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The concept of conjugate directions came from classical geometry of ellipse. For an ellipse, two semi-axes center are mutually conjugate with respect to the ellipse iff the lines are parallel to the tangent bounding parallelogram, as pictured.
Conjugate gradient, assuming exact arithmetic, converges in at most n steps, where n is the size of the matrix of the system (here n = 2). In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-semidefinite.
Whereas linear conjugate gradient seeks a solution to the linear equation =, the nonlinear conjugate gradient method is generally used to find the local minimum of a nonlinear function using its gradient alone. It works when the function is approximately quadratic near the minimum, which is the case when the function is twice differentiable at ...
In differential geometry, conjugate points or focal points [1] [2] are, roughly, points that can almost be joined by a 1-parameter family of geodesics. For example, on a sphere, the north-pole and south-pole are connected by any meridian. Another viewpoint is that conjugate points tell when the geodesics fail to be length-minimizing.
The ellipse, parabola, and hyperbola are viewed as conics in projective geometry, and each conic determines a relation of pole and polar between points and lines. Using these concepts, "two diameters are conjugate when each is the polar of the figurative point of the other." [5] Only one of the conjugate diameters of a hyperbola cuts the curve.
In projective geometry, the harmonic conjugate point of a point on the real projective line with respect to two other points is defined by the following construction: Given three collinear points A, B, C , let L be a point not lying on their join and let any line through C meet LA, LB at M, N respectively.
As isogonal conjugation is a function, it makes sense to speak of the isogonal conjugate of sets of points, such as lines and circles. For example, the isogonal conjugate of a line is a circumconic ; specifically, an ellipse , parabola , or hyperbola according as the line intersects the circumcircle in 0, 1, or 2 points.
In fact, every equational law satisfied by conjugation in a group follows from the quandle axioms. So, one can think of a quandle as what is left of a group when we forget multiplication, the identity, and inverses, and only remember the operation of conjugation. Every tame knot in three-dimensional Euclidean space has a 'fundamental quandle'.
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