Numerical algorithms for problems in analysis and algebra, scientific computation
2
votes
0answers
79 views
How to get an expression for this integral(Numerically/Analytically)
I have the following problem:
I need to evaluate the integral $$\int_{\cos(\alpha)}^{1} P_l(t)P_{l'}(t) dt $$ for $\alpha \in [0,\pi]$ and each combination of $l$ and $l'$, where $P_l$ is the l-th ...
1
vote
0answers
15 views
In what paper was the shrinkage parameter introduced to the nelder-mead simplex direct search algorithm?
I have read lots of papers referencing a 4th shrinkage parameter when talking about the Nelder Mead Simplex method. However, I cannot see any shrinkage parameter in the flow chart of the original ...
3
votes
1answer
45 views
Practical error-estimates for (adaptive) Newton-Cotes Quadrature
I am looking for practical error estimates for Newton-Cotes Quadrature rules.
Most books on numerical methods I have found mainly deal with theoretical error bounds/estimates for the respective ...
6
votes
2answers
126 views
Finding a low-degree polynomial vanishing on half the zeroes of a polynomial system
Let $f(x)$ be a real polynomial of degree $2d$ without real roots. Let the complex roots be $z_1$, $\bar{z_1}$, $z_2$, $\bar{z_2}$, ..., $z_d$, $\bar{z_d}$ with $z_i$ in the upper half plane. Let ...
4
votes
0answers
80 views
Pair of two-variable polynomial equations of high order
I have the following pair of equations to be solved for two variables $\rho$ and $D$ resulting from a certain Maximum Likelihood Estimation for a time series $X_n > 0$, $n=0, \ldots, N+1$ with $N ...
1
vote
1answer
42 views
Parameter estimation for stochastic differential equation from discrete observations
Suppose we have a time-series $x(t_i)$ at discrete times $t_i$ and we want to estimate the parameters of an underlying SDE corresponding to this time-series:
$$dx_t = f(x_t,\theta)dt + ...
-2
votes
0answers
71 views
What's the solution to this Navier Stokes equation--initial,boundary value problem [closed]
I have a flow in a pipe, the flow is viscous, its initial velocity is (1,0,0) (towards x direction, the same for the pipe) everywhere. The inlet's constant velocity is (1,0,0), the outlet's constant ...
2
votes
1answer
113 views
Possible pathological properties of positive definite matrix
Suppose $A$ is a positive definite matrix such that
$$I \preceq A \preceq 1.01I.$$
Is it possible that
$$\sum_{i=1}^n A_{1i}$$
can be arbitrarily large?
Thanks,
Jack
1
vote
0answers
50 views
A question on discrete numerical simulation on fluids mechanics
I read a paper "stable, circulation-preseving simplicial fuids" by Elcott,.etc. http://www.cs.jhu.edu/~misha/Fall09/Elcott07.pdf. It gives a structure preseving discretion of fluids. I have a ...
5
votes
3answers
281 views
Square Root Algorithm
I would like an efficient algorithm for square root of a positive integer. Is there a reference that compares various square root algorithms?
0
votes
0answers
44 views
Convergence acceleration of series by suitable dummy variables
We can show that if $|\frac{x}{x+y}| < 1$; then,
$$
\sum_{n=0}^{\infty} a_n x^n = \Big(\frac{y}{x+y}\Big)^{r+1}
\sum_{n=0}^{\infty}\Big(\frac{x}{x+y}\Big)^n \sum_{k=0}^{n} \binom {n+r} {k+r}a_k ...
7
votes
3answers
337 views
accelerating convergence of a class of sequences
Do any of the standard methods of acceleration convergence of series, when applied to
the series $1 - 1 + 1/2 - 1/2 + 1/3 - 1/3 + ...$, give convergence to 0 with error $o(1/n)$?
I tried applying ...
1
vote
1answer
134 views
What is the definition of an antilimit?
I've seen some references to antilimits in the numerical analysis literature, but no definition of the term. The impression I get is that in specific contexts where every sequence $x_0,x_1,x_2,\dots$ ...
1
vote
1answer
57 views
Understanding the rationale behind “batch means” estimation
Hello all,
I am implementing an MCMC algorithm for my work, and I've come upon something in the literature which I just can't understand.
Specifically, I am attempting to estimate the amount of ...
3
votes
0answers
100 views
Are there some numerical test to check if a map is a contraction?
Let's say I have a multivariate function
$$
f:D \to D, D \subset \mathbb R ^n, D \text{ compact},
$$
for which there is no closed form.
That is the only way to evaluate the function is to do it ...
