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Recently, some authors (Li, Yang and Wu, 2014) studied the parameterized preconditioned HSS (PPHSS) method for solving saddle point problems. In this short note, we further discuss the PPHSS method for solving singular saddle point problems. We prove the semi-convergence of the PPHSS method under some conditions. Numerical experiments are given to illustrate the efficiency of the method with appropriate parameters.

We consider the iterative solution of the following linear system:

where

We review the Hermitian and skew-Hermitian splitting (HSS) [

where

The PPHSS Iteration Method ([

where

matrix C is Hermitian positive definite.

Evidently, the iteration scheme (2) of PPHSS method can be rewritten as

here,

with

Evidently, the matrix

where

Owing to the similarity of the matrices

As the coefficient matrix A is singular, then the iteration matrix T has eigenvalue 1, and the spectral radius of matrix T cannot be small than 1. For the iteration matrix T of the singular linear systems, we introduce its pseudo-spectral radius

where

For a matrix

Lemma 2.1 ([

Lemma 2.2 ([

Theorem 2.3. Assume that B and C be Hermitian positive definite, E be of rank-deficient. Then

Proof. The proof is similar to the proof of Lemma 2.8 in [

Lemma 2.4 ([

Let

Lemma 2.5. The eigenvalues of the iteration matrix

, (8)

Proof. Notice the similarity of matrices

Lemma 2.6. If

Proof. If

here,

If

Lemma 2.7 ([

Theorem 2.8. If the iteration parameters

, (9)

then, the pseudo-spectral radius of the PPHSS method satisfies

Proof. Using condition (9), it follows that

and

By Lemma 2.7, for the eigenvalues

If

Theorem 2.9. Let

and correspondingly,

Proof. According to Lemma 2.5 and Lemma 2.6, we know that the eigenvalues of the iteration matrix

, (11)

If

In this section, we use an example to demonstrate the numerical results of the PPHSS method as a solver by comparing its iteration steps (IT), elapsed CPU time in seconds (CPU) and relative residual error (RES) with other methods. The iteration is terminated once the current iterate satisfies

Example 3.1 ([

where symbol

the right-hand side vector b is chosen by

For the Example 3.1, we choose

clear to see that the pseudo-spectral radius of the PPHSS and the SPPHSS methods are much smaller than of the PHSS method when the optimal parameters are employed. In

Method | 8 | 16 | 24 | 32 | |
---|---|---|---|---|---|

PHSS | 1.6328 | 2.1999 | 2.6511 | 3.0367 | |

0.6756 | 0.8112 | 0.8667 | 0.8969 | ||

SPPHSS | 1.0216 | 1.0059 | 1.0027 | 1.0015 | |

0.4947 | 0.4985 | 0.4993 | 0.4996 | ||

PPHSS | 1.9815 | 2.6990 | 2.5976 | 2.9953 | |

0.6853 | 0.7845 | 1.0336 | 1.1234 | ||

0.5209 | 0.5258 | 0.5340 | 0.5452 |

Method | 8 | 16 | 24 | 32 | |
---|---|---|---|---|---|

PHSS | IT | 26 | 37 | 47 | 54 |

CPU | 0.399 | 1.548 | 7.286 | 27.178 | |

RES ( | 6.6914 | 7.2250 | 7.3711 | 9.3294 | |

SPPHSS | IT | 26 | 26 | 26 | 26 |

CPU | 1.075 | 1.583 | 4.879 | 8.610 | |

RES ( | 5.3781 | 6.7198 | 7.0689 | 7.2124 | |

PPHSS | IT | 16 | 16 | 16 | 16 |

CPU | 0.220 | 1.008 | 3.620 | 12.558 | |

RES ( | 7.8783 | 7.6704 | 7.7535 | 8.1446 | |

GMRES | IT | 883 | 2560 | 5450 | 10376 |

CPU | 0.524 | 3.179 | 12.572 | 16.264 | |

RES ( | 9.8243 | 9.9925 | 9.9903 | 9.9950 | |

PHSS-GMRES | IT | 22 | 35 | 44 | 50 |

CPU | 0.111 | 0.467 | 1.649 | 3.751 | |

RES ( | 9.6710 | 8.0874 | 9.3990 | 9.8588 | |

SPPHSS-GMRES | IT | 10 | 11 | 13 | 14 |

CPU | 0.082 | 0.372 | 1.831 | 3.892 | |

RES ( | 4.7178 | 6.7487 | 6.2498 | 4.2212 | |

PPHSS-GMRES | IT | 11 | 13 | 13 | 16 |

CPU | 0.070 | 0.428 | 1.564 | 3.366 | |

RES ( | 5.4534 | 2.3975 | 9.8449 | 9.8690 |

IT, CPU and RES of the texting methods with different problem sizes l. We see that the PPHSS and SPPHSS methods with appropriate parameters always outperforms the PHSS method both as a solver and as a preconditioner for GMRES in iteration steps and CPU times. Notice

where

Yueyan Lv,Naimin Zhang, (2016) A Note on Parameterized Preconditioned Method for Singular Saddle Point Problems. Journal of Applied Mathematics and Physics,04,608-613. doi: 10.4236/jamp.2016.44067