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# Tag Info

## Hot answers tagged sparse-matrices

11 votes
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### What does LSQR stand for

You’re right, the name is not explained anywhere, but there is some logic behind it. When Chris Paige visited Stanford in 1972, we started several methods that needed names. LSQR means that it’s for ...
9 votes
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### Solve this specific large sparse system of linear equations

The key is the fact that $C_i = - I$, i.e., the appropriately size identity matrix is trivially nonsingular and can be used to nullify the blocks above each copy. We have the following row equivalence ...
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7 votes
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• 2,549
3 votes
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### Diagonalization of very large (but very simple) sparse matrix

As $A$ is symmetric, we have $\max|\lambda_i|=\|A\|_2$ and $\|A\|_1 = \|A\|_{\infty}.$ Furthermore $\|A\|_2 \leq \sqrt{\|A\|_1\|A\|_{\infty}}$, see here. If we put all this together, we get an easy-to-...
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3 votes

### Diagonalization of very large (but very simple) sparse matrix

You can use Gershgorin Circle Theorem to get a lower bound $\lambda_{GCT}$ for $\lambda_{min}$ and then use that as a starting point for inverse iteration (https://en.wikipedia.org/wiki/...
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3 votes
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### Relationship between matrices whose singular values are the same

So, if they are just close but not exactly the same, I am not so sure there is much that we can say. However, for the purpose of exploration suppose that the singular values of both matrices are the ...
• 370
3 votes

### Eigenvalues of a sparse 8x8 matrix

Since $P^3=\frac18ee^T$, $1$ is a simple eigenvalue of $P$ and $0$ is an eigenvalue of algebraic multiplicity $7$.
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3 votes

### Sparse Cholesky decomposition of factorized matrix

You'll find the LSQR as an option for solving large sparse linear systems that max be over or under determined. This is included in scipy.sparse.linalg.lsqr. ...
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2 votes
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### Choosing $\lambda$ to yield sparse solution

It's not hard to show (Hint: show that the dual problem is a projection of $\frac{1}{\lambda}(Ax_0 + z)$ onto the polyhedron $\mathcal P := \{\theta \text{ s.t }\|A^T\theta\|_\infty \le 1\}$, and then ...
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2 votes
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### What techniques are useful for solving this sparse symmetric matrix?

You want the Tridiagonal matrix algorithm, also known as the Thomas algorithm.
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2 votes
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• 67.5k
2 votes

### compressive sensing

The compressive sensing theory basically says, a signal can be (even fully perhaps) recovered even when it is extremely under-sampled. Normally, it is an inverse problem, meaning that if you have a ...
• 466
2 votes
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### Making a matrix as sparse as possible

You can pose the following optimization problem: $$min_{C,B} \quad \|C\|_1 \qquad \text{s.t. } \quad C = BA.$$ An easy way to solve this problem is the ADMM approach. In summary, you add the augmented ...
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