The Conjugate Gradient Decomposition And Its Application

The conjugate gradient decomposition (CGD) was introduced for symmetric and positivedefinite matrices based on the conjugate gradient method. In fact, the decomposition is not newfrom theory of linear algebra, but it gives some special interpretation of the classic result aboutmatrix decomposition. The CGD has some similar properties of the singular valuedecomposition(SVD), but loses uniqueness and some orthogonal projection property. From thecomputational point of view, the CGD is much cheaper than the SVD.The properties of the CGD arestudied also. At the same time, we generalize the CG decomposition for every matrix which is notnecessarily a square matrix. Finally some applications of the CGD are given which illustrate thefeasibility of the CGD.we use some examples to illustrate that the CGD can be used to solute somepractical problem.In theory, we use the conjugate gradient decomposition algorithm to solute subproblem of trustregion method, it can not only reach the effect similar to SVD, but in very short.In engineering fields,we use the conjugate gradient decomposition algorithm to estimate the harmonics of signals in powersystem, it not only reduces the computation of SVD algorithm, but also improves the precision ofFFT, and can be applied to the serious distortion of periodic signal estimation.The numericalsimulation results indicate that the algorithm is effective.