By Giorgio Pauletto

This booklet is the results of my doctoral dissertation learn on the division of Econometrics of the collage of Geneva, Switzerland. This learn used to be additionally partly financed by way of the Swiss nationwide technological know-how origin (grants 12- 31072.91 and 12-40300.94). initially, I desire to show my private gratitude to Professor Manfred Gilli, my thesis manager, for his consistent aid and support. i might additionally wish to thank the president of my jury, Professor Fabrizio Carlevaro, in addition to the opposite participants of the jury, Professor Andrew Hughes Hallett, Professor Jean-Philippe Vial and Professor Gerhard Wanner. i'm thankful to my colleagues and acquaintances of the Departement of Econometrics, particularly David Miceli who supplied consistent aid and sort knowing in the course of all of the levels of my study. i'd additionally wish to thank Pascale Mignon for proofreading my textual content and im­ proving my English. ultimately, i'm vastly indebted to my mom and dad for his or her kindness and inspire­ ments with no which i'll by no means have accomplished my objectives. Giorgio Pauletto division of Econometrics, college of Geneva, Geneva, Switzerland bankruptcy 1 creation the aim of this booklet is to offer the to be had methodologies for the answer of large-scale macroeconometric types. This paintings stories classical resolution equipment and introduces newer options, reminiscent of parallel com­ puting and nonstationary iterative algorithms.

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The largest decrease in q at x(O) is obtained by choosing an update in the direction - \7q(x(O)) = b-Ax(O). We see that the direction of maximum decrease is the residual of x(O) defined by r(O) = b - Ax(O). We can look for the optimum step length in the direction r(O) by solving the line search problem min q(x(O) a + ar(O)) . As the derivative with respect to a is \7 a q(x(O) + ar(O)) x(O)' Ar(O) + ar(O)' Ar(O) - (x(O)' A - b')r(O) = r(O)'r(O) b'r(O) + ar(O)' Ar(O) + ar(O)' Ar(O) , the optimal a is aO = r(O)' r(O) r(O)' Ar(O) The method described up to now is just a steepest descent algorithm with exact line search on q.

5 BiConjugate Gradient Method The BiConjugate Gradient method (BiCG) takes a different approach based upon generating two mutually orthogonal sequences of residual vectors {f{i)} and {r(j)} and A-orthogonal sequences of direction vectors {p( i)} and {p(j)}. A Review of Solution Techniques 32 The interpretation in terms of the minimization ofthe residuals r( i) is lost. The updates for the residuals and for the direction vectors are similar to those of the CG method but are performed not only using A but also A'.

G. YI2MA, UMFPACK, SuperLU, SPARSE). 4 17 Stationary Iterative Methods Iterative methods form an important class of solution techniques for solving large systems of equations. They can be an interesting alternative to direct methods because they take into account the sparsity of the system and are moreover easy to implement. Iterative methods may be divided into two classes: stationary and nonstationary. The former rely on invariant information from an iteration to another, whereas the latter modify their search by using the results of previous iterations.

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