Questions tagged [linear-algebra]

Questions on the algorithmic/computational aspects of linear algebra, including the solution of linear systems, least squares problems, eigenproblems, and other such matters.

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Analytic formula for $\arg\max_{\|z\|_\infty \le 1}z^T A z$, where $A=uu^T+vv^T$

Let $u$ and $v$ be column vectors of size $n \gg 1$ (not both zero), and consider the matrix $A:=uu^T+vv^T$ Question What is an analytic formula for $\arg\max_{\|z\|_\infty \le 1}z^TAz=\arg\max_{\|z\...
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1answer
119 views

Analytic formula for leading eigenvector of $uu^T + vv^T$?

Let $u$ and $v$ be nonzero column vectors of size $n$ and consider the $n \times n$ positive-definite matrix $A:=uu^T + vv^T$. In this post https://math.stackexchange.com/a/112201/168758, the ...
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1answer
89 views

Efficient computation of leading eigenvector of a matrix product of the form $ADA^T$, where $D$ is diagonal

Let $A=[A_1|\ldots|A_m] \in \mathbb R^{n \times m}$ with $n \gg m \gg 1$ and $D=\text{diag}(d_1,\ldots,d_m)$ where $d_1,\ldots,d_m > 0$, and consider the $n\times n$ positive-definite matrix $X=\...
4
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1answer
124 views

Do most statistical packages and libraries in high-level programming languages rely on LAPACK for their matrix inversion operations?

Possible an open-ended question, but I am wondering if most statistical packages and libraries, for instance, Stata, R, Python's NumPy and MATLAB rely on LAPACK algorithms to perform matrix operations,...
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66 views

implementation for coppersmith matrix multiplication

Is there any online implementation for the coppersmith matrix multiplication I have searched alot but can not find any? and if there is not any why is that Isn't this algotithm much faster than ...
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209 views

What algorithm do BLAS and ATLAS use for matrix multiplication?

I have searched and what I understood was that they use the naive one with several memory and cache optimizations. However, I wanted to know whether they are using the Strassen or the Coppersmith-...
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1answer
109 views

Lapack symmetric update $B^{-1}AB^{-T}$

Does Lapack have a routine that, given symmetric $A=A^T$ and $B$, computes the symmetric matrix $B^{-1}AB^{-T}$ (while preserving symmetry exactly)? It would be enough to have this routine for ...
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1answer
95 views

Vectorization of Jacobi iteration

Assume I have a linear system of $A x = b$ which I want to solve with Jacobi iteration. Matrix $A$ is given in CSR format. The vectors are dense. The code for Jacobi iteration is quite clear and can ...
4
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1answer
61 views

Value of $\gamma$ in the H-infinity norm

Suppose I have the system: $$\dot{x} = Ax+Bu\\ y=Cx+Du$$ and the following Hamiltonian matrix: $$H=\begin{pmatrix} A & \frac{1}{2}B^TB\\ -CC^T&-A \end{pmatrix}$$ I want to find the ...
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188 views

Solving $AXB + X\odot C = D$ matrix equation

Can anyone see a way to solve this equation efficiently? $$AXB + X\odot C = D$$ I tried a straightforward solution that involved vectorizing $X$ but that turned out too expensive for my application -...
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1answer
91 views

Using LAPACK to compute $B^{-1}AB^{-T}$ for thin $B$

How can I use BLAS/LAPACK to compute $$ B^{-1}AB^{-T} $$ where $A\in\mathbb{R}^{n,n}$, $B\in\mathbb{R}^{m,n}$ is full rank matrix with $m>n$, and $B^{-1}y:=\arg \min_{x} \|Bx-y\|_{2}$. In theory, ...
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1answer
161 views

Computing square root of diag(u)-uu'?

I need an efficient way to take square root of a matrix which is a sum of diagonal matrix and rank-1 matrix. More specifically it's the following matrix $$A=D-uu'=\text{diag}(u)-uu'$$ Where entries ...
2
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1answer
103 views

Efficiently finding binary vectors satisfying multiple conditions

I am trying to solve the following problem: Given a binary matrix $\mathbf{A} \in \{0,1\}^{m \times n}$ and a vector $\mathbf{b} \in \mathbb N^n$, does there exist a binary vector $\mathbf{c} \in \{0,...
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1answer
63 views

How to avoid unnecessary checks when inverting this LU decomposition

Background for the question I am currently working on a Matlab code in which the systems of linear equations $Ax_1 = b_1$, $Ax_2 = b_2$, ... have to be solved. As the matrix $A$ is constant during ...
4
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1answer
105 views

Weighted QR Implementation

Say I want a QR decomposition of matrix $A$, where orthogonality of $Q$ is with respect to a generic non-degenerate positive-definite bilinear form $\phi$ (in my case, $\phi$ is "defined" by a finite-...
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0answers
94 views

Small residual but wrong results

When I use BiCGStab to solve a linear matrix system, I use the relative residual to exit the iteration and output the results. For calculating the relative residual I divide the norm of vector $r$ ...
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2answers
536 views

Fast and accurate eigenvalue computation for 3x3 posdef matrices

I am looking for a very fast and efficient algorithm for the computation of the eigenvalues of a $3\times 3$ symmetric positive definite matrix. The algorithm will be part of a massive computational ...
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2answers
145 views

Singular values of $X$ in $AX+XA=C$?

Suppose I have semi-positive definite matrices $A$ and $C$, is there an efficient approach to get top singular values of X entering the following expression? $$ AX+XA=C $$ My matrices are 4k-by-4k ...
4
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1answer
341 views

Diagonalize a unitary matrix with orthogonal matrices using numpy

An important component of the Cartan KAK decomposition for 2 qubit operations is to diagonalize a 4x4 unitary matrix using orthogonal (not unitary, purely real orthogonal) matrices. That is to say, ...
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2answers
327 views

Fastest algorithm for pseudoinverse of skinny matrices

For a performance-sensitive problem, I need to compute the pseudoinverse of a skinny matrix (#rows = 1000–10000, #cols= 10–20). I already employ the traditional SVD econ method. For some problem ...
2
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1answer
347 views

Implementation of Jacobi iteration

I have implemented the Jacobi iteration in C++ using a dense vector and a sparse matrix in CSR format. The code is as follows: ...
2
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1answer
473 views

Cholesky for ill-conditioned/singular covariance matrices

Can someone suggest a way to get Cholesky factorization of a singular covariance matrix? I need it to match Cholesky on full-rank matrices, ie coordinate order should be preserved. My attempt below ...
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0answers
41 views

Inverses of many standard subspaces of one large matrix

i have a large rectangular invertible matrix M (about 5000x5000) and i have a loop in which i do the following for each iteration i (there are about 6000 iterations): i am given a subspace S_i (which ...
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118 views

Implementation of Lanczos method that returns tridiagonal matrix

The Lanczos method can be used to obtain extremal eigenpairs of sparse symmetric or hermitian matrices. I know there are several implementations of the Lanczos method (as well as Arnoldi, Davidson, ...
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1answer
117 views

Incomplete LU decomposition of sparse matrix

I have a sparse matrix stored in CSR format. For this matrix, I would like to get the incomplete LU decomposition. I tried to find algorithms which can utilize the CSR format but I could not find ...
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1answer
145 views

How to use QZ decomposition for single matrix in Matlab?

Can I use QZ decomposition on a single square matrix in Matlab? Like, [Aa,Q,Z]=qz(A);
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2answers
95 views

Fast iterative approximate order-oblivious Orthogonalization algorithm?

I have set of N m-dimensional vectors $\{\phi_i\}$ which gradually loose mutual orthogonality in an algorithm. => I have to re-orthogonalize them every few iterations. But if I do e.g. Gram–Schmidt ...
3
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1answer
75 views

Why are the round-off errors when solving the linear system $Ax = b$ of order $\varepsilon_\text{mach} x_j$?

I was reading a paper on arXiv where, in Section 2.4, the authors are discussing the error that arises in the solution of a linear system $$Ax = b,$$ or, to match up better with the paper, $$\Phi \...
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1answer
403 views

Is there a library that allows einstein summation on dense, sparse, and LinearOperator type tensors

Numpy's einsum only works with dense tensors. Is there an alternative that also works with sparse tensors and linear operators? For example, I might have a ...
3
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1answer
161 views

Matrix condition number and reordering

Does the condition number change when a matrix is reordered by e.g. Cuthill Mckee or some other method?
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0answers
253 views

Is there any catch on using `zgemm3m` vs regular `zgemm`?

I've just (to my embarrassment) encountered a BLAS-like extension of a matrix-matrix product subroutine gemm in Intel MKL: gemm3m...
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1answer
92 views

How to implement this even-odd matrix decomposition efficiently?

Note: This question has also been asked on stackoverflow - see https://stackoverflow.com/questions/57197910/how-to-implement-this-even-odd-matrix-decomposition-efficiently?noredirect=1#...
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0answers
48 views

Does Boost provide a template implementation of the wedge product?

Does the boost C++ library implement the computation of the wedge product? The wedge product is mentioned here, but it is not very clear (to me at least) whether there is a template implementation of ...
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0answers
81 views

What's the more efficient way to solve this matrix equation?

This is intended to be a more generic question not about a specific system. Given a hermitian matrix $H(x_1,\dots,x_n)$ depending non-linearly on some real parameters $x_1,\dots,x_n$. We want these to ...
2
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1answer
56 views

How to pass matrices to parallel workers quickly in matlab?

I am trying to solve many different linear systems in parallel in matlab. The problem is, each linear system has entirely different parts and are fairly large, so passing the information to each of ...
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1answer
119 views

Sparse matrix inversion

I have the impedance matrix $Y$, formulated from an electrical network by augmented nodal analysis. The matrix $Y$ is shown as an image to illustrate its feature visually, where all the white blocks ...
2
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1answer
60 views

numpy.outer without flatten

$x$ is an $N \times M$ matrix. $y$ is a $1 \times L$ vector. I want to return "outer product" between $x$ and $y$, let's call it $z$. z[n,m,l] = x[n,m] * y[l] ...
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1answer
83 views

what does “D = diag(W.1)” means?

, what does “D = diag(W.1)” means?on page #2, just below equation (6) PFA screenshot and here is the link of the paper - original paper
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1answer
350 views

Fast algorithm for computing cofactor matrix

I wonder if there is a fast algorithm, say ($\mathcal O(n^3)$) for computing the cofactor matrix (or conjugate matrix) of an $N\times N$ square matrix. And yes, one could first compute its determinant ...
2
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1answer
113 views

Null space for smoothed aggregation algebraic multigrid

I do not really get the point of null space usage for creating the prolongation operator for smoothed aggregation algebraic multigrid. I know what the null space is per definition and I know that the ...
2
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0answers
164 views

Regularized least squares with QR factorization

Consider the regularized least squares problem $$ \min_x || b - A x ||^2 + \lambda^2 ||x||^2 $$ which is equivalent to $$ \min_x \left|\left| \pmatrix{b \\ 0} - \pmatrix{A \\ \lambda I} x \right|\...
5
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1answer
147 views

Complex differentiation of linear solvers

I have a linear system $$Ax=b$$ which I'm solving approximately, and I need to take the frechet derivative of x with respect to z. Were I solving the problem exactly (either analytically or to machine ...
3
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1answer
179 views

Optimality of block-Jacobi preconditioner

For a dense $N \times N$ matrix $A$, is the block-Jacobi preconditioner comprising the inverse of the diagonal blocks of $A$ the optimal block-diagonal preconditioner? Could there exist another matrix ...
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0answers
56 views

Kinetic preconditioning

Publication arXiv:0804.2583 describes a method for doing self-consistent iteration without having to diagonalize the Hamiltonian operator at every step. IX. PRECONDITIONING As already mentioned, ...
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34 views

A preconditioner for self-consistent iteration

I tried to derive a preconditioner for self-consistent iteration similar to section IX in arXiv:0804.2583. For simplicity, consider here only one orbital (one or two electrons) systems. Suppose that ...
3
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1answer
128 views

Whitening transformation does NOT return a unit covariance matrix

For this question, I am using the following Wiki definition of Matrix whitening: Suppose $X$ is a random (column) vector with non-singular covariance matrix $\Sigma$ and mean 0. Then the ...
2
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1answer
149 views

In iterative methods, are matrix decompositions considered useful for implementation?

When we study an iterative method from textbooks, for example, see the Gauss-Seidel Method, the given matrix is decomposed with suitable splittings. In the example, $A = L+U$. So we can proceed with ...
5
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2answers
398 views

Python: vectorizing a structured linear system solve

Overview I am looking for a way to solve a structured linear system in Python without using a for loop (preferably using vectorization, if possible). Background Consider the following linear system:...
2
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1answer
185 views

Re-using LU factorization within iterative (?) setup for a sum of two matrices

So, I would love to make at least some use of my preexisting data, no matter how small, and just out of ideas. Maybe I am just a prisoner of a Kahneman-like theatre-ticket paradox, and don't know ...
4
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1answer
251 views

Accurate way of getting the square root inverse of a positive definite symmetric matrix

What is the most accurate algorithm to get the square root inverse of a positive definite symmetric matrix? I am not looking as much for efficiency, though using quadruple precision computation is out ...

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