1291:
1286:
1278:
1268:
1264:
1263:
1260:
1253:
1252:
1243:
1238:
1237:
1235:
1232:
1231:
1209:
1206:
1205:
1189:
1186:
1185:
1169:
1167:
1164:
1163:
1147:
1144:
1143:
1120:
1109:
1101:
1098:
1097:
1080:
1075:
1074:
1066:
1064:
1061:
1060:
1044:
1035:
1030:
1029:
1021:
1018:
1017:
1014:
958:
929:
928:
914:Integration Bee
889:
886:
879:
878:
854:
851:
844:
843:
816:
813:
806:
805:
787:Volume integral
722:
717:
710:
709:
626:
621:
614:
613:
583:
502:
497:
490:
489:
481:Risch algorithm
456:Euler's formula
342:
337:
330:
329:
311:General Leibniz
194:generalizations
176:
171:
164:
150:Rolle's theorem
145:
120:
56:
50:
45:
39:
36:
35:
17:
12:
11:
5:
8645:
8635:
8634:
8629:
8624:
8619:
8614:
8597:
8596:
8594:
8593:
8588:
8583:
8568:
8565:
8564:
8562:
8561:
8556:
8551:
8546:
8544:Perfect matrix
8541:
8536:
8531:
8526:
8520:
8518:
8514:
8513:
8511:
8510:
8505:
8500:
8495:
8490:
8485:
8480:
8475:
8470:
8465:
8460:
8455:
8450:
8444:
8442:
8438:
8437:
8435:
8434:
8429:
8424:
8419:
8414:
8409:
8404:
8399:
8393:
8391:
8384:
8383:
8381:
8380:
8375:
8370:
8365:
8360:
8355:
8350:
8345:
8340:
8335:
8329:
8327:
8320:
8319:
8317:
8316:
8314:Transformation
8311:
8306:
8301:
8296:
8291:
8286:
8281:
8276:
8271:
8266:
8261:
8256:
8251:
8246:
8241:
8236:
8231:
8226:
8221:
8216:
8211:
8206:
8201:
8196:
8191:
8186:
8181:
8176:
8171:
8166:
8160:
8158:
8154:
8153:
8151:
8150:
8145:
8140:
8135:
8130:
8125:
8120:
8115:
8110:
8105:
8100:
8091:
8085:
8083:
8072:
8071:
8069:
8068:
8063:
8058:
8053:
8051:Diagonalizable
8048:
8043:
8038:
8033:
8027:
8025:
8021:Conditions on
8018:
8017:
8015:
8014:
8009:
8004:
7999:
7994:
7989:
7984:
7979:
7974:
7969:
7963:
7961:
7957:
7956:
7954:
7953:
7948:
7943:
7938:
7933:
7928:
7923:
7918:
7913:
7908:
7903:
7901:Skew-symmetric
7898:
7896:Skew-Hermitian
7893:
7888:
7883:
7878:
7873:
7868:
7863:
7858:
7853:
7848:
7843:
7838:
7833:
7828:
7823:
7818:
7813:
7808:
7803:
7798:
7793:
7788:
7783:
7778:
7773:
7768:
7763:
7758:
7753:
7748:
7743:
7738:
7733:
7731:Block-diagonal
7728:
7723:
7718:
7713:
7708:
7706:Anti-symmetric
7703:
7701:Anti-Hermitian
7698:
7693:
7687:
7685:
7681:
7680:
7672:
7671:
7664:
7657:
7649:
7643:
7642:
7623:
7603:
7602:External links
7600:
7599:
7598:
7592:
7579:
7573:
7554:
7551:
7548:
7547:
7540:
7516:
7509:
7487:
7465:
7432:
7397:(3): 103–112.
7377:
7334:
7315:(1): 147–160.
7292:
7285:
7273:Nocedal, Jorge
7264:
7257:
7239:
7232:
7212:
7205:
7185:
7170:
7148:
7147:
7145:
7142:
7141:
7140:
7135:
7126:
7120:
7113:
7112:
7096:
7093:
7081:
7078:
7075:
7070:
7066:
7062:
7059:
7056:
7053:
7050:
7047:
7044:
7041:
7038:
7035:
7032:
7029:
7026:
7023:
7020:
7017:
7014:
7011:
7008:
7005:
6995:
6992:
6989:
6986:
6983:
6980:
6975:
6971:
6967:
6964:
6961:
6958:
6955:
6952:
6949:
6946:
6943:
6940:
6937:
6934:
6908:
6903:
6900:
6896:
6873:
6869:
6865:
6862:
6857:
6853:
6849:
6845:
6836:
6832:
6828:
6823:
6820:
6812:
6807:
6804:
6800:
6796:
6788:
6784:
6780:
6775:
6771:
6767:
6762:
6757:
6753:
6745:
6741:
6736:
6732:
6728:
6724:
6718:
6714:
6710:
6704:
6699:
6695:
6688:
6684:
6680:
6677:
6674:
6671:
6668:
6665:
6644:
6639:
6635:
6631:
6610:
6607:
6604:
6601:
6598:
6595:
6592:
6589:
6586:
6583:
6580:
6577:
6574:
6571:
6565: by
6560:
6556:
6551:
6547:
6543:
6540:
6535:
6531:
6526:
6522:
6519:
6516:
6513:
6510:
6507:
6504:
6483:
6479:
6476:
6473:
6470:
6446:
6422:
6419:
6416:
6413:
6410:
6398:
6395:
6378:
6358:
6350:
6345:
6341:
6335:
6330:
6326:
6322:
6317:
6312:
6308:
6284:
6280:
6274:
6270:
6266:
6263:
6260:
6255:
6251:
6246:
6242:
6222:
6218:
6214:
6209:
6204:
6199:
6196:
6180:
6177:
6165:
6162:
6159:
6139:
6135:
6131:
6126:
6122:
6118:
6114:
6110:
6107:
6104:
6101:
6096:
6092:
6088:
6084:
6080:
6077:
6072:
6068:
6064:
6060:
6055:
6051:
6048:
6044:
6040:
6036:
6014:
5993:
5969:
5966:
5963:
5943:
5939:
5935:
5931:
5927:
5922:
5918:
5914:
5911:
5908:
5905:
5901:
5897:
5892:
5888:
5884:
5881:
5877:
5873:
5868:
5864:
5859:
5855:
5852:
5848:
5844:
5840:
5819:
5814:
5809:
5804:
5799:
5794:
5789:
5785:
5761:
5749:
5746:
5733:
5730:
5727:
5707:
5702:
5698:
5694:
5691:
5688:
5678:
5664:
5659:
5656:
5653:
5649:
5645:
5642:
5639:
5629:
5611:
5608:
5605:
5585:
5582:
5579:
5576:
5556:
5553:
5550:
5547:
5527:
5524:
5521:
5518:
5498:
5495:
5492:
5489:
5469:
5466:
5442:
5436:
5432:
5428:
5423:
5419:
5415:
5412:
5409:
5404:
5400:
5396:
5391:
5387:
5382:
5378:
5358:
5355:
5350:
5346:
5342:
5337:
5333:
5329:
5324:
5320:
5298:
5292:
5288:
5284:
5279:
5275:
5271:
5266:
5262:
5257:
5253:
5233:
5230:
5227:
5207:
5187:
5184:
5181:
5157:
5136:
5133:
5129:
5124:
5117:
5111:
5085:
5065:
5045:
5042:
5039:
5019:
4995:
4983:
4978:
4973:
4968:
4963:
4959:
4951:
4946:
4940:
4932:
4928:
4923:
4920:
4913:
4908:
4907:
4899:
4895:
4890:
4887:
4880:
4878:
4875:
4874:
4872:
4867:
4862:
4850:
4845:
4841:
4837:
4832:
4827:
4823:
4815:
4813:
4810:
4802:
4798:
4794:
4788:
4784:
4780:
4775:
4770:
4766:
4758:
4750:
4746:
4742:
4736:
4732:
4728:
4723:
4718:
4714:
4706:
4698:
4694:
4690:
4685:
4682:
4675:
4674:
4671:
4668:
4666:
4663:
4661:
4658:
4656:
4653:
4651:
4648:
4647:
4638:
4634:
4630:
4624:
4620:
4616:
4611:
4606:
4602:
4594:
4592:
4589:
4581:
4576:
4572:
4568:
4563:
4558:
4554:
4546:
4538:
4534:
4530:
4524:
4520:
4516:
4511:
4506:
4502:
4494:
4486:
4482:
4478:
4473:
4470:
4463:
4462:
4453:
4449:
4445:
4439:
4435:
4431:
4426:
4421:
4417:
4409:
4407:
4404:
4396:
4392:
4388:
4382:
4378:
4374:
4369:
4364:
4360:
4352:
4344:
4339:
4335:
4331:
4326:
4321:
4317:
4309:
4301:
4297:
4293:
4288:
4285:
4278:
4277:
4268:
4264:
4260:
4255:
4252:
4245:
4243:
4240:
4232:
4228:
4224:
4219:
4216:
4209:
4201:
4197:
4193:
4188:
4185:
4178:
4176:
4173:
4172:
4170:
4165:
4160:
4148:
4143:
4138:
4133:
4128:
4124:
4116:
4111:
4105:
4097:
4093:
4090:
4087:
4082:
4077:
4073:
4065:
4060:
4059:
4051:
4047:
4044:
4041:
4036:
4031:
4027:
4019:
4011:
4007:
4003:
3998:
3993:
3989:
3981:
3980:
3978:
3973:
3970:
3967:
3964:
3960:
3940:
3937:
3934:
3931:
3928:
3924:
3920:
3917:
3914:
3911:
3908:
3905:
3901:
3897:
3894:
3891:
3888:
3885:
3882:
3878:
3874:
3871:
3848:
3845:
3842:
3839:
3835:
3831:
3828:
3808:
3788:
3777:
3770:
3767:
3765:
3762:
3729:
3726:
3701:
3698:
3695:
3690:
3668:
3647:
3644:
3641:
3636:
3631:
3627:
3623:
3619:
3615:
3612:
3609:
3606:
3603:
3599:
3595:
3592:
3588:
3584:
3581:
3578:
3574:
3568:
3565:
3560:
3556:
3552:
3548:
3544:
3540:
3519:
3516:
3511:
3507:
3503:
3498:
3493:
3490:
3486:
3482:
3479:
3476:
3473:
3470:
3466:
3462:
3459:
3455:
3451:
3448:
3445:
3442:
3438:
3434:
3430:
3426:
3422:
3418:
3415:
3411:
3407:
3404:
3400:
3396:
3392:
3388:
3384:
3380:
3376:
3372:
3368:
3364:
3343:
3340:
3319:
3315:
3312:
3308:
3304:
3282:
3277:
3273:
3268:
3264:
3261:
3258:
3253:
3248:
3244:
3240:
3236:
3232:
3228:
3224:
3220:
3217:
3213:
3209:
3206:
3203:
3200:
3197:
3193:
3189:
3186:
3182:
3178:
3175:
3172:
3152:
3149:
3145:
3141:
3137:
3095:loss functions
3080:
3075:
3071:
3067:
3063:
3042:
3038:
3029:
3025:
3021:
3016:
3013:
3007:
3004:
3001:
2993:
2989:
2985:
2980:
2977:
2970:
2946:
2943:
2922:
2918:
2914:
2910:
2906:
2902:
2895:
2889:
2884:
2878:
2875:
2870:
2866:
2862:
2856:
2851:
2846:
2842:
2839:
2836:
2833:
2830:
2826:
2822:
2819:
2816:
2813:
2809:
2805:
2802:
2798:
2794:
2791:
2788:
2785:
2761:
2758:
2736:, because its
2715:
2712:
2702:
2688:
2685:
2675:
2661:
2658:
2648:
2633:
2607:
2582:
2578:
2567:
2561:
2559:critical point
2545:
2525:
2521:
2500:
2484:
2481:
2468:
2465:
2462:
2430:
2410:
2375:
2372:
2348:
2324:
2321:
2301:
2281:
2278:
2254:
2251:
2231:
2211:
2208:
2181:
2171:critical point
2155:Main article:
2152:
2149:
2137:
2117:
2077:
2074:
2071:
2047:
2035:
2032:
2030:
2027:
2015:
2010:
2006:
2002:
1998:
1994:
1991:
1988:
1985:
1981:
1977:
1974:
1971:
1967:
1963:
1960:
1957:
1953:
1932:
1904:
1891:
1860:
1852:
1848:
1844:
1838:
1834:
1830:
1825:
1820:
1816:
1809:
1804:
1801:
1798:
1794:
1788:
1783:
1778:
1750:
1745:
1733:
1728:
1724:
1720:
1715:
1710:
1706:
1698:
1696:
1693:
1685:
1681:
1677:
1671:
1667:
1663:
1658:
1653:
1649:
1641:
1633:
1629:
1625:
1619:
1615:
1611:
1606:
1601:
1597:
1589:
1588:
1585:
1582:
1580:
1577:
1575:
1572:
1570:
1567:
1566:
1557:
1553:
1549:
1543:
1539:
1535:
1530:
1525:
1521:
1513:
1511:
1508:
1500:
1495:
1491:
1487:
1482:
1477:
1473:
1465:
1457:
1453:
1449:
1443:
1439:
1435:
1430:
1425:
1421:
1413:
1412:
1403:
1399:
1395:
1389:
1385:
1381:
1376:
1371:
1367:
1359:
1357:
1354:
1346:
1342:
1338:
1332:
1328:
1324:
1319:
1314:
1310:
1302:
1294:
1289:
1285:
1281:
1276:
1271:
1267:
1259:
1258:
1256:
1251:
1246:
1241:
1219:
1216:
1213:
1193:
1172:
1151:
1127:
1123:
1119:
1116:
1112:
1108:
1105:
1083:
1078:
1073:
1069:
1047:
1043:
1038:
1033:
1028:
1025:
1013:
1010:
970:Hessian matrix
960:
959:
957:
956:
949:
942:
934:
931:
930:
927:
926:
921:
916:
911:
909:List of topics
906:
901:
896:
890:
885:
884:
881:
880:
877:
876:
871:
866:
861:
855:
850:
849:
846:
845:
840:
839:
838:
837:
832:
827:
817:
812:
811:
808:
807:
802:
801:
800:
799:
794:
789:
784:
779:
774:
769:
761:
760:
756:
755:
754:
753:
748:
743:
738:
730:
729:
723:
716:
715:
712:
711:
706:
705:
704:
703:
698:
693:
688:
683:
678:
670:
669:
665:
664:
663:
662:
657:
652:
647:
642:
637:
627:
620:
619:
616:
615:
610:
609:
608:
607:
602:
597:
592:
587:
581:
576:
571:
566:
561:
553:
552:
546:
545:
544:
543:
538:
533:
528:
523:
518:
503:
496:
495:
492:
491:
486:
485:
484:
483:
478:
473:
468:
466:Changing order
463:
458:
453:
435:
430:
425:
417:
416:
415:Integration by
412:
411:
410:
409:
404:
399:
394:
389:
379:
377:Antiderivative
371:
370:
366:
365:
364:
363:
358:
353:
343:
336:
335:
332:
331:
326:
325:
324:
323:
318:
313:
308:
303:
298:
293:
288:
283:
278:
270:
269:
263:
262:
261:
260:
255:
250:
245:
240:
235:
227:
226:
222:
221:
220:
219:
218:
217:
212:
207:
197:
184:
183:
177:
170:
169:
166:
165:
163:
162:
157:
152:
146:
144:
143:
138:
132:
131:
130:
122:
121:
109:
106:
103:
100:
97:
94:
91:
88:
85:
82:
79:
76:
72:
69:
66:
62:
59:
53:
48:
44:
34:
31:
30:
24:
23:
15:
9:
6:
4:
3:
2:
8644:
8633:
8630:
8628:
8625:
8623:
8620:
8618:
8615:
8613:
8610:
8609:
8607:
8592:
8589:
8587:
8584:
8582:
8581:
8576:
8570:
8569:
8566:
8560:
8557:
8555:
8552:
8550:
8549:Pseudoinverse
8547:
8545:
8542:
8540:
8537:
8535:
8532:
8530:
8527:
8525:
8522:
8521:
8519:
8517:Related terms
8515:
8509:
8508:Z (chemistry)
8506:
8504:
8501:
8499:
8496:
8494:
8491:
8489:
8486:
8484:
8481:
8479:
8476:
8474:
8471:
8469:
8466:
8464:
8461:
8459:
8456:
8454:
8451:
8449:
8446:
8445:
8443:
8439:
8433:
8430:
8428:
8425:
8423:
8420:
8418:
8415:
8413:
8410:
8408:
8405:
8403:
8400:
8398:
8395:
8394:
8392:
8390:
8385:
8379:
8376:
8374:
8371:
8369:
8366:
8364:
8361:
8359:
8356:
8354:
8351:
8349:
8346:
8344:
8341:
8339:
8336:
8334:
8331:
8330:
8328:
8326:
8321:
8315:
8312:
8310:
8307:
8305:
8302:
8300:
8297:
8295:
8292:
8290:
8287:
8285:
8282:
8280:
8277:
8275:
8272:
8270:
8267:
8265:
8262:
8260:
8257:
8255:
8252:
8250:
8247:
8245:
8242:
8240:
8237:
8235:
8232:
8230:
8227:
8225:
8222:
8220:
8217:
8215:
8212:
8210:
8207:
8205:
8202:
8200:
8197:
8195:
8192:
8190:
8187:
8185:
8182:
8180:
8177:
8175:
8172:
8170:
8167:
8165:
8162:
8161:
8159:
8155:
8149:
8146:
8144:
8141:
8139:
8136:
8134:
8131:
8129:
8126:
8124:
8121:
8119:
8116:
8114:
8111:
8109:
8106:
8104:
8101:
8099:
8095:
8092:
8090:
8087:
8086:
8084:
8082:
8078:
8073:
8067:
8064:
8062:
8059:
8057:
8054:
8052:
8049:
8047:
8044:
8042:
8039:
8037:
8034:
8032:
8029:
8028:
8026:
8024:
8019:
8013:
8010:
8008:
8005:
8003:
8000:
7998:
7995:
7993:
7990:
7988:
7985:
7983:
7980:
7978:
7975:
7973:
7970:
7968:
7965:
7964:
7962:
7958:
7952:
7949:
7947:
7944:
7942:
7939:
7937:
7934:
7932:
7929:
7927:
7924:
7922:
7919:
7917:
7914:
7912:
7909:
7907:
7904:
7902:
7899:
7897:
7894:
7892:
7889:
7887:
7884:
7882:
7879:
7877:
7874:
7872:
7869:
7867:
7866:Pentadiagonal
7864:
7862:
7859:
7857:
7854:
7852:
7849:
7847:
7844:
7842:
7839:
7837:
7834:
7832:
7829:
7827:
7824:
7822:
7819:
7817:
7814:
7812:
7809:
7807:
7804:
7802:
7799:
7797:
7794:
7792:
7789:
7787:
7784:
7782:
7779:
7777:
7774:
7772:
7769:
7767:
7764:
7762:
7759:
7757:
7754:
7752:
7749:
7747:
7744:
7742:
7739:
7737:
7734:
7732:
7729:
7727:
7724:
7722:
7719:
7717:
7714:
7712:
7709:
7707:
7704:
7702:
7699:
7697:
7696:Anti-diagonal
7694:
7692:
7689:
7688:
7686:
7682:
7677:
7670:
7665:
7663:
7658:
7656:
7651:
7650:
7647:
7638:
7637:
7632:
7629:
7624:
7620:
7616:
7615:
7610:
7606:
7605:
7595:
7593:0-471-98633-X
7589:
7585:
7580:
7576:
7570:
7565:
7564:
7563:Matrix Theory
7557:
7556:
7543:
7537:
7533:
7529:
7528:
7520:
7512:
7506:
7502:
7498:
7491:
7483:
7476:
7469:
7460:
7455:
7451:
7447:
7443:
7436:
7428:
7424:
7420:
7416:
7412:
7408:
7404:
7400:
7396:
7392:
7388:
7381:
7372:
7367:
7362:
7357:
7353:
7349:
7345:
7338:
7330:
7326:
7322:
7318:
7314:
7310:
7303:
7296:
7288:
7282:
7278:
7274:
7268:
7260:
7254:
7250:
7243:
7235:
7233:9788847021136
7229:
7225:
7224:
7216:
7208:
7202:
7198:
7197:
7189:
7181:
7177:
7173:
7167:
7163:
7159:
7153:
7149:
7139:
7136:
7130:
7127:
7124:
7121:
7119:
7115:
7114:
7110:
7104:
7099:
7092:
7079:
7073:
7068:
7057:
7054:
7051:
7045:
7042:
7036:
7033:
7027:
7024:
7021:
7012:
7006:
7003:
6990:
6987:
6984:
6981:
6978:
6973:
6962:
6956:
6953:
6950:
6941:
6935:
6932:
6924:
6906:
6901:
6898:
6871:
6867:
6863:
6860:
6855:
6851:
6847:
6843:
6834:
6830:
6821:
6810:
6805:
6802:
6794:
6786:
6782:
6773:
6769:
6760:
6755:
6743:
6739:
6734:
6730:
6726:
6722:
6716:
6712:
6708:
6702:
6697:
6686:
6678:
6672:
6666:
6663:
6642:
6637:
6633:
6629:
6608:
6605:
6602:
6596:
6593:
6584:
6578:
6572:
6569:
6558:
6554:
6549:
6545:
6541:
6538:
6533:
6529:
6524:
6517:
6511:
6505:
6502:
6474:
6471:
6468:
6460:
6436:
6417:
6414:
6411:
6394:
6392:
6376:
6356:
6343:
6339:
6328:
6324:
6315:
6310:
6282:
6278:
6272:
6268:
6264:
6261:
6258:
6253:
6249:
6244:
6240:
6220:
6207:
6197:
6194:
6186:
6176:
6163:
6160:
6157:
6137:
6133:
6124:
6120:
6108:
6105:
6102:
6094:
6090:
6078:
6070:
6066:
6053:
6049:
5991:
5983:
5967:
5964:
5961:
5941:
5937:
5920:
5916:
5912:
5909:
5906:
5890:
5886:
5882:
5866:
5862:
5857:
5853:
5817:
5812:
5797:
5787:
5775:
5772:is instead a
5759:
5745:
5731:
5728:
5725:
5705:
5700:
5692:
5689:
5676:
5662:
5657:
5654:
5651:
5643:
5640:
5627:
5625:
5609:
5606:
5603:
5583:
5580:
5577:
5574:
5554:
5551:
5548:
5545:
5525:
5522:
5519:
5516:
5496:
5493:
5490:
5487:
5467:
5464:
5455:
5440:
5434:
5430:
5426:
5421:
5417:
5413:
5410:
5407:
5402:
5398:
5394:
5389:
5385:
5380:
5376:
5356:
5353:
5348:
5344:
5340:
5335:
5331:
5327:
5322:
5318:
5296:
5290:
5286:
5282:
5277:
5273:
5269:
5264:
5260:
5255:
5251:
5231:
5228:
5225:
5205:
5185:
5182:
5179:
5170:
5134:
5131:
5097:
5083:
5063:
5043:
5040:
5037:
5017:
5008:
4993:
4981:
4961:
4938:
4921:
4911:
4888:
4876:
4870:
4865:
4860:
4848:
4843:
4839:
4825:
4811:
4800:
4796:
4786:
4782:
4768:
4748:
4744:
4734:
4730:
4716:
4696:
4692:
4683:
4669:
4664:
4659:
4654:
4649:
4636:
4632:
4622:
4618:
4604:
4590:
4579:
4574:
4570:
4556:
4536:
4532:
4522:
4518:
4504:
4484:
4480:
4471:
4451:
4447:
4437:
4433:
4419:
4405:
4394:
4390:
4380:
4376:
4362:
4342:
4337:
4333:
4319:
4299:
4295:
4286:
4266:
4262:
4253:
4241:
4230:
4226:
4217:
4199:
4195:
4186:
4174:
4168:
4163:
4158:
4146:
4126:
4103:
4088:
4075:
4063:
4042:
4029:
4009:
4005:
3991:
3976:
3971:
3938:
3932:
3929:
3915:
3909:
3906:
3892:
3889:
3883:
3880:
3862:
3846:
3843:
3840:
3826:
3806:
3786:
3778:
3775:
3761:
3759:
3755:
3751:
3747:
3743:
3739:
3735:
3725:
3722:
3718:
3713:
3696:
3666:
3642:
3629:
3625:
3610:
3604:
3593:
3590:
3579:
3572:
3566:
3563:
3558:
3517:
3509:
3505:
3491:
3477:
3471:
3460:
3457:
3446:
3440:
3416:
3413:
3405:
3386:
3341:
3338:
3313:
3310:
3293:
3275:
3246:
3218:
3204:
3198:
3184:
3173:
3150:
3127:
3122:
3120:
3116:
3112:
3108:
3104:
3100:
3096:
3092:
3078:
3073:
3069:
3065:
3040:
3036:
3027:
3023:
3014:
3005:
3002:
2999:
2991:
2987:
2978:
2968:
2960:
2944:
2876:
2873:
2868:
2837:
2831:
2817:
2814:
2800:
2789:
2786:
2783:
2775:
2771:
2767:
2757:
2755:
2750:
2745:
2743:
2739:
2735:
2731:
2726:
2713:
2710:
2700:
2686:
2683:
2673:
2659:
2656:
2646:
2622:
2596:
2580:
2566:
2563:
2560:
2557:
2543:
2523:
2498:
2490:
2480:
2466:
2463:
2460:
2452:
2447:
2444:
2428:
2408:
2400:
2395:
2393:
2387:
2373:
2370:
2362:
2346:
2338:
2322:
2319:
2299:
2279:
2276:
2268:
2252:
2249:
2229:
2209:
2206:
2198:
2193:
2179:
2172:
2168:
2164:
2158:
2148:
2135:
2115:
2107:
2103:
2099:
2095:
2091:
2075:
2072:
2069:
2061:
2045:
2026:
2013:
2008:
1989:
1975:
1958:
1930:
1922:
1918:
1902:
1893:
1889:
1887:
1882:
1880:
1876:
1871:
1858:
1850:
1846:
1836:
1832:
1823:
1818:
1807:
1802:
1799:
1796:
1786:
1769:th column is
1748:
1743:
1731:
1726:
1722:
1713:
1708:
1694:
1683:
1679:
1669:
1665:
1656:
1651:
1631:
1627:
1617:
1613:
1604:
1599:
1583:
1578:
1573:
1568:
1555:
1551:
1541:
1537:
1528:
1523:
1509:
1498:
1493:
1489:
1480:
1475:
1455:
1451:
1441:
1437:
1428:
1423:
1401:
1397:
1387:
1383:
1374:
1369:
1355:
1344:
1340:
1330:
1326:
1317:
1312:
1292:
1287:
1283:
1274:
1269:
1254:
1249:
1244:
1217:
1214:
1211:
1191:
1149:
1141:
1125:
1117:
1103:
1081:
1071:
1036:
1026:
1023:
1009:
1007:
1003:
999:
995:
991:
987:
983:
982:square matrix
979:
975:
971:
967:
955:
950:
948:
943:
941:
936:
935:
933:
932:
925:
922:
920:
917:
915:
912:
910:
907:
905:
902:
900:
897:
895:
892:
891:
883:
882:
875:
872:
870:
867:
865:
862:
860:
857:
856:
848:
847:
836:
833:
831:
828:
826:
823:
822:
821:
820:
810:
809:
798:
795:
793:
790:
788:
785:
783:
780:
778:
777:Line integral
775:
773:
770:
768:
765:
764:
763:
762:
758:
757:
752:
749:
747:
744:
742:
739:
737:
734:
733:
732:
731:
727:
726:
720:
719:Multivariable
714:
713:
702:
699:
697:
694:
692:
689:
687:
684:
682:
679:
677:
674:
673:
672:
671:
667:
666:
661:
658:
656:
653:
651:
648:
646:
643:
641:
638:
636:
633:
632:
631:
630:
624:
618:
617:
606:
603:
601:
598:
596:
593:
591:
588:
586:
582:
580:
577:
575:
572:
570:
567:
565:
562:
560:
557:
556:
555:
554:
551:
548:
547:
542:
539:
537:
534:
532:
529:
527:
524:
522:
519:
516:
512:
509:
508:
507:
506:
500:
494:
493:
482:
479:
477:
474:
472:
469:
467:
464:
462:
459:
457:
454:
451:
447:
443:
442:trigonometric
439:
436:
434:
431:
429:
426:
424:
421:
420:
419:
418:
414:
413:
408:
405:
403:
400:
398:
395:
393:
390:
387:
383:
380:
378:
375:
374:
373:
372:
368:
367:
362:
359:
357:
354:
352:
349:
348:
347:
346:
340:
334:
333:
322:
319:
317:
314:
312:
309:
307:
304:
302:
299:
297:
294:
292:
289:
287:
284:
282:
279:
277:
274:
273:
272:
271:
268:
265:
264:
259:
256:
254:
253:Related rates
251:
249:
246:
244:
241:
239:
236:
234:
231:
230:
229:
228:
224:
223:
216:
213:
211:
210:of a function
208:
206:
205:infinitesimal
203:
202:
201:
198:
195:
191:
188:
187:
186:
185:
181:
180:
174:
168:
167:
161:
158:
156:
153:
151:
148:
147:
142:
139:
137:
134:
133:
129:
126:
125:
124:
123:
104:
98:
95:
89:
83:
80:
77:
74:
67:
60:
57:
51:
46:
42:
33:
32:
29:
26:
25:
21:
20:
8622:Morse theory
8571:
8503:Substitution
8389:graph theory
8248:
7886:Quaternionic
7876:Persymmetric
7634:
7612:
7583:
7562:
7526:
7519:
7499:. New York:
7496:
7490:
7481:
7468:
7449:
7445:
7435:
7394:
7390:
7380:
7351:
7347:
7337:
7312:
7308:
7295:
7276:
7267:
7248:
7242:
7222:
7215:
7195:
7188:
7161:
7158:Binmore, Ken
7152:
6400:
6182:
5774:vector field
5751:
5456:
5171:
5098:
5009:
3774:
3772:
3731:
3714:
3294:
3123:
3115:quasi-Newton
3105:, and other
2766:optimization
2763:
2746:
2730:Morse theory
2727:
2645:is called a
2621:discriminant
2486:
2448:
2396:
2392:Morse theory
2388:
2361:saddle point
2194:
2160:
2108:has at most
2037:
2029:Applications
1894:
1883:
1872:
1204:is a square
1015:
994:scalar field
978:Hesse matrix
977:
973:
969:
963:
796:
438:Substitution
200:Differential
173:Differential
8478:Hamiltonian
8402:Biadjacency
8338:Correlation
8254:Householder
8204:Commutation
7941:Vandermonde
7936:Tridiagonal
7871:Permutation
7861:Nonnegative
7846:Matrix unit
7726:Bisymmetric
3754:normal mode
3750:scale space
3354:this gives
3099:neural nets
2742:eigenvalues
2595:determinant
2443:determinant
2337:eigenvalues
1943:; that is:
1886:determinant
966:mathematics
894:Precalculus
887:Miscellanea
852:Specialized
759:Definitions
526:Alternating
369:Definitions
182:Definitions
8606:Categories
8378:Transition
8373:Stochastic
8343:Covariance
8325:statistics
8304:Symplectic
8299:Similarity
8128:Unimodular
8123:Orthogonal
8108:Involutory
8103:Invertible
8098:Projection
8094:Idempotent
8036:Convergent
7931:Triangular
7881:Polynomial
7826:Hessenberg
7796:Equivalent
7791:Elementary
7771:Copositive
7761:Conference
7721:Bidiagonal
7482:Iowa State
7452:: 104849.
7361:1806.03674
6233:and write
3819:such that
874:Variations
869:Stochastic
859:Fractional
728:Formalisms
691:Divergence
660:Identities
640:Divergence
190:Derivative
141:Continuity
8559:Wronskian
8483:Irregular
8473:Gell-Mann
8422:Laplacian
8417:Incidence
8397:Adjacency
8368:Precision
8333:Centering
8239:Generator
8209:Confusion
8194:Circulant
8174:Augmented
8133:Unipotent
8113:Nilpotent
8089:Congruent
8066:Stieltjes
8041:Defective
8031:Companion
8002:Redheffer
7921:Symmetric
7916:Sylvester
7891:Signature
7821:Hermitian
7801:Frobenius
7711:Arrowhead
7691:Alternant
7636:MathWorld
7631:"Hessian"
7619:EMS Press
7411:0175-7571
7180:717598615
7065:∇
7052:−
7007:
6994:⟩
6982:
6970:∇
6966:⟨
6936:
6895:Γ
6861:⊗
6827:∂
6819:∂
6799:Γ
6795:−
6779:∂
6766:∂
6752:∂
6723:⊗
6694:∂
6683:∇
6667:
6600:∇
6591:∇
6588:∇
6573:
6550:∗
6542:⊗
6534:∗
6521:Γ
6518:∈
6506:
6478:→
6445:∇
6349:¯
6334:∂
6321:∂
6307:∂
6262:…
6213:→
6198::
6106:…
5965:×
5910:…
5830:that is,
5803:→
5690:−
5641:−
5607:−
5427:−
5414:−
5229:−
5183:−
5041:×
4972:∂
4967:Λ
4958:∂
4927:∂
4919:∂
4894:∂
4886:∂
4836:∂
4831:Λ
4822:∂
4812:⋯
4793:∂
4779:∂
4774:Λ
4765:∂
4741:∂
4727:∂
4722:Λ
4713:∂
4689:∂
4681:∂
4670:⋮
4665:⋱
4660:⋮
4655:⋮
4650:⋮
4629:∂
4615:∂
4610:Λ
4601:∂
4591:⋯
4567:∂
4562:Λ
4553:∂
4529:∂
4515:∂
4510:Λ
4501:∂
4477:∂
4469:∂
4444:∂
4430:∂
4425:Λ
4416:∂
4406:⋯
4387:∂
4373:∂
4368:Λ
4359:∂
4330:∂
4325:Λ
4316:∂
4292:∂
4284:∂
4259:∂
4251:∂
4242:⋯
4223:∂
4215:∂
4192:∂
4184:∂
4137:∂
4132:Λ
4123:∂
4092:∂
4089:λ
4086:∂
4081:Λ
4072:∂
4046:∂
4043:λ
4040:∂
4035:Λ
4026:∂
4006:λ
4002:∂
3997:Λ
3988:∂
3966:Λ
3930:−
3910:λ
3884:λ
3870:Λ
3740:(see the
3608:∇
3605:−
3577:∇
3530:that is,
3475:∇
3472:−
3444:∇
3379:Δ
3303:Δ
3272:‖
3263:Δ
3260:‖
3239:Δ
3202:∇
3188:Δ
3171:∇
3062:Θ
3020:∂
3012:∂
3003:…
2984:∂
2976:∂
2942:∇
2917:Δ
2883:Δ
2861:Δ
2835:∇
2815:≈
2804:Δ
2464:×
1987:∇
1843:∂
1829:∂
1815:∂
1719:∂
1705:∂
1695:⋯
1676:∂
1662:∂
1648:∂
1624:∂
1610:∂
1596:∂
1584:⋮
1579:⋱
1574:⋮
1569:⋮
1548:∂
1534:∂
1520:∂
1510:⋯
1486:∂
1472:∂
1448:∂
1434:∂
1420:∂
1394:∂
1380:∂
1366:∂
1356:⋯
1337:∂
1323:∂
1309:∂
1280:∂
1266:∂
1215:×
1118:∈
1072:∈
1042:→
998:curvature
864:Malliavin
751:Geometric
650:Laplacian
600:Dirichlet
511:Geometric
96:−
43:∫
8617:Matrices
8387:Used in
8323:Used in
8284:Rotation
8259:Jacobian
8219:Distance
8199:Cofactor
8184:Carleman
8164:Adjugate
8148:Weighing
8081:inverses
8077:products
8046:Definite
7977:Identity
7967:Exchange
7960:Constant
7926:Toeplitz
7811:Hadamard
7781:Diagonal
7419:25538002
7095:See also
6921:are the
3295:Letting
2959:gradient
2489:gradient
1921:gradient
1016:Suppose
990:function
904:Glossary
814:Advanced
792:Jacobian
746:Exterior
676:Gradient
668:Theorems
635:Gradient
574:Integral
536:Binomial
521:Harmonic
386:improper
382:Integral
339:Integral
321:Reynolds
296:Quotient
225:Concepts
61:′
28:Calculus
8488:Overlap
8453:Density
8412:Edmonds
8289:Seifert
8249:Hessian
8214:Coxeter
8138:Unitary
8056:Hurwitz
7987:Of ones
7972:Hilbert
7906:Skyline
7851:Metzler
7841:Logical
7836:Integer
7746:Boolean
7678:classes
7621:, 2001
7427:2945423
7329:1251969
5677:minimum
5628:maximum
2957:is the
2487:If the
2339:, then
2104:that a
2088:is the
1919:of the
1877:by the
974:Hessian
899:History
797:Hessian
686:Stokes'
681:Green's
513: (
440: (
384: (
306:Inverse
281:Product
192: (
8407:Degree
8348:Design
8279:Random
8269:Payoff
8264:Moment
8189:Cartan
8179:Bézout
8118:Normal
7992:Pascal
7982:Lehmer
7911:Sparse
7831:Hollow
7816:Hankel
7751:Cauchy
7676:Matrix
7590:
7571:
7538:
7507:
7425:
7417:
7409:
7327:
7283:
7255:
7230:
7203:
7178:
7168:
6886:where
6706:
6461:. Let
5982:tensor
2934:where
2770:Newton
2738:kernel
2556:has a
2451:minors
2096:. The
968:, the
741:Tensor
736:Matrix
623:Vector
541:Taylor
499:Series
136:Limits
8468:Gamma
8432:Tutte
8294:Shear
8007:Shift
7997:Pauli
7946:Walsh
7856:Moore
7736:Block
7478:(PDF)
7423:S2CID
7356:arXiv
7325:S2CID
7305:(PDF)
7144:Notes
6433:be a
3721:up to
2672:or a
2568:) at
2536:then
2359:is a
2292:then
2222:then
2092:of a
2058:is a
992:, or
980:is a
564:Ratio
531:Power
450:Euler
428:Discs
423:Parts
291:Power
286:Chain
215:total
8274:Pick
8244:Gram
8012:Zero
7716:Band
7588:ISBN
7569:ISBN
7536:ISBN
7505:ISBN
7415:PMID
7407:ISSN
7281:ISBN
7253:ISBN
7228:ISBN
7201:ISBN
7176:OCLC
7166:ISBN
7004:Hess
6979:grad
6933:Hess
6664:Hess
6570:Hess
6503:Hess
6457:its
6437:and
6401:Let
3748:and
3736:and
3119:BFGS
3113:and
2740:and
2732:and
2593:The
2562:(or
2397:The
2363:for
1884:The
645:Curl
605:Abel
569:Root
8363:Hat
8096:or
8079:or
7532:386
7454:doi
7450:188
7399:doi
7366:doi
7352:801
7317:doi
6369:If
5752:If
5147:if
3097:of
2756:.)
2703:of
2676:of
2649:of
2269:at
2199:at
2165:is
2038:If
1184:of
1142:of
964:In
276:Sum
8608::
7633:.
7617:,
7611:,
7534:.
7480:.
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7364:.
7350:.
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7323:.
7311:.
7307:.
7174:.
6585::=
6164:1.
6026::
3773:A
3121:.
3101:,
2394:.
2136:3.
1892:.
1881:.
1008:.
972:,
448:,
444:,
8493:S
7951:Z
7668:e
7661:t
7654:v
7639:.
7596:.
7577:.
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7513:.
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7456::
7429:.
7401::
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7368::
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7331:.
7319::
7313:6
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7074:Y
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7028:Y
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7019:(
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6991:Y
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6963:=
6960:)
6957:Y
6954:,
6951:X
6948:(
6945:)
6942:f
6939:(
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6770:x
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6756:2
6744:(
6740:=
6735:j
6731:x
6727:d
6717:i
6713:x
6709:d
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6687:i
6679:=
6676:)
6673:f
6670:(
6643:}
6638:i
6634:x
6630:{
6609:,
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6579:f
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6555:M
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6512:f
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6418:g
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6221:,
6217:C
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6158:m
6138:.
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6050:=
6047:)
6043:f
6039:(
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5968:n
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5934:)
5930:x
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5913:,
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5904:)
5900:x
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5891:2
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5883:,
5880:)
5876:x
5872:(
5867:1
5863:f
5858:(
5854:=
5851:)
5847:x
5843:(
5839:f
5818:,
5813:m
5808:R
5798:n
5793:R
5788::
5784:f
5760:f
5732:0
5729:=
5726:m
5706:.
5701:m
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5693:1
5687:(
5663:.
5658:1
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5652:m
5648:)
5644:1
5638:(
5610:m
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5584:,
5581:m
5578:+
5575:n
5555:1
5552:+
5549:m
5546:2
5526:2
5523:+
5520:m
5517:2
5497:1
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5491:m
5488:2
5468:m
5465:2
5441:)
5435:2
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5422:1
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5399:x
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5386:x
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5377:f
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5354:=
5349:3
5345:x
5341:+
5336:2
5332:x
5328:+
5323:1
5319:x
5297:)
5291:3
5287:x
5283:,
5278:2
5274:x
5270:,
5265:1
5261:x
5256:(
5252:f
5232:m
5226:n
5206:m
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5180:n
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5135:0
5132:=
5128:z
5123:H
5116:T
5110:z
5084:m
5064:m
5044:m
5038:m
5018:m
4994:]
4982:2
4977:x
4962:2
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4931:x
4922:g
4912:(
4898:x
4889:g
4877:0
4871:[
4866:=
4861:]
4849:2
4844:n
4840:x
4826:2
4801:2
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4783:x
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4693:x
4684:g
4637:n
4633:x
4623:2
4619:x
4605:2
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4169:[
4164:=
4159:]
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4110:T
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4096:x
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4064:(
4050:x
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3992:2
3977:[
3972:=
3969:)
3963:(
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3936:]
3933:c
3927:)
3923:x
3919:(
3916:g
3913:[
3907:+
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3900:x
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3890:=
3887:)
3881:,
3877:x
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3847:,
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3841:=
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3834:x
3830:(
3827:g
3807:g
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3700:)
3697:r
3694:(
3689:O
3667:r
3646:)
3643:r
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3506:r
3502:(
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3406:r
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2253:.
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2207:x
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2116:9
2076:0
2073:=
2070:f
2046:f
2014:.
2009:T
2005:)
2001:)
1997:x
1993:(
1990:f
1984:(
1980:J
1976:=
1973:)
1970:)
1966:x
1962:(
1959:f
1956:(
1952:H
1931:f
1903:f
1859:.
1851:j
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1744:]
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