Ad
related to: orthogonal techniques examples problems solver
Search results
Results From The WOW.Com Content Network
In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations.The idea is to choose a finite-dimensional space of candidate solutions (usually polynomials up to a certain degree) and a number of points in the domain (called collocation points), and to select that solution which satisfies the ...
Orthogonal decomposition methods of solving the least squares problem are slower than the normal equations method but are more numerically stable because they avoid forming the product X T X. The residuals are written in matrix notation as = ^.
The orthogonal Procrustes problem [1] is a matrix approximation problem in linear algebra. In its classical form, one is given two matrices A {\displaystyle A} and B {\displaystyle B} and asked to find an orthogonal matrix Ω {\displaystyle \Omega } which most closely maps A {\displaystyle A} to B {\displaystyle B} .
To solve the underdetermined (<) linear problem = where the matrix has dimensions and rank , first find the QR factorization of the transpose of : =, where Q is an orthogonal matrix (i.e. =), and R has a special form: = [].
When the object is three-dimensional, the optimum rotation is represented by a 3-by-3 rotation matrix R, rather than a simple angle, and in this case singular value decomposition can be used to find the optimum value for R (see the solution for the constrained orthogonal Procrustes problem, subject to det(R) = 1).
The conjugate gradient method can also be used to solve unconstrained optimization problems such as energy minimization. It is commonly attributed to Magnus Hestenes and Eduard Stiefel , [ 1 ] [ 2 ] who programmed it on the Z4 , [ 3 ] and extensively researched it.
In numerical analysis, a multigrid method (MG method) is an algorithm for solving differential equations using a hierarchy of discretizations. They are an example of a class of techniques called multiresolution methods, very useful in problems exhibiting multiple scales of behavior.
Hilbert matrix — example of a matrix which is extremely ill-conditioned (and thus difficult to handle) Wilkinson matrix — example of a symmetric tridiagonal matrix with pairs of nearly, but not exactly, equal eigenvalues; Convergent matrix — square matrix whose successive powers approach the zero matrix; Algorithms for matrix multiplication: