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They proved that the maximum number of points in the grid with no three points collinear is (). Similarly to Erdős's 2D construction, this can be accomplished by using points ( x , y , x 2 + y 2 {\displaystyle (x,y,x^{2}+y^{2}} mod p ) {\displaystyle p)} , where p {\displaystyle p} is a prime congruent to 3 mod 4 . [ 20 ]
Given an n × n square matrix A of real or complex numbers, an eigenvalue λ and its associated generalized eigenvector v are a pair obeying the relation [1] =,where v is a nonzero n × 1 column vector, I is the n × n identity matrix, k is a positive integer, and both λ and v are allowed to be complex even when A is real.l When k = 1, the vector is called simply an eigenvector, and the pair ...
This happens if and only if the triangle vertices aren't collinear and the ray isn't parallel to the plane. The algorithm can use Cramer's Rule to find the t {\displaystyle t} , u {\displaystyle u} , and v {\displaystyle v} values for an intersection, and if it lies within the triangle, the exact coordinates of the intersection can be found by ...
The NLEVP collection of nonlinear eigenvalue problems is a MATLAB package containing many nonlinear eigenvalue problems with various properties. [ 6 ] The FEAST eigenvalue solver is a software package for standard eigenvalue problems as well as nonlinear eigenvalue problems, designed from density-matrix representation in quantum mechanics ...
Householder reflection for QR-decomposition: The goal is to find a linear transformation that changes the vector into a vector of the same length which is collinear to . We could use an orthogonal projection (Gram-Schmidt) but this will be numerically unstable if the vectors x {\displaystyle \mathbf {x} } and e 1 {\displaystyle \mathbf {e} _{1 ...
The examples in this article apply to active rotations of vectors counterclockwise in a right-handed coordinate system (y counterclockwise from x) by pre-multiplication (the rotation matrix R applied on the left of the column vector v to be rotated).
A demo of Graham's scan to find a 2D convex hull. Graham's scan is a method of finding the convex hull of a finite set of points in the plane with time complexity O(n log n). It is named after Ronald Graham, who published the original algorithm in 1972. [1] The algorithm finds all vertices of the convex hull ordered along its boundary.
In matrix notation, = /, where has orthonormal basis vectors {} and the matrix is composed of the given column vectors {}. The matrix G − 1 / 2 {\displaystyle G^{-1/2}} is guaranteed to exist. Indeed, G {\displaystyle G} is Hermitian, and so can be decomposed as G = U D U † {\displaystyle G=UDU^{\dagger }} with U {\displaystyle U} a unitary ...