Connecting Quadratics, Line Segments, Continued Fractions, and Matrices

  • Connecting Quadratics, Line Segments, Continued Fractions, and Matrices

    Lee Melvin M. Peralta
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    Joseph Rizcallah - 10/15/2019 3:37:57 PM

    Upon reading the article one gets the impression that the author has aborted the task he set out to accomplish. Namely, by the logic of the exposition the author had to show (or at least convince the reader) that the quotient p_n/q_n converges to a root (the positive one) of the original equation, but the matrix approach he followed forced him to abandon his goal in order to evade the necessary use of similarity transformation. Instead, the author was contented with the statement that the secular equation of the eigenvalue problem is identical to the original quadratic equation. This does not seem to be a good connection to warrant the effort in the first place. Incidentally, the fact that the eigenvalues are real shows that the matrix A involves no rotation, contrary to the author’s claim on page 548.

    A more feasible approach is provided by recursive sequences. In the author’s notation, the numerator p_n and denominator q­n satisfy the recurrent relations p_n = 5p_(n-1) +14q_(n-1) and q_n = p_(n-1), which can be combined into one linear recursion (akin to the Fibonacci’s): pn = 5p_(n-1) +14p_(n-2). Now one proceeds as follows. Rewriting this relation twice as

    p_n + 2p_(n-1)=7(p_(n-1) + 2p_(n-2)) and p_n - 7p_(n-1) = -2(p_(n-1) -7p_(n-2)),

    and introducing a_n = p_n + 2p_(n-1) and b_n = p_n - 7p_(n-1) one easily sees that a_n and b_n are geometric sequences given by a_n =(7^n)(p_1 + 2p_0) = (7^n)49 and bn =((-2)^n) (p_1 - 7p_0) = ((-2)^n)4. Solving for p_n in terms of a_n and b_n one finds

    p_n = (7a_n + 2b_n)/9 = (7^(n+1)49 - ((-2)^(n+1)4)/9.

    Now the sought for quotient is p_n/q_n = p_n/p_(n-1) = (7^(n+1)49 - (-2)^(n+1)4)/ ((7^n)49 - ((-2)^n)4) whose limit as n tends to infinity is 7 (the other root -2 is obtained as n tends to - infinity). To emphasize the connection with the given quadratic equation, note that both a_n and b_n are linear combinations of the form p_n - lambda p_(n-1), with lambda = -2 for the former and lambda = 7 for the latter, i.e. the roots of the quadratic equation. The reader can readily generalize this procedure to a generic quadratic equation and its associated linear recursion.