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Eigen Value of Gradient of a vector

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How does the gradient of a vector(delta V) become a 3x3 matrix? And how do you compute it's eigen value efficiently? Is there c++ library that can do this (can the c++ library Eigen do this

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"Gradient" is not quite the right word here. If you have a differentiable mapping from R^3 to R^3, you can compute how the vector changes as you move infinitesimally in each of the axes, and this indeed is a 3x3 matrix, but it's called the Jacobian. There are lots of libraries that compute eigenvalues efficiently, but for a 3x3 matrix you can probably use just about any method. What are you trying to do exactly?

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I'm trying to calculate a scalar field for marching cubes based on an equation in this paper https://www.researchgate.net/publication/220357083_A_unified_particle_model_for_fluid-solid_interactions. I need to find the biggest Eigen value efficiently. It says the gradient of a vector.

Because of curvature, some grid points in the field maybe moving faster than the point being sampled. So need to need Eigen values to correct this.

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I would probably try Armadillo first. It's generally easy to work with and it has served me well in the past. I don't have direct experience with the Armadillo functions that compute eigenvalue and eigenvectors, though.

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