[VA] SRC #012b - Then and Now: Root

[J-F Garnier]

Thanks Werner and C.Ret to put me back on the right track, so we are looking for complex roots.
But I didn't completely loose my time, just see:

(11-09-2022 09:02 AM)Werner Wrote:  There are indeed no real roots.

(11-09-2022 10:19 AM)C.Ret Wrote:  But I suddenly discover what already discover Werner and J.-F. Garnier; polynomials with only positive coefficients and even degree have no real root.

This is wrong! There are (at least) two real roots !
I used brute force, using HTBasic (1999 version) which is a HP BASIC compatible programming environment on PC, that still runs fine on my W10 computer in 2022.
I know this is slightly outside the rules, but not so much, HTBasic is to the HP-71B what Free42 is to the HP-42S: a native (not emulated) compatible language running on PC, with the notable difference is that HTBasic is not free, actually quite expensive being a professional tool.
So the below program, that searches where the polynomial sign changes, could in principle, be run on a HP-71B:

100 ! RE-STORE "src12b"
105 ! SRC12B, MoHPC, 8nov2022
110 ! quite 'force brut' approach
115 !
120 OPTION BASE 0
125 DIM P(10000)
130 !
135 N=10000
140 READ P(*) ! READ P() on the 71B
145 !
150 ! find out where the sign changes:
155 Y1=2 ! polynomial value at X=0
160 FOR X=-.01 TO -1.01 STEP -.001
165   GOSUB Evalpoly
170   IF SGN(Y)<>SGN(Y1) THEN
175      PRINT X1,Y1 ! print the interval where sign changes
180      PRINT X,Y
185      PRINT "------"
190   END IF
195   X1=X ! save X,Y for next iteration
200   Y1=Y
205 NEXT X
210 STOP
215 !
225 Evalpoly: ! evaluate polynomial at X, result in Y
230 Y=0
235 FOR I=0 TO N
240   IF (I*LGT(-X))>-300 THEN Y=Y+X^I*P(I)
245 NEXT I
250 RETURN
255 !
290 !
295 DATA 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29
300 DATA 31 , 37 , 41 , 43 , 47 , 53 , 59 , 61 , 67 , 71
305 DATA 73 , 79 , 83 , 89 , 97 , 101 , 103 , 107 , 109 , 113
...
5290 DATA 104677 , 104681 , 104683 , 104693 , 104701 , 104707 , 104711 , 104717 , 104723 , 104729
5295 DATA 104743
5300 END

X, polynomial value:
-.996 .9768...
-.997 -5.9593...
------
-.999 -39.8287...
-1 52726
------

So there is a real root between -.996 and -.997 and another one between -.999 and -1, in accordance to my guess that real roots (if existing) would be close to -1.

Now, I will go back to the actual question, using the 71B or a HP calculator (promised!), but I thought this unexpected result was worth to be reported here.

J-F