# 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 ...

### 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 ...
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### 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 ...

### Numerical stability of higher order Zernike polynomials

A possible solution (suggested by @gammatester) is to use Jacobi polynomials. This circumvents the problem of catastrophic cancellation in adding the large polynomial coefficients by 'naive' ...
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### Polynomial approximation for floating-point arithmetic

The sine is an odd function, so you want that also in an approximation. A polynomial with $p(0)=0$ can be factored as $p(x)=xq(x)$, so $q(x)\approx \frac{\sin(x)}{x}$. Each interval $[2^n,2^{n+1})$ ...
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### Numerical stability of higher order Zernike polynomials

In this paper, Honarvar and Paramesran derive an interesting method to compute the radial Zernike polynomials in a very nice recursive way. The recursion formula is surprisingly straightforward, ...
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### Numerically stable computation of the Characteristic Polynomial of a matrix for Cayley-Hamilton Theorem

Generally speaking, $P(A)$ would not be close to the zero matrix: if you compute the polynomial $P$ to a relative accuracy of $\epsilon$ (which is $10^{-16}$ in double-precision), then $P(A)$ would be ...

### Benchmarks for Gröbner bases and polynomial system solution

Always keep in mind that the results of any benchmark will depend, in addition to the problem's size, on the base field over which the polynomial ring is defined (rational numbers or integers modulo ...

### Is there any way/any python function to calculate the condition number of the roots of a polynomial directly?

The condition number of a root $r$ of a polynomial $p$ is $$\kappa := \frac{\left\| p \right\|}{|rp'(r)|}$$ There is some arbitrariness in the choice of norm which affects the definition of 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 ...

### 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 ...

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### Polynomial order of an approximation of a section of sine and numerical accuracy

By using orthonormal polynomial basis functions, then you do not need to solve a linear set of equations in order to get the polynomial coefficients, similar to the Fourier series coefficients. ...

### Find the roots of a complicated polynomial

One technique is to use a library with arbitrary large integers and use fixed point arithmetic. Just scale up x by some integer factor, then compute your polynomial in fixed-point, keeping only the ...

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

Once you're at the level of a long list of expressions, there is little you can still do other than hope that the compiler finds opportunities for optimization at the assembly level. This may bring ...

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

You have demonstrated that the polynomial has only a single positive root (I assume here that you are only interested in positive real roots). Using Cauchy's upper bound for polynomial roots 1+ \max\...
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### Fitting a monotonically increasing spline function

A smoothing spline might be good enough in your case. For example, scipy.interpolate.UnivariateSpline implements this. You can use it in the following way: ...