# 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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### 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 ...
• 131
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### 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, ...
• 171
1 vote
302 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 ...
• 818
1 vote
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### 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);
234 views

### Fast iterative approximate order-oblivious Orthogonalization algorithm?

I have set of N m-dimensional vectors $\{\phi_i\}$ which gradually loses mutual orthogonality in an algorithm. => I have to re-orthogonalize them every few iterations. But if I use the Gram–Schmidt ...
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• 956
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### 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 ...
• 1,902
264 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 ...
• 653
1 vote
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### 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, ...
• 331
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### 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 ...
• 331
245 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 ...
• 133
154 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 ...
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477 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:...
344 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 ...
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### 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 ...
453 views

### Compiled c++ code runs much faster with double than float. Explanation?

I am still rather new on here and I hope question is suitable for this forum otherwise please help me migrate it to greener pastures. I am an electrical engineer specializing in applying mathematics ...
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### CHOLMOD condition number estimate

The CHOLMOD library provides a CHOLMOD_rcond function that estimates the reciprocal condition number (in the one norm) of a symmetric positive definite matrix from ...
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### How to efficiently invert $K \otimes M+I_T\otimes \Sigma$?
I'm looking for a way to efficiently invert $$K \otimes M+I_T\otimes \Sigma$$ where the inverses for $M,K$ exist. $I_T$ is the identity matrix of dimension $T$, and $\Sigma$ is a diagonal matrix, with ...