QUAD solver missing solutions for specific hyperbolic equation

[wasdmco]

I'm trying to solve 26x^2-91xy-26y^2-58x-59y-71=0
The solver (directly running the C version or running through webassembly) returns

x0 = -2
y0 = -1

with 2 recurrence equation and 2 sets of parameter:

Recursive solutions:

xn+1 = P ⁢xn + Q ⁢yn + K
yn+1 = R ⁢xn + S ⁢yn + L

where:

P = -20 959202 739850 883948 245647 530545 (32 digits)
Q = -5 566018 877141 698132 634154 763072 (31 digits)
K = -8 718342 976376 307924 004248 001090 (31 digits)
R = -5 566018 877141 698132 634154 763072 (31 digits)
S = -1 478136 669854 940484 026105 859793 (31 digits)
L = -2 315281 844744 993080 179339 614324 (31 digits)

and also:

P = -1 478136 669854 940484 026105 859793 (31 digits)
Q = 5 566018 877141 698132 634154 763072 (31 digits)
K = 3 912238 321752 930146 204026 268958 (31 digits)
R = 5 566018 877141 698132 634154 763072 (31 digits)
S = -20 959202 739850 883948 245647 530545 (32 digits)
L = -14 731785 493753 396606 824761 778100 (32 digits)

When using a different solver, I get the following results which are not found here:

x = 4611818748
y = 1224735049

and also:

x = -487959346597421219678
y = 1837442362285426324051

I tried using older version of the quad.c based solver and as far as I can see, none are finding it.
I did try with the java solver, and that one seems to find them.

I've forked your repo and created a branch where I have an html file with the output of running both quad.c and quad.java (with teach/step enabled):

• https://raw.githack.com/sdemarre/calculators/refs/heads/experiments-to-debug-quad-solver-issue/solve_c.html
• https://raw.githack.com/sdemarre/calculators/refs/heads/experiments-to-debug-quad-solver-issue/OldApplets/solve_java.html

According to this, it looks like the java solver does find the second solution, but apparently not the third.

[alpertron]

Thanks for sending me this error. I will analyze the code.

[alpertron]

Notice that your second solution is not independent from your first one as shown by the following PARI-GP session:

(22:12) gp > x = 4611818748
%256 = 4611818748
(22:12) gp > y = 1224735049
%257 = 1224735049
(22:12) gp > P = -1 478136 669854 940484 026105 859793
%258 = -1478136669854940484026105859793
(22:12) gp > Q = 5 566018 877141 698132 634154 763072
%259 = 5566018877141698132634154763072
(22:12) gp > K = 3 912238 321752 930146 204026 268958
%260 = 3912238321752930146204026268958
(22:12) gp > R = 5 566018 877141 698132 634154 763072
%261 = 5566018877141698132634154763072
(22:12) gp > S = -20 959202 739850 883948 245647 530545
%262 = -20959202739850883948245647530545
(22:12) gp > L = -14 731785 493753 396606 824761 778100
%263 = -14731785493753396606824761778100
(22:12) gp > P*x+Q*y+K
%264 = -487959346597421219678
(22:13) gp > R*x+S*y+L
%265 = 1837442362285426324051

Anyway, one of these solutions should be displayed by the application.