USE OF QMR ALGORITHM TO CALCULATE VERY SLOW-MOTIONAL SPECTRA

这个子项目是许多研究子项目中的一个 由NIH/NCRR资助的中心赠款提供的资源。子项目和 研究者（PI）可能从另一个NIH来源获得主要资金， 因此可以在其他CRISP条目中表示。列出的机构是 中心，不一定是研究者的机构。 利用Lanczos算法（LA）对随机Liouville（SL）矩阵进行对角化，并借助共轭（CG）方法计算慢运动状态下的二维ELDOR谱。在LA递归的每一步中，根据CG迭代中计算的残差范数来监控收敛。因此，CG和LA的方法可以耦合在一起，有效地三对角化一个大型的对称和复杂的稀疏矩阵。LA-CG已经非常成功地用于在慢运动状态下三对角化SLE矩阵。 然而，由于在LA矢量的正交性的损失，我们发现，LA-CG方法可以打破在非常慢的运动制度的计算光谱。这主要是由于CG需要残差范数来计算每次迭代中的解方向。这表明，在一定数量的LA投影之后，由于正交性的损失，整个算法将被破坏。当计算需要大量LA投影以充分获得良好本征值的非常慢的运动谱时，这将是特别真实的。 准最小残差（QMR），这是最初开发的是一个线性方程求解器，已被改编为确定在我们的程序中的LA投影的数量。在QMR方法中，通过使用QR分解最小化准残差范数来获得解向量。用QMR代替CG来确定LA投影的数量对我们来说是很有好处的，因为在每次迭代中，CG方法不迭代地使用残差范数来确定解的方向。因此，我们利用的这种LA-QMR方法提供了避免上述故障的优点。

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.