# Tag Info

### 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 ...
• 11.4k
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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 ...
• 8,515

### 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' ...
• 273
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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 $[2^n,2^{n+1})$ ...
• 6,109
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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, ...
• 571
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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 ...
• 11.4k

### 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 ...
• 2,155

### 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 ...
• 6,149

### 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 ...
• 3,283

### 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 (...
• 201
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### 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 $a_1, a_2, x_1, x_2$.
• 3,028
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 ...
• 2,535

### Accurate Polynomial Evaluation in Floating Point

The stability of Horner can be improved by subtracting a constant from each $x$ in the Horner form. This is described in "Stable Evaluation of Polynomials" by C. Mesztenyi and C. Witzgall, ...

• 11.4k
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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. ...
• 450

• 11.4k
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### Polynomial Interpolation with Matlab polyfit

According with the documentation polyfit is a function that it can make an approximation in the sense of least square. For this reason you can choose the last input parameter, i.e. the degree. ...
• 1,340
Let me answer you using a more general concept. We seek a space of functions (not only polynomials) in which a function may be described. This space of functions $V$ is defined by a basis. The number ...