Iterative Methods. Most iterative methods can be viewed as a proper decomposition of A, then solve an important and treat the rest as a perturbation term. Jacob method In Xxxxxx method, wer decompose A = D + B where D is the diagonal part and B is the off diagonal part. Since A is diagonally dominant, we may approximate x by the sequence xn, where xn is defined by the following iteration scheme: Dxn+1 + Bxn = b. Let the error en := xn+1 − xn. Then Or Let us define the maximum norm Den = −Ben—1 en = −D—1Ben—1. j j en := max |en|
Iterative Methods. When an efficient decomposition of the convolution matrix is not available, for example when zero boundary conditions are used, we have to resort to iterative methods. We have implemented the following iterative methods in PYRET: CGLS, LSQR, MR2, MRNSD and their preconditioned versions. Conjugate gradient least squares (CGLS) [6] treats (3.2) as a least squares problem, and is mathematically equivalent to applying the conjugate gradi- ent method to the normal equations. LSQR is another popular method for solving least squares problems. Its use of Lanczos bidiagonalization makes it more stable for ill-conditioned matrices. For details of LSQR, see [56]. MR2 [34] is a modification of the the classical minimum residual method of Paige and Xxxxxxxx [55] that is more appropriate for ill-posed inverse prob- lems when the matrix is symmetric, and possibly indefinite. MRNSD [46] is a modification of the standard residual norm steepest descent method [61] that enforces a nonnegativity constraint on the solution. Some good references for iterative methods are [12, 20, 61].
