By Guoyong Shi, Sheldon X.-D. Tan, Esteban Tlelo Cuautle (auth.)
This e-book offers entire insurance of the new advances in symbolic research innovations for layout automation of nanometer VLSI platforms. The presentation is equipped in components of basics, simple implementation equipment and functions for VLSI layout. subject matters emphasised comprise statistical timing and crosstalk research, statistical and parallel research, functionality certain research and behavioral modeling for analog built-in circuits. one of the contemporary advances, the Binary selection Diagram (BDD) established ways are studied intensive. The BDD-based hierarchical symbolic research ways, have basically damaged the analog circuit dimension barrier.
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Extra info for Advanced Symbolic Analysis for VLSI Systems: Methods and Applications
An,n xn bn (12) Assuming det(A) ⊕= 0, the Cramer’s rule says that the unknown xk can be solved as xk = det(Ak ) . det(A) (13) 20 2 Symbolic Analysis Techniques in a Nutshell where Ak denotes the n × n matrix A whose kth column has been replaced by the column b. The Cramer’s rule tells us that any unknown x1 , . . , xn can be solved explicitly as a ratio of two determinants. If we expand the determinant det(Ak ) along the kth column, then the unknown xk can be expressed in the following form xk = n i=1 bi (−1)i+k det(Aai,k ) , det(A) (14) where det(Aai,k ) is the minor of det(A) with respect to element ai,k , called a firstorder minor.
Fortunately, a simple modification to the graph manipulation quickly solves this problem. Since we wish the subgraphs generated by the edge operations can be compared and shared, we should reduce a graph by collapsing the edge whenever this edge is to be included. In this way, after (n − 1) edges have been collapsed for a connected graph containing n vertices, the graph must have been reduced into a single node and the collapsed edges form one spanning tree. In the modified Minty algorithm, we simply delete the edge when an edge is excluded.
In addition, it takes considerable knowledge of control theory and numerical procedures to implement balanced truncation in a stable way [102, 176]. Especially for nonstandard systems, additional decompositions and special treatments are required [91, 151, 220]. To remedy this problem, several gramian approximation methods have been proposed [110, 152, 263, 207, 266], where the approximated dominant subspace of a gramian can be obtained in a variety of efficient ways. However, no rigorous error bounds were derived for gramian approximation methods.
Advanced Symbolic Analysis for VLSI Systems: Methods and Applications by Guoyong Shi, Sheldon X.-D. Tan, Esteban Tlelo Cuautle (auth.)