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Topic: Jacobian matrix

Related Topics

Jacobian determinant

Hessian matrix

Power series

Connected space

Submersion

Differential of the first kind

Jordan curve

Proof that holomorphic functions are analytic

Singularity theory

Critical point

Equilibrium constant

Gibbs free energy

Length scale

	jacobian -- the Jacobian matrix of partial derivatives (Site not responding. Last check: 2007-10-11)
		-- calculates the Jacobian matrix of the matrix f, which will normally be a matrix with one row.
		For a one row matrix, the derivatives w.r.t.
		Caveat: if a matrix or ideal over a quotient polynomial ring S/J is given, then only the derivatives of the given elements are computed and NOT the derivatives of elements of J. The function
	www.math.temple.edu /computing/Macaulay2/0551.html (116 words)

	PlanetMath: Jacobian matrix
		, it is easy to show that the effect of a change of coordinates on volume forms is a local scaling of the volume form by the determinant of the Jacobian matrix of the derivative of the backwards change of coordinates, which is called the inverse Jacobian.
		The determinant of the inverse Jacobian is thus commonly seen in integration over a change of coordinates.
		This is version 14 of Jacobian matrix, born on 2001-11-14, modified 2007-01-14.
	planetmath.org /encyclopedia/JacobianMatrix.html (278 words)

	PlanetMath: chain rule (several variables)
		The most general, and conceptually clear approach to the multi-variable chain is based on the notion of a differentiable mapping, with the Jacobian matrix of partial derivatives playing the role of generalized derivative.
		The derivative of such mappings is no longer a function, but rather a matrix of partial derivatives, customarily called the Jacobian matrix.
		This form of the chain rule also generalizes quite nicely to the even more general setting where one is interested in describing the derivative of a composition of mappings between manifolds.
	planetmath.org /encyclopedia/ChainRuleSeveralVariables.html (314 words)

	Nonlinear Analysis & Chaos II, 4397 Solution to Examples Sheet V
		This matrix has strictly negative eigenvalues and so it follows from a theorem from the course that the equilibrium at the origin of (a) is asymptotically stable.
		(b) The Jacobian matrix of (b) at the origin is the same as the matrix of the ODE in Q1(a).
		This matrix has strictly negative eigenvalues and so it follows from a theorem from the course that the equilibrium at the origin of (b) is asymptotically stable.
	nothung.math.uh.edu /~mike/NLA2/NLAS5 (321 words)

	Method and apparatus for substantially improving the throughput of circuit simulators - Patent 4918643 (Site not responding. Last check: 2007-10-11)
		The (p,q) entry of the square matrix J(x.sub.i) is equal to the partial derivative of the pth component in the function F with respect to the qth component in the argument vector x.sub.i.
		The Jacobian matrix J(x.sub.i) is, in essence, a measure of the sensitivity of the steady-state equilibrium condition with respect to the variation in the initial state vector x.sub.i.
		The cyclic Jacobian matrix for the entire periodic cycle is computed and utizlied in an iteration procedure to adjust a value of the initial state vector.
	www.freepatentsonline.com /4918643.html (6773 words)

	2.4.7 Calculation of the Jacobian matrix (j) (Site not responding. Last check: 2007-10-11)
		The latter must be confirmed because the matrix may be quite large.
		The matrix is compared to the exact Jacobian calculated by tangent-linear or adjoint model runs.
		The Jacobian matrix calculated so far is deleted and restored from a file written in previous model run.
	w3g.gkss.de /G/imashome/manual/node17.html (1088 words)

	Configuration control of seven degree of freedom arms - Patent 5430643
		The hand Jacobian matrix is computed using the vector cross-product form [14] ##EQU5## where z.sub.i is the unit vector along the z-axis of link frame {i}, and P.sup.i is the position vector from the origin O.sub.i of link frame {i}to the origin of hand frame {7}.
		Therefore, the extended Jacobian method is retrieved as a special case of the configuration control approach when the latter is implemented as a differential kinematic controller with an optimization additional task.
		In the present implementation, because of the slow sampling rate of 25 milliseconds, the Jacobian matrix J is computed using the desired joint angles instead of the actual joint angles.
	www.freepatentsonline.com /5430643.html (8250 words)

	Homework 5
		H is the Jacobian matrix of partial derivatives of h with respect to x:
		V is the Jacobian Matrix of partial derivatives of h with respect to
		The model noise is represented by the Q matrix, were the diagonal terms represent the noise of each state variable respectively and the terms below and above the diagonal are the covariance of the noise (in case there is some dependence).
	www.frc.ri.cmu.edu /~fcalder/HW5 (1775 words)

	InfoBasedCriteria
		DEX - This criterion is the condition number of the jacobian matrix.
		DYN - This criterion attempts to minimize joint torques in terms of inertial loading by minimizing the maximum eigenvalue of the inertial matrix divided by the minimum singular value of the jacobian matrix.
		GEN - This criterion minimizes the euclidean norm of the eigenvalues of the stiffness matrix.
	www.robotics.utexas.edu /applications/oscar_help/oscadw.htm (483 words)

	JACOBIAN Statement
		The m rows of the Jacobian matrix must correspond to the order of the m function names listed in the MIN, MAX, or LSQ statement.
		To specify the values of nonzero derivatives, the variables specified in the JACOBIAN statement have to be defined in the left-hand side of algebraic expressions in programming statements.
		In the case of least-squares minimization, the cross-product Jacobian is used as an approximative Hessian matrix.
	www.okstate.edu /sas/v7/sashtml/books/ormp/chap5/sect15.htm (244 words)

	SBML ODE Solver Library API: Jacobian Matrix: J = df(x)/dx
		Constructs and returns the determinant of the jacobian matrix.
		At the moment this matrix is freed together with the ODE model.
		Returns NULL if either the jacobian has not been constructed yet, or if the v1 or vi2 are not ODE variables.
	www.tbi.univie.ac.at /~raim/odeSolver/doc/api/group__jacobian.html (218 words)

	Wen Chen
		Some primary works show that this new framework is very promising in nonlinear stability analysis, uncoupling computation, evaluation of Jacobian matrix, and construction of simple, efficient and robust iteration formulas.
		The approach produces the exact Jacobian matrix in the chain rules similar to those in differentiation of a scalar function.
		This relationship formula is in fact a straightforward extension of differentiation of a scalar power function to the Jacobian derivative matrix of a polynomial function vector.
	folk.uio.no /wenc/html/nonlinear.htm (1059 words)

	General Least-Squares - Direct Solutions and Bundle Adjustments
		The Jacobian matrix, J, is the matrix of the partial differentials of each equation with respect to each unknown.
		The dimensions and orientations of the ellipse are calculated from the coefficients of the covariance matrix.
		The ellipse matrix is composed of entries from the covariance matrix.
	www.orbitals.com /self/least/least.htm (2327 words)

	Steady states, eigenvalues and circuits
		Stability at stationary states is usually studied in terms of eigenvalues of the Jacobian matrix.
		This is the most appropriate approach from the mathematical point of view, but not from the biological one, from which one considers interactions between different components, and especially feedback circuits.
		This allows to derive constraints on the weight of positive feedback circuits at stable stationary states, but one must bear in mind that the weight of feedback circuits cannot usually be altered without displacing stationary states.
	cinquin.org.uk /node4.html (575 words)

	SYM - Software package for symbolic robot modeling (Site not responding. Last check: 2007-10-11)
		Two types of Jacobian matrices are available: The Jacobian matrix with respect to the base frame, and the Jacobian matrix expressed with respect to the end-effector coordinate frame.
		For example, for the computation of the Jacobian matrix with respect to the base frame for 6 degree-of-freedom PUMA robot, the C-code program requires 71 floating point operations, while the computation of the homogeneous transformation matrix requires 69 floating point operations.
		There are five types of robot dynamic models that SYM can generate: complete driving torques in the manipulator joints, the inertial matrix, the vector of gravitational forces, the vector of Coriolis and centrifugal terms, the vector of gravitational, Coriolis and centrifugal terms.
	www.imp.bg.ac.yu /prez/lab150/sympacke.htm (437 words)

	Class optimization.Minpack_f77
		The fdjac2 method computes a forward-difference approximation to the m by n Jacobian matrix associated with a specified problem of m functions in n variables.
		Column j of P is column ipvt[j] of the identity matrix.
		The qrfac_f77 method uses Householder transformations with column pivoting (optional) to compute a QR factorization of the m by n matrix A. That is, qrfac_f77 determines an orthogonal matrix Q, a permutation matrix P, and an upper trapezoidal matrix R with diagonal elements of nonincreasing magnitude, such that AP = QR.
	www1.fpl.fs.fed.us /optimization.Minpack_f77.html (4467 words)

	odeset :: Functions (MATLAB Function Reference)
		The solver uses this sparsity pattern to generate a sparse Jacobian matrix numerically.
		If the Jacobian matrix is large and sparse, this can greatly accelerate execution.
		This allows the solver to reduce the number of function evaluations required to compute all the columns of the Jacobian matrix, and might significantly reduce solution time.
	www.mathworks.com /access/helpdesk/help/techdoc/ref/odeset.html (2386 words)

	RRFKJacobian member TOC
		Returns a const pointer to the Jacobian matrix.
		This method will compute the value of the jacobian and Jacobian transpose first using supplied joint configuration to determine the linkToBase transformation matrices.
		This method overides the FKPosition method of the same name to ensure that the Jacobian is of the proper size if the active coordinates change.
	www.robotics.utexas.edu /applications/oscar_help/osca70.htm (120 words)

	Computational Geometry Package: geom::SimplexAdjJac< N, T > Class Template Reference
		Implements operations for the adjoint Jacobian matrix of a simplex.
		In this context, the adjoint is the scaled inverse of a matrix.
		The adjoint of the Jacobian matrix of a simplex is used in the condition number quality metric.
	www.cacr.caltech.edu /~sean/projects/stlib/html/geom/classgeom_1_1SimplexAdjJac.html (234 words)

	Identification in Structural Equation Models
		First, the rank of the information matrix is only evaluated after the parameters have been estimated, and the evaluation only applies at that point in parameter space.
		Their Jacobian is augmented because it also includes equations representing restrictions, such as equality constraints, imposed on the values of the parameters.
		Using modern computer algebra techniques, Bekker, Merckens and Wansbeek (1994) show that the identification of the model can be assessed by evaluating the rank of a subset of this augmented Jacobian matrix, and that this evaluation can be conducted symbolically, before the parameters are estimated, and thus independently of any particular set of parameter values.
	www.gsu.edu /~mkteer/identifi.html (1431 words)

	Achieving finite element mesh quality via optimization of the Jacobian matrix norm and associated quantities. Part ... (Site not responding. Last check: 2007-10-11)
		Because certain matrix norm identities which hold for 2×2 matrices do not hold for 3×3 matrices, significant differences arise between surface and volume mesh optimization objective functions.
		Matrix norm identities are used to systematically construct dimensionally homogeneous groups of objective functions.
		The concept of an ideal minimizing matrix is introduced for both hexahedral and tetrahedral elements.
	www.andrew.cmu.edu /user/sowen/abstracts/Kn856.html (314 words)

	Differential Equations (Mathematics)
		A constant matrix or a function that evaluates the Jacobian.
		It is used to generate a sparse Jacobian matrix numerically.
		In some ODE problems the times of specific events are important, such as the time at which a ball hits the ground, or the time at which a spaceship returns to the earth.
	www.cs.berkeley.edu /titan/sww/software/matlab/techdoc/math_anal/diffeq7.html (2096 words)

	fsolve (Optimization Toolbox)
		is a matrix that has the same number of rows as there are dimensions in the problem.
		However, if the Jacobian of the system is singular, the algorithm may converge to a point that is not a solution of the system of equations (see Limitations and Diagnostics below).
		This algorithm is a subspace trust region method and is based on the interior-reflective Newton method described in [1],[2].
	www.weizmann.ac.il /matlab/toolbox/optim/fsolve.html (1360 words)

	Command >> JACOBIAN (Site not responding. Last check: 2007-10-11)
		This is the parent command of a number of subcommands that deal with the calculation of the Jacobian matrix J. The elements of the Jacobian matrix, calculated at the calibration points, are the partial derivatives of the system response with respect to the parameters to be estimated:
		The Jacobian matrix discussed here must be distinguished from the one calculated in the simulation program TOUGH2.
		The latter is used to solve the set of non-linear algebraic iterations arising at each time step; its elements are the partial derivatives of the mass residuals with respect to the primary variables, and its numerical computation is controlled by the TOUGH2 variable DFAC.
	www-esd.lbl.gov /ITOUGH2/Command/JACOBIAN_2.HTML (125 words)

	OOMPH-LIB: The Object-Oriented Multi-Physics finite-element LIBrary
		Indeed, the assembly of the Jacobian matrix in step 4, and the solution of the linear system in step 5 have an "optimal" computational complexity: their cost increases linearly with the number of unknowns in the problem.
		The Jacobian matrix (13) that arises from the symmetric form of the weak problem is symmetric for any choice of basis functions whereas the Jacobian matrix (15) that arises from the original form is symmetric only for certain types of basis functions.
		The entries in the Element's residual vector and its Jacobian matrix are labelled by local equation numbers, and each element stores a lookup scheme for the correspondence between local and global equation numbers.
	www.maths.manchester.ac.uk /~mheil/oomph-lib_link/intro/html/index.html (7921 words)

	Solver Advanced Tutorial - Handling Sparsity in the Jacobian Directly
		A matrix with many zero elements is called sparse; a matrix with mostly nonzero elements is called dense.
		A matrix of a few hundred, or even a thousand rows and columns doesn't take that much memory, by modern standards.
		But a 10,000 by 10,000 matrix of double precision numbers (8 bytes each) would require 800 megabytes of RAM -- more than the memory available on most PCs today -- if all of the zero elements were stored explicitly.
	www.solver.com /tutorialadv3.htm (485 words)

	Going through each Step (Site not responding. Last check: 2007-10-11)
		Then the transimpedance matrix is derived by exciting one by one the port nodes of the MNA matrix with unity current.
		After that the transadmittance matrix is calculated by inverting the transimpedance matrix.
		One further thing must be mentioned: Because the non-linear components and the sources are missing in the linear MNA matrix, there are often components that are completely disconnected from the rest of the circuit.
	qucs.sourceforge.net /tech/node35.html (1672 words)

	Nonlinear Solver Project: NOX::Epetra::Group Class Reference
		The Group constructor with separate entries for the Jacobian and preconditioner objects must be used.
		Since IFPACK preconditioners require matrix information, the Jacobian must be derived from an Epetra_RowMatix (NOX will test for this and throw an error if it is not).
		Applies the inverse of the Jacobian matrix to the given input vector and puts the answer in result.
	software.sandia.gov /nox/classNOX_1_1Epetra_1_1Group.html (2962 words)

	linf05fin
		are spanned by the columns of the matrix
		is differentiable, and moreover, that the Jacobian matrix is
		Note:Although possibly confusing, the determinant of the Jacobian matrix
	www-math.cudenver.edu /~rrosterm/linf05fin/linf05fin.html (423 words)

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