Conjugate gradients algorithm

McIntosh, A. Fitting Linear Models An Application of Conjugate Gradient Algorithms, Springer-Verlag, New York (1982). 

Although the overall cost of the conjugate gradient algorithm may be higher than that of some of the iterative algorithms described in Section III.B, the algorithm allows us easily to restrict the spectral and temporal stmcture of optimal pulses and enables us to incorporate the exact form of the laser-molecule interactions. 

EXAMPLE 6.2 APPLICATION OF THE FLETCHER-REEVES CONJUGATE GRADIENT ALGORITHM... 

X. Gonze, "First-principle responses of solids to atomic displacements and homogeneous electric fields a conjugate-gradient algorithm, used with plane waves and pseudopotentials," Phys. Rev. B 55 (1997), 10337-10354. 

A. Cohen, SIAM J. Numer. Anal., 9, 248 (1972). Rate of Convergence of Several Conjugate Gradient Algorithms. 

Mailer, M. (1993). A scaled conjugate gradient algorithm for fast supervised learning. Neural Networks 6,525-33. 

M. E. Davis and J. A. McCammon, J. Comput. Chem., 10, 386 (1989). Solving the Finite Difference Linearized Poisson-Boltzmann Equation A Comparison of Relaxation and Conjugate Gradient Algorithms. 

The conjugate gradient algorithm of Hestenes and Stiefel [70] and the biconjugate gradient algorithm of Fletcher [52] and their preconditioned versions [166] are presented in the following sections. ... 

For N parameters the least-squares equations lead to an V x AT normal matrix. Because the restraints involve only near neighbour atoms, the matrix is sparse, with the majority of non-diagonal terms being zero and less than 1% of the elements nonzero [117]. For n atoms and m distance restraints the number of elements to be stored is 6n-l-9m. For example, with a small protein of 812 atoms and 2030 restraints (approximately 3 x the number of atoms), the number of elements is 23 142. For phosphorylase b with 6640 atoms there are 26 561 parameters and some 229451 nonzero elements on the normal matrix, which is still only 0.03% of the total matrix elements. In the restrained least-squares refinement (and many of the other refinement methods) the normal equations are solved by the conjugate-gradient algorithm [129]. 

Symbolically, the conjugate gradient algorithm can be written in terms of the parameters 8 and a in the form... 

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