5 Iterative Techniques in Matrix Algebra

Direct methods cost about \(\frac {n^3}{3}\) operations and fill in the zeros of a matrix as they eliminate. The systems that arise from differential equations and networks are often very large and sparse: most entries are zero. For them, iterative methods, which generate a sequence of approximations \(\mathbf x^{(0)},\mathbf x^{(1)},\dots \) using only products with \(A\), are often far cheaper. To say when such a sequence converges, and how close an approximation is, we first need a way to measure the size of vectors and matrices.

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