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Topic: Hessian matrix

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Mixed derivatives

Jacobian matrix

Morse theory

Derivative

Partial derivative

Total derivative

Positive definite

Optimization (mathematics)

Schwarzian derivative

List of matrices

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	Hessian matrix -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		The term Hessian was coined by (Click link for more info and facts about James Joseph Sylvester) James Joseph Sylvester, named for German mathematician (Click link for more info and facts about Ludwig Otto Hesse) Ludwig Otto Hesse, who had used the term functional determinants.
		The mixed derivatives of f are the entries off the (The diagonal of a square matrix running from the upper left entry to the lower right entry) main diagonal in the Hessian.
		If the Hessian has both positive and negative (Click link for more info and facts about eigenvalue) eigenvalues then x is a (Click link for more info and facts about saddle point) saddle point for f (this is true even if x is degenerate).
	www.absoluteastronomy.com /encyclopedia/h/he/hessian_matrix.htm (482 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		Date: Wed, 16 Dec 1998 07:38:06 -0600 (CST) To: materiaux@ecole-debroglie.fr Subject: Re: "Hessian" matrix for f : Rn -> Rp >Considering a function f : Rn -> Rp of n variables, the matrix of df >(differential of f) is the jacobian matrix.
		Q is the matrix of a quadratic form >the signature of which may give information about the extremums of f.
		Indeed, the Hessian is only well- defined after bases have been picked, while the function f itself and its quadratic approximation are well-defined without reference to bases.
	www.math.niu.edu /~rusin/known-math/98/hessian (365 words)

	Forcing Hessian Matrix to be Positive Definite Mini
		The Mini-Project of Forcing the Hessian Matrix to be positively definite is objected to understand the methods of forcing the Hessian Matrix to be positively definite of Modified Marquardt and Cholesky Factorization including their basic ideas and theorem.
		The magnitude of D is limited by Gershgorin circle, which defined that each element in diagonal m x m matrix subtracted by Z is less than summation of the rest elements as equation (8.2.1).
		This restriction means that the maximum value of elements d in matrix D is equal to the summation of every element in column m and row j at all j not m.
	www.geocities.com /professorjeab/ForcingHessianPositive-Suphannee.htm (1386 words)

	Z-Matrix Manipulation (Site not responding. Last check: 2007-10-08)
		command, causes the hessian element for the O-H stretch to be 0.3, and the hessian elements associated with the H-O-H bend to be determined by the two-point finite-difference formula.
		However, as well as changing the z-matrix, it is frequently useful to make corresponding changes to the hessian matrix used in the geometry optimisation procedures, enlarging or reducing the size of this matrix and maintaining the correct correspondence with the z-matrix variable ordering.
		To modify an existing hessian matrix in a corresponding manner, it is necessary to pass either two or three extra arguments to z_var.
	www.cse.clrc.ac.uk /qcg/chemshell/manual/zmatrix.html (2025 words)

	Computation - Operations Research Models and Methods (Site not responding. Last check: 2007-10-08)
		The matrix of second derivatives is the Hessian matrix.
		For the quadratic function the Hessian is the same as the Q matrix and does not depend on the decision vector.
		The add-in diagonalizes the Hessian matrix using a linear transformation of the variables.
	www.me.utexas.edu /~jensen/ORMM/computation/unit/functions/differentiate.html (661 words)

	Initial hessians, sparsity and storage (Site not responding. Last check: 2007-10-08)
		with a small squared Hessian for a core and a diagonal vector for the environment was suitable for a very localized reaction.
		The Hessian is calculated numerically in Cartesian coordinates representation.
		When there are many significative elements out of the diagonal the initial Hessian matrix represented as a small matrix plus a vector is not the most suitable.
	klingon.uab.es /prat/Thesis/node99.html (772 words)

	Surface Evolver Documentation
		H is a square matrix of second derivatives, the Hessian.
		Anyone who uses "hessian" on a surface with very many facets quickly finds that it is very difficult to get near enough to a minimum for "hessian" to work.
		The advantages of hessian_normal is that "hessian" converges much better and the dimension of the configuration space is cut by a factor of 3 (or whatever the ambient space dimension is).
	www.geom.uiuc.edu /software/evolver/html/eigentut.htm (4302 words)

	34.2.8 Approximating hessian matrix elements (HESSIAN) (Site not responding. Last check: 2007-10-08)
		For minimizations, the Model Hessian provides very good approximations the Hessian matrix, improving convergence rapidly, so it was chosen as default.
		If the Model Hessian is disabled, the initial hessian is diagonal, with values 1hartreebohr(-2) for all lengths, 1 hartreeradian**(-2) for all angles.
		It is also possible to use the numerical hessian or the hessian from a frequency calculation in minimizations.
	www.lrz-muenchen.de /services/software/chemie/molpro/molpro.2000.9/doc/manual/node471.html (193 words)

	Second Order Conditions for Optimization, Constrained and Unconstrained: The Hessian and Bordered Hessian Matrix
		For the unconstrained case the conditions are stated in terms of the matrix of second derivatives called the Hessian matrix.
		Generation after generation of applied mathematics students have accepted the bordered Hessian without a clue as to why it is the relevant entity.
		The conditon for a matrix to be positive definite is that its principal minors all be positive.
	www.sjsu.edu /faculty/watkins/2ndOrdergen.htm (705 words)

	Exploiting Eigenvalues of the Hessian Matrix for Volume Decimation
		In recent years the Hessian matrix and its eigenvalues became important in pattern recognition.
		We recall the relationship between the eigenvalues of Hessian matrix and the 2nd order edge detection filter, show the usefulness of treating them separately and exploit these facts to design a combined threshold operation to generate sparse data sets.
		The Laplacian (c) is a sum of the Hessian matrix eigenvalues (d) + (e).
	www.cg.tuwien.ac.at /research/vis/vismed/evDecimation (260 words)

	Geometry optimization (Site not responding. Last check: 2007-10-08)
		Thus, the user might consider the generation of a Hessian matrix for the input geometry by performing a semiempirical calculation (if the semi-empirical parameters are available for all atoms of the molecular structure under study).
		An alternative is the generation of Hessian matrix with DGauss using a small basis and a coarse grid.
		For simplicity, this Hessian is also given in Cartesian coordinates and the program then transforms it into the appropriate internal coordinates as specified in the geometry input.
	www.cesup.ufrgs.br /unichem/5505/5505_92.html (1733 words)

	8.13.2 Hessian matrix in FORCE calculations (Site not responding. Last check: 2007-10-08)
		The Hessian matrix is the matrix of second derivatives of the energy with respect to geometry.
		Normal modes are expressed as Cartesian displacements, consequently the Hessian is based on Cartesian rather than internal coordinates.
		The Hessian is quite sensitive to geometry, and should only be evaluated at stationary points.
	home.att.net /~mopacmanual/node583.html (173 words)

	Posterior Errors of the Estimated State (Site not responding. Last check: 2007-10-08)
		R.Giering discussed the computation of posterior errors of the estimated ocean state from properties of the Hessian matrix of the objective function.
		Error estimates of derived quantities (e.g., meridional heat flux) are projections of the inverse Hessian (i.e., the error covariance matrix).
		Firstly the calculation of the covariance matrix times a vector which requires many evaluations of the Hessian times a vector and is suitable for systems of moderate complexity.
	puddle.mit.edu /~detlef/OSE/GODAE_WS/node6.html (657 words)

	Maximising the Likelihood
		SABRE uses two methods to form the Hessian matrix of second derivatives of the likelihood with respect to the model parameters.
		The first uses an approximation which is essentially the variance-covariance matrix of the first derivatives whilst the second method uses the true matrix of second derivatives.
		The disadvantage is that it may not be a consistent estimator of the Hessian (and hence the parameter standard errors) unless the observations are independent and identically distributed.
	www.cas.lancs.ac.uk /software/sabre3.1/sabre/node3.html (226 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		This option is useful in those situations where an initial hessian calculation is performed as a preface to a geometry optimization or a transition state search.
		If a GAMESS hessian is selected, and if the rotation matrix was not found in the archive (as is most often the case), you will be given the chance to enter it manually.
		In the former case, because of the possible presence of dummy atoms in the Z matrix, it is more difficult to determine the number of atoms so the user is asked to enter this information.
	www.osc.edu /PET/CCM/skeleton/software/tested/source/grabff/doc.html (2508 words)

	Technical Program Paper Detail - AIChE
		This enables it to introduce a shift to the diagonal elements of the Hessian of f(x) and yield a positive definite matrix for the underestimating function L(x).
		The Hessian matrix of the new convexification function is a constant non-diagonal matrix.
		We use semidefinite programming techiques to construct a symmetric perturbation matrix which is added to the Hessian matrix to render a positive semidefinite quadratic function.
	www.aiche.org /conferences/techprogram/paperdetail.asp?PaperID=1957&DSN=Annual03 (492 words)

	GAUSS's Maximum Likelihood Procedure
		This will be most useful if the inversion of the hessian fails, which is indicated when MAXLIK returns a missing value for the covariance matrix of the parameters.
		The procedure returns a single output argument, an NxK matrix of gradients of the log- likelihood function with respect to the parameters evaluated at the vector of parameter values.
		The procedure returns a single output argument, a KxK Hessian matrix of the function with respect to the parameters.
	faculty.washington.edu /ezivot/maxlik.htm (1775 words)

	Lesson 10
		# The several variable version of this uses the Hessian matrix, # which is the matrix of second partial derivatives of the function.
		The first input is an expression in several variables, # and the second is a list of the variables.
		So for m and n both even # the Hessian is negative definite and the critical point # is a local maximum, while for m and n both odd # the Hessian is positive definite and the critical point # is a local minimum.
	www.ugrad.math.ubc.ca /coursedoc/m210/lesson10.html (1282 words)

	Constrained Maximum Likelihood Reference
		K x K matrix, the Hessian used to compute the covariance matrix of the parameters is stored in _cml_FinalHess.
		This is most useful if the inversion of the hessian fails, which is indicated when CML returns a missing value for the covariance matrix of the parameters.
		The procedure returns a single output argument, an N x K matrix of gradients of the log-likelihood function with respect to the parameters evaluated at the vector of parameter values.
	www.la.utexas.edu /ICPSR/gausscml/node48.html (5672 words)

	Command >>> HESSIAN (Site not responding. Last check: 2007-10-08)
		This command computes a finite difference Hessian matrix H for the error analysis following optimization.
		By default, the Hessian matrix, which is the inverse of the parameter covariance matrix, is approximated by
		However, inclusion of the second-derivative term may yield a Hessian matrix that is not positive definite due to the presence of outliers, strong non-linearities, or the fact that the minimum has not been detected accurately, i.e., when the positive and negative residuals r do not cancel each other.
	www-esd.lbl.gov /ITOUGH2/Command/HESSIAN_3.HTML (179 words)

	Appendix A: The Lanczós algorithm (Site not responding. Last check: 2007-10-08)
		This algorithm is a recursion method based on the repeated application of the local Hessian on a random displacement vector [24].
		Applying the Hessian matrix H onto this vector is equivalent to computing the difference between the force at
		The application of the Hessian to the position requires a single force evaluation and is an
	www.phy.ohiou.edu /~malek/protein7/node14.html (256 words)

	Variance Estimation (Site not responding. Last check: 2007-10-08)
		Two different procedures are provided in MARK to estimate the variance-covariance matrix of the estimates.
		The first is the inverse of the Hessian matrix obtained as part of the numerical optimization of the likelihood function.
		Because the rank of the variance-covariance matrix is used to determine the number of parameters that were actually estimated, using different methods will sometimes result in a different number of parameters estimated, and hence a very different value of the AIC and AICc.
	www.cnr.colostate.edu /~gwhite/mark/markhelp/varianceestimation.htm (173 words)

	Critical Points and Generic Surfaces (Site not responding. Last check: 2007-10-08)
		A smooth surface is non-degenerate at a point if it has two principal directions of curvature and it is degenerate if it is locally flat or has only one principal direction of curvature.
		A smooth surface is locally approximated at a non-degenerate critical point by a quadratic form of the Hessian matrix (Wylie 1975, chap.
		In the extremum case, the Hessian matrix has two eigenvalues of the same sign while in the saddle case their signs are different.
	www.ensc.sfu.ca /people/grad/brassard/personal/THESIS/node182.html (248 words)

	World Web Math: Vector Calculus: The Hessian (Site not responding. Last check: 2007-10-08)
		To introduce this test, we first must define a structure called the Hessian Matrix.
		A technical point to notice is that the Hessian matrix is not symmetrical unless the partial drivatives f
		For two-variable functions, our Hessian matrix will be a 2 by 2 matrix.
	web.mit.edu /wwmath/vectorc/minmax/hessian.html (347 words)

	HESS=n (Site not responding. Last check: 2007-10-08)
		The initial Hessian is set equal to a diagonal matrix, with the diagonal terms set to 1000 kcal/mol/Å
		The full Hessian matrix is constructed using single-sided derivatives, see Hessian Matrix, using the same density matrix throughout the entire construction of the Hessian.
		The Hessian matrix from an earlier run can be used to start the current job.
	www.cachesoftware.com /mopac/Mopac2002manual/node193.html (175 words)

	4.24 slapaf
		use the eigenvectors of the approximative Hessian expressed in cartesians as the definition of the internal coordinates.
		The resulting force constant matrix is used for an analysis of the harmonic frequences.
		Observe that in case of the use of internal coordinates defined as cartesian coordinates that these has to be linear combinations which are free from translational and rotational components for the harmonic frequency analysis to be valid.
	vergil.chemistry.gatech.edu /manuals/molcas5/users.guide/node41.html (2393 words)

	Exact Calculation of the Product of the Hessian Matrix of Feed-Forward Network Error Functions and a Vector in O(N) ... (Site not responding. Last check: 2007-10-08)
		Although it is possible to calculate the Hessian matrix exactly it is often not desirable because of the computation and memory requirements involved.
		This paper presents a method to calculate the Hessian matrix times a vector in O(N) time, where N is the number of variables...
		29 Exact Calculation of the Hessian Matrix for the Multilayer P..
	citeseer.ist.psu.edu /92594.html (387 words)

	Eigenvector Following routine (2) (Site not responding. Last check: 2007-10-08)
		HESS=0 may also be useful for very large molecules, for which the calculation of the partial Hessian matrix may be very time-consuming.
		Because the estimated initial Hessian is probably not a very close approximation to the real one, optimizations using HESS=0 usually require far more cycles than the default method.
		The entire Hessian matrix is calculated in the first optimization cycle.
	www.ccc.uni-erlangen.de /clark/MML/MMLFolien/Vamp65Man/USING/uv_03b.htm (284 words)

	Optimization Algorithms
		If the Hessian is not positive definite, a multiple of the identity matrix is added to the Hessian matrix to make it positive definite (Eskow and Schnabel 1991).
		However, if the computation of the Hessian matrix is computationally expensive, one of the (dual) quasi-Newton or conjugate gradient algorithms may be more efficient.
		Instead of updating an approximate Hessian matrix, this algorithm uses the dual BFGS (or DFP) update that updates the Cholesky factor of an approximate Hessian.
	www.caspur.it /risorse/softappl/doc/sas_docs/ormp/chap5/sect27.htm (3664 words)

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