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Results tagged with na.numerical-analysis
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user 5963
Numerical algorithms for problems in analysis and algebra, scientific computation
11
votes
Accepted
Are the banded versions of a positive definite matrix positive definite?
No. The matrix
$M = \begin{bmatrix}5 & 4 & 4 \\\\ 4 & 5 & 4 \\\\ 4 & 4 & 5\end{bmatrix} = \begin{bmatrix}2 & 2 & 2\end{bmatrix}\begin{bmatrix}2 \\\\ 2 \\\\ 2\end{bmatrix} + I$
is positive definite, …
- 8,501
9
votes
Accepted
minimize the sum of absolute eigenvalues
The sum of the absolute value of the eigenvalues is the same (since the matrix is real and symmetric) as the sum of the singular values. This sum is called the nuclear norm of the matrix. So what yo …
- 8,501
3
votes
Do runtimes for P require EXP resources to upper-bound? ... are concrete examples known? (an...
I'm not entirely sure I understand your question, but you may be interested in the Robertson-Seymour Theorem. It shows that any family of graphs satisfying a certain property (being "minor-closed") h …
- 8,501
2
votes
Accepted
Approximation by polynom 1) with respect to supremum-norm 2) I need F_{approx} > F_{exact}
The framework of sum ofsquares / semidefinite programming (SOS / SDP) allows one to compute arbitrarily good approximations to such problems. The general idea is as follows.
First, we write the prob …
- 8,501
2
votes
Solving a System of Quadratic Equations
What you have is an instance of a quadratically constrained quadratic program (QCQP). These problems are NP-hard in general (though it's possible your particular type of instance is not hard as fedja …
- 8,501
0
votes
Accepted
Best constant in a convex polynomial inequality.
This is just an answer to the new question added in the edit: no, $p\geq q$ need not hold everywhere. Let $p(x) = x^2 + (1+\alpha) x^6$ for some $\alpha\in (0,\frac{1}{4})$ and $q(x) = x^4 + x^6$. T …
- 8,501