USE OF QMR ALGORITHM TO CALCULATE VERY SLOW-MOTIONAL SPECTRA

这个子项目是许多研究子项目中利用 资源由NIH/NCRR资助的中心拨款提供。子项目和 调查员(PI)可能从NIH的另一个来源获得了主要资金， 并因此可以在其他清晰的条目中表示。列出的机构是 该中心不一定是调查人员的机构。 利用共轭(CG)方法在慢动区计算二维Eldor谱，优化了基于Lanczos算法的随机Liouville(SL)矩阵的对角化。在LA递归的每一步中，根据在CG迭代中计算的残差范数来监控收敛。因此，CG和LA方法可以有效地结合起来对大型对称复杂稀疏矩阵进行三对角化。LA-CG已被成功地用于慢运动区SLE矩阵的三对角化。 然而，由于LA矢量的正交性的丧失，我们发现LA-CG方法在很慢的运动区计算光谱时会崩溃。这主要是因为CG在每次迭代中都需要残差范数来计算解的方向。这表明，在一定数量的LA投影后，由于失去正交性，整个算法将被破坏。当一个人计算一个非常慢的运动频谱时，这将是特别正确的，它需要大量的LA投影才能充分地获得良好的特征值。 准最小残差(QMR)最初是一种线性方程求解器，现已被用来确定LA投影的数目。在QMR方法中，解向量是通过QR分解使拟剩余范数最小化而得到的。用QMR代替CG来确定LA投影的个数对我们有很大的好处，因为在每次迭代中不像CG方法那样迭代地使用残差范数来确定解的方向。因此，我们使用的这种LA-QMR方法提供了避免上述故障的优势。

This subproject is one of many research subprojects utilizing the resources provided by a Center grant funded by NIH/NCRR. The subproject and investigator (PI) may have received primary funding from another NIH source, and thus could be represented in other CRISP entries. The institution listed is for the Center, which is not necessarily the institution for the investigator. The diagonalization of the stochastic Liouville (SL) matrix using the Lanczos algorithm (LA) is optimized with the aid of the conjugate (CG) method for calculating 2D-ELDOR spectra in the slow-motional regime. In each step of the LA recursion, the convergence is monitored according to the residual norm calculated in the CG iterations. Thus the methods of the CG and LA can be coupled together to tri-diagonalize a large symmetric and complex sparse matrix efficiently. The LA-CG has been very successfully used to tri-diagonalize the SLE matrix in the slow-motional regime. However, due to the loss of orthogonality in the LA vectors, we found that the LA-CG method can break down in calculating spectra in the very slow-motional regime. This is mainly due to the fact that CG requires the residual norm to calculate the solution direction in each iteration. This indicates that the whole algorithm would be spoiled due to the loss of orthogonality after a certain number of LA projections. This would be particularly true when one calculates a very slow-motional spectrum that requires a large number of LA projections in order to adequately obtain good eigenvalues. The quasi-minimal residual (QMR), which was originally developed to be a linear equation solver, has been adapted for determining the number of LA projections in our program. In the QMR method, the solution vector is obtained by minimizing the quasi-residual norm, using QR factorization. It is of great advantage to us to replace CG with QMR to determine the number of LA projections, because the residual norm is NOT used iteratively as in the CG method to determine the solution direction in each iteration. Therefore, this LA-QMR method that we utilize provides an advantage of avoiding the breakdown noted above.