By Giorgio Pauletto
This e-book is the results of my doctoral dissertation examine on the division of Econometrics of the college of Geneva, Switzerland. This learn was once additionally partly financed through the Swiss nationwide technology origin (grants 12- 31072.91 and 12-40300.94). at first, I desire to convey my private gratitude to Professor Manfred Gilli, my thesis manager, for his consistent help and support. i'd additionally prefer to thank the president of my jury, Professor Fabrizio Carlevaro, in addition to the opposite contributors of the jury, Professor Andrew Hughes Hallett, Professor Jean-Philippe Vial and Professor Gerhard Wanner. i'm thankful to my colleagues and associates of the Departement of Econometrics, specifically David Miceli who supplied consistent aid and sort knowing in the course of all of the levels of my study. i'd additionally prefer to thank Pascale Mignon for proofreading my textual content and im proving my English. eventually, i'm enormously indebted to my mom and dad for his or her kindness and inspire ments with no which i may by no means have completed my objectives. Giorgio Pauletto division of Econometrics, college of Geneva, Geneva, Switzerland bankruptcy 1 advent the aim of this e-book is to give the on hand methodologies for the answer of large-scale macroeconometric versions. This paintings stories classical answer equipment and introduces newer concepts, reminiscent of parallel com puting and nonstationary iterative algorithms.
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Extra resources for Computational Solution of Large-Scale Macroeconometric Models
The pseudo-code for this method is given hereafter. 11 Preconditioned GMRES(m). Choose an initial guess x(O) and initialize an (m + 1) x m matrix Hm to hij = 0 for k = 1,2, ... until convergence Solve for r(k-l) Mr(k-l) = b _ Ax(k-l) /3 = IIr(k-l%; vel) = r(k-l) //3; q = m for j = 1, ... ,m Solve for w Mw = Av U ) for i = 1, ... ,j Orthonormal basis by modified Gram-Schmidt hij = W'V(i) W = W - hijV(i) end hj+1,j = IIwl12 if hj+l,j is sufficiently small then q=j exit from loop on j end v U+1) = W/hj+l,j end Vm = [v(1) ...
3 for i = 1, ... ,n (HI) _ end Xi - (k) Xi + I (XSOR,i (HI) _ (k») Xi end This method may be seen as a second-order method, since it uses a SOR iterate as an intermediate step to compute its next guess, and that the SOR already uses the information from a GS step. It is easy to see that when 'Y = 1, we find the SOR method. Like w in the SOR part, the choice of the value for I is not straightforward. For some problems, the optimal choice of w can be explicitly found (this is discussed in Hageman and Young ).
Given F : IRn --+ IRn continuously differentiable and a starting point for k = 0,1,2, ... 1 x(O) E IRn Convergence of the Simplified Newton Method The kind of simplification presented leads to a degradation of the speed of convergence as the Jacobian matrix is not updated at each step. However, one may note that for some macroeconometric models the nonlinearities are often such that this type of techniques may prove advantageous compared to the classical Newton iterations because of the computational savings that can be made.
Computational Solution of Large-Scale Macroeconometric Models by Giorgio Pauletto