Texas Instruments TI-89 Tip List page 275

We find the function values at the solutions with
which returns (approximately)
These values may not seem 'close' to zero, still, they may be the best roots as defined above. The first
step is to test each root. Command-line and TI Basic calculations on the TI-89 / TI-92 Plus are
performed with a 14-digit mantissa, so we need to find the value of e for each root. If a root is
expressed in scientific notation as a.bEc, where a.b is the mantissa, a is not equal to 0 and c is the
exponent, then e = 10^(c-13). For example, for the first root of -4.217E17, e = 10^(17 - 13) = 1E4. The
table below shows the verification results for all three roots.
Since the signs of y1(x-e) and y1(x+e) are different for the first two roots, they are at least plausible.
However, the third root of zero fails the test. To verify that zero is not a root, we plot the function over
the range of x = -200 to x = 200:
The graph shows the root at about x = -141.8, but x = 0 is clearly not a root.
The sign test is automated with the following function proot_t1(). The arguments are the list of
polynomial coefficients and a list of at least one root, and it returns a list of boolean true and false
results: true if the corresponding root passes the sign test, and false if it fails.
proot_t1(c,r)
Func
©(coefList,rootList)
©Sign test for polynomial roots
©Calls math\mantexp()
©14jun02/dburkett@infinet.com
local i,o,e
{}→o
for i,1,dim(r)
10^(math\mantexp(r[i])[2]-13)→e
© Test passes if f(x)=0, or f(x-e), f(x+e) have opposite sign or are zero
polyeval(c,r[i])=0 or polyeval(c,r[i]-e)*polyeval(c,r[i]+e)≤0→o[i]
endfor
return o
EndFunc
root y1(x-e)
-4.217E17 -1.610E39
-141.8196 9643 465 7E9
0 -6.494E22
© Initialize result list
© Loop to test each root
© Find least significant digit
y1({-4.217E17, -141.81969643465, 0})
{1.67879E38, 1E9, -6.494E22}
y1(x+e)
1.946E39
-3E9
-6.494E22
6 - 117