In theory, the Lanczos algorithm generates an orthogonal basis of the corresponding Krylov subspace. However, in finite precision arithmetic, the orthogonality and linear independence of the computed Lanczos vectors is usually lost quickly. In this paper we study a class of matrices and starting vectors having a special nonzero structure that guarantees exact computations of the Lanczos algorithm whenever floating point arithmetic satisfying the IEEE 754 standard is used. Analogous results are formulated also for a variant of the conjugate gradient method that produces then almost exact results. The results are extended to the Arnoldi algorithm, the nonsymmetric Lanczos algorithm, the Golub-Kahan bidiagonalization, the block-Lanczos algorithm and their counterparts for solving linear systems.

翻译：理论上, 兰佐斯 算法生成了相应的 Krylov 子空间的正弦基数。 但是, 在有限的精确算法中, 计算出的朗佐斯 矢量的正弦和线性独立性通常会很快消失。 在本文中, 我们研究的是一组具有特殊的非零结构的矩阵和起始矢量, 保证在使用浮点算法达到 IEEE 754 标准时精确计算兰佐斯算法。 模拟结果还针对类似梯度法的变体, 该变体产生几乎准确的结果。 结果被扩展至阿诺迪算法、 非对称兰佐斯算法、 Golub- Kahan 梯形化法、 区- Lanczo 算法及其用来解决线性系统的对应方 。