Advanced Model Order Reduction Techniques in VLSI Design by Sheldon Tan, Lei He

By Sheldon Tan, Lei He

Version order aid (MOR) recommendations decrease the complexity of VLSI designs, paving tips to larger working speeds and smaller function sizes. This booklet offers a scientific creation to, and therapy of, the main MOR tools hired as a rule linear circuits, utilizing real-world examples to demonstrate the benefits and downsides of every set of rules. Following a assessment of conventional projection-based options, assurance progresses to complicated 'state-of-the-art' MOR tools for VLSI layout, together with HMOR, passive truncated balanced attention (TBR) tools, effective inductance modeling through the VPEC version, and structure-preserving MOR suggestions. the place attainable, numerical equipment are approached from the CAD engineer's standpoint, heading off advanced arithmetic and permitting the reader to tackle actual layout difficulties and advance more suitable instruments. With useful examples and over a hundred illustrations, this ebook is appropriate for researchers and graduate scholars of electric and machine engineering, in addition to practitioners operating within the VLSI layout undefined.

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O = Σ, with Σ again being ˜c = X We may find a coordinate system in which X ˜ ˜ diagonal. In this coordinate system, the matrices A, B, C˜ may be partitioned and truncated, similar to the standard TBR procedure. Here is the outline of this algorithm [87]. 2: passive truncated balanced realization 1. Solve Lur’e equations or AREs for Xc and Xo . 2. 1, substituting Xc for Wc and Xo for Wo . 6 Hybrid TBR and combined TBR-Krylov subspace methods A hybrid TBR method was proposed in [87] where the classic TBR method is tried first.

In the following subsections, we show two numerically stable algorithms, the Arnoldi method and the Lanczos method to find the Krylov subspaces. The Arnoldi method is an one-side method while Lanczos method is a two-side method. The fundamental idea is to orthonormalize the basis vectors in the Krylov subspaces defined in colspV = Kq (A−1 E, A−1 b) or colspW = Kq (A−T E T , A−T l). Such orthogonal vectors contain much less numerical noise compared with the circuit moments as lower-order moment vectors are subtracted during the orthonormalization process.

Let Z be a matrix whose columns are zk , and W a diagonal matrix with diagonal √ entries Wkk = wk . The above equation can be written more compactly as ˆ = ZW 2 Z H = (ZW )(W Z H ) = (ZW )(ZW )H . 8 Computational complexities of TBR methods 47 Following the POD method, we perform singular value decomposition of ZW to perform the model reduction. 55) with SZ real diagonal, V and U unitary matrices. As expected, we have the eigenˆ decomposition of the approximate Gramian X ˆ = X = = = (ZW )(ZW )T (V SU )(V SU )T V SU U T SV T V S 2V T .

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