The Lanczos and conjugate gradient (CG) algorithm is a numerical algorithm interesting. He details the nature of mathematics from both algorithms and shows how the CG algorithm derived from the Lanczos algorithm. Loss of orthogonality involved with using the Lanczos algorithm, the ways to improve the accuracy achieved maximum CG calculation, and what modifications need to be done if the CG method is used with a preconditioner are discussed. It will be interesting to engineers and scientists use the Lanczos algorithm to compute eigenvalues and CG algorithms for solving linear systems, and researchers in the Krylov subspace method for symmetric matrices, especially those concerned with the analysis of floating point error. Contents Preface, Chapter 1: The Lanczos algorithm in exact arithmetic; Chapter 2: The CG algorithm in exact arithmetic; Chapter 3: A historical perspective on the Lanczos algorithm in the precision up to; Chapter 4: Lanczos algorithm in the precision up to; Chapter 5: CG algorithm in the precision up to; Chapter 6: The maximum accuracy is achieved, Chapter 7: Approximate norm error in the precision up to; Chapter 8: The algorithm CG asphalt; Chapter 9: Miscellaneous, Appendix, Bibliography, Index.