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When m = n, the Jacobian matrix is square, so its determinant is a well-defined function of x, known as the Jacobian determinant of f. It carries important information about the local behavior of f .
The standard convergence condition (for any iterative method) is when the spectral radius of the iteration matrix is less than 1: ((+)) < A sufficient (but not necessary) condition for the method to converge is that the matrix A is strictly or irreducibly diagonally dominant. Strict row diagonal dominance means that for each row, the absolute ...
For a system of the form (,, ′) =, some sources also require that the Jacobian matrix (,,) be non-singular in order to call this an implicit ODE [system]; an implicit ODE system satisfying this Jacobian non-singularity condition can be transformed into an explicit ODE system.
Difficult integrals may also be solved by simplifying the integral using a change of variables given by the corresponding Jacobian matrix and determinant. [1] Using the Jacobian determinant and the corresponding change of variable that it gives is the basis of coordinate systems such as polar, cylindrical, and spherical coordinate systems.
The Jacobian of this system is the bordered matrix [ F u F λ u ˙ ∗ λ ˙ ] {\displaystyle \left[{\begin{array}{cc}F_{u}&F_{\lambda }\\{\dot {u}}^{*}&{\dot {\lambda }}\\\end{array}}\right]} At regular points, where the unmodified Jacobian is full rank, the tangent vector spans the null space of the top row of this new Jacobian.
Newton's method for solving f(x) = 0 uses the Jacobian matrix, J, at every iteration. However, computing this Jacobian can be a difficult and expensive operation; for large problems such as those involving solving the Kohn–Sham equations in quantum mechanics the number of variables can be in the hundreds of thousands. The idea behind Broyden ...
A Jacobi operator, also known as Jacobi matrix, is a symmetric linear operator acting on sequences which is given by an infinite tridiagonal matrix. It is commonly used to specify systems of orthonormal polynomials over a finite, positive Borel measure. This operator is named after Carl Gustav Jacob Jacobi.
In mathematics, a Jacobian, named for Carl Gustav Jacob Jacobi, may refer to: Jacobian matrix and determinant (and in particular, the robot Jacobian ) Jacobian elliptic functions