大型稀疏矩阵的LU分解及特征值求解-Grusoftcom.pptx

2013. 7. 20;稀疏矩阵求解的广泛应用;稀疏矩阵复杂、多变 ;求解器的飞速发展;稀疏LU分解算法的关键;稀疏排序;近似最小度排序算法 – 商图;为何需要密集子块(Dense Matrix);多波前法(multifrontal)简介;LU分解形成frontal;Frontal的装配，消去，更新过程;消去树;GSS简介;user;GSS – 加权消去树;GSS -双阈值列选主元算法;GSS - CPU/GPU混合计算;GSS – 求解频域谱元方法生成的矩阵;对比测试;大型稀疏矩阵的特征值求解;幂迭代 -- Power iteration;幂迭代 -- 一个形象的解释;瑞利商迭代 -- Rayleigh Quotient iteration;子空间迭代;Krylov子空间;Arnoldi 迭代;Arnoldi 迭代的基本算法;Arnoldi 迭代求解特征值;Krylov分解 -- 基于酉相似变换(unitary similarity );实测更效率 实测迭代次数，运行时间都减少约1/3。（与ARPACK对比）;Krylov-schur及其重启;收敛速度更快;最新算法 Subspace iteration with approximate spectral projection;shift-invert变换 ;Cayley 变换;Filter polynomial;Aleksei Nikolaevich Krylov (1863–1945) showed in 1931 how to use sequences of the form {b, Ab, A2b, . . .} to construct the characteristic polynomial of a matrix. Krylov was a Russian applied mathematician whose scientific interests arose from his early training in naval science that involved the theories of buoyancy, stability, rolling and pitching, vibrations, and compass theories. Krylov served as the director of the Physics–Mathematics Institute of the Soviet Academy of Sciences from 1927 until 1932, and in 1943 he was awarded a “state prize” for his work on compass theory. Krylov was made a “hero of socialist labor,” and he is one of a few athematicians to have a lunar feature named in his honor—on the moon there is the “Crater Krylov.”;附二 参考文献;;内容总结