Deflated Iterative Methods for Linear Equations with Multiple Right-Hand Sides

Deflated and restarted symmetric Lanczos methods for eigenvalues and linear equations with multiple right-hand sides

A deflated restarted Lanczos algorithm is given for both solving symmetric linear equations and computing eigenvalues and eigenvectors. The restarting limits the storage so that finding eigenvectors is practical. Meanwhile, the deflating from the presence of the eigenvectors allows the linear equations to generally have good convergence in spite of the restarting. Some reorthogonalization is necessary to control roundoff error, and several approaches are discussed. The eigenvectors generated while solving the linear equations can be used to help solve systems with multiple right-hand sides. Ex…

Deflated GMRES for Systems with Multiple Shifts and Multiple Right-Hand Sides

We consider solution of multiply shifted systems of nonsymmetric linear equations, possibly also with multiple right-hand sides. First, for a single right-hand side, the matrix is shifted by several multiples of the identity. Such problems arise in a number of applications, including lattice quantum chromodynamics where the matrices are complex and non-Hermitian. Some Krylov iterative methods such as GMRES and BiCGStab have been used to solve multiply shifted systems for about the cost of solving just one system. Restarted GMRES can be improved by deflating eigenvalues for matrices that have a…

Deflated Iterative Methods for Linear Equations with Multiple Right-Hand Sides

A new approach is discussed for solving large nonsymmetric systems of linear equations with multiple right-hand sides. The first system is solved with a deflated GMRES method that generates eigenvector information at the same time that the linear equations are solved. Subsequent systems are solved by combining restarted GMRES with a projection over the previously determined eigenvectors. This approach offers an alternative to block methods, and it can also be combined with a block method. It is useful when there are a limited number of small eigenvalues that slow the convergence. An example is…

Volterra型非线性积分微分方程的单调区域分解迭代方法及其收敛性

Monotone Domain Decomposition Iterative Method and its Convergence for a Nonlinear Integro-Differential Equation of Volterra Type

针对Volterra型非线性积分微分方程，提出单调区域分解迭代方法，通过添加项构造线性方程，证明迭代解的存在性、唯一性和收敛性。