Updating eigenvectors

The conclusion is obvious instead of using a fixed orthogonalising matrix V we update the orthogonalising matrix to be the current eigenvector matrix. Thus the transformation to the orthogonal basis using the previous C matrix generates... 

By using eigenvalues and eigenvectors of the rank-2 matrix Q , which is useful in updating the Hessian, a more stable and accurate procedure is obtained. 

Update the matrix C if by adding the value Xee using the eigenvalue and eigenvector from the preceding step. 

A two-term approximation is used in (10.5.2), the first column of B (i.e. bj) being the current approximation to the required eigenvector, while the second column is chosen to be a single element (52) = d, - and therefore represents the ith basis vector in the full-CI space. The procedure is to perform a whole cycle of corrections to bi by letting i run through the entire space and then to use the fully updated bj as the starting approximation for the next cycle. This is such a simple process that it can be performed very rapidly. The term relaxation is commonly used to describe such processes, in which one coordinate at a time (or a block of coordinates) is improved by relaxing constraints. 

The coefficients are compnted from (4.138) by substimting the Hermite interpolation polynomial (4.134) into the update formula (4.130). An efficient implementation of BDF methods is presented below in the discnssion of DAE systems. For the test equation jc = Ax, Ah> = we again expand the states in the numerical trajectory in eigenvectors of A, and define the growth coefficient for each mode as... 

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