Search results
Results from the WOW.Com Content Network
Carl Friedrich Gauss in 1810 devised a notation for symmetric elimination that was adopted in the 19th century by professional hand computers to solve the normal equations of least-squares problems. [6] The algorithm that is taught in high school was named for Gauss only in the 1950s as a result of confusion over the history of the subject. [7]
NOTE: Gauss's method is a preliminary orbit determination, with emphasis on preliminary. The approximation of the Lagrange coefficients and the limitations of the required observation conditions (i.e., insignificant curvature in the arc between observations, refer to Gronchi [2] for more details) causes inaccuracies.
At any step in a Gauss-Seidel iteration, solve the first equation for in terms of , …,; then solve the second equation for in terms of just found and the remaining , …,; and continue to . Then, repeat iterations until convergence is achieved, or break if the divergence in the solutions start to diverge beyond a predefined level.
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 ...
To quote: "It appears that Gauss and Doolittle applied the method [of elimination] only to symmetric equations. More recent authors, for example, Aitken, Banachiewicz, Dwyer, and Crout … have emphasized the use of the method, or variations of it, in connection with non-symmetric problems …
In this example, the Gauss–Newton algorithm will be used to fit a model to some data by minimizing the sum of squares of errors between the data and model's predictions. In a biology experiment studying the relation between substrate concentration [S] and reaction rate in an enzyme-mediated reaction, the data in the following table were obtained.
Examples of such matrices commonly arise from the discretization of 1D Poisson equation and natural cubic spline interpolation. Thomas' algorithm is not stable in general, but is so in several special cases, such as when the matrix is diagonally dominant (either by rows or columns) or symmetric positive definite ; [ 1 ] [ 2 ] for a more precise ...
Thus, Gauss–Jacobi quadrature can be used to approximate integrals with singularities at the end points. Gauss–Legendre quadrature is a special case of Gauss–Jacobi quadrature with α = β = 0. Similarly, the Chebyshev–Gauss quadrature of the first (second) kind arises when one takes α = β = −0.5 (+0.5).