Iterative Methods Sample Clauses
The Iterative Methods clause defines the process by which parties will repeatedly refine or improve a product, service, or deliverable through a series of cycles or stages. Typically, this involves regular feedback, testing, and adjustments at each stage, allowing for incremental progress and continuous improvement. For example, in software development, this might mean delivering working versions at set intervals and incorporating user feedback before moving to the next phase. The core function of this clause is to ensure flexibility and adaptability in project execution, reducing the risk of major errors and aligning the final outcome more closely with the parties' evolving needs.
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 ▇▇▇▇▇▇▇▇ [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].
Algorithm 3.1 Conjugate Gradient Least Squares (CGLS) x0 is an initial guess 0 s0 y Ax0; r0 AT s0; p0 r0; ρ0 rT r0; p 1 0; β 1 0 for i 0, 1, 2, . . . do pi ri βi 1pi 1 qi Api ρi qT qi
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. In ▇▇▇▇▇▇ 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. ǁ := max |e | ǁe