By N. BOURBAKI

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**Extra resources for Elements de Mathematique. Integration. Chapitre 9**

**Sample text**

The PRIMA method, in full passive reduced-order interconnect macromodeling algorithm, builds upon the same Krylov space as in the Arnoldi method and PVL, using the Arnoldi method to generate an orthogonal basis for the Krylov space. The fundamental difference with preceding methods is, however, that the projection of the matrices is done explicitly. This is in contrast with PVL and Arnoldi, where the tridiagonal or the Hessenberg matrix is used for this purpose. In other words, the following matrix is formed: Aq = VqT AVq , where Vq is the matrix containing an orthonormal basis for the Krylov space.

In this chapter we will ﬁrst discuss brieﬂy some standard techniques for solving linear systems and for matrix eigenvalue problems. We will mention some relevant properties, but we refer the reader for background and more references to the standard text by Golub and van Loan [3]. We will then focus our attention on subspace techniques and highlight ideas that are relevant and can be carried over to Model Order Reduction approaches for other sorts of problems. 34 H. 1 Some Basic Properties We will consider linear systems Ax = b, where A is usually an n by n matrix: A ∈ Rn×n .

Orthogonalization of such a set of ill-conditioned set of vectors may lead to a correct projection process, but most often it leads to a loss of information and loss of efﬁciency. Using the iteration vectors xi or ri is not a good alternative, because they also may suffer from near dependency. It is much better to generate an orthogonal basis for the Krylov subspace (or any other appropriate subspace) right from the start. We will explain later how to do that for the Krylov subspace. For standard eigenproblems Ax = λx, the subspace approach is even more obvious.