[6.62] Try Asymptotic Expansion For Functions Of Large Arguments - Texas Instruments TI-89 Tip List

No actual contents here, but an exhaustive bibliography which would be even more helpful with a little
annotation. The categories are
Bracketing methods (real roots only). Newton's method. Simultaneous root-finding methods. Graeffe's method.
Integral methods, esp. Lehmer's. Bernoulli's and QD method. Interpolation methods such as secant, Muller's.
Minimization methods. Jenkins-Traub method. Sturm sequences, greatest common divisors, resultants. Stability
questions (Routh-Hurwitz criterion, etc.). Interval methods. Miscellaneous. Lin and Bairstow methods. Methods
involving derivatives higher than first. Complexity, convergence and efficiency questions. Evaluation of
polynomials and derivatives. A priori bounds. Low-order polynomials (special methods). Integer and rational
arithmetic. Special cases such as Bessel polynomials. Vincent's method. Mechanical devices. Acceleration
techniques. Existence questions. Error estimates, deflation, sensitivity, continuity. Roots of random polynomials.
Relation between roots of a polynomial and those of its derivative. Nth roots.
Polynomial sweep at the WWW Interactive Mathematics Server
http://wims.unice.fr/wims/wims.cgi?session=RNC912CC8A.3&+lang=en&+module=tool%2Falgebra%2
Fsweeppoly.en
This is a very illuminating animation of polynomial roots plotted in the complex plane. The polynomial is
expressed parametrically, then the polynomial and its roots are plotted as the parameter varies. It is
fascinating to watch the roots move as the polynomial changes. You can use example polynomials, or
enter your own. Very cool!

[6.62] Try asymptotic expansion for functions of large arguments

This tip shows a function to generate an asymptotic series expansion for a function. As the name
implies, the asymptotic series is useful for evaluating functions at large arguments. The present
example shows that asymptotic expansion can fix a severe round-off error problem.
In June 2002, a post on the TI-89 / TI-92 Plus discussion group reported problems evaluating this
function for large arguments:
f ( x ) = 1 − x + 1 2 x − 1 x +
This function resulted from a tidal force analysis. Plotting the function from x = 100 to x = 1000
confirms there is definitely a problem:
4
6x+3 ( x−1 )( x+1 ) $ln 2x
x+1 −1
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