# Questions tagged [matrix-factorization]

Decomposition of a matrix into a product of matrices with special properties. Common matrix factorizations include LU, QR, SVD, and Cholesky.

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### Efficient schemes for solving the extended Saddle point problem

I am interested in knowing some efficient techniques for solving the following extended Saddle point problem. \begin{align} \begin{bmatrix} A & B^T & C^T \\ B & 0 & 0 \\ C & ...
129 views

### Is there a simple way to add a sparse matrix to an LU decomposition of a dense matrix?

I am solving a parabolic equation in the form: $$\left( {M \over\tau_j} + A \right) u^{j+1} = M f^j + {u^j \over \tau_j},$$ where $A$ and $f$ are a dense stiffness matrix and the right hand side of ...
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### Accurate Way to Calculate Matrix Powers and Matrix Exponential for Sparse Positive Semidefinite Matrices

I do need to numerically calculate the following forms for any $x\in\mathbb{R}^n$, possibly in python: $x^T M^k x$, where $M\in\mathbb{R^{n\times n}}$ is a PSD sparse matrix, $n$ can be quite large ...
64 views

### Solve for large array of PD matrices

I have N matrices that are positive definite, and I have to solve for a M vectors. As M is large in my case, doing all solves simultaneously using np.linalg.solve ...
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### Nonsymmetric permutations for LU factorisation of symmetric matrix

Let $A$ be a symmetric matrix. It is then well known that computing the LU factorisation of $PAP^T$ instead of $A$ for a suitably chosen permutation matrix $P$ can greatly reduce fill-in. My question ...
122 views

### Quick way to find a common basis of eigenvectors between 2 matrices : valid or not?

Following the advise of @Federico Polonion a previous post, one suggested, to find a basis of common eigen vectors between 2 matrices, to simply do : Generate 2 ...
101 views

### Efficiently compute a projection matrix from Householders reflectors

Let $A \in \mathbb{R}^{m \times n}$ where $m \geq n$. Let $B$ and $\tau$ be the result of applying LAPACK's dgeqrfp method (R on the upper right triangle, and the ...
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### Givens rotation vs 2x2 Householder reflection

The usual story of Givens rotations vs Householder reflections is that Householder reflections are better if you want to map a long vector to $e_1$, while Givens is better if you want to map a 2-...
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### Clever ways to update LU factorization for ridge regression [duplicate]

Ridge regression can be posed as minimizing the following objective function (over $x$): $$\frac{1}{2} \lVert Ax - b \lVert_2^2 ~+ \frac{\lambda}{2} \lVert x \lVert_2^2$$ Which has a closed form ...
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### “WY” representation of QR factorization — implementations?

I have a matrix $A \in \mathbb{R}^{m \times n}$ where $m \gg n$ and I want to compute the full QR decomposition $A = QR$. Where $Q$ is an orthogonal $m \times m$ matrix. Bishof & Van Loan (1987) ...
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### Methods to improve the efficiency and the memory requirement of LU factorization for complex symmetric system matrix

I want to solve a linear set of equations (Ax=b) using LU decomposition. My "A" matrix is a complex matrix which is ...
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### Computing Singular Value Decomposition of small ($4\times 4$) matrices

I need to compute the Singular Value Decomposition (SVD) of many $4 \times 4$ matrices. I'm looking for SVD algorithms specialized for small matrices. I've read that the ...
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### Binary data clustering by Matrix factorization [closed]

I have read an article talking about binary clustering using Matrix factorization(see attached), but i would like to understand some optimization concepts: Is it reasonable to use a Frobenius norm in ...
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### Robust algorithm for $2 \times 2$ SVD

What is a simple algorithm for computing the SVD of $2 \times 2$ matrices? Ideally, I'd like a numerically robust algorithm, but I'll like to see both simple and not-so-simple implementations. C code ...
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### Implementation of sparse matrix SVD for GPU

I have a sparse matrix $W$ which is almost-squared ($N+1 \times N$) and I would like to know the eigenvalues of $A = W^T W$. $A$ is Hermitian so the eigenvalues are real-positive valued. The usual ...
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### Efficient way to solve a set of linear equations $Ax=b$ when $A$ is sparse and some elements of $b$ are equal to zero

I have a set of linear equations, $Ax=b$. And about half of the elements in the right-hand side (vector $b$) are equal to zero. My system matrix $A$ is a sparse complex matrix. And $A$ is in the size ...
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### Functions from Scipy, Blas, or Lapack that compute only upper triangular matrix

My goal is to transform a matrix into upper triangular form in Python. I know the function scipy.linalg.lu will do LU decomposition and get both upper and lower ...
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### Inverting a matrix from LU decomposition

The LAPACK routines xGETRI compute the inverse of a matrix $A = PLU$ in its LU decomposed form by first computing $U^{-1}$, and then solving the system: $$(A^{-1} P) L = U^{-1}$$ My question is: ...
238 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 ...
I am solving nonlinear least squares problems with the normal equations approach, so on each iteration, I need to solve: $$J^T J \delta = -J^T f$$ for the step $\delta$, where $J$ is a large (...