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In numerical linear algebra, the Jacobi method (a.k.a. the Jacobi iteration method) is an iterative algorithm for determining the solutions of a strictly diagonally dominant system of linear equations. Each diagonal element is solved for, and an approximate value is plugged in. The process is then iterated until it converges.
The absolute value of the Jacobian determinant at p gives us the factor by which the function f expands or shrinks volumes near p; this is why it occurs in the general substitution rule. The Jacobian determinant is used when making a change of variables when evaluating a multiple integral of a function over a region within its domain. To ...
Relaxation methods are used to solve the linear equations resulting from a discretization of the differential equation, for example by finite differences. [ 2 ] [ 3 ] [ 4 ] Iterative relaxation of solutions is commonly dubbed smoothing because with certain equations, such as Laplace's equation , it resembles repeated application of a local ...
It was first conjectured in 1939 by Ott-Heinrich Keller, [1] and widely publicized by Shreeram Abhyankar, as an example of a difficult question in algebraic geometry that can be understood using little beyond a knowledge of calculus. The Jacobian conjecture is notorious for the large number of attempted proofs that turned out to contain subtle ...
In algebraic geometry a generalized Jacobian is a commutative algebraic group associated to a curve with a divisor, generalizing the Jacobian variety of a complete curve. They were introduced by Maxwell Rosenlicht in 1954, and can be used to study ramified coverings of a curve, with abelian Galois group .
The conformal property may be described in terms of the Jacobian derivative matrix of a coordinate transformation. The transformation is conformal whenever the Jacobian at each point is a positive scalar times a rotation matrix (orthogonal with determinant one). Some authors define conformality to include orientation-reversing mappings whose ...
The following algorithm is a description of the Jacobi method in math-like notation. It calculates a vector e which contains the eigenvalues and a matrix E which contains the corresponding eigenvectors; that is, e i {\displaystyle e_{i}} is an eigenvalue and the column E i {\displaystyle E_{i}} an orthonormal eigenvector for e i {\displaystyle ...
Jacobian variety. Generalized Jacobian; Moduli of algebraic curves; Hurwitz's theorem on automorphisms of a curve; Clifford's theorem on special divisors; Gonality of an algebraic curve; Weil reciprocity law; Algebraic geometry codes