If we want gradients for only a few g's, the second approach is clearly most efficient. Arora and Haug make the assumption that NG is always small compared to NDV*NLC. In a real design situation where we are setting up an approximate design problem, either for hand calculations or for optimization, this may not be true since NG will be the number of constraints retained for design consideration. In any case, the message is that the order of the matrix operation should be determined at the time it is done, based upon a comparison of NG and NDV*NLC and the computational cost of each ordering. In terms of total computational effort to analyze the structure and compute gradients, the time difference should not be as significant as we are led to believe. Based upon the discussion, the following conclusions seem to be in order; 1) A gradient is a gradient, no matter how it is calculated. In suggesting that their method is more general, Arora and Haug are only saying that they retain more terms. Certainly the referenced investigators are qualified to do the same if they need the additional terms. 2) The conclusion that their method is up to ten times faster is valid so long as we understand that they are talking about one part of the total computational effort. 3) It is important to remember that the order of a matrix operation is criticial to the computational efficiency of performing that operation.
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