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Topic: Method of steepest descent

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Cousin problems

Residue theorem

Elliptic function

Exponential function

Theta function

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Schwarzian derivative

Gamma function

The analytization trick

Riemann zeta function

	Gradient descent - Wikipedia, the free encyclopedia
		Gradient descent is an optimization algorithm that approaches a local minimum of a function by taking steps proportional to the negative of the gradient (or the approximate gradient) of the function at the current point.
		Gradient descent is also known as steepest descent, or the method of steepest descent, not to be confused with the method for approximating integrals with the same name, see method of steepest descent.
		A more powerful algorithm is given by the BFGS method which consists in calculating on every step a matrix by which the gradient vector is multiplied to go into a "better" direction, combined with a more sophisticated linear search algorithm, to find the "best" value of γ.
	en.wikipedia.org /wiki/Gradient_descent (435 words)

	Method of steepest descent - Wikipedia, the free encyclopedia
		In mathematics, the steepest descent method or saddle-point approximation is a method used to approximate integrals of the form
		In extensions of this method, complex analysis is used to find a contour of steepest descent for an equivalent integral, expressed as a path integral.
		The method of the steepest descent can be used to derive Stirling's approximation
	en.wikipedia.org /wiki/Method_of_steepest_descent (393 words)

	Justification that steepest descent paths are paths with constant imaginary part.
		Justification that steepest descent paths are paths with constant imaginary part.
		The approximation based on the method of steepest descent is derived as follows.
		The method of the stationary phase is equivalent to the method of the steepest descent.
	www.uic.edu /classes/eecs/eecs526/steepestdescent/node2.html (176 words)

	PlanetMath: conjugate gradient algorithm
		The conjugate gradient method was developed in 1952 by Hestenes and Stiefel as an improvement to the steepest descent method.
		Whereas steepest descent approaches the solution asymptotically, the conjugate gradient method will find the solution in n iterations (assuming no roundoff error).
		The conjugate gradient method has been generalized to the case where the function being minimized is only approximately quadratic.
	planetmath.org /encyclopedia/ConjugateGradientAlgorithm.html (323 words)

	What Direction to Shift?
		The steepest descent method is very good as long as the curvature for each parameter is the equal, but this is far from true in crystallographic refinement.
		With steepest descent and conjugate gradient the large difference in curvature between the different types of parameters results in the requirement that they each be refined separately.
		If a method equivalent to conjugate gradient existed which used the gradient/curvature method as its base instead of steepest descent the off-diagonal elements would be ``learned'' from the history of the refinement.
	www.uoxray.uoregon.edu /tnt/manual/node14.html (858 words)

	RHPH project (Site not responding. Last check: 2007-10-29)
		The steepest descent method gave rise to remarkably strong asymptotic results for orthogonal polynomials as the degree tends to infinity [5], [6].
		The steepest descent method was also used by Baik, Deift, and Johansson in their proof of the distribution of the longest increasing subsequence of a random permutation [1].
		The aim of the project is to apply the steepest descent method to a number of problems arising in approximation theory and mathematical physics.
	www.cs.kuleuven.ac.be /~ade/nalag/research/projects/RHPH.shtml (1932 words)

	Method of Steepest Descent
		The method of Steepest Descent is the simplest of the gradient methods.
		This implementation of the Steepest Descent method are often referred to as the optimal gradient method.
		In other words, the Steepest Descent method can be used where one has an indication of where the minimum is, but is generally considered to be a poor choice for any optimization problem.
	trond.hjorteland.com /thesis/node26.html (568 words)

	The MOE Nonlinear Optimization Library (Site not responding. Last check: 2007-10-29)
		Among the methods contained in the library are Steepest Descent, Conjugate Gradient, and Truncated Newton.
		Each of the methods is suitable for large-scale problems; that is, each of the methods requires memory linear in the number of free variables.
		The method of Steepest Descent is the simplest of the methods and is most often run for a few iterations before using one of the more powerful methods.
	www.chemcomp.com /Journal_of_CCG/Articles/opt.htm (932 words)

	[No title]
		The LMS algorithm is based on the Steepest Descent algorithm where the MSE descents along the performance surface towards the minimum point.
		Steepest Descent method is an iterative technique that search the minimum point of performance surface by following the direction in which the performance surface has the greatest rate of decrease or whose path follows the negative gradient of the performance surface (Sen and Dennis, 1996).
		Since R and P are usually unknown, therefore the method of Steepest Descent must be modified in order to be useful.
	www.geocities.com /fangyih/project/lms.html (645 words)

	Steepest Descent Method for Representing Spatially Correlated Uncertainty in GIS (Site not responding. Last check: 2007-10-29)
		Steepest Descent Method for Representing Spatially Correlated Uncertainty in GIS
		In this research, those methods are presented and compared in terms of computation complexity for the particular system.
		The writer presents the steepest descent method as the best possible method with linear complexity.
	www.pubs.asce.org /WWWdisplay.cgi?0305310 (188 words)

	1 (Site not responding. Last check: 2007-10-29)
		In order to demonstrate the convergence of the steepest descent method, the mean square error (MSE) was calculated and plotted against the number of iterations as shown in Fig 3.
		As in the steepest descent method, the number of iterations were varied to various value and the restored images were save separately in the binary format (*.mat).
		As in the first part, in order to demonstrate the convergence of the steepest descent method, the mean square error (MSE) was calculated and plotted against the number of iterations as shown in Fig 5.
	users.ece.gatech.edu /~gt0371a/proj/image/imagr_report.html (838 words)

	The method of steepest descent (Site not responding. Last check: 2007-10-29)
		The method of steepest descent is a technique to compute the asymptotic behavior of integrals of the form
		The analyticity of the integrand is exploited to deform the integration contour
		Therefore, by integrating along a steepest path the only contribution to the integral come from small neighborhood around local maxima of
	www.uic.edu /classes/eecs/eecs526/steepestdescent/node1.html (82 words)

	Chapter TWO (Site not responding. Last check: 2007-10-29)
		This is a method based on expanding the model equation(s) of interest in a Taylor series [Hartley, 1961].
		The Marquardt method is intermediate between the Taylor series (Gauss-Newton) method and the method of steepest descent (gradient method).
		In the method of steepest descent the new parameter values are calculated in the direction of the negative slope of the WSS with respect to each of the parameters.
	www.boomer.org /Manual/ch02.html (385 words)

	Steepest Descent -Gradient Search
		The influence of the largest eigenvalues on the numerical convergence of the conjugate gradient method.
		Steepest descent evolution equations: asymptotic behavior of solutions and rate of convergence.
		The asymptotic behaviour of the three-step method of steepest descent.
	math.fullerton.edu /mathews/n2003/gradientsearch/GradientSearchBib/Links/GradientSearchBib_lnk_3.html (1072 words)

	No Title
		In the method of steepest descent, we make an initial estimate of the solution x to Ax = b and then proceed down gradients of Q to arrive at the minimum of Q and thus arrive at the solution to Ax = b.
		The method of steepest descent is easy to program, but it often converges slowly.
		The main reason for its slow convergence is that the method may spend time minimizing Q along parallel or nearly parallel search directions.
	cauchy.math.colostate.edu /Resources/SD_CG/sd (273 words)

	Steepest Descent (Site not responding. Last check: 2007-10-29)
		One way to minimize a general function of several variables is to use the method of steepest descent.
		This method is based on the fact that the gradient of a function points opposite the ``fall-line'', the path of steepest descent.
		The problem with this method is that we don't know how far to go along the line of steepest descent, i.e.
	www.physics.utah.edu /~detar/phycs6720/handouts/curve_fit/curve_fit/node5.html (169 words)

	Conjugate Gradient Method
		The conjugate gradient method is based on the idea that the convergence to the solution could be accelerated if we minimize Q over the hyperplane that contains all previous search directions, instead of minimizing Q over just the line that points down gradient.
		From the discussion of the method of steepest descent, we have
		Having developed the CG method for one step it is easy to see that successive iterates are defined as follows.
	cauchy.math.colostate.edu /Resources/SD_CG/conjgrad/node1.html (559 words)

	OhioLINK ETD: PARTHASARATHY, NAVITHA
		The two algorithms used to determine the cylindricity tolerance are the minimum zone method and the least square method.
		Here, the orientation of the axis and the location of a point on the axis are iteratively changed using steepest descent method until the least possible value for the difference between the two coaxial cylinders is found.
		The steepest descent method (SDM) moves the axis at each iteration in the direction of the steepest gradient of the non-linear Cylindricity Error Function to get a new axis.
	rave.ohiolink.edu /etdc/view?acc_num=ucin1093005770 (260 words)

	The Newton-Raphson Method
		The method differs from the Steepest Descent and Conjugate Gradients method, which both are of category (2), in that the information of the second derivative is used to locate the minimum of the function
		The method is then no longer guaranteed to proceed toward a minimum; it may end up at any other critical point, whether it be a saddle point or a maximum point.
		The resultant algorithm for the Newton-Raphson method using backtracking is given in Algorithm 4.3.
	trond.hjorteland.com /thesis/node28.html (829 words)

	Steepest Descent (Site not responding. Last check: 2007-10-29)
		This module demonstrates the method of steepest descent for minimizing a nonlinear function.
		The steps of the method of steepest descent are then carried out sequentially by repeatedly clicking on NEXT or on the currently highlighted step.
		If the starting guess is close enough to the true minimum, then the method of steepest descent converges to it, typically with a linear convergence rate.
	www.cse.uiuc.edu /eot/modules/optimization/SteepestDescent (230 words)

	Feasible Newton's Steepest Descent Method For Linear Programming (ResearchIndex) (Site not responding. Last check: 2007-10-29)
		Abstract: this report we will consider the new version of the primal-dual method in which the pure without perturbation system of optimality conditions is solved by Newton's method.
		The step length is chosen from the steepest descent approach basing on minimization of the dual gap.
		We have not use the safety factor and allow trajectories in the primal and dual spaces to move along the boundaries of the feasible sets.
	citeseer.ist.psu.edu /276527.html (418 words)

	Homework 4
		ANSWER: They are different, in the case of the steepest descent you need the eigenvalues of R to calculate the Tao's.
		You can find information about the time constants in pages 82-86 from W&S. For the case of steeepst descent, the time constants depend on the eigenvalues of R, which in that problem is a 1x1 matrix equal to the averaged square inoput signal.
		This seems to be violating to the fact that > the Newton's method in general converges faster than the steepest descent > method.
	www.stanford.edu /class/ee373a/hw4.html (892 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		%We shall use the method of steepest descent, and %the simplex method as implemented by the matlab %routine fmins.
		pause %Let's plot up the contour lines of this function: xx1=[.5:.1:1.5]; xx2=[0:.1:2]; for i=1:length(xx2) for j=1:length(xx1) z(i,j)=fex([xx1(j),xx2(i)]); end end contour(xx1,xx2,z,10) hold on pause %The first method we shall try is the method of %steepest descent.
		We pick the descent direction %as the negative of the gradient of the function.
	www.nd.edu /~dtl/cheg258/cheg258-2000/cheg258-1999/cheg258-1998/cheg258-1997/cheg258-1996/notes/examples/example22.m (203 words)

	Unconstrained Optimal Method
		The difference between this method and last method is the choice of search direction.
		This method modifies the diagonal elements of [H] as
		decreases from a large value to zero, the characteristics of the search method change from those of a steepest descent method to those of the Newton method.
	www.ecs.umass.edu /mie/courses/761/xliu/webpage.html (237 words)

	A Toy Model of Stationary Phase (Site not responding. Last check: 2007-10-29)
		in an unambiguous fashion: an introduction to moving the contour of integration and the Method of Steepest Descent.
		Note that this steepest descent path is also one of stationary phase.
		, the steepest descent line is perpendicular to the lines of constant Ref(z), and is therefore a line of constant Imf(z), that is, constant phase of
	landau1.phys.virginia.edu /classes/752.mf1i.spring03/StatPhase101.htm (579 words)

	Preconditioned Steepest Ascent/Descent Methods
		These methods are similar to Algorithm 11.5 in the previous subsection, but the shifts are chosen in the process of iterations and do not require knowledge of any bounds.
		By construction, the steepest ascent (descent) method provides maximization (minimization) of the Rayleigh quotient on every iteration as compared with the corresponding iterative Algorithm 11.5.
		The main advantages of the preconditioned steepest descent/ascent method are its relative simplicity and a minimal cost of every iteration compared to other preconditioned eigensolvers considered below.
	www.cs.utk.edu /~dongarra/etemplates/node414.html (265 words)

	Levenberg-Marquardt Method
		The problem with overshooting can be solved by a method of Levenberg and Marquardt that combines steepest descent with Newton.
		which is the steepest descent method but with an explicit choice for the step length in each direction.
		However, the estimate of the error in the parameters through the formula (39) is often quite sensitive to how close we are to the true minimum, so it is a good idea to adjust the stopping criterion for the minimization in accordance with the resulting effect on the error estimate.
	www.physics.utah.edu /~detar/phycs6720/handouts/curve_fit/curve_fit/node7.html (513 words)

	test1_635 (Site not responding. Last check: 2007-10-29)
		c) from the above formula find the upper bound for the step size of the steepest descent method (note that the eigenvalues of R can be calculated very easily).
		b) write or derive the formulas for the learning curve in both the Newton and the steepest descent method.
		is the geometric ratio of the Newton method, that you are supposed to be able to calculate from the data of this problem.
	bass.gmu.edu /ececourses/ece635/test1_635.htm (403 words)

	[No title]
		In this method you can move in the opposite direction of the initial gradient until f(x,y) stops decreasing, that is, becomes level along your direction of travel.
		The steepest descent method to find the minimum can be applied to solve a system of nonlinear equations of the form f1(x1, x2, (, xn) = 0, f2(x1, x2, (, xn) = 0, EMBED Equation.3 EMBED Equation.3 fn(x1, x2, (, xn) = 0.
		The method will converge even with a poor initial guess therefore it is often used to find starting approximation for other techniques that can converge faster.
	www.csupomona.edu /~tknguyen/egr511/Notes/Ch2-4d2.doc (949 words)

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