.. Welcome in the year 1838 for time-passengers ..
; Thorvald Nicolai Thiele (1838 - 1910)
Thiele was a Danish astronomer and the director of the Copenhagen Observatory (from 1875 until his retirement in 1907). Thiele's father was highly educated and served as the private librarian to King Christian VIII of Denmark. Thorvald Thiele was the co-founder of the Danish Mathematical Society and the founder of a Danish insurance company. He also made significant contributions to numerical analysis (especially interpolation theory) and was an excellent chess player. Thiele was a personality considered to be ahead of his time; it is alleged that he had only a few students, and he was rarely fully understood even by his colleagues. He was engaged in important social issues such as the pension system and disability insurance. Famous is Thiele's differential equation from actuarial mathematics, which expresses the net premium reserve for life insurance at time t:
dV(t)/dt = π + δV - μ(1 - V), and where μ(t) is the mortality rate, δ the interest and π premium per time. Thiele never published his equation himself; it was first published posthumously in his obituary in 1910 by the Danish mathematician Jørgen Pedersen Gram. Jørgen Gram was Thiele's friend and is well known for the Gram-Schmidt orthogonalization process.
Motivation:
Let be:
A(x) = a0xn + a1xn-1 + ... + an and B(x) = b0xm + b1xm-1 + ... + bm
T(x) := A(x)/B(x) and m + n = s
We want to construct such a T(x) where T(xk) = yk and k ∈ {0,1,...,s}, using Thiele's continued fraction method of inverted divided differences
The inverted divided differences;
Θr(xj,xr-1,...,x0) := (xj - xr-1) / [Θr-1(xj,xr-2,...,x0) - Θr-1(xr-1,xr-2,...,x0)] and where j ∈ {r, r+1,...,s} and r ∈ {2,3,...,s}
Q snippets ..
a:InverseDividedDifferences[x;y];
alph:reverse 0f+first each 1_a;
P:(enlist 1f;(b,alph[1]+b*-1f*first xv:0f+reverse -1_x));
fu:{$[1=count x;R:(first x)*y;R:((x 0)*y 0),(sum (+)x*/:1_(,':)y),(last x)*last y];:R}
...
Q:addPolynomials[Idd[alph k+1;last P];fu[(1f,neg xv k);first P]];
P:(last P;Q);
alph:reverse 0f+first each 1_a;
P:(enlist 1f;(b,alph[1]+b*-1f*first xv:0f+reverse -1_x));
fu:{$[1=count x;R:(first x)*y;R:((x 0)*y 0),(sum (+)x*/:1_(,':)y),(last x)*last y];:R}
...
Q:addPolynomials[Idd[alph k+1;last P];fu[(1f,neg xv k);first P]];
P:(last P;Q);
example:
.. unknown function F .. we want F-1(1.45) --> extrapolated inverse
x0 = 0.00 y0 = 0.0000000000
x1 = 0.25 y1 = 0.0151940192
x2 = 0.40 y2 = 0.0590409758
x3 = 1.25 y3 = 0.7913208285
x4 = 1.70 y4 = 1.1299911167
x5 = 2.10 y5 = 1.3263227904
Then F-1 (rounded to 5 decimals) ~ (0.07357x3 - 0.31092x2 - 0.05503x)/(0.10686x3 - 0.14703x2 - 0.15296x - 0.00127)
So F-1(1.45) ~ 2.4669 (the exakt value is 2.490662, the test-function F was ln(1+x3)/(1+x2)1/3
NB: In the construction of these inverted divided differences, you always end up dividing a polynomial of degree c by a polynomial of degree c or c-1.
;Jørgen Pedersen Gram (1850 - 1916)
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