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# 数学代写|数值分析代写Numerical analysis代考|STAT721 Multiple-vector iterations

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## 数学代写|数值分析代写Numerical analysis代考|Multiple-vector iterations

The single-vector iteration methods in the original form can find only one eigenpair. In many situations, however, we need more eigenpairs with similar spectral characteristics, that is, several dominant eigenpairs or a dozen eigenpairs around a specified shift $\sigma \in \mathbb{C}$. Iterations using multiple-vectors can be used in this setting, and are a natural extensions of the single-vector methods.

The subspace iteration (also called simultaneous iteration) is a widely used extension of this type. We provide the outline of a basic version of this method. One can see easily that it is a straightforward extension of the power method, and the only essential difference is that the single-vector iterate $\mathbf{x}_k$ (with $\left|\mathbf{x}_k\right|_2=1$ ) of the power method is replaced with a block of $p$ vectors $\mathbf{X}_k$ with orthonormal columns. In addition, suppose that we want individual dominant eigenpairs (instead of an orthonormal basis of this invariant subspace), a post-processing step is needed to retrieve these eigenpairs of $\mathbf{A}$ from $\mathbf{X}_k$ and the block Rayleigh quotient $\mathbf{M}_k=\mathbf{X}_k^H \mathbf{A} \mathbf{X}_k$.

## 数学代写|数值分析代写Numerical analysis代考|Finding all eigenvalues and eigenvectors of a matrix

All the algorithms introduced in previous sections can be used to compute one or a few eigenvalues of a small dense or a large and typically sparse matrix. Notably absent from this chapter is a discussion of finding all eigenvalues and eigenvectors of a matrix. In MATLAB, this can be done with the eig command, and most students use this command in their introductory linear algebra class.

We have omitted discussion of this topic for two reasons. First, there are additional technical details to efficiently extend the ideas above for subspace iteration to the whole space to make converge rapidly. Second, it is not a practical algorithm for very large matrices: it is slow $\left(O\left(n^2\right)\right.$ flops for real symmetric or complex Hermitian matrices and $O\left(n^3\right)$ flops for general nonsymmetric matrices), and is storage-consuming since the eigenvectors will form a dense matrix the same size as the input matrix. The algorithm is practical for matrices of size up to approximately twenty two thirty thousand on a personal computer (though many hours needed). As this is an overview course, we choose to leave the material to a first graduate course in numerical linear algebra.

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