.. Welcome in the year 1792 for time-passengers ..
Nikolai Ivanovich Lobachevsky (1792 - 1856);
Lobachevsky was a russian mathematician and the founder of the non-Euclidean geometry. He developed it independently from Karl Friedrich Gauss(1777 - 1855) and Janos Bolyai(Hungarian mathematician, 1802 - 1860). Lobachevsky's geometry represented a crucial framework for Albert Einstein's curved space-time model. Lobachevsky was a person of strong character and, despite Tsar Alexander I's (1777 - 1825) distrust of modern science and philosophy - which led to a decline in academic standards - Lobachevsky remained unaffected and continued to make significant scientific contributions. In 1834, Nikolai Lobachevsky developed a method to approximate the roots of algebraic equations. This technique was independently discovered in 1837 by the German mathematician Karl Heinrich Graeffe (1799 - 1873) and the French mathematician Germinal Pierre Dandelin (1794 - 1847). Today, it is regarded as a very useful numerical method:
Issue point is the existence of a dominant root x1 in P(x) = xn + a1xn-1 + a2xn-2 + ... + an = 0. All ak are real. Let's assume xj are the roots, and let be |xk| < |xk-1| ∀ k and so ∑ xkn → x1n
For practical reasons we take for n powers of 2
sample 1:
4x3 + 131x2 - 1820x - 11172 = 0
n:3;p:4 131 -1820 -11172.
Then:
(signum (*)r)*(*){x xexp 1%2 xexp n} abs r:n {a1:x 0;a2:x 1;a3:x 2;:0f+(((a2*2)-a1*a1),((a2*a2)-2*a1*a3),(neg a3*a3))}/1_p%(*)p
Then:
(signum (*)r)*(*){x xexp 1%2 xexp n} abs r:n {a1:x 0;a2:x 1;a3:x 2;:0f+(((a2*2)-a1*a1),((a2*a2)-2*a1*a3),(neg a3*a3))}/1_p%(*)p
will return -42.0008, the other 2 roots will be then obtained directly: 14.00004 and -4.749896
The underlying polynomial was (rounded):
t3 - 9.68 × 1012t2 + 1.43 × 1022t - 3.70 × 1027t
For n=4 we would have obtained the exact solutions: -42, 14 and -4.75
NOTE:
It is possible to apply this concept, to some extent, also in the case of complex roots. However, additional steps are required, one of which is to compare the absolute value of the complex roots with that of the real roots to determine whether it is larger or smaller. This method has the significant advantage that it provides all solutions simultaneously - whether approximated or exact. Once we have an approximation, we can employ specific methods to refine it. In our example, increasing n led to a more accurate solution, but this is not generally the case.
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