DirectSolver yields incorrect result for random and close to random matrices

I was considering swapping cupy sparse solvers for nvmath Python/cuDSS DirectSolver, but I am unable to obtain correct results even for small problem instances.

Minimal example

In example01_cupy.py replace

n = 8
...
a += sp.diags([2.0] * n, format="csr", dtype="float64")

with

n = 100
...
# a += sp.diags([2.0] * n, format="csr", dtype="float64")

The same remains true, if I used small, but not tiny values for the diagonal like 0.01.

Example observation 1

||A||: 	40.65133430135082
||b||: 	14.142135623730951

det(A): 7.262125991606542e+28

02-10 17:35:07 userlogger   INFO     = SPECIFICATION PHASE =
02-10 17:35:07 userlogger   INFO     The LHS package is cupyx.
02-10 17:35:07 userlogger   INFO     The RHS package is cupy.
02-10 17:35:07 userlogger   INFO     The device_id=0, dtype = float64, index type = int32.
02-10 17:35:07 userlogger   INFO     The number of equations = 100.
02-10 17:35:07 userlogger   INFO     The operands' memory space is cuda, and the execution space is on device 0.
02-10 17:35:07 userlogger   INFO     The specified stream for the DirectSolver ctor is <cuda.core.experimental._stream.Stream object at 0x14b6709b87c0>.
02-10 17:35:07 userlogger   INFO     The library handle has been created: 94793135484384.
02-10 17:35:07 userlogger   INFO     The sparse direct solver operation has been created.
02-10 17:35:07 userlogger   INFO     Starting solver phase ANALYSIS...
02-10 17:35:07 userlogger   INFO     This call is non-blocking and will return immediately after the operation is launched on the device.
02-10 17:35:07 userlogger   INFO     Starting solver phase FACTORIZATION...
02-10 17:35:07 userlogger   INFO     This call is non-blocking and will return immediately after the operation is launched on the device.
02-10 17:35:07 userlogger   INFO     Starting solver phase SOLVE...
02-10 17:35:07 userlogger   INFO     This call is non-blocking and will return immediately after the operation is launched on the device.
02-10 17:35:07 userlogger   INFO     The DirectSolver object's resources have been released.

 CuDSS:
||x||: 		29391823863.993862
||Ax - b||: 	92575181911.9934

CuPy:
||x||: 		10.647411227999223
||Ax - b||: 	4.289313285599112e-14

NumPy:
||x||: 		10.647411227999223
||Ax - b||: 	4.29904389535207e-14

Note that I computed the determinant to check whether A is invertible.

Example observation 2

Vice versa I made the observation that DirectSolver sometimes returns not nan even though the determinant is 0. Note that here b = cp.ones((n, 1), order="F") was only a vector

||A||: 	5.6922584694712395
||b||: 	10.0

det(A): 0.0

02-10 17:30:48 userlogger   INFO     = SPECIFICATION PHASE =
02-10 17:30:48 userlogger   INFO     The LHS package is cupyx.
02-10 17:30:48 userlogger   INFO     The RHS package is cupy.
02-10 17:30:48 userlogger   INFO     The device_id=0, dtype = float64, index type = int32.
02-10 17:30:48 userlogger   INFO     The number of equations = 100.
02-10 17:30:48 userlogger   INFO     The operands' memory space is cuda, and the execution space is on device 0.
02-10 17:30:48 userlogger   INFO     The specified stream for the DirectSolver ctor is <cuda.core.experimental._stream.Stream object at 0x14c6b6504580>.
02-10 17:30:48 userlogger   INFO     The library handle has been created: 94064601263200.
02-10 17:30:48 userlogger   INFO     The sparse direct solver operation has been created.
02-10 17:30:48 userlogger   INFO     Starting solver phase ANALYSIS...
02-10 17:30:48 userlogger   INFO     This call is non-blocking and will return immediately after the operation is launched on the device.
02-10 17:30:48 userlogger   INFO     Starting solver phase FACTORIZATION...
02-10 17:30:48 userlogger   INFO     This call is non-blocking and will return immediately after the operation is launched on the device.
02-10 17:30:48 userlogger   INFO     Starting solver phase SOLVE...
02-10 17:30:48 userlogger   INFO     This call is non-blocking and will return immediately after the operation is launched on the device.
02-10 17:30:48 userlogger   INFO     The DirectSolver object's resources have been released.

 CuDSS:
||x||: 		1.5379474527090538e+102
||Ax - b||: 	1.5379474527090537e+89

CuPy:
||x||: 		nan
||Ax - b||: 	nan

NumPy:
||x||: 		nan
||Ax - b||: 	nan

Inizia con example01_cupy.py e riproduci i casi segnalati variando n e il valore diagonale. Confronta i risultati di DirectSolver con gli output di CuPy e NumPy, incluse le norme e i residui mostrati nell’issue. Il lavoro è completato quando la causa dei risultati errati per le matrici non singolari e singolari è stata identificata e corretta.