Schubert s formula

Schubert s formula is, therefore, used only for the numerical derivatives, whereas the known or analytic derivatives are evaluated at the (i + l)th point. 

It is possible to update the Hessian as if it were the Jacobian of gradient of the objective function. When the Jacobian of a system of equations is updated, it is possible to preserve its sparsity by means of the Schubert s formula (7.103). 

If the system has been calculated in another point, xi, leading to the residuals f i, the Jacobian can be evaluated in such a point or updated using Schubert s formula (see Section 7.16) also the Hessian of each system function can be updated using the Schubert s formula, treating the Hessian itself as the Jacobian of function gradient, analogously to what we mentioned above for the Hessian of the objective function ... 

The updating of the Hessian of the Lagrange function is often computationally ( ) infeasible because of matrix filling due to the BFGS formula. Conversely, the updating of each Hessian with Schubert s formula minimizes the overall matrix filling. 

In the formulation of Schubert s method,21 it is convenient to denote the kth approximation of the jacobian by G(k), where the iteration number is carried as a superscript enclosed by parentheses. Then Broyden s formula for computing the next approximation of the jacobian is given by... 

Schubert s updating formula is particularly useful in the case of large-scale ( ) sparse systems, since it preserves Jacobian sparsity and executes only the calculations for nonzero derivatives. 

Smith, E. J. The Wiswesser line-formula chemical notation. New York McGraw-Hill 1968 Feldmann, R. J., in Computer representation and manipulation of chemical information, (eds. W. T. Wipke, S. R. Heller, R. J. Feldmann, E. Hyde). London Wiley 1974, p. 55 Feldmann, R. J., Gasteiger, J., Schubert, W., Thoma, J. in preparation Gasteiger, J., Schubert, W Thoma, J. unpublished results Viehe, H. G. Angew. Chem. 77, 768 (1965)... 

Infants raised on cow s milk may obtain as little as 1% of their calories from linoleic acid, whereas commercial infant formulas presently in common use normally contain vegetable oils and may provide more than 20% of total calories as linoleic acid (Schubert, 1973). Human milk, which is accepted as the optimal nutritional standard in infancy (Macy Kelly, 1961 Jelliffe Jelliffe, 1971) normally provides about 4% of total calories as linoleic acid (Schubert, 1973), but the percentage of linoleic acid in human milk may vary anywhere from 1% to more than 15% (Cuthbertson, 1976). Although various S3rmptoms attributed to dietary deficiencies of essential fatty acids have been observed by a number of investigators, they occur only rarely, and Cuthbertson (1976) considers that the minimum requirement for children is, in fact, less than 0.5% of calories. 

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