I'd like to submit you the program attached (Hp42_11_EV.raw), which attempts to find all real eigenvalues of a real (square) matrix, using SOLVE & doing automatically all the necessary guesses.

It's based on a personal algorithm which could also be used for finding roots of a function: it "scans" intervals (Guess=middle of interval, then SOLVE) and uses a stack (feeded with intervals to scan later (PUSH) or discarded if to scan now (POP))

1) Input the initial interval [L,U] (which should contain all roots)  (For the matrix eigenvalues, I used [-FNRM,FNRM] automatically )
2) Input the minimum difference in X ("Diff X") (in order to reject a proposed root (not enough different) or to consider that an interval is too little to explore, if difference is below or equal)
    (e.g. 1E-03)
3) Input the maximum difference in f(X) ("Diff Y")  (to detect an extremum OR to round f(X) to zero, more quickly)
    (e.g. 1E-15)
    if ABS(f(X))<=DiffY => return f(X)=0
    if ABS(f(X)-old f(X))<=DiffY => Set Flag 1 and return f(X)=0  (Extremum detected)
    Set Old f(X) = f(X)

4) If Interval length (U-L) <= DiffX => too little interval => POP other interval =>Goto 4)   or STOP if Stack empty
5) Solve, i.e.: Scan Interval  using Guess G=(L+U)/2   (Result X OK if return code 0, (if <>0=> POP other interval
    => Goto 4  or STOP if Stack empty))
6) if Result X >=(U-DiffX) => Upper half  [G,U] is empty => Scan Lower Half  [L,G]   => Goto 4)
    if Result X<=(L+DiffX) => Lower half  [L,G] is empty => Scan Upper Half  [G,U]  => Goto 4)
    if Result X>(L+DiffX) AND X<(U-DiffX)
    =>   Root found ! (Display value if not extremum; if extremum (Flag 1 set) => treat it as a root, without displaying it)
            if Result X>G =>  Upper part [G,X] is empty, PUSH  Lower half [L,G] and Scan now Upper part [X,U] => Goto 4)
            if Result X<=G => Lower part [X,G] is empty, PUSH  Lower part [L,X] and Scan now Upper Half [G,U] => Goto 4)

- The stack begins at Register R03 and uses R01 as stack pointer:
e.g. R01=8 => there are 3 intervals in the stack:  [L1,U1], [L2,U2], [L3,U3] => R03=L1  R04=U1  R05=L2  R06=U2  R07=L3  R08=U3
if POP interval => L=R07 (L3), U=R08 (U3),  Set R01=6  (8-2)
if PUSH a 4th interval [L4, U4] => R09=L4  R10=U4, Set R01=10  (8+2)
- The automatic extremum detection furnished by SOLVE (return code 2) caused problems sometimes (Stuck => I did my own (see above, using Flag 1))
- I had to round f(X) to 0 more quickly than SOLVE (otherwise was stuck sometimes)

Pierre GILLET (pierregillet AT swing DOT be)