DirectSolver yields incorrect result for random and close to random matrices #56
Description
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