# Questions tagged [numerical-linear-algebra]

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

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### LU decomposition of banded matrix with partial pivoting

Disclaimer: I'm rusty as can be in this department. I'm looking into how to implement a banded matrix LU decomposition with partial pivoting ($PA = LU$). So to start with I implemented regular matrix ...
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### Derive conditional number for eigenvalue [closed]

How to derive the absolute conditional number, $$\kappa \le \frac{\|x\|_2\|y\|_2}{|y^*x|}$$ where x is the right eigen vector of matrix A and y is the left eigenvector. I tried to perturbed the system ...
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### Product between matrix-polynomial and vector

I was wondering if it is possible to optimize the evalutaion of the product of a matrix polynomial and a vector. $$\vec{y} = \left( \sum_{i=0}^{n}a_iM^i \right)\vec{x}$$ Matrix size is maybe ...
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### Gauss-Seidel and bound of iterations using the residual norm

I would like to find the least $k\in \mathbb{N}$ such that $$Res=\frac{\|b-Ax^{(k)}\|}{\|b\|}\leq \epsilon$$ for given $\epsilon$, when using the Gauss-Seidel iteration. I calculated the norm (...
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### Using underdetermined system for dimensionality reduction.

I have an underdetermined linear system $A\vec{x} = \vec{b}$. I.e. A is $M$x$N$ and $M < N$. Now, I want to be able to calculate $M$ of the values of $\vec{x}$ from the $N$-$M$ rest. I know how to ...
1 vote
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### Finding Matrix of Minimum 2-norm to obtain singular matrix

Let $A,X \in \mathbb{R}^{m \times m}$ with rank$(A) = m$. Find $X$ of minimum 2-norm for $A + X$ to be singular. My thoughts: Since we want $A + X$ to be singular, we want to show that it has a zero ...
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### Numerical Inversion of Elliptic Operator

I am trying to solve the following elliptic differential equation. $$\frac{\partial^2\psi}{\partial R^2} + \frac{\partial^2\psi}{\partial Z^2} - \frac{1}{R}\frac{\partial\psi}{\partial R} = S(R,Z)$$ ...
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### SVD-based low-rank approximation doesn't work

Let $M$ be a matrix. Consider the singular value decomposition $$M = U \Sigma V^*.$$ I've often been told that you can get the best low-rank approximation of $M$ using the SVD. Namely, you split up ...
1 vote
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### Approximating a First Order PDE using Finite Difference Method

I'm attempting to approximate the solution $u(x,t)$ to the PDE $x\cdot u_x=t\cdot u_t$ with the initial condition $u(x,t_0)=f(x)$, boundary conditions $u(x_0,t)=g_1(t)$ and $u(x_n,t)=g_2(t)$, and ...
### Iterative algorithm for computing $\Sigma^{1/2} x$
Say I have a PSD matrix $\Sigma$ and a vector $x$, is there an iterative algorithm (faster than computing $\Sigma^{1/2}$ using Cholesky decomposition) for computing $\Sigma^{1/2} x$? (In this ...
Why is in Matlab $$e^{\pi\cdot\frac{\sqrt{163}}{3}} - 640320 = -2.3283 * 10^{-10}$$ exp(pi * sqrt(163)/3) - 640320 ans =-2.3283e-10 I know that it does have ...