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1500 questions
10
votes
2 answers
About faster approximation of log(x)
I had written a code a while ago which attempted to calculate $log(x)$ without using library functions. Yesterday, I was reviewing the old code, and I tried to make it as fast as possible, (and correct). Here's my attempt so far:
const double ee =…
sarker306
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10
votes
2 answers
Higher-order numerical integration on a triangle/tetrahedron/simplex
Let $T$ be a triangle and let $f$ be a smooth function on $T$.
We can use mid-point quadrature $\int f dx \approx |T|\cdot f(x_M)$, where $x_M$ is the middle-point of $T$.
Can you provide me with (a reference for) higher-order formulas on a simplex?
shuhalo
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10
votes
2 answers
scipy.optimize.fmin_bfgs: "Desired error not necessarily achieved due to precision loss"
I am getting the warning in the post subject when attempting to optimize a function in Python with the scipy.optimize.fmin_bfgs function.
The complete output:
Warning: Desired error not necessarily achieveddue to precision loss
Current…
ACEG
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10
votes
1 answer
What iterative method can effectively solve a linear system with this kind of spectrum
I have a linear system with matrix which eigenvalues are uniformly distributed on the unit circle like this:
Is it possible to solve this kind of system effectively by iterative method, maybe with some preconditioner?
faleichik
- 1,832
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10
votes
1 answer
Order of operations, numerical algorithms
I have read that
(1) Ill conditioned operations should be performed before well conditioned ones.
As an example, one should calculate $xz-yz$ as $(x-y)z$ since subtraction is ill conditioned while multiplication isn't.
However, a first-order…
Bananach
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10
votes
3 answers
How to sample points in hyperbolic space?
Hyperbolic space in the Poincaré upper half space model looks like ordinary $\Bbb R^n$ but with the notion of angle and distance distorted in a relatively simple way. In Euclidean space I can sample a random point uniformly in a ball in several…
doetoe
- 593
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10
votes
4 answers
Are DAXPY, DCOPY, DSCAL overkills?
I have implemented CG in FORTRAN by linking it to Intel MKL.
When there are statements like:
(Refer Wikipedia)
p=r;
x=x+alpha*p
r=r-alpha*Ap;
or similar ones in QMR (in much greater quantity)
v_tld = r;
y = v_tld;
rho = norm( y );
w_tld = r;
z…
Inquest
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10
votes
1 answer
Can the method of lines be used to discretize all PDEs?
I have found that the method of lines is a very natural way to think about the discretization of PDE's. Therefore I always default to that mindset when presented with a new set of equations. I have never seen a PDE where this would not work.
What…
Godric Seer
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10
votes
1 answer
How to find the interior eigenvalues by krylov subspace method?
I am wondering how to find the eigenvalues of some sparse matrix in given interval [a, b] by iterative method. To my personal understanding, it is more obvious to use Krylov subspace method to find the extreme eigenvalues rather than the interior…
Willowbrook
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10
votes
2 answers
Finite elements $W^{1,\infty}$ error estimates
Are there finite element method setups that provide error estimates in the $W^{1,\infty}$ norm (i.e., bounds on $\|u'_h - u'\|_\infty$)? Which families of elements can be used for implementing them?
(Crossposted from MathOverflow, where it…
Federico Poloni
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10
votes
2 answers
What about this simple error estimate for linear PDE?
Let $\Omega$ be a convex polygonally bounded Lipschitz domain in $\mathbb R^2$, let $f \in L^2(\Omega)$.
Then the solution of the Dirichlet problem $\Delta u = f$ in $\Omega$, $\operatorname{trace} u = 0$ on $\partial\Omega$ has a unique solution in…
shuhalo
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10
votes
1 answer
How can I determine the initial values of pseudo-random number generator if the sequence is given?
Suppose I knew that a random number sequence was generated by a linear congruential generator. That is,
$x_{n+1}=(aX_n+c) \bmod m$
If I am given the entire period (or at least a large contiguous subsequence of it), how can I reconstruct the…
Paul
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10
votes
4 answers
Nonlinear least squares with box constraints
What are recommended ways of doing nonlinear least squares,
min $\sum err_i(p)^2$,
with box constraints $lo_j <= p_j <= hi_j$ ?
It seems to me (fools rush in) that one could
make the box constraints quadratic, and minimize
$$ \sum_i err_i(p)^2 + C *…
denis
- 932
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10
votes
3 answers
Numerical integration of compactly supported function on a triangle
as the title suggests I'm trying to compute the integral of a compactly supported function (Wendland's quintic polynomial) on a triangle. Notice, that the center of the function is somewhere in 3-D space. I integrate this function on an arbitrary,…
Azrael3000
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10
votes
5 answers
Symbolic solution of a system of 7 nonlinear equations
I've got a system of ordinary differential equations - 7 equations, and ~30 parameters governing their behavior as part of a mathematical model of disease transmission. I'd like to find the steady states for those equations Changing dx/dt = rest of…
Fomite
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