# Tag Info

### Rule of thumb for sparse vs dense matrix storage

For what it is worth, for random sparse matrices of size 10,000 by 10,000 vs. dense matrices of the same size, on my Xeon workstation using MATLAB and Intel MKL as the BLAS, the sparse matrix-vector ...
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### Rule of thumb for sparse vs dense matrix storage

All matrix operations are memory bound (and not compute bound) on today's processors. So basically, you have to ask which format stores fewer bytes. This is easy to compute: For a full matrix, you ...
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### Why does sparse linear algebra have a low arithmetic intensity?

BLAS1-operations, BLAS2-operations, and sparse-operations share the same curse of low arithmetic intensity, that they perform $O(1)$ flops for each memory read (contrast this to a BLAS3-operation like ...
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### Is there an iterative solver for dense matrices with possible zero diagonal entries?

Iterative Krylov-subspace solvers generally only require matrix-vector products and don't care whether or where there are zeros in the matrix. In your case, unless you have other information about the ...
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### Rule of thumb for sparse vs dense matrix storage

Even if a matrix is very sparse, its matrix product with itself can be dense. Take for example a diagonal matrix and fill its first row and column with nonzero entries; its product with itself will be ...
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### Matrix exponential of a Hamiltonian matrix

Very quick answer... The exponential of a Hamiltonian matrix is symplectic, a property that you probably wish to preserve, otherwise you would simply use a non-structure-preserving method. Indeed, ...
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### Tools to compare two matrices with same dimensions

Welcome to Scicomp! You might be interested in the field of (multimodal) medical image registration. In medical contexts one often wants to register a CT image with an MR or PET-Scan. The amplitudes ...
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### Solving a linear system whose coefficient matrix is dense but symmetric

It can be lowered; it is called packed storage and Lapack has some functions to deal with it, e.g., ?PPSVX, ?SPSVX. As this ...
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### Trace of inverse from LU decomposition

Base case: Computing the factorization requires $\frac 2 3n^3$ operations and the inverse requires $\frac 4 3n^3$ operations. Computing the trace adds $O(n)$. Let's round that to $2n^3$. LU case: ...
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### Library to solve dense linear system with GMRES

PETSC might be a good option. Not super user friendly, but its good with lots of options.
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### Matrix exponential of a Hamiltonian matrix

You might have an option to use hierarchical matrices ($\mathcal H$-matrices) and the corresponding functionality of the libraries that support them. Effectively, if every matrix $A$, $G$, and $Q$ ...
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### Discretized matrix from the integral kernel function

(1) It appears that $(i_1,i_2),(j_1,j_2)$ are coordinate pairs, such that $(j_1,j_2)$ are the integer coordinates on the source grid and $(i_1,i_2)$ are the integer coordinates on the observer grid. ...
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### Preconditioning of two step iteration for dense matrices

I would try to circumvent the problem entirely by the substitution $x=By$. Then your equation reduces to $$\left[ (\gamma^+ + \gamma^-) AB + \frac{1}{2} (\gamma^+-\gamma^-) I \right] y = b.$$ If $AB$ ...
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### Computing sparse matrix products into a dense result

Since the question was asked a long time ago, I would give a detailed overview answer. The first question is how the assembled matrix $A$ will be used later, as the construction of the matrix itself ...
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