Gram - Schmidt orthogonalization process

In this formulation the unknown coefficients can be treated as the densities of the different blocks  

Thus we have a system of linear equations with respect to the densities of the blocks  

Sometimes it is difficult to solve the system of equations (B.6). Evidently, this system can be very much simplified if the set of elements dj, d2, dg,. ..d are orthonormal. Fortunately we can exhibit a constructive procedure for converting this set into the 

Operators in the spaces of models and data Vertical component of gravity field 

The next step is to construct the vector d2—(d2,ei)e, which is orthogonal to ei, and then to normalize it  

---

Gram - Schmidt orthogonalization process Thus, equation (B.6) takes the form... 

It is evident from these discussions that population balance equations are important in the description of dispersed-phase systems. However, they are still of limited use because of difficulties in obtaining solutions. In addition to the numerical approaches, solution of the scalar problem has been via the generation of moment equations directly from the population balance equation (H2, H17, R6, S23, S24). This approach has limitations. Ramkrishna and co-workers (H2, R2, R6) presented solutions of the population balance equation using the method of weighted residuals. Trial functions used were problem-specific polynomials generated by the Gram-Schmidt orthogonalization process. Their approach shows promise for future applications. 

The simple Gram-Schmidt ON-process could have been applied to any initial basis set to find a set of solutions to (3.1.15). The equations above are all equivalent. Any two solutions to these equations are related by some orthogonal matrix V ... 

The Gram-Schmidt orthogonalization of the frequency independent vectors (j), A(j) produces the orthonormal basis qi, , qm by the Lanczos process so that... 

The process of orthgonalization is somewhat analogous to the orthgonalization of vectors, known as the Gram-Schmidt procedure. Construction of orthogonal descriptors 2, 2, 2,... from given descriptors D, D, D,... is accomplished by calculating the residuals between the descriptors. " By definition, the residual of a correlation between two descriptors is the part of one descriptor that does not correlate with the other. 

---