|Abstract: ||本論文是研究求不良(ill-conditioned) 矩陣之逆矩陣的演算法和病態(ill-posed) 線性系統之解法. 對於一個病態線性系統表成Ax = b, 其中 A 是不良矩陣, 欲求其逆矩陣和近似解, 一般的方法(如高斯消去法或其他改進之方法) 很難求得精確之結果. 而應用共軛梯度法(Conjugate Gradient) 解一般的線性系統問題很受歡迎, 但對於病態系統亦無良好結果. 若將其轉換成矩陣型的共軛梯度法, 則對於求不良矩陣之逆矩陣和解病態線性系統有稍好的效果. 然而其計算效率很差, 收斂也很慢, 且其精確度不高. 因此, 本論文提出一種新的演算法: 稱為預處理的矩陣型雙共軛梯度法(Matrix Version of Preconditioned Biconjugate Gradient Method,縮寫為MPBGM), 來求不良矩陣之逆矩陣和解病態線性系統. 依據我們實驗測試之結果, 此新方法效果良好, 不只提高精確度且提升計算效率和穩定性. 此方法與共軛梯度法和修正後的矩陣型共軛梯度法比較都有顯著的進步, 具有相當的競爭力|
This paper mainly investigates numerical algorithms for finding the inverses of ill-conditioned matrices and solving the ill-posed linear systems.The ill-posed linear system is difficult to solve by a general method, such as the class of Gaussian Elimination methods. Also, the class of conjugate gradient methods (CGM) is very popular in the applications of science and engineering.However, it can not reach good results for ill-posed linear systems. A recent modification of CGM is recasted to be a matrix form, that is the so called the Matrix Version of Conjugate Gradient Method (MCGM). The MCGM is useful for finding its inverse of the cofficient matrix. However, the accuracy and efficiency of its performance are still poor. In this paper, we present a new algorithm, called as the Matrix Version of Preconditioned Biconjugate Gradient Method (MPBGM) for fiding the inverses of ill-conditioned matrices and solving ill-posed linear systems. This new approach performs not only accurately but also efficiently. The comparison of our new method with the famous CGM and Modification of MCGM are good evidence to show that our scheme works well and competitively.