To summarize, the computed bounds cause the sequence of nSolve steps to take a somewhat longer time
to settle on a "candidate" solution that is produced by roundoff noise due to catastrophic cancelation. While
this "candidate" solution is not the best available, it is a good approximation with a very small relative error.
Given the catastrophic cancelation and the fact that the slope of the curve is extremely small in the
neighborhood of the solution, the reported "candidate" solution is a very good result despite the
conservative "Questionable Accuracy" warning. Moreover, the computing time is quite modest for such an
example."
I followed TI's advice and plotted the function, which looks like this:
As TI wrote and this plot shows, the difference between vfts120(x) and vfts120(100.17) is not a smooth
curve. The 'catastrophic cancellation' to which TI refers is also called destructive cancellation, and
refers to computation errors that result from subtracting two nearly equal numbers. This results in a
loss of significant digits during the calculation.
Note also that the y-scale for this plot is very small, ±4E-13, so the plot shows effects that are not
usually evident with the display resolution of 12 significant digits.
In summary, be aware that nSolve() may return the "Questionable Accuracy" warning even for
solutions that are fairly good. And, in situations like this, nSolve() will take slightly longer to find the
solution.
[11.10] solve() may return false solutions for trigonometric expressions
Executing the solve() function as shown:
solve(tan(x-1)/(x-1)=0,x)
returns
x = @n1*Π+ 1
The 89/92+ use the notation @n1 to indicate an arbitrary integer, which usually includes 0. In this
case, x = 1 is not a solution, since the expression is undefined at x = 1.
solve() does not even return the correct result for the limit as x approaches 1, since
lim
x
d
1
tan(x−1)
x−1
= 1
This effect is also shown if tan() is replaced with sin().
11 - 10
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