Scaling of Matrix

In the second method, row scaling is preceded by column scaling. Neither of these scaling procedures yields an optimum solution to the scaling problem as pointed out by Tewarson7 who also provides several references to more advanced scaling methods. 

This procedure is developed for the case where the problem is to be solved by use of the Newton-Raphson method. For the general case of n independent equations in n unknowns, n equations are obtained upon application of the Newton-Raphson method. The equations so obtained may be represented by the following matrix equation 

The notation B(X ) is used to emphasize that the jacobian matrix B is to be evaluated at X = X.  

Let R be a diagonal matrix whose elements ru are just greater than the absolute value of the corresponding elements of X, that is, 

After the multiplication implied by Eq. (15-7) has been carried out, form the diagonal matrix Mk whose elements mn are selected such that for each row 

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In this chapter, topics in matrix operation and numerical methods for solving multivariable problems are presented and examined. In Sec. 15-1, the use of linked lists in the storage of large sparse matrices is presented, and the scaling of matrices is presented in Sec. 15-2. The presentation and examination of selected numerical methods are presented in Sec. 15-3. 

The complex coordinate rotation (CCR) or complex scaling method (5,6,10,19) is directly based on the ABCS theory (1-3), therefore Reinhardt (5) also called it the direct approach. A complex rotated Hamiltonian, H 0), is obtained from the electron Hamiltonian of the atom, H, by replacing the radial coordinates r by re, where 0 is a real parameter. The eigenproblem of this non-Hermitian operator is solved variationally in a basis of square-integrable functions. The matrix representation of H ) is obtained by simple scaling of matrices T and V representing the kinetic and Coulomb potential part of the unrotated Hamiltonian H,... 

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