By Le Bellac M.

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**Additional resources for A short introduction to quantum information and quantum computation: solutions of exercises**

**Sample text**

Accomplished as follows. , an, aq and transformed by the product P p - 1 P p - 2 (Sp_ I I' I ~ then T \%1 ] "'" PO to obtain a new is triangularized as usual. p-1 This method allows Q to be kept in product form always, and there is no accumulation of errors. Of course, if p = I the complete decomposition must be re-done and since with m ~ a lot of work. n the work is roughly proportional to (m-n/3)n 2 this can mean But if p A n/2 on the average, then only about I/8 of the original work must be repeated each updating.

181-217. [25] Shanno, D. C. "Parameter Selection for Modified Newton Methods for Function Minimization", J. SIAM, Numer. , Ser. B,7 (1970). , "On the Numerical Solution of Constrained Least Squares Problems", (private communication), 1970. [27] Tewarson, R. , "The Gaussian Elimination and Sparse Systems", Proceedings of the Symposium on Sparse Matrices and Their Applications~ T. J. Watson 39 Research Publication RA1, no. 11707, 1969. [28] Wilkinson, J. , "Error Analysis of Direct Methods of Matrix Inversion", J.

H. Wilkinson, "Eigenvalues of Ax = kB x with Band Symmetric A and B", Comput. , 12 (1969), pp. 398-404. , "Rank One Methods for Unconstrained Optimization", T. P. 372, Atomic Energy Research Establishment, Harwell, England, (1969). [24] Rosen, J. , "Gradient Projection Method for Non-linear Programming. I. Part Linear Constraints", J. SIAM, 8 (1960), pp. 181-217. [25] Shanno, D. C. "Parameter Selection for Modified Newton Methods for Function Minimization", J. SIAM, Numer. , Ser. B,7 (1970). , "On the Numerical Solution of Constrained Least Squares Problems", (private communication), 1970.

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