.. Welcome in the year 1880 ..

Leonhard Bairstow (1880 - 1963), who later became an English aeronautical engineer. He played a crucial role in investigating the crash of the British airship R101 in turbulent weather on October 5, 1930, in France, shortly after 2 a.m. near Beauvais, approximately 100 km north of Paris. At that time, the R101 was the largest flying craft, measuring over 220 meters in length. Forty-eight people died, which was more than in the famous Hindenburg disaster in 1937.
Bairstow helped identify the structural failure that led to the crash, and he was awarded a knighthood by King George VI. However, the king died before the ceremony could take place, making Bairstow one of the first individuals to be knighted by Queen Elizabeth II (in 1952).
Additionally, Leonhard Bairstow became also renowned for developing a numerical method for finding the roots of polynomials with real coefficients (as presented below). Bairstow published 1920 the book "Applied aerodynamics" (2^nd enlarged edition in 1939). Famous was the dispute (regarding the effects of viscosity) between Bairstow and the German fluid dynamicist and aerospace scientist, Ludwig Prandtl (1875 - 1953).

For any real polynomial P(x) of grade > 1 holds:

a₀xⁿ + a₁x^n-1 + ... + a_n-1x + a_n = (x² - px - q)(b₀x^n-2 + b₁x^n-3 + ... + b_n-3x + b_n-2) + b_n-1(x - p) + b_n

If that linear remainder polynomial vanishes, we have found a quadratic factor. A simple compair of coefficients we find without difficulties:

b₀ = a₀
b₁ = a₁ + pb₀
b_k = a_k + pb_k-1 + qb_k-2

Assuming θ and ξ are solutions for f(x,y) = 0 and g(x,y) = 0. So, θ = x_k + ζ_k and ξ = y_k + η_k and where ζ_k,η_k are the "corrections", thats where Newton steps in by linearizing:

f(θ,ξ) = f(x_k + ζ_k,y_k + η_k) ~ f(x_k,y_k) + f_xζ_k + f_yη_k
g(θ,ξ) = g(x_k + ζ_k,y_k + η_k) ~ g(x_k,y_k) + g_xζ_k + g_yη_k

Let's define:

Λ_p := ∂/∂p, and c_k-1 := Λ_p(b_k)
Λ_q := ∂/∂q, and c_k-2 := Λ_q(b_k)

leading to:

c_k = b_k + pc_k-1 + qc_k-2, c_-2 = 0, c_-1 = 0, c₀ = b₀, c₁ = pc₀ + b₁

fu:{x1,y+((z 0)*x1:x 1)+(z 1)*x 0}[;;p,q];
b:(b0;b1:(p*b0)+a 1);
/b(k)=a(k)+pb(k-1)+qb(k-2);
bk:b,last each b fu1\2 _ a;

/c(k)=b(k)+pc(k-1)+qc(k-2);
c:(b0,b1+p*b0);
ck:c,last each c fu\2_bk

example:

P(x) = 48x⁵ - 746x⁴ - 1019x³ - 695x² + 9881x + 14212

1^st run: p = -2, q = -3 (start values)
After 7 iterations. p will converge to -7/3 and q against -17/3, the quotient polynomial is: 8x³ - 858x² + 711x + 2508

2^nd run: p = 11, q = 4 (start values)
After 10 iterations. p will converge to 31/2 and q agaist 22, the quatient polynomial is: 48x - 114

So, P(x) = (8x - 19)(3x² + 7x + 17)(2x² - 31x - 44)

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