## Matrix computations. 4th ed.(English)Zbl 1268.65037

Johns Hopkins Studies in the Mathematical Sciences. Baltimore, MD: The Johns Hopkins University Press (ISBN 978-1-4214-0794-4/hbk; 978-1-4214-0859-0/ebook). xxi, 756 p. (2013).
For the first edition see [Oxford: North Oxford Academic; Baltimore, Maryland: The Johns Hopkins University Press (1983; Zbl 0559.65011)]; for the second edition see [Johns Hopkins Series in the Mathematical Sciences, 3. Baltimore etc.: The Johns Hopkins University Press (1989; Zbl 0733.65016)]; for the third edition see [Baltimore, MD: The Johns Hopkins Univ. Press (1996; Zbl 0865.65009)].
In this fourth edition, the authors have added new sections on “fast transforms, parallel LU, fast methods for circulant systems and discrete Poisson systems, Hamiltonian and product eigenvalue problems, pseudospectra, the matrix sign, square root, and logarithm functions, Lanczoz and quadrature, large scale singular value decompositon, Jacobi-Davidson, sparse direct methods, multigrid, low displacement rank system, structure-rank systems, Kronecker product problems, tensor contractions, and tensor decompositions”. New topics found in subsections are: “recursive block LU, rook pivoting, tournament pivoting, diagonal dominance, recursive block structures, band matrix inverse properties, divide-and-conquer strategies for block tridiagonal systems, the cross product and various point/plane least squares problems, the polynomial eigenvalue problem, and the structured quadratic eigenvalue problem”. Substantial upgrades include “the treatment of floating-point arithmetic, LU roundoff error analysis, LS sensitivity analysis, the generalized singular eigenvalue decomposition, and the CS decomposition. For space limitation, the 66 page master bibliography has been moved to http://www.cs.cornell.edu/cv/GVL4/golubandvanloan.htm. This fourth edition is a summit in the 20th century matrix computations. According to D. L. Donoho and E. J. Candès, the 21st century is turning to compressed sensing and decoding by linear programming in the $$\ell_1$$ norm, as mentioned on p. 302 about the last two references.

### MSC:

 65Fxx Numerical linear algebra 65-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis 65Y05 Parallel numerical computation 15A18 Eigenvalues, singular values, and eigenvectors 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory

### Citations:

Zbl 0559.65011; Zbl 0733.65016; Zbl 0865.65009