By W. J. Thron
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Additional resources for Analytic Theory of Continued Fractions II
An extended discussion of these theorems can be found in Hairer et al. . Another topic that was only partially investigated concerns methods for stiﬀ equations. 5). 3, the solution starts near zero, gradually increases, and then levels oﬀ at one. If the value of λ is increased the transition from zero to one occurs much faster, and for large values of λ it looks almost vertical. Such rapid changes in the solution are characteristic of stiﬀ equations, and what this means is that the logistic equation becomes stiﬀer as λ increases.
G) d dt √ y 1+(y )2 + y + y = 0, where y(0) = 0, y (0) = 1. (h) θ1 = −µθ1 + λθ2 and θ2 = λθ1 − µθ2 , where θ1 (0) = 1, θ1 (0) = θ2 (0) = θ2 (0) = 0. (i) yey = t for t ≥ 0. t (j) y(t) = (1 + t2 ) 0 (y(s) − sin(s2 ))ds for t ≥ 0. t (k) y = a(t) − y 3 , where a(t) = 0 e2(t−s) y(s)ds and y(0) = −1. ∞ (l) Calculating the value of y(t) = n=0 2n1n! t2n , for t ≥ 0. 5. 18) involves picking a value of θ satisfying 0 ≤ θ ≤ 1 and then using the diﬀerence equation yj+1 = yj + kθfj+1 + k(1 − θ)fj . (a) What method is obtained if θ = 0, or θ = 21 , or θ = 1?
Before doing this, however, there is another useful bit of information from linear algebra related to the eigenvalues of a tridiagonal matrix. For this it is assumed that the entries of A are constant, that is, it has the form ⎛ ⎞ a c ⎜b a c 0 ⎟ ⎜ ⎟ ⎜ ⎟ b a c ⎜ ⎟ A=⎜ . 23) . . .. 15), for smaller values of h the coeﬃcients satisfy bc > 0. 28) √ λi = a + 2 bc cos iπ N +1 , for i = 1, 2, . . , N. 24) The usefulness of this result comes from the fact that the condition number cond2 (A) of the matrix, using the Euclidean norm, is the ratio of the largest 52 2 Two-Point Boundary Value Problems and smallest eigenvalues (in absolute value).
Analytic Theory of Continued Fractions II by W. J. Thron