An indefinite LOBPCG type of algorithm for detecting a definite Hermitian matrix pair: MATLAB codes

For a given pair of Hermitian matrices $(A,B)$ of order $n$ with indefinite $B$, this algorithm declares whether this matrix pair is definite or not. A pair of Hermitian matrices is called definite if there exist a real linear combination of these matrices that is a positive definite matrix. Otherwise, it is called indefinite pair.
Our algorithm is suitable for medium-size or large and sparse matrix pairs, especially for banded matrix pairs. The proposed algorithm iteratively project a Hermitian matrix pair $(A,B)$ to small dimensional subspaces of $\mathbb{C}^n$ and detect the (in)definiteness of the resulting projected pair. Especially, our algorithm make use of the fact that if the projected pair $(U^HAU, U^H BU)$ is indefinite, then the original pair is also indefinite. The algorithm also include some other conditions to confirm indefiniteness of the given matrix pair or to confirm that the given matrix pair is close to some indefinite matrix pair. On the other hand, the decision that $(A,B)$ is definite is given by finding $\nu\in\R$ such that $A-\nu B$ is a positive definite or negative definite matrix by successfully completed Cholesky factorization. In the algorithm candidates for such $\nu$ are formed from the Ritz values, i.e., from the eigenvalues of the projected pair.

The data set contains the MATLAB codes for the algorithm and for the experiments from the paper.

针对给定的$n$阶埃尔米特矩阵（Hermitian matrix）对$(A,B)$（其中$B$为不定矩阵），本算法将判定该矩阵对是否为正定对。若存在该矩阵的实线性组合为正定矩阵，则称该埃尔米特矩阵对为正定对；反之则为不定矩阵对。
本算法适用于中等规模、大规模稀疏矩阵对，尤其适用于带状矩阵对。所提算法通过迭代将埃尔米特矩阵对$(A,B)$投影至$mathbb{C}^n$的低维子空间，并检测所得投影矩阵对的（不）定性。特别地，本算法利用如下结论：若投影矩阵对$(U^mathrm{H}AU, U^mathrm{H}BU)$为不定矩阵对，则原矩阵对亦为不定矩阵对。本算法还辅以若干其他条件，以判定给定矩阵对的不定性，或判定给定矩阵对与某不定矩阵对近似。
另一方面，若要判定$(A,B)$为正定对，则可通过寻找实数$
uinmathbb{R}$，使得$A-
u B$可成功完成乔利斯基分解（Cholesky factorization），进而成为正定或负定矩阵。本算法中，此类$
u$的候选值由里斯值（Ritz value）生成，即由投影矩阵对的特征值生成。
本数据集包含本算法及论文中相关实验的MATLAB代码。