# Tag Info

Accepted

### Computational method to compute both the (log) determinant and inverse of a matrix

The LU decomposition will give you what you want with only $\tfrac{2}{3}n^3 + \mathcal{O}(n^2)$ FLOPs. The linear system is solved by solving two triangular systems. The determinant is the product ...

### Robust algorithm for $2 \times 2$ SVD

I needed an algorithm that has little branching (hopefully CMOVs) no trigonometric function calls high numerical accuracy even with 32 bit floats We want to calculate $c_1, s_1, c_2, s_2, \sigma_1$ ...

### Rapidly determining whether or not a dense matrix is of low rank

The problem, of course, is that computing the true rank (e.g., via a QR decomposition) is not really any cheaper than computing a low-rank representation of the matrix. The best you can probably do ...
Accepted

### Solving $A=B+AB$ without matrix inverse

If $A$ and $B$ are real symmetric, then $A=B+AB$ if and only if the product $AB$ is also real symmetric. In turn, $AB=BA$ holds if any only if $A$ and $B$ share a common eigendecomposition. This ...
Accepted

### Cholesky for ill-conditioned/singular covariance matrices

If your covariance matrix is singular, then you really should consider why the matrix is singular and come up with a higher-level approach that avoids the singularity. However, if you insist on ...
Accepted

### Does a symmetric positive definite matrix also have $\mathbf{A} = \mathbf{L}^T\mathbf{L}$ (where $\mathbf{L}$ is a lower triangular matrix)?

Let $P$ be the anti-diagonal permutation matrix, $$P = \begin{bmatrix} & & & 1 \\ & & 1 \\ & 1 \\ 1 \end{bmatrix}$$ so that $PAP$ is the version of $A$ with rows and columns ...
Accepted

### Bareiss algorithm vs. LU-decomposition

Bareiss' algorithm is a better choice if you have to compute exactly determinants of integer matrices, as noted in the comments. When it comes to real (floating-point) or complex matrices, the main ...
Accepted

What is the size of your $A$ matrix? Is $A$ sparse? Does $A$ have some other special structure? How many values of $\lambda$ do you want to try? Normally, you'd use the Cholesky factorization of $... 7 votes ### Rapidly determining whether or not a dense matrix is of low rank There is a neat trick I have recently learned from this paper. You start doing rank-revealing QR, and stop after the first$k$Householder reflections, when you have a matrix of the form $$\begin{... 7 votes ### Supernodal vs multifrontal factorizations The scopes of these terms are (in my opinion) somewhat different. I'd use "supernodal" as a broad term, to describe any sparse direct solver that applies sufficient intelligence during the symbolic ... 7 votes Accepted ### Term for the typical "linear in the larger dimension, quadratic in the smaller" cost for linear algebra The book "Introduction to Applied Linear Algebra" by Boyd and Vandenberghe has an appendix about complexity of basic operations in linear algebra and they call this case big-times-small-... 7 votes Accepted ### Numerically find the nearest positive semi definite matrix to a symmetric matrix After you compute Q and D, form D'=\max(D,0), and compute A'=QD'Q^\top, the algorithms involved in multiplying those matrices do not promise that A' will be exactly QD'Q^\top. Most ... 7 votes Accepted ### Do most statistical packages and libraries in high-level programming languages rely on LAPACK for their matrix inversion operations? tl;dr Yes. But your question doesn't make it clear that you understand what LAPACK is about. LAPACK is both a software as well as an interface. That is, the operations that LAPACK defines are standard ... 7 votes Accepted ### Algorithm to factorize matrix whose many rows are already of upper triangular form? I believe you can accomplish what you want efficiently using the recursive LU algorithm. In brief, recursive LU on a M \times N matrix A proceeds by partitioning the matrix into 4 blocks: \begin{... 7 votes Accepted ### Functions from Scipy, Blas, or Lapack that compute only upper triangular matrix I think you are overestimating the overhead of computing L. There are zero extra operations needed; the only additional cost is writing to RAM some numbers that you ... 7 votes Accepted ### Accurate Way to Calculate Matrix Powers and Matrix Exponential for Sparse Positive Semidefinite Matrices This is a slightly modified version of my response on math.stackexchange. One standard approach to computing matrix functions times a vector f(M)x or quadratic forms x^Tf(M)x when M is symmetric ... 6 votes Accepted ### Fastest way to solve a sparse unsymmetric system many times Generally speaking, for many right-hand side (RHS) problems, a direct solver is a more feasible solution for several reasons: Major computations are performed during the factorization step (which is ... 6 votes Accepted ### Ways to solve Ax=b for a sparse (banded) A with updates If the only non-zero entries of A_{ij} have j in \{i - 1, i, i + 1\}, then A is a banded matrix with bandwidth 1. More generally, you can talk about matrices of bandwidth k where k is any ... 6 votes ### Does a symmetric positive definite matrix also have \mathbf{A} = \mathbf{L}^T\mathbf{L} (where \mathbf{L} is a lower triangular matrix)? Yes, for an SPD matrix \mathbf A there are a variety of Cholesky-like decompositions, you can derive the \mathbf A = \mathbf L^T \mathbf L variant by first writing down an educated/structured ... 6 votes Accepted ### Block-matrix: optimal fill-in reduction for LU factorization Unfortunately, optimal reordering is NP-complete, so all the algorithms in this space are kinda heuristic. At first glance my instinct would be to try some variant of minimum degree, mainly because ... 6 votes Accepted ### Numerical Linear Algebra: When to use Direct methods versus iterative methods to solve a linear system - for PDEs in particular It's a complicated question, which is why I've recorded a whole bunch of video lectures on the topic :-) Take a look at lectures 34 and following here: https://www.math.colostate.edu/~bangerth/videos.... 6 votes Accepted ### Algorithm for solving systems which are nearly symmetric/adjoint? Let's express the matrix A \in \mathbb{R}^{n \times n} with which we want to solve linear systems as$$ A = S + U V $$where S is a symmetric matrix, U \in \mathbb{R}^{n \times r}, and V \in \... 6 votes ### QR decomposition with only diagonal elements changing Variants of this question get asked regularly on [scicomp.se], but unfortunately the answer is always no: one cannot update easily any matrix factorizations after a full-rank diagonal update. Some ... 6 votes ### Is there a way to generate a matrix-free decomposition for a matrix-free operator? This is not efficient, primarily because (i) the factors of sparse matrices are, in general, not sparse themselves, and (ii) when computing matrix factorizations, you don't just need A_{ij} but you ... 5 votes Accepted ### Does \log(\det(A)) equals sum of log of diagonal elements of D in LDLT decomposition? Yes, it is possible. By construction, the L matrix is lower triangular with all its diagonal entries equal to one. Therefore, \det(L) = \det(L^T) = 1 and, consequently, \det(L D L^T) = \det(L) \,... 5 votes ### The fast, and The Backward-Stable (left) 3\times 3 matrix inverse I will try to give my thought on the first question regarding fast 3\times 3 inverse. Consider$$ A=\left[ \begin{array}{ccc} a & d & g\\ b & e & h\\ c & f & i \end{array}\... 5 votes ### Rapidly determining whether or not a dense matrix is of low rank Another approach, which might be of interest to you is randomized sampling. This is of particular interest if you can quickly compute matrix-vector products$x\rightarrow Ax$and$x\rightarrow A^* x$. ... 5 votes ### inertia count sparse matrix with dense low-rank perturbation Upon further examination, I do think the Woodbury identity can be used to solve this problem. With it we can write:$\left( \mathbf K - \sigma \mathbf Z \mathbf Z^T \right)^{-1} = \mathbf K^{-1} - \...
It is enough if you consider a $D$ that uses forward or backward differences with reflecting boundaries: \begin{equation} D_f = \frac{1}{h}\begin{bmatrix} -1 & 1 & & \\ & \ddots &\...