The null space problem I. complexity
SIAM Journal on Algebraic and Discrete Methods
Computing a sparse basis for the null space
SIAM Journal on Algebraic and Discrete Methods
The null space problem II. Algorithms
SIAM Journal on Algebraic and Discrete Methods
SIAM Journal on Scientific and Statistical Computing
The probability of large diagonal elements in the QR factorization
SIAM Journal on Scientific and Statistical Computing
Algorithm 694: a collection of test matrices in MATLAB
ACM Transactions on Mathematical Software (TOMS)
Some applications of the rank revealing QR factorization
SIAM Journal on Scientific and Statistical Computing
Estimating the largest eigenvalues by the power and Lanczos algorithms with a random start
SIAM Journal on Matrix Analysis and Applications
Iterative SVD-based methods for ill-posed problems
SIAM Journal on Scientific Computing
Matrix computations (3rd ed.)
Sparse Multifrontal Rank Revealing QR Factorization
SIAM Journal on Matrix Analysis and Applications
Applied numerical linear algebra
Applied numerical linear algebra
The symmetric eigenvalue problem
The symmetric eigenvalue problem
Computing rank-revealing QR factorizations of dense matrices
ACM Transactions on Mathematical Software (TOMS)
Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion
Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion
LAPACK Users' guide (third ed.)
LAPACK Users' guide (third ed.)
LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares
ACM Transactions on Mathematical Software (TOMS)
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
A Rank-Revealing Method with Updating, Downdating, and Applications
SIAM Journal on Matrix Analysis and Applications
Algorithm 853: An efficient algorithm for solving rank-deficient least squares problems
ACM Transactions on Mathematical Software (TOMS)
Direct Methods for Sparse Linear Systems (Fundamentals of Algorithms 2)
Direct Methods for Sparse Linear Systems (Fundamentals of Algorithms 2)
On the Computation of Null Spaces of Sparse Rectangular Matrices
SIAM Journal on Matrix Analysis and Applications
The university of Florida sparse matrix collection
ACM Transactions on Mathematical Software (TOMS)
Algorithm 915, SuiteSparseQR: Multifrontal multithreaded rank-revealing sparse QR factorization
ACM Transactions on Mathematical Software (TOMS)
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The SPQR_RANK package contains routines that calculate the numerical rank of large, sparse, numerically rank-deficient matrices. The routines can also calculate orthonormal bases for numerical null spaces, approximate pseudoinverse solutions to least squares problems involving rank-deficient matrices, and basic solutions to these problems. The algorithms are based on SPQR from SuiteSparseQR (ACM Transactions on Mathematical Software 38, Article 8, 2011). SPQR is a high-performance routine for forming QR factorizations of large, sparse matrices. It returns an estimate for the numerical rank that is usually, but not always, correct. The new routines improve the accuracy of the numerical rank calculated by SPQR and reliably determine the numerical rank in the sense that, based on extensive testing with matrices from applications, the numerical rank is almost always accurately determined when our methods report that the numerical rank should be correct. Reliable determination of numerical rank is critical to the other calculations in the package. The routines work well for matrices with either small or large null space dimensions.