# Tag Info

Accepted

### Benchmarks for Gröbner bases and polynomial system solution

I posted some benchmarks here: http://www.cecm.sfu.ca/~rpearcea/mgb.html (archived copy) These are for total degree orders. To solve systems you typically need to do more work. Timings are for a ...

### Positive root of $x^q + bx - b$

According to Wolfram Alpha, $x^5+3(x-1)=0$ has no closed-form solution, so you can forget about a nice closed-form expression. :) I see nothing wrong with Newton's method; it should be quick and ...
Accepted

### Interpolating a mathematical function using a Hermite Cubic Finite Element Space

Short answer You are missing the Jacobian of the transformation for the derivatives. Long answer The conditions that you propose for your interpolator translate into the following system of ...

### Benchmarks for Gröbner bases and polynomial system solution

Googling benchmark polynomial systems leads to a few hits, including the University of Mannheim's Computer Algebra Benchmark Initiative. Sadly, most of these are ...
Accepted

### Accurate Polynomial Evaluation in Floating Point

Horner is indeed the most stable way to evaluate a polynomial (and you get the bonus of evaluating its derivatives with not too much extra cost). Higham presents a nice error analysis of the algorithm ...

### Building Gaussian-type quadrature schemes with Zernike polynomials

Unfortunately the 1D reasoning will not directly carry into 2D and beyond. Depending on the domain for your integral, you will have considerable latitude in the node choice - but of course the ...

### Building Gaussian-type quadrature schemes with Zernike polynomials

Yes, one can extend the Gaussian quadrature (in one dimension) to multiple dimensions. Often this is called cubature. Much work on this has been done by Ronald Cools and collaborators (see his ...

### Fast evaluation functions given by straight-line programs

OK, you have a very nice problem, I tried to run some benchmarks. First, I don't have your parameters so I used your small example. Second, since you do not specify the language, I used C + GSL (...
Accepted

### How to robustly and numerically expand a $k$-order polynomial in two variables defined on a polygon domain?

The problem that you are seeing is a well known problem of these basis functions: $$\phi_{m,n}(x,y) = x^{m}y^{n}$$ I quote from here: The reason why the coefficient matrix is nearly singular and ...

### Gauss Integration of $\sqrt(x)$

Quadrature points $x_1,x_2$ are also unknown. You need two more equations corresponding to $x^2$ and $x^3$. Then you solve for $a_1, a_2, x_1, x_2$.
Accepted

Choose four collocation points in the interval $[a,b]$, e.g., $x_0=a\\ x_1=a + (1/3)(b-a)\\ x_2=a + (2/3)(b-a)\\ x_3=b$ and form a matrix $M$ \begin{bmatrix} 1 & x_0 & x_0^2 & x_0^3 \\ 1 ...

### Problems Implementing the Remez Algorithm

Let's use the example (approximate the square root function on $[0.25,1.0]$ with a quartic polynomial) to step through your calculations. I suspect that the code is going to work with only modest ...
Accepted

### Polynomial approximation

You can easily do this with for loops. Just use n to define your loop limits. Here's a fully functional solution for MATLAB (just define n and x first): ...