On sampling algorithms in multilevel QR factorization method for magnetoquasistatic analysis of integrated circuits over multilayered lossy substrates

This paper proposes improvements in the row and column samplings of the multilevel QR factorization method known as IES³, a fast integral equation solver previously proposed for efficient three-dimensional (3-D) parameter extraction. First, a rigorous Gram-Schmidt row sampling is developed to replace the row sampling algorithm in IES³, leading to a more stable algorithm. Second, to further enhance the efficiency of column sampling, a new scheme based on the idea of locating the interpolation points is presented. Error analyses indicate that the proposed schemes have higher accuracies than the original sampling in IES³, especially when the number of sampled points is small. The IES³ that uses one of these improved algorithms is called improved multilevel matrix QR factorization (IMLMQRF). These IMLMQRFs are applied in the magnetoquasistatic analysis of printed circuits on multilayered lossy medium for extractions of inductances and resistances. The frequency dependency of such parameters is also illustrated