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Results tagged with na.numerical-analysis
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user 11142
Numerical algorithms for problems in analysis and algebra, scientific computation
1
vote
How can I find minimum and maximum eigenvalue of non-positive define matrix
Find your greatest in absolute value eigenvalue, call it $E.$ If it is positive, the matrix $M + 2 EI$ is positive definite, so do whatever you do for positive definite matrices. If it is negative, $M …
- 96.4k
8
votes
Accepted
Automatic vs numerical differentiation of a function known from samples
Automatic differentiation needs the structure of the function ( computation graph, or preferably a straight line program).
In your case, I am not sure how numeric differentiation helps to get a rel …
- 96.4k
2
votes
Numerically solving for pseudo inverse of non-squared Vandermonde matrix
Read the ancient paper entitled "Inverse of the Vandermonde Matrix with Applicataions", by L. Richard Turner, and be enlightened.
- 96.4k
3
votes
complexity of computing the singular vector corresponding to the smallest singular value
The question has been studied at some length. See, for example,
Hubert Schwetlick and Uwe Schnabel, MR 1997360 Iterative computation of the smallest singular value and the corresponding singular vec …
- 96.4k
6
votes
show that $ \frac{\Gamma(\frac{1}{24})\Gamma(\frac{11}{24})}{\Gamma(\frac{5}{24})\Gamma(\fra...
This follows from the discussion at and preceding page 31 in Campbell's book.
- 96.4k
2
votes
Fast Fourier Transforms for non-trigonometric bases
Related questions have been considered in some depth by Dan Rockmore and collaborators. For more, check out Rockmore's web page.
- 96.4k
14
votes
Accepted
How to compute $\sum_{x \in \mathbb{Z}^n} e^{-x^TMx}$ efficiently
You are trying to compute a multi-dimensional theta function, and this question is studied in depth in this 2003 Math. Comp. article by Deconinck, Heil, Bobenko, van Hoeij,and Schmies.
- 96.4k
3
votes
Solving over-determined system of polynomials
I am not sure I understand what the ellipsis $\dots$ means in the last set of equations, since it seems that you only have pairs $l_i, u_i.$ If that is true, that means that there are $2^n$ possible v …
- 96.4k
1
vote
Multiprecision numerical evaluation of integral: Sage vs. PARI/GP vs. mpmath
Not really appropriate for this site, but Mathematica gives the below (after complaining about convergence problems). On the other hand, integrating your function from $-1$ to $1-\epsilon,$ and using …
- 96.4k
1
vote
Numerical integration of legendre polynomials
See:
@article {MR1078802,
AUTHOR = {Alpert, Bradley K. and Rokhlin, Vladimir},
TITLE = {A fast algorithm for the evaluation of {L}egendre expansions},
JOURNAL = {SIAM J. Sci. Statist. Com …
- 96.4k
1
vote
Numerical calculation of Fourier transform with a nice error bound
This is a subject of a very nice paper by Charlie Epstein (2004)
- 96.4k
1
vote
Accepted
Finding a low-degree polynomial vanishing on half the zeroes of a polynomial system
As @Federico notes (but does not say explicitly), the magic words are "spectral factorization" -- for an algorithm see here. I should say that it is very far from clear to what extent the fancy algori …
- 96.4k
1
vote
What software one needs to solve a big linear system on a small computer?
Mathematica will do this with no problems (actually, you can use your GPU to do it REALLY fast). Unlike Maple (apparently) there is no problem getting mathematica to use floating point computation, bu …
- 96.4k
0
votes
Characterizing convex polynomials
To add to @coudy's answer: a polynomial is nonnegative if and only if it is a sum of squares, which gives a slightly different set of equations to be satisfied by the second derivative.
- 96.4k
2
votes
Estimating the volume of a semialgebraic set from above
Here is an algorithm: partition $\mathbb{R}^n$ into cubes of side $1/k.$ For each cube $C_i$, use your favorite quantifier elimination algorithm to check whether the set $S$ intersects it. Then, your …
- 96.4k