In numerical optimization , the nonlinear conjugategradientmethod generalizes the conjugategradientmethod to nonlinear optimization . For a quadratic function math displaystyle f x math math displaystyle f x Ax b 2 math The minimum of math f math is obtained when the gradient is 0 math nabla x f 2 A top Ax b 0 math . Whereas linear conjugategradient seeks a solution to the linear equation math displaystyle A top Ax A top b math , the nonlinear conjugategradientmethod is generally used to find the local minimum of a nonlinear function using its gradient math nabla x f math alone. It works ... conjugategradientmethod but have been obtained with line searches. The conjugategradientmethod can follow narrow ill conditioned valleys where the steepest descent method slows down and follows a criss ... based methods Newton Raphson Algorithm , Quasi Newton methods e.g., BFGS method tend to converge in fewer iterations, although each iteration typically requires more computation than a conjugategradient iteration as Newton like methods require computing the Hessian matrix of second derivatives in addition to the gradient. Quasi Newton methods also require more memory to operate see also the limited memory L BFGS method . External links http www.cs.cmu.edu quake papers painless conjugate gradient.pdf An Introduction to the ConjugateGradientMethod Without the Agonizing Pain by Jonathan ... , chapter 10, section 6 ConjugateGradient Methods in Multidimensions William H. Press Editor ... to minimize, its gradient math nabla x f math indicates the direction of maximum increase. One ... of moving along a subsequent conjugate direction math displaystyle Lambda x n math , where math ... x n math , Compute math displaystyle beta n math according to one of the formulas below, Update the conjugate ... resetting every iteration turns the method into steepest descent . The algorithm stops when it finds ... Press 2nd edition 1992 . Category Optimization algorithms and methods Category Gradient methods ... more details
definite matrix positive definite . The conjugategradientmethod is an iterative method , so it can ... equation s. The conjugategradientmethod can also be used to solve unconstrained Mathematical ... provides a generalization to non symmetric matrices. Various nonlinear conjugategradientmethod ... sub sub . The conjugategradientmethod as a direct method We say that two non zero vectors u and v ... k sub . The conjugategradientmethod as an iterative method If we choose the conjugate vectors p ... x sub sub . So, we want to regard the conjugategradientmethod as an iterative method. This also allows ... will be conjugate to the gradient, hence the name conjugategradientmethod . Let r sub k ... towards the conjugategradientmethod. However, it requires storage of all the previous searching ... for math beta k math is also used in the Fletcher Reeves nonlinear conjugategradientmethod . Example ... r rsnew rsold p rsold rsnew end end source Numerical example To illustrate the conjugategradientmethod ... bmatrix , math we will perform two steps of the conjugategradientmethod beginning with the initial ... properties of the conjugategradientmethod The conjugategradientmethod can theoretically ... conjugategradientmethod See also Preconditioner In most cases, preconditioning is necessary to ensure fast convergence of the conjugategradientmethod. The preconditioned conjugategradient ... is equivalent to applying the conjugategradientmethod without preconditioning to the system ref label ... of the preconditioned conjugategradientmethod may become unpredictable. The flexible preconditioned conjugategradientmethod In numerically challenging applications, sophisticated preconditioners ... of the algorithm presented above. Using the nonlinear conjugategradientmethod Polak Ribi re formula ... r k math instead of the nonlinear conjugategradientmethod Fletcher Reeves formula math beta ... ConjugateGradientMethod with Inner Outer Iteration year 1999 last1 Golub first1 Gene H. last2 ... more details
In numerical linear algebra , the conjugategradientmethod is an iterative method for numerically solving the System of linear equations linear system math boldsymbol Ax boldsymbol b math where math boldsymbol A math is Symmetric matrix symmetric Positive definite matrix positive definite . The conjugategradientmethod can be derived from several different perspectives, including specialization of the conjugate direction method for Optimization mathematics optimization , and variation of the Arnoldi ... is to document the important steps in these derivations. Derivation from the conjugate direction method Expand section date April 2010 The conjugategradientmethod can be seen as a special case of the conjugate direction method applied to minimization of the quadratic function math f boldsymbol ... above straightforwardly lead to the direct Lanczos method, which turns out to be slightly more complex. The conjugategradientmethod from imposing orthogonality and conjugacy If we allow math ... . math The conjugate direction method In the conjugate direction method for minimizing math f boldsymbol ..., boldsymbol p 2, ldots math are a series of mutually conjugate directions, i.e., math boldsymbol p i mathrm T boldsymbol Ap j 0 math for any math i neq j math . The conjugate direction method is imprecise ... p 1, boldsymbol p 2, ldots math . Specific choices lead to various methods including the conjugategradientmethod and Gaussian elimination . Derivation from the Arnoldi Lanczos iteration see Arnoldi iteration Lanczos iteration The conjugategradientmethod can also be seen as a variant of the Arnoldi Lanczos iteration applied to solving linear systems. The general Arnoldi method In the Arnoldi ... V i boldsymbol y i math . The direct Lanczos method For the rest of discussion, we assume that math .... R. authorlink1 David Hestenes last2 Stiefel first2 E. authorlink2 Eduard Stiefel title Methods of conjugate ... and methods Category Gradient methods Category Articles containing proofs zh ... more details
In optimization mathematics optimization , gradientmethod is an algorithm to solve problems of the form math min x in mathbb R n f x math with the search directions defined by the gradient of the function at the current point. Examples of gradientmethod are the gradient descent and the conjugategradient . See also col begin col break Gradient descent methodConjugategradientmethod Derivation of the conjugategradientmethod Nonlinear conjugategradientmethod Biconjugate gradientmethod Biconjugate gradient stabilized method References cite book year 1997 title Optimization Algorithms and Consistent Approximations publisher Springer Verlag isbn 0 387 94971 2 author Elijah Polak Optimization algorithms DEFAULTSORT GradientMethod Category First order methods Category Optimization algorithms and methods Category Numerical linear algebra Category Gradient methods fr Algorithme du gradient ja ... more details
Multiple issues confusing January 2010 tone January 2010 unreferenced January 2010 wikify January 2011 The conjugate method is a multi faceted method of rotating and linking special exercises that are close in nature to one another. Description of the method The most common template for this method revolves around three methods of weight training used in conjunction with one another. These three methods are Overcoming maximal resistance that causes maximal or near maximal muscle tension maximal effort method . Using considerably less than maximal resistance until fatigue causes one to fail repeated effort method . Using sub maximal weights accompanied by maximal speed dynamic method . External links http www.westside barbell.com Category Weightlifting Category Powerlifting ... more details
The conjugate residual method is an iterative numeric method used for solving systems of linear equations . It s a Krylov subspace method very similar to the much more popular conjugategradientmethod , with similar construction and convergence properties. This method is used to solve linear equations of the form math mathbf A mathbf x mathbf b math where A is an invertible and Hermitian matrix , and b is nonzero. The conjugate residual method differs from the closely related conjugategradientmethod primarily in that it involves somewhat more computation but is applicable to problems that aren t positive definite in fact the only requirement besides the obvious invertible A and nonzero b is that A be Hermitian matrix Hermitian or, with real numbers, symmetric . This makes the conjugate residual method applicable to problems which intuitively require finding saddle points instead of minima, such as numeric optimization with Lagrange multiplier constraints. Given an arbitrary initial estimate of the solution math mathbf x 0 math , the method is outlined below math mathbf x 0 text Some initial guess , math math mathbf r 0 mathbf b mathbf A x 0 , math math mathbf p 0 mathbf r 0 , math math text Iterate, with k text starting at 0 math math alpha k frac mathbf r k mathrm T mathbf A r k mathbf A p k mathrm T mathbf A p k , math math mathbf x k 1 mathbf x k alpha k mathbf p k , math math mathbf r k 1 mathbf r k alpha k mathbf A p k , math math beta k frac mathbf r k 1 mathrm T mathbf A r k 1 mathbf r k mathrm T mathbf A r k , math math mathbf p k 1 mathbf r k 1 beta k mathbf p k ... this and the conjugategradientmethod is the calculation of math alpha k math and math beta ... By making a few substitutions and variable changes, a preconditioned conjugate residual method may be derived in the same way as done for the conjugategradientmethod math mathbf x 0 text Some initial ... 978 0898715347. Jonathan Richard Shewchuck, An Introduction to the ConjugateGradientMethod Without ... more details
Orphan date May 2011 File elastic load method full .png thumb right 300px 0 real beam, 1 shear and moment, 2 conjugate beam, 3 slope and displacement The conjugate beam method is an engineering method to derive the slope and displacement of a beam. The conjugate beam method was developed by H. M ller ... beam method. Conjugate beam Draw the conjugate beam for the real beam. This beam has the same ... to determine a beam s slope or deflection however, this method relies only on the principles of statics ... for the method comes from the similarity of Eq. 1 and Eq 2 to Eq 3 and Eq 4. To show this similarity ... beam, but referred here as the conjugate beam. The conjugate beam is loaded with the M EI diagram ... to the conjugate beam ref cite book last Hibbeler first R.C. title Structural Analysis year 2009 ... in the real beam is numerically equal to the shear at the corresponding point in the conjugate beam ... point tn the conjugate beam. ref cite book last Hibbeler first R.C. title Structural Analysis year 2009 publisher Pearson location Upper Saddle River, NJ pages 328 335 ref Conjugate beam supports When drawing the conjugate beam it is important that the shear and moment developed at the supports of the conjugate beam account for the corresponding slope and displacement of the real beam ... Theorems 1 and 2, the conjugate beam must be supported by a pin or a roller, since this support ... and displacement are zero. Here the conjugate beam has a free end, since at this end there is zero shear and zero moment. Corresponding real and conjugate supports are shown below. Note that, as a rule, neglecting axial forces, statically determinate real beams have statically determinate conjugate beams and statically indeterminate real beams have unstable conjugate beams. Although this occurs, the M EI loading will provide the necessary equilibrium to hold the conjugate beam stable. ref cite ... Saddle River, NJ pages 328 335 ref class wikitable Real support vs Conjugate support ref name okamura ... more details
gradientmethod is numerical stability numerically unstable Citation needed date September 2009 compare to the biconjugate gradient stabilized method , but very important from theoretical ... k r j left 1 P k right math where math i,j k math . The biconjugate gradientmethod now makes a special ..., and the algorithm takes the form stated above. Properties If math A A , math is Conjugate transpose self adjoint , math x 0 x 0 math and math b b math , then math r k r k math , math p k p k math , and the conjugategradientmethod produces the same sequence math x k x k math at half the computational ... deg left P i right k math , then math v i P i left AM 1 right r k 0 math . see also Conjugategradientmethod Biconjugate gradient stabilized method References cite journal first R. last Fletcher year 1976 title Conjugategradient methods for indefinite systems journal Numerical Analysis volume ... , and math overline alpha math is the complex conjugate . Unpreconditioned version of the algorithm ... otimes v k A left 1 P k right over v k A left 1 P k right u k . math A relation to Quasi Newton method ... algebra Category Gradient methods de BiCG Verfahren fr M thode du gradient biconjugu ... more details
also Biconjugate gradientmethodConjugategradient squared methodConjugategradientmethod References ...In numerical linear algebra , the biconjugate gradient stabilized method , often abbreviated as BiCGSTAB , is an iterative method developed by Henk van der Vorst H. A. van der Vorst for the numerical solution of nonsymmetric System of linear equations linear system s. It is a variant of the biconjugate gradientmethod BiCG and has faster and smoother convergence than the original BiCG as well as other variants such as the conjugategradient squared method CGS . It is a Krylov subspace method. Algorithmic steps Unpreconditioned BiCGSTAB To solve a linear system math var Ax var var b var , BiCGSTAB starts with an initial guess math var x var sub 0 sub and proceeds as follows math var r var sub 0 sub var b var var Ax var sub 0 sub Choose an arbitrary vector math var r var sub 0 sub such that math var r var sub 0 sub , var r var sub 0 sub 0 , e.g., math var r var sub 0 sub var r var sub 0 sub math var var sub 0 sub var var var var sub 0 sub 1 math var v var sub 0 sub var p var sub 0 sub 0 For math var i var 1, 2, 3, math var sub i sub var var r var sub 0 sub , var r var sub var i var 1 sub math var var var sub i sub var var var sub var i var 1 sub var var var var sub var i var 1 sub math var p sub i sub var var r var sub var i var 1 sub var var var p var sub var i var 1 sub var var sub var i var 1 sub var v var sub var i var 1 sub math var v sub i sub var var Ap var sub var i var sub math var var var sub i sub var var r var sub 0 sub , var v sub i sub var math var s var var r var sub var i var 1 sub var v sub i sub var math var t var var As var math var sub i sub var var t var , var s var var t var , var t var math var x sub i sub var var x var sub var i var 1 sub var p sub i sub var var sub i sub s var If math var x sub i sub var is accurate ... Numerical linear algebra Category Numerical linear algebra Category Gradient methods Category Articles ... more details
Wiktionarypar gradient The Gradient is the rate of variation of a numerical quantity. It may refer to Slope or Grade slope grade , referring to the inclination of a road or other geographic features Image gradient , a gradual change or blending of color Color gradient Texture gradient Mathematics Gradient , in vector calculus, a vector field representing the maximum rate of increase of a scalar field or a multivariate function and the direction of this maximal rate. Gradient descent Gradient theorem GradientmethodConjugategradientmethod Nonlinear conjugategradientmethod Stochastic gradient descent Biology diffusion Concentration gradient , the ratio of solute concentration between two adjoining regions Potential gradient , the difference in electric charge between two adjoining regions Fluid dynamics and earth science Density gradient Pressure gradient Temperature gradient Geothermal gradient Sound speed gradient Wind gradient See also lookfrom gradient intitle gradient Graduation disambiguation Fade disambiguation disambig bs Gradijent vor cs Gradient de Gradient eo Gradiento io Gradiento homonimo ja pl Gradient simple Gradient disambiguation tl Gradient ... more details
of steepness of a line Slope other uses In vector calculus , the gradient of a scalar field is a vector ... magnitude mathematics magnitude is that rate of increase. A generalization of the gradient ... corresponding gradient is represented by blue arrows. Interpretations File Gradient of a Function.tif thumb 350px Gradient of the 2 d function math f x,y xe x 2 y 2 math is plotted as blue arrows over ... time. At each point in the room, the gradient of math T math at that point will show the direction the temperature rises most quickly. The magnitude of the gradient will determine how fast the temperature ..., y math is math H x, y math . The gradient of math H math at a point is a vector pointing in the direction ... is given by the magnitude of the gradient vector. The gradient can also be used to measure how ... as follows. If the hill height function math H math is differentiable function differentiable , then the gradient ..., the dot product of the gradient of math H math with a given unit vector is equal to the directional ... 350px The gradient of the function f x , y &minus cos sup 2 sup x cos sup 2 sup y sup 2 sup depicted as a projected vector field on the bottom plane The gradient or gradient ... , del . The notation math operatorname grad f math is also commonly used for the gradient. The gradient ... v f x . math In a rectangular coordinate system, the gradient is the vector field whose components ... directions. When a function also depends on a parameter such as time, the gradient often ... unit vectors. For example, the gradient of the function math f x,y,z 2x 3y 2 sin z math ... to represent the gradient as a row vector or column vector of its components in a rectangular coordinate system. Gradient and the derivative or differential Linear approximation to a function The gradient ... nabla f x 0 cdot x x 0 math for math x math close to math x 0 math , where math nabla f x 0 math is the gradient ... of math f math at math x math . The gradient is therefore related to the differential by the formula ... more details
, gradient descent is rarely used, with the conjugategradientmethod being one of the most popular ... of math A math , while the speed of convergence of conjugategradientmethodconjugate gradients ... no extra heavy computation, yet yields faster convergence. See also Conjugategradientmethod Stochastic gradient descent Rprop Delta rule Wolfe conditions Preconditioning Nelder Mead method ... 400px The Zig Zagging nature of the method is also evident below, where the gradient ascent method ... and inversion of the Hessian matrix Hessian using conjugategradient techniques can be better ... is the BFGS method which consists in calculating on every step a matrix by which the gradient vector ... of BFGS or the steepest descent. Gradient descent can be viewed as Euler s method for solving ordinary ... step math k math by the gradient descent method will be Big O notation bounded by math mathcal O ...About the mathematical analysis analytical method called steepest descent Method of steepest descent Gradient descent is a First order approximation first order Mathematical optimization optimization algorithm . To find a local minimum of a function using gradient descent, one takes steps proportional to the negative of the gradient or of the approximate gradient of the function at the current point. If instead one takes steps proportional to the positive of the gradient , one approaches a local maximum of that function the procedure is then known as gradient ascent . Gradient descent is also known as steepest descent , or the method of steepest descent . When known as the latter, gradient descent should not be confused with the method of steepest descent for approximating integrals. Description Image gradient descent.png thumb right 350px Illustration of gradient descent Gradient descent ... of the negative gradient of math F math at math mathbf a math , math nabla F mathbf a math . It follows ... in this case gradient descent can converge to the global solution. This process is illustrated in the picture ... more details
Image Conjugate Diameters.svg thumb 300px right Two conjugate diameters of an ellipse . Each edge of the bounding parallelogram is Parallel geometry parallel to one of the diameters. In geometry , two diameter s of a conic section are said to be conjugate if each chord geometry chord parallel geometry parallel to one diameter is bisection bisected by the other diameter. For example, two diameters of a circle are conjugate if and only if they are perpendicular . For an ellipse , two diameters are conjugate if and only if the tangent line to the ellipse at one endpoint of a diameter is parallel to the tangent at the second endpoint. Each pair of conjugate diameters of an ellipse has a corresponding tangent parallelogram , sometimes called the bounding parallelogram . In his manuscript De motu corporum in gyrum , and in the Philosophi Naturalis Principia Mathematica Principia , Isaac Newton cites as a lemma mathematics lemma proved by previous authors that all bounding parallelograms for a given ... an ellipse from any pair of conjugate diameters, or from any bounding parallelogram. For example, in proposition 14 of Book VIII of his Collection , Pappus of Alexandria gives a method for constructing the axes of an ellipse from a given pair of conjugate diameters. File Drini conjugatehyperbolas.svg thumb right Blue and green hyperbolas are conjugate. A diameter from x,y to &minus x ,&minus y is conjugate to the one from y,x to &minus y ,&minus x . Two hyperbola s are conjugate if they are images ... hyperbola is conjugate to its reflection in the asymptote, which is a diameter of the other hyperbola. They are hyperbolic orthogonal to each other. Conjugate diameters of hyperbolas are useful for stating ... can be formulated Any pair of conjugate diameters of conjugate hyperbolas can be taken for the axes .... References PlanetMath urlname ConjugateRadii title Conjugate Diameters of Ellipse http www.cut the knot.org Curriculum Geometry ConjugateDiameters.shtml Conjugate Diameters in Ellipse at cut the knot.org. ... more details
In chemistry , a Lewis conjugate might mean The conjugate base of a Lewis acid or the conjugate acid of a Lewis base A molecule having a conjugated system of bonds in its Lewis structure Disambig ... more details
File Complex conjugate picture.svg right thumb Geometric representation of z and its conjugate z in the complex plane In mathematics , complex conjugates are a pair of complex number s, both having the same real number real part, but with imaginary number imaginary parts of equal magnitude and opposite sign mathematics sign s. ref MathWorld ComplexConjugate Complex Conjugates ref ref MathWorld ImaginaryNumber Imaginary Numbers ref For example, 3 4i and 3 &minus 4i are complex conjugates. The conjugate ... 7 7 math math overline i i. math An alternative notation for the complex conjugate is math z math . However, the math bar z math notation avoids confusion with the notation for the conjugate transpose ... plane about the Re axis. In Polar coordinate system Complex numbers polar form , the conjugate of math ... to a problem, so does its conjugate, such as is the case for complex solutions of the quadratic ..., the conjugate of the conjugate of a complex number z is again that number math z 1 frac overline z left z right 2 math if z is non zero The latter formula is the method of choice to compute the inverse .... Thus, non real roots of real polynomials occur in complex conjugate pairs see Complex conjugate ... z rho e i theta math is given, its conjugate is sufficient to reproduce the parts of the z variable ... determines the line through math z 0 , math in the direction of u. These uses of the conjugate of z ... A B math . Taking the conjugate transpose or adjoint of complex matrix mathematics matrices generalizes ... s. One may also define a conjugation for quaternion s and coquaternion s the conjugate of math a bi ..., p. 29 ref One example of this notion is the conjugate transpose operation of complex matrices ... canonical notion of complex conjugation. See also Complex conjugate vector space Real structure Complex ... Conjugate Category Complex numbers ar bs Konjugovano kompleksan broj ca Conjugat cs ... conjugate zh ... more details
Unreferenced date December 2009 In differential geometry , conjugate points are, roughly, points that can almost be joined by a 1 parameter family of geodesic s. For example, on a Spherical geometry sphere , the north pole and south pole are connected by any Meridian geography meridian . Definition Suppose p and q are points on a Riemannian manifold , and math gamma math is a geodesic that connects p and q . Then p and q are conjugate points along math gamma math if there exists a non zero Jacobi field along math gamma math that vanishes at p and q . Recall that any Jacobi field can be written as the derivative of a geodesic variation see the article on Jacobi field s . Therefore, if p and q are conjugate along math gamma math , one can construct a family of geodesics which start at p and almost end at q . In particular, if math gamma s t math is the family of geodesics whose derivative in s at math s 0 math generates the Jacobi field J , then the end point of the variation, namely math gamma s 1 math , is the point q only up to first order in s . Therefore, if two points are conjugate, it is not necessary that there exist two distinct geodesics joining them. Examples On the sphere math S 2 math , antipodal point s are conjugate. On math mathbb R n math , there are no conjugate points. On Riemannian manifolds with non positive sectional curvature , there are no conjugate points. See also Cut locus Riemannian manifold cut locus Jacobi field DEFAULTSORT Conjugate Points Category Riemannian geometry ... more details
Image Isogonal Conjugate.svg 200px right thumb Isogonal coniugate of P . Image Isogonal Conjugate transform.svg 200px right thumb Isogonal coniugate transformation over the points inside the triangle. In geometry , the isogonal conjugate of a point geometry point P with respect to a triangle ABC is constructed by reflection mathematics reflecting the lines PA , PB , and PC about the angle bisectors of A , B , and C . These three reflected lines concurrent lines concur at the isogonal conjugate of P . This definition applies only to points not on a sideline of triangle ABC . The isogonal conjugate of a point P is sometimes denoted by P . The isogonal conjugate of P is P . The isogonal conjugate of the incentre I is itself. The isogonal conjugate of the orthocentre H is the circumcentre O . The isogonal conjugate of the centroid G is by definition the symmedian symmedian point K . The isogonal conjugates of the Fermat point s are the isodynamic point s and vice versa. The Brocard points are isogonal conjugates of each other. In trilinear coordinates , if X x y z is a point not on a sideline of triangle ABC , then its isogonal conjugate is 1 x 1 y 1 z . For this reason, the isogonal conjugate of X is sometimes denoted by X sup &minus 1 sup . The set S of triangle centers under trilinear product, defined by p q r u v w pu qv rw , is a commutative group, and the inverse of each X in S is X sup &minus 1 sup . 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 ... according as the line intersects the circumcircle in 0, 1, or 2 points. The isogonal conjugate ... cubic, Neuberg cubic are self isogonal conjugate, in the sense that if X is on the cubic, then X sup &minus 1 sup is also on the cubic. See also Isotomic conjugate External links http www.uff.br trianglecenters isogonal conjugate en.html Interactive Java Applet illustrating isogonal conjugate ... more details
In group theory , the conjugate closure of a subset S of a group mathematics group G is the subgroup of G generating set of a group generated by S sup G sup , i.e. the closure of S sup G sup under the group operation, where S sup G sup is the Conjugate group theory conjugates of the elements of S S sup G sup g sup &minus 1 sup sg g &isin G and s &isin S The conjugate closure of S is denoted S sup G sup or S sup G sup . The conjugate closure of any subset S of a group G is always a normal subgroup of G in fact, it is the smallest by inclusion normal subgroup of G which contains S . For this reason, the conjugate closure is also called the normal closure of S or the normal subgroup generated by S . The normal closure can also be characterized as the intersection set theory intersection of all normal subgroups of G which contain S . Any normal subgroup is equal to its normal closure. The conjugate closure of a singleton set singleton subset a of a group G is a normal subgroup generated by a and all elements of G which are conjugate to a . Therefore, any simple group is the conjugate closure of any non identity group element. The conjugate closure of the empty set math varnothing math is the trivial group . Contrast the normal closure of S with the normalizer of S , which is for S a group the largest subgroup of G in which S itself is normal. This need not be normal in the larger group G , just as S need not be normal in its conjugate normal closure. References cite book title Handbook of Computational Group Theory author Derek F. Holt coauthors Bettina Eick, Eamonn A. O Brien publisher CRC Press year 2005 isbn 1 58488 372 3 pages 73 Category Group theory Abstract algebra stub zh ... more details
In mathematics , two real number s math p, q 1 math are called conjugate indices if math frac 1 p frac 1 q 1. math Formally, we will also define math q infty math as conjugate to math p 1 math and List of Latin phrases V vice versa vice versa . Conjugate indices are used in H lder s inequality . Also, if math p, q 1 math are conjugate indices, the spaces L sup p sup and L sup q sup are dual space dual to each other see Lp space L sup p sup space . See also Beatty s theorem References A B Antonevich, Linear Functional Equations , Birkh user, 1999. ISBN 3 7643 2931 9. PlanetMath attribution id 2051 title Conjugate index Category Functional analysis de Konjugation Reelle Zahlen zh ... more details
About binomial conjugates in algebra Conjugate disambiguation merge Difference of two squares date January 2012 In algebra , a conjugate is a binomial formed by taking the opposite of the second term of a binomial. The conjugate of math x y , math is math x y , math , where x and y are real number s. If y is imaginary number imaginary , the process is termed complex conjugation . The complex conjugate of a bi is a bi . Differences of squares main Difference of two squares An expression of the form math a 2 b 2 , math can be factored to give math a b a b , math where one factor is the conjugate of the other. This can be useful when trying to rationalize a denominator containing radicals. Rationalizing radicals in denominator main Rationalisation mathematics An irrational number irrational binomial can sometimes be made rational by multiplying by its conjugate. When rationalizing a denominator, the numerator may remain irrational, though. In order to keep the value of the fraction the same, it is multiplied by the conjugate divided by itself, as shown in the examples below. math left frac 1 a sqrt b right left frac a sqrt b a sqrt b right frac a sqrt b a 2 b , math math frac 1 2 2 sqrt 3 frac 2 2 sqrt 3 2 2 sqrt 3 frac 2 2 sqrt 3 2 2 2 2 3 frac 2 sqrt 3 2 8 frac sqrt 3 1 4 , math See also Difference of two squares Conjugate element field theory External links http www.mathwords.com r rationalizing the denominator.htm Rationalizing the Denominator from Mathwords.com http www.blc.edu fac rbuelow common glossarya m.htm conjugate Math glossary from Bethany Lutheran College Category Algebra nn Konjugert i matematikk sv Konjugat algebra ... more details
Gradient noise is a type of noise commonly used as a procedural texture primitive in computer graphics. It is conceptually different, and often confused with value noise . This method consists of a creation of a lattice of random gradients , which are then interpolated to obtain values in between the lattices. An artifact of some implementations of this noise is that the returned value at the lattice points is 0. Unlike the value noise, gradient noise has more energy in the high frequencies. The first implementation of a gradient noise function is credited to Ken Perlin , who published the description of it in 1985. This noise is now commonly known as the Perlin noise . ref David Ebert, Kent Musgrave, Darwyn Peachey, Ken Perlin, and Worley. Texturing and Modeling A Procedural Approach. Academic Press, October 1994. ISBN 0 12 228760 6 ref See also Value noise Perlin noise Simplex noise Wavelet noise Worley noise References reflist Category Noise Category Computer graphic techniques graphics software stub ru ... more details
For conjugate variables in context of thermodynamics Conjugate variables thermodynamics Conjugate variables are pairs of variables mathematically defined in such a way that they become Fourier transform dual mathematics duals of one another, ref http www.aip.org history heisenberg p08a.htm Heisenberg Quantum Mechanics, 1925 1927 The Uncertainty Relations ref ref http www.springerlink.com content r40472577250313r Some remarks on time and energy as conjugate variables ref or more generally are related through Pontryagin duality . The duality relations lead naturally to an uncertainty in physics called the Heisenberg uncertainty principle relation between them. In mathematical terms, conjugate variables are part of a symplectic basis, and the uncertainty principle corresponds to the symplectic form . Examples There are many types of conjugate variables, depending on the type of work a certain system is doing or is being subjected to . Examples of canonically conjugate variables include the following Time and frequency the longer a musical note is sustained, the more precisely we know its frequency but it spans more time . Conversely, a very short musical note becomes just a click, and so one can t know its frequency very accurately. Doppler effect Doppler and range the more we know about how far away a radar target is, the less we can know about the exact velocity of approach or retreat, and vice versa. In this case, the two dimensional function of doppler and range is known as a radar ambiguity function or radar ambiguity diagram . Surface energy dA surface tension A surface area . Elastic stretching FdL F elastic force L length stretched . Derivatives of action In classical physics, the derivatives of action physics action are conjugate variables to the quantity with respect ... Canonical coordinates Notes Reflist DEFAULTSORT Conjugate Variables Category Classical mechanics Category Quantum mechanics ar fr Variables conjugu es pl Zmienne sprz one simple Conjugate ... more details
In geometry , the isotomic conjugate of a point P with respect to a triangle ABC is another point, defined from P and ABC . Construction We assume that P is not collinear with two of the vertices of ABC . Let A nowiki nowiki , B nowiki nowiki , C nowiki nowiki be the points in which the lines AP , BP , CP meet the lines BC , CA , AB , respectively. Reflect A nowiki nowiki B nowiki nowiki C nowiki nowiki in the midpoints of sides BC , CA , AB to obtain points A , B , C , respectively. The isotomic lines AA , BB , CC meet at a point this can be proved using Ceva s theorem , and this point is called the isotomic conjugate of P . Coordinates If Trilinear coordinates trilinears for P are p q r , then trilinears for the isotomic conjugate of P are a sup &minus 2 sup p sup &minus 1 sup b sup &minus 2 sup q sup &minus 1 sup c sup &minus 2 sup r sup &minus 1 sup . Properties The isotomic conjugate of the centroid of triangle ABC is the centroid itself. The isotomic conjugate of the symmedian point is the third Brocard point , and the isotomic conjugate of the Gergonne point is the Nagel point . Isotomic conjugates of lines are circumconics, and conversely, isotomic conjugates of circumconics are lines. This property holds for isogonal conjugates as well. References Robert Lachlan, An Elementary Treatise on Modern Pure Geometry , Macmillan and Co., 1893, page 57. Category Triangle geometry de Isotomisch konjugierte Punkte nl Isotomische verwantschap zh ... more details
for geometric conjugate points Projective harmonic conjugates In mathematics , a function math u x, ,y math defined on some open domain math Omega subset R 2 math is said to have as a conjugate a function math v x, ,y math if and only if they are respectively real and imaginary part of a holomorphic function math f z math of the complex variable math z x iy in Omega. math That is, math v math is conjugate to math u math if math f z u x,y iv x,y math is holomorphic on math Omega. math As a first consequence of the definition, they are both harmonic function harmonic real valued functions on math Omega math . Moreover, the conjugate of math u, math if it exists, is unique up to an additive constant. Also, math u math is conjugate to math v math if and only if math v math is conjugate to math u math . Equivalently, math v math is conjugate to math u math in math Omega math if and only if math u math and math v math satisfy the Cauchy Riemann equations in math Omega. math As an immediate consequence of the latter equivalent definition, if math u math is any harmonic function on math Omega subset R 2, math the function math u y math is conjugate to math u x math , for then the Cauchy Riemann ... in math Omega, math in which case a conjugate of math u math is, of course, math scriptstyle mathrm Im ,f x iy . math So any harmonic function always admits a conjugate function whenever its domain is simply connected , and in any case it admits a conjugate locally at any point of its domain. There is an operator ... conjugate v putting e.g. v x sub 0 sub 0 on a given x sub 0 sub in order to fix the indeterminacy of the conjugate up to constants . This is well known in applications as essentially the Hilbert ... operator s. Conjugate harmonic functions and the transform between them are also one of the simplest ... conjugate of x is y , and the lines of constant x and constant y are orthogonal. Conformality ... occurrence of the term harmonic conjugate in mathematics , and more specifically ... more details
differentiable loss function . Gradient boosting method can be also used for classification machine learning classification problems by reducing them to regression with a suitable loss function. The method ... Function Approximation A Gradient Boosting Machine. February 1999 ref introduced the method, and the second ... Gradient boosting method assumes a real valued y and seeks an approximation math hat F x math in the form ... equation above. In pseudocode, the generic gradient boosting method is ref name Friedman1999a ref name ... trees of a fixed size as base learners. For this special case Friedman proposes a modification to gradient boosting method which improves the quality of fit of each base learner. Generic gradient ... other regularization techniques are used. Shrinkage An important part of gradient boosting method .... Usage Recently, gradient boosting method has gained some popularity in learning to rank field ... Model . Sometimes the method is referred to as functional gradient boosting , Gradient Boosted ...Gradient boosting is a machine learning technique for Regression machine learning regression problems ... Stochastic Gradient Boosting. March 1999 ref described an important tweak to the algorithm, which improves its accuracy and performance. Gradient boosting In many supervised learning problems one has ... risk minimization principle, the method tries to find an approximation math hat F x math that minimizes ... allows us to generalize , we ll just choose the one that most closely approximates the gradient of L ... F m x F m 1 x gamma m h m x . math Output math F M x . math frame footer Gradient tree boosting Gradient ... math J math , the number of terminal nodes in trees, is the method s parameter which can be adjusted .... One natural regularization parameter is the number of gradient boosting iterations M i.e. the number ... yields dramatic improvements in model s generalization ability over gradient boosting without shrinking ... both during training and querying lower learning rate requires more iterations. Stochastic gradient ... more details
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