# Tagged Questions

Questions on the various algorithms used in linear algebra computations (matrix computations).

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### Fastest way to solve linear system with block symmetric banded/Toeplitz matrix

I have a matrix of the following form: The size of the matrix may grow to be large, but the general pattern of being blockwise symmetric and banded (with 5 bands) will always hold. What is the ...
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### Finding the Coefficient Matrix of a Spring-Mass System

So as part of a class in numerical linear algebra, we're exploring the topic of banded matrix system. I've come across a problem that involves Hooke's Law, but I'm having a little difficulty ...
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### SVD for Seam Carving

Could SVD be used for Seam Carving ? I am making a small program for a uni course and I'm looking for different ways to calculate pixel energy; which made me come across SVD. Among others, I have ...
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### How to determine if an algorithm converges for solving Ax = b in practice?

Suppose I am using iterative refinement. I am updating the solution $$x^{(k+1)} = x^{(k)} +p$$ where $Ap = b - Ax^{(k)}$. If $A$ is ill-conditioned, then we can not simply use residual norm or the ...
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### How to find a number greater than the largest (smaller than the smallest) eigen value of a matrix efficiently?

I have a symmetric matrix $A$, I want to find a number say $\lambda_1$ which is greater than $\lambda$,i.e. the largest eigen value of the matrix $A$. It doesn't need to be equal to $\lambda$. But it'...
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### $LU$ Factorization, improving upon stability

I was wondering when we add partial pivoting to an $LU$ factorization to a matrix $A$ it supposedly changes the data structure but improves the overall algorithm since we get better numerical ...
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### Iterative methods: What happens when the spectral radius of a matrix is exactly 1?

I know that an iterative method (I'm using Jacobi and Gauss-Seidel in this case) will converge iff the spectral radius (max absolute value of eigenvalues) of its iterative matrix is strictly less than ...
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### What is the function of the pivot index vector in Gauss Jordan Elimination with full pivoting?

In numerical recipes, on page 39 (page 4 of the pdf) the following algorithm has been suggested for finding a pivot: ...