When does the Lanczos algorithm compute exactly?

In theory, the Lanczos algorithm generates an orthogonal basis of corresponding Krylov subspace. However, in finite precision arithmetic orthogonality and linear independence computed vectors is usually lost quickly. this paper we study a class matrices starting having special nonzero structure that guarantees exact computations whenever floating point satisfying IEEE 754 standard used. Analogous results are formulated also for implementation conjugate gradient method called cgLanczos. This then computes approximations agree with their counterparts to relative accuracy given by machine condition number system matrix. The extended Arnoldi algorithm, nonsymmetric Golub-Kahan bidiagonalization, block-Lanczos solving systems.