Matrix Computations, Third Edition

By Gene H. Golub, Charles F. Van Loan. 1996

Subfields: Numerical Methods & Scientific Computing

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Numerical Methods & Scientific Computing

Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “For d = 2 all these schemes boil down to the well known reduced singular value decomposition (SVD) of matrices [97]; however, they differ signiﬁcantly for tensors of order d ≥ 3 .”
From Frequency to Meaning: Vector Space Models of Semantics (2010, background): “Minimizing the the Frobenius norm ‖X̂−X‖F will minimize the noise, if the noise has a Gaussian distribution.”
Finite Discrete Gabor Analysis (2007, methodology): “The op counts for EIG and SVD methods are only approximations, because eigenvalues and singular values can be calculated by many di erent methods with di erent op counts, and because the process usually involves an iterative step, see [40].”
Fast High-dimensional Kernel Summations Using the Monte Carlo Multipole Method (2008, methodology): “The subspace of each leaf node is computed using PCABASE which can use the exact PCA [3] or a stochastic one [2].”
A nonparametric probabilistic approach for quantifying uncertainties in low‐dimensional and high‐dimensional nonlinear models (2017, methodology): “For example, one can note the construction procedure based on the economy-size Q R decomposition [67], which is computationally efficient for N very large andn << N .”
Updating the QR factorization and the least squares problem (2008, background): “Now, it may be required to update the least squares solution in the case where one or more observations (rows of A) are added or deleted.”
Direct Solve of Electrically Large Integral Equations for Problem Sizes to 1 M Unknowns (2008, background): “The SVD of A can also be written as the sum of r rank-one outer product matrices due to the properties of Σ [14, 15]”
On Multivariate Interpolation (2006, methodology): “Interestingly, the rather extensive numerical literature on practical solutions to Vandermonde systems, including [3, 25, 32, 33, 48, 59], fails to reveal the explicit factorization formula.”
Signal encoding in magnetic particle imaging: properties of the system function (2009, methodology): “Optimally, to take into account noise in the system function as well as in the measured object data, image reconstruction should be based on the total least squares approach [11].”
A High-Performance Semi-Supervised Learning Method for Text Chunking (2005, methodology): “…of the predictor distribution in approximation of U in the least square sense is obtained by projecting U onto the row space of (cid:2) ; see e.g. Golub and Loan (1996) Let the rows of (cid:2) be the h left singular vectors of U corresponding to the h largest singular values. until converge the…”
Observable Operator Models for Discrete Stochastic Time Series (2000, background): “Estimating the true rank of a noisy matrix is a common task in numerical linear algebra, for which heuristic solutions are known (Golub & Loan, 1996).”