# Tag Info

Accepted

### why is A*v+B*v faster than (A+B)*v?

Except for code which does a significant number of floating-point operations on data that are held in cache, most floating-point intensive code is performance limited by memory bandwidth and cache ...
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### Why do we usually not want the eigenvalues of non-symmetric matrices?

Stability under perturbations Let $E$ be a perturbation such that $\|E\| \leq \varepsilon$. If $A$ is symmetric, then the eigenvalues of $A+E$ are at a distance $\varepsilon$ from those of $A$. (Bauer-...

### why is A*v+B*v faster than (A+B)*v?

Your code is limited by memory bandwidth. For trivial math, it's often better to count memory accesses rather than flops. You'll get the following table: ...
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### In FEM, why is the stiffness matrix positive definite?

The property follows from the property of the corresponding (weak form of the) partial differential equation; this is one of the advantages of finite element methods compared to, e.g., finite ...
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Accepted

### Finding the matrix inverse given a solver for the matrix equation $Ax=b$

Your two ideas make it much too complicated. If $X$ is the inverse of $A$, $$AX=I,$$ and $x_i$ is the $i$-th column of $X$ and $e_i$ is the $i$-th column of the identity matrix $I$ ($e_i$ is a ...
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### Stabilizing a 3x3 real symmetric matrix eigenvalue calculation

This is trying to compute the eigenvalues by computing the roots of the characteristic polynomial. In this case, the characteristic polynomial is $p(t) = t^3-2t^2x$, $x=1.25\times 10^6$, and zero is a ...

### Compute $x = B^{-1}(2A+I)(C^{-1}+A)b$ without calculating matrix inverses

As was mentioned in the comment, calculating $x=M^{-1}y$ is equivalent to solving $Mx=y$. Here is the full solution: First, you can reformulate the equation to: $Bx=(2A+I)(C^{-1}+A)b$, and by ...
### Methods for solving $Ax=b$, small and sparse A
By direct substitution, trivially.  \begin{bmatrix} 0\\f \end{bmatrix} = \begin{bmatrix} M & -I\\ I & 0 \end{bmatrix} \begin{bmatrix} y\\z \end{bmatrix} = \begin{bmatrix} My-z\\ y \end{...