The reduction of a large-scale symmetric linear discrete ill-posed problem with multiple right-hand sides to a smaller problem with a symmetric block tridiagonal matrix can easily be carried out by the application of a small number of steps of the symmetric block Lanczos method. We show that the subdiagonal blocks of the reduced problem converge to zero fairly rapidly with increasing block number. This quick convergence indicates that there is little advantage in expressing the solutions of discrete ill-posed problems in terms of eigenvectors of the coefficient matrix when compared with using a basis of block Lanczos vectors, which are simpler and cheaper to compute. Similarly, for nonsymmetric linear discrete ill-posed problems with multiple right-hand sides, we show that the solution subspace defined by a few steps of the block Golub–Kahan bidiagonalization method usually can be applied instead of the solution subspace determined by the singular value decomposition of the coefficient matrix without significant, if any, reduction of the quality of the computed solution.

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关于应用于离散不适定问题的块 Lanczos 和块 Golub-Kahan 约简方法

通过应用对称块 Lanczos 方法的少量步骤，可以轻松地将具有多个右侧的大规模对称线性离散不适定问题简化为具有对称块三对角矩阵的较小问题. 我们表明，随着块数的增加，简化问题的次对角块很快收敛到零。这种快速收敛表明，与使用块 Lanczos 向量的基相比，用系数矩阵的特征向量来表达离散不适定问题的解决方案几乎没有优势，后者更简单且计算成本更低。类似地，对于具有多个右手边的非对称线性离散不适定问题，