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Results tagged with rootfinding
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user 11260
Algorithms to approximate numerically a root of a nonlinear equation or system: for instance, Newton's method, secant method, bisection, etc.
1
vote
Accepted
How to solve for $a$ in $\sum_{j=i}^n (a -j) \binom{n}{j} y^j (1-y)^{n-j} = 0$
$$\sum_{j=i}^n (a -j) \binom{n}{j} y^j (1-y)^{n-j}
=0$$
$$\Rightarrow a=i-\frac{y \binom{n}{i+1} \, _2F_1\left(2,i-n+1;i+2;\frac{y}{y-1}\right)}{(y-1) \binom{n}{i} \, _2F_1\left(1,i-n;i+1;\frac{y}{y-1 …
- 188k
14
votes
Accepted
Solution to sixth order equation
For $x,y>0$ there is a unique solution $y(x)$ to $x = \frac{1}{3}y(y+1)(2y+1)^2(2y^2+2y+1)$ given by
$$y=\tfrac{1}{2} 3^{-1/3} \sqrt{\frac{\left(\sqrt{11664 x^2-3}+108 x\right)^{2/3}+3^{1/3}}{\bigl(\s …
- 188k
1
vote
Searching the roots of a self-consistent transcendental equation
There does not seem to be something as simple as a single cutoff point $T_c$. For any given real $T$, there is an odd number of real solutions $M_1,M_2,M_3,\dots M_{2p+1}$. There are critical $T$'s wh …
- 188k