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Progressive Iterative Approximation with Preconditioners

Author

Listed:
  • Chengzhi Liu

    (School of Mathematics and Statistics, Central South University, Changsha 410083, China
    School of Mathematics and Finance, Hunan University of Humanities, Science and Technology, Loudi 417000, China)

  • Zhongyun Liu

    (School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410076, China)

Abstract

The progressive iterative approximation (PIA) plays an important role in curve and surface fitting. By using the diagonally compensated reduction of the collocation matrix, we propose the preconditioned progressive iterative approximation (PPIA) to improve the convergence rate of PIA. For most of the normalized totally positive bases, we show that the presented PPIA can accelerate the convergence rate significantly in comparison with the weighted progressive iteration approximation (WPIA) and the progressive iterative approximation with different weights (DWPIA). Furthermore, we propose an inexact variant of the PPIA (IPPIA) to reduce the computational complexity of the PPIA. We introduce the inexact solver of the preconditioning system by employing some state-of-the-art iterative methods. Numerical results show that both the PPIA and the IPPIA converge faster than the WPIA and DWPIA, while the elapsed CPU times of the PPIA and IPPIA are less than those of the WPIA and DWPIA.

Suggested Citation

  • Chengzhi Liu & Zhongyun Liu, 2020. "Progressive Iterative Approximation with Preconditioners," Mathematics, MDPI, vol. 8(9), pages 1-16, September.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:9:p:1503-:d:408893
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