Enhancing the Stability of Lanczos-type Algorithms by Embedding Interpolation and Extrapolation for the Solution of Systems of Linear Equations

Maharani Maharani (School of Informatics and Applied Mathematics, University of Malaysia Terengganu, Gong Badak, Kuala Nerus 21030, Terengganu)
Abdelah Salhi (Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester CO43SQ, Essex)
Rifka Amelia Suharto (Department of Mathematics, University of Jenderal Soedirman, Jl.Dr.Soeparno No.61, Purwokerto55281, Jawa Tengah)

Journal of Mathematical and Fundamental Sciences
Vol. 50 No. 2 (2018)

A new method to treat the inherent instability of Lanczos-type algorithms is introduced. It enables us to capture the properties of the sequence of iterates generated by a Lanczos-type algorithm by interpolating on this sequence of points. The interpolation model found is then used to generate a point that is outside the range. It is expected that this new point will link up the rest of the sequence of points generated by the Lanczos-type algorithm if breakdown does not occur. However, because we assume that the interpolation model captures the properties of the Lanczos sequence, the new point belongs to that sequence since it is generated by the model. This paper introduces the so-called Embedded Interpolation and Extrapolation Model in Lanczos-type Algorithms (EIEMLA). The model was implemented in algorithms A13/B6 and A13/B13, which are new variants of the Lanczos algorithm. Individually, these algorithms perform badly on high dimensional systems of linear equations (SLEs). However, with the embedded interpolation and extrapolation models, EIEM A13/B6 and EIEM A13/B13, a substantial improvement in the performance on SLEs with up to 105 variables can be achieved.