[petsc-users] Seeking help for analyzing non-convergence (KSP)

Mon Nov 7 06:13:23 CST 2016

On Mon, Nov 7, 2016 at 3:53 AM, Florian Lindner <mailinglists at xgm.de> wrote:

> Hey Barry!
>
> Am 04.11.2016 um 18:42 schrieb Barry Smith:
> >
> >   Try running on one process with the SuperLU direct solver (note
> superlu NOT superlu_dist). You need to ./configure PETSc with
> --download-superlu then run your code with -pc_type lu
> -pc_factor_mat_solver_package superlu
>
> I am having some problems making superlu work, when configuring with
> --download-superlu.
>
> Firstly, I was getting "libsuperlu.so.5: cannot open shared object file:
> No such file or directory" errors.
> They vanished after adding petsc/arch-linux2-c-debug/lib to
> LD_LIBRARY_PATH.
>
> But my application (C and python) still do not find the solver:
>
> Traceback (most recent call last):
>   File "RBF_Load.py", line 46, in <module>
>     main()
>   File "RBF_Load.py", line 38, in main
>     solver.solve(b, x)
>   File "PETSc/KSP.pyx", line 384, in petsc4py.PETSc.KSP.solve
> (src/petsc4py.PETSc.c:154741)
> petsc4py.PETSc.Error: error code 92
> [0] KSPSolve() line 600 in /data/scratch/lindnefn/
> software/petsc/src/ksp/ksp/interface/itfunc.c
> [0] KSPSetUpOnBlocks() line 220 in /data/scratch/lindnefn/
> software/petsc/src/ksp/ksp/interface/itfunc.c
> [0] PCSetUpOnBlocks() line 1001 in /data/scratch/lindnefn/
> software/petsc/src/ksp/pc/interface/precon.c
> [0] PCSetUpOnBlocks_ASM() line 428 in /data/scratch/lindnefn/
> software/petsc/src/ksp/pc/impls/asm/asm.c
> [0] KSPSetUp() line 390 in /data/scratch/lindnefn/
> software/petsc/src/ksp/ksp/interface/itfunc.c
> [0] PCSetUp() line 968 in /data/scratch/lindnefn/
> software/petsc/src/ksp/pc/interface/precon.c
> [0] PCSetUp_LU() line 134 in /data/scratch/lindnefn/
> software/petsc/src/ksp/pc/impls/factor/lu/lu.c
> [0] MatGetFactor() line 4244 in /data/scratch/lindnefn/
> software/petsc/src/mat/interface/matrix.c
> [0] See http://www.mcs.anl.gov/petsc/documentation/linearsolvertable.html
> for possible LU and Cholesky solvers
> [0] Could not locate a solver package. Perhaps you must ./configure with
> --download-<package>
>

It sounds like you have more than one PETSC_ARCH, or more than one
PETSC_DIR, and they are getting mixed up.
You can see what module gets loaded for petsc4py, and then run ldd on it to
see what PETSc libraries it links.

   Matt

> I configured with
>
> python2 configure --with-debugging=1 --download-petsc4py --download-mpi4py
> --download-superlu
>
> on a recent checkout of the maint branch.
>
> make PETSC_DIR=/data/scratch/lindnefn/software/petsc
> PETSC_ARCH=arch-linux2-c-debug test reported no errors.
>
> Best Thanks,
> Florian
>
> >
> >    Barry
> >
> >> On Nov 4, 2016, at 10:44 AM, Florian Lindner <mailinglists at xgm.de>
> wrote:
> >>
> >> Hello,
> >>
> >> I have a matrix C that is the result of an RBF interpolation.
> >>
> >> It is constructured like that:
> >>
> >> c_ij = phi( |x_i - x_j| )
> >>
> >> x are supporting points. phi is the radial basis functions, here it is
> a Gaussian: phi(r) = exp( -(1.5*r)^2 ).
> >>
> >> The system is augmented by a global polynomial, which results in the
> first 3 rows being dense.
> >>
> >> The matrix is symmetric and of size 2309. It is also sparse since the
> basis functions are decaying very rapidly. A
> >> picture of the sparsity pattern I have uploaded at [1]
> >>
> >> I am having big trouble solving the system.
> >>
> >> Using default settings on 10 procs I'm not achieving convergence after
> 40k iterations:
> >>
> >> 40000 KSP preconditioned resid norm 2.525546749843e+01 true resid norm
> 5.470906445602e+01 ||r(i)||/||b|| 3.122197420093e-02
> >>
> >> I have uploaded a small python script using petsc4py that simply loads
> the matrix and the rhs and tries to solve it. It
> >> is bundled with the matrix and rhs and available at [2] (650 KB). Some
> paths in the Python source need to be adapted.
> >>
> >> The condition number using -pc_type svd -pc_svd_monitor did not work,
> due to incompatible matrix formats.
> >>
> >> The condition number estimate using -pc_type none -ksp_type gmres
> -ksp_monitor_singular_value -ksp_gmres_restart 1000
> >> has been running for over half an hour now, its current line is:
> >>
> >> 5655 KSP preconditioned resid norm 9.481821216790e-02 true resid norm
> 9.481821212512e-02 ||r(i)||/||b|| 5.411190635748e-05
> >> 5655 KSP Residual norm 9.481821216790e-02 % max 2.853532248325e+02 min
> 1.073886912921e-06 max/min 2.657199947212e+08
> >>
> >> The Eigenvalues, computed with -ksp_compute_eigenvalues
> -ksp_gmres_restart 1000 -pc_type none did not converge after 40k
> >> iterations, bit they eigenvalues were printed. They are all real, none
> is zero, but some are pretty close. I've pasted
> >> them at the end of the mail.
> >>
> >> I tried a bunch of different PCs, but I'm somehow helpless on how to
> analyse that problem more systematically.
> >>
> >> I know that RBF interpolation tends to produce badly conditioned
> matrices, but this seems to be (almost) singular. I
> >> have scrutinized my algorithms and in most cases it works just fine,
> converges and the output results are sane.
> >>
> >> I am grateful for any help!
> >>
> >> Best,
> >> Florian
> >>
> >>
> >> [1] http://xgm.de/upload/sparsity.png
> >> [2] http://xgm.de/upload/RBF.tar.bz2
> >>
> >> Eigenvalues:
> >>
> >> Iteratively computed eigenvalues
> >> -276.515 + 0.i
> >> -276.348 + 0.i
> >> -43.5577 + 0.i
> >> 3.27753e-09 + 0.i
> >> 5.09352e-06 + 0.i
> >> 1.56431e-05 + 0.i
> >> 2.79046e-05 + 0.i
> >> 0.000145571 + 0.i
> >> 0.000184333 + 0.i
> >> 0.000283665 + 0.i
> >> 0.000395128 + 0.i
> >> 0.000488579 + 0.i
> >> 0.000568982 + 0.i
> >> 0.000704054 + 0.i
> >> 0.000864538 + 0.i
> >> 0.00101309 + 0.i
> >> 0.00119913 + 0.i
> >> 0.00142954 + 0.i
> >> 0.00156907 + 0.i
> >> 0.00178931 + 0.i
> >> 0.00200295 + 0.i
> >> 0.00223425 + 0.i
> >> 0.00246876 + 0.i
> >> 0.00276749 + 0.i
> >> 0.00303141 + 0.i
> >> 0.00326647 + 0.i
> >> 0.00346721 + 0.i
> >> 0.00437335 + 0.i
> >> 0.0047525 + 0.i
> >> 0.0049999 + 0.i
> >> 0.0051701 + 0.i
> >> 0.00556462 + 0.i
> >> 0.00605938 + 0.i
> >> 0.00628264 + 0.i
> >> 0.00697463 + 0.i
> >> 0.00756472 + 0.i
> >> 0.00783031 + 0.i
> >> 0.00822392 + 0.i
> >> 0.00868428 + 0.i
> >> 0.00928107 + 0.i
> >> 0.00954981 + 0.i
> >> 0.0104767 + 0.i
> >> 0.0108791 + 0.i
> >> 0.0111159 + 0.i
> >> 0.0119535 + 0.i
> >> 0.0125402 + 0.i
> >> 0.0131083 + 0.i
> >> 0.0135975 + 0.i
> >> 0.0142472 + 0.i
> >> 0.0148179 + 0.i
> >> 0.0156007 + 0.i
> >> 0.0160567 + 0.i
> >> 0.0169899 + 0.i
> >> 0.0174895 + 0.i
> >> 0.0182683 + 0.i
> >> 0.0189543 + 0.i
> >> 0.0196781 + 0.i
> >> 0.0206413 + 0.i
> >> 0.0212257 + 0.i
> >> 0.0220118 + 0.i
> >> 0.0229829 + 0.i
> >> 0.023606 + 0.i
> >> 0.0244077 + 0.i
> >> 0.0253428 + 0.i
> >> 0.0263089 + 0.i
> >> 0.0269905 + 0.i
> >> 0.0278572 + 0.i
> >> 0.028664 + 0.i
> >> 0.0293435 + 0.i
> >> 0.0306571 + 0.i
> >> 0.0315797 + 0.i
> >> 0.0320843 + 0.i
> >> 0.0334732 + 0.i
> >> 0.0344048 + 0.i
> >> 0.0349148 + 0.i
> >> 0.0361261 + 0.i
> >> 0.037176 + 0.i
> >> 0.0382401 + 0.i
> >> 0.0390156 + 0.i
> >> 0.0404156 + 0.i
> >> 0.0413084 + 0.i
> >> 0.0421763 + 0.i
> >> 0.0434521 + 0.i
> >> 0.0452247 + 0.i
> >> 0.0460417 + 0.i
> >> 0.0467954 + 0.i
> >> 0.0479258 + 0.i
> >> 0.0489899 + 0.i
> >> 0.0506182 + 0.i
> >> 0.0512171 + 0.i
> >> 0.0529938 + 0.i
> >> 0.0537732 + 0.i
> >> 0.0549977 + 0.i
> >> 0.056272 + 0.i
> >> 0.0574546 + 0.i
> >> 0.0585867 + 0.i
> >> 0.0595438 + 0.i
> >> 0.0614507 + 0.i
> >> 0.0625339 + 0.i
> >> 0.0634786 + 0.i
> >> 0.0651556 + 0.i
> >> 0.066393 + 0.i
> >> 0.067585 + 0.i
> >> 0.0701521 + 0.i
> >> 0.0709748 + 0.i
> >> 0.0725257 + 0.i
> >> 0.073676 + 0.i
> >> 0.0739372 + 0.i
> >> 0.0758159 + 0.i
> >> 0.078728 + 0.i
> >> 0.0803127 + 0.i
> >> 0.0813565 + 0.i
> >> 0.0819371 + 0.i
> >> 0.0824692 + 0.i
> >> 0.0847166 + 0.i
> >> 0.0878818 + 0.i
> >> 0.0901822 + 0.i
> >> 0.0906192 + 0.i
> >> 0.0912192 + 0.i
> >> 0.0921386 + 0.i
> >> 0.0964252 + 0.i
> >> 0.0978271 + 0.i
> >> 0.0991256 + 0.i
> >> 0.100197 + 0.i
> >> 0.101757 + 0.i
> >> 0.103163 + 0.i
> >> 0.104427 + 0.i
> >> 0.105316 + 0.i
> >> 0.108255 + 0.i
> >> 0.110149 + 0.i
> >> 0.111963 + 0.i
> >> 0.113533 + 0.i
> >> 0.114857 + 0.i
> >> 0.11807 + 0.i
> >> 0.119241 + 0.i
> >> 0.121196 + 0.i
> >> 0.124063 + 0.i
> >> 0.124824 + 0.i
> >> 0.125867 + 0.i
> >> 0.129153 + 0.i
> >> 0.129692 + 0.i
> >> 0.130804 + 0.i
> >> 0.13186 + 0.i
> >> 0.136671 + 0.i
> >> 0.137159 + 0.i
> >> 0.139948 + 0.i
> >> 0.140338 + 0.i
> >> 0.143836 + 0.i
> >> 0.145398 + 0.i
> >> 0.146868 + 0.i
> >> 0.14813 + 0.i
> >> 0.149113 + 0.i
> >> 0.152292 + 0.i
> >> 0.154569 + 0.i
> >> 0.1592 + 0.i
> >> 0.159868 + 0.i
> >> 0.16113 + 0.i
> >> 0.163854 + 0.i
> >> 0.16612 + 0.i
> >> 0.167395 + 0.i
> >> 0.169089 + 0.i
> >> 0.171137 + 0.i
> >> 0.173693 + 0.i
> >> 0.176847 + 0.i
> >> 0.177794 + 0.i
> >> 0.178951 + 0.i
> >> 0.179342 + 0.i
> >> 0.185936 + 0.i
> >> 0.18772 + 0.i
> >> 0.189272 + 0.i
> >> 0.191293 + 0.i
> >> 0.193085 + 0.i
> >> 0.195212 + 0.i
> >> 0.197571 + 0.i
> >> 0.199457 + 0.i
> >> 0.201105 + 0.i
> >> 0.204097 + 0.i
> >> 0.207915 + 0.i
> >> 0.208748 + 0.i
> >> 0.212964 + 0.i
> >> 0.214298 + 0.i
> >> 0.217664 + 0.i
> >> 0.218679 + 0.i
> >> 0.220846 + 0.i
> >> 0.222277 + 0.i
> >> 0.226707 + 0.i
> >> 0.228239 + 0.i
> >> 0.229806 + 0.i
> >> 0.231871 + 0.i
> >> 0.235329 + 0.i
> >> 0.237857 + 0.i
> >> 0.241059 + 0.i
> >> 0.242546 + 0.i
> >> 0.244337 + 0.i
> >> 0.246314 + 0.i
> >> 0.248005 + 0.i
> >> 0.250223 + 0.i
> >> 0.252628 + 0.i
> >> 0.255812 + 0.i
> >> 0.256945 + 0.i
> >> 0.258131 + 0.i
> >> 0.266436 + 0.i
> >> 0.269103 + 0.i
> >> 0.270135 + 0.i
> >> 0.271536 + 0.i
> >> 0.273719 + 0.i
> >> 0.279941 + 0.i
> >> 0.28122 + 0.i
> >> 0.281853 + 0.i
> >> 0.285513 + 0.i
> >> 0.28672 + 0.i
> >> 0.28773 + 0.i
> >> 0.291966 + 0.i
> >> 0.296344 + 0.i
> >> 0.299661 + 0.i
> >> 0.303254 + 0.i
> >> 0.304844 + 0.i
> >> 0.306891 + 0.i
> >> 0.309339 + 0.i
> >> 0.311826 + 0.i
> >> 0.313826 + 0.i
> >> 0.315959 + 0.i
> >> 0.31859 + 0.i
> >> 0.319327 + 0.i
> >> 0.322979 + 0.i
> >> 0.329764 + 0.i
> >> 0.330846 + 0.i
> >> 0.335162 + 0.i
> >> 0.336212 + 0.i
> >> 0.338197 + 0.i
> >> 0.34279 + 0.i
> >> 0.345071 + 0.i
> >> 0.34884 + 0.i
> >> 0.34954 + 0.i
> >> 0.354234 + 0.i
> >> 0.355576 + 0.i
> >> 0.359178 + 0.i
> >> 0.361957 + 0.i
> >> 0.364873 + 0.i
> >> 0.367973 + 0.i
> >> 0.370043 + 0.i
> >> 0.3732 + 0.i
> >> 0.375047 + 0.i
> >> 0.376219 + 0.i
> >> 0.38884 + 0.i
> >> 0.390239 + 0.i
> >> 0.392451 + 0.i
> >> 0.394641 + 0.i
> >> 0.39603 + 0.i
> >> 0.40011 + 0.i
> >> 0.402863 + 0.i
> >> 0.40477 + 0.i
> >> 0.407166 + 0.i
> >> 0.409253 + 0.i
> >> 0.41114 + 0.i
> >> 0.417392 + 0.i
> >> 0.419475 + 0.i
> >> 0.421597 + 0.i
> >> 0.42226 + 0.i
> >> 0.425458 + 0.i
> >> 0.432817 + 0.i
> >> 0.434498 + 0.i
> >> 0.437448 + 0.i
> >> 0.440073 + 0.i
> >> 0.441784 + 0.i
> >> 0.44471 + 0.i
> >> 0.449445 + 0.i
> >> 0.450416 + 0.i
> >> 0.454865 + 0.i
> >> 0.456899 + 0.i
> >> 0.459119 + 0.i
> >> 0.464975 + 0.i
> >> 0.466739 + 0.i
> >> 0.470461 + 0.i
> >> 0.473138 + 0.i
> >> 0.473366 + 0.i
> >> 0.482849 + 0.i
> >> 0.487335 + 0.i
> >> 0.487996 + 0.i
> >> 0.490971 + 0.i
> >> 0.492149 + 0.i
> >> 0.498317 + 0.i
> >> 0.49924 + 0.i
> >> 0.501193 + 0.i
> >> 0.502928 + 0.i
> >> 0.505282 + 0.i
> >> 0.507326 + 0.i
> >> 0.5094 + 0.i
> >> 0.510887 + 0.i
> >> 0.52977 + 0.i
> >> 0.529973 + 0.i
> >> 0.534443 + 0.i
> >> 0.536762 + 0.i
> >> 0.539341 + 0.i
> >> 0.542418 + 0.i
> >> 0.543103 + 0.i
> >> 0.545815 + 0.i
> >> 0.547044 + 0.i
> >> 0.548941 + 0.i
> >> 0.553764 + 0.i
> >> 0.558755 + 0.i
> >> 0.55932 + 0.i
> >> 0.566552 + 0.i
> >> 0.566763 + 0.i
> >> 0.571683 + 0.i
> >> 0.571809 + 0.i
> >> 0.587266 + 0.i
> >> 0.58805 + 0.i
> >> 0.588659 + 0.i
> >> 0.58981 + 0.i
> >> 0.593885 + 0.i
> >> 0.598992 + 0.i
> >> 0.60122 + 0.i
> >> 0.602449 + 0.i
> >> 0.604899 + 0.i
> >> 0.607548 + 0.i
> >> 0.609311 + 0.i
> >> 0.611774 + 0.i
> >> 0.621952 + 0.i
> >> 0.623849 + 0.i
> >> 0.624084 + 0.i
> >> 0.624863 + 0.i
> >> 0.635872 + 0.i
> >> 0.63847 + 0.i
> >> 0.639046 + 0.i
> >> 0.646496 + 0.i
> >> 0.650915 + 0.i
> >> 0.652575 + 0.i
> >> 0.654514 + 0.i
> >> 0.656047 + 0.i
> >> 0.658775 + 0.i
> >> 0.66205 + 0.i
> >> 0.664198 + 0.i
> >> 0.666337 + 0.i
> >> 0.670879 + 0.i
> >> 0.675989 + 0.i
> >> 0.677792 + 0.i
> >> 0.678938 + 0.i
> >> 0.6845 + 0.i
> >> 0.684931 + 0.i
> >> 0.687205 + 0.i
> >> 0.692041 + 0.i
> >> 0.692177 + 0.i
> >> 0.693601 + 0.i
> >> 0.711189 + 0.i
> >> 0.716514 + 0.i
> >> 0.71922 + 0.i
> >> 0.72388 + 0.i
> >> 0.72405 + 0.i
> >> 0.730072 + 0.i
> >> 0.730319 + 0.i
> >> 0.732633 + 0.i
> >> 0.73299 + 0.i
> >> 0.737779 + 0.i
> >> 0.74204 + 0.i
> >> 0.744661 + 0.i
> >> 0.746251 + 0.i
> >> 0.746947 + 0.i
> >> 0.751408 + 0.i
> >> 0.751786 + 0.i
> >> 0.753859 + 0.i
> >> 0.761784 + 0.i
> >> 0.762905 + 0.i
> >> 0.765961 + 0.i
> >> 0.767479 + 0.i
> >> 0.769471 + 0.i
> >> 0.769529 + 0.i
> >> 0.774108 + 0.i
> >> 0.794173 + 0.i
> >> 0.799031 + 0.i
> >> 0.799111 + 0.i
> >> 0.811928 + 0.i
> >> 0.812351 + 0.i
> >> 0.813483 + 0.i
> >> 0.815524 + 0.i
> >> 0.815749 + 0.i
> >> 0.826079 + 0.i
> >> 0.826832 + 0.i
> >> 0.831323 + 0.i
> >> 0.832443 + 0.i
> >> 0.8415 + 0.i
> >> 0.842963 + 0.i
> >> 0.843567 + 0.i
> >> 0.84377 + 0.i
> >> 0.847571 + 0.i
> >> 0.8493 + 0.i
> >> 0.849347 + 0.i
> >> 0.85882 + 0.i
> >> 0.859196 + 0.i
> >> 0.862321 + 0.i
> >> 0.863661 + 0.i
> >> 0.867476 + 0.i
> >> 0.8677 + 0.i
> >> 0.884245 + 0.i
> >> 0.884266 + 0.i
> >> 0.893719 + 0.i
> >> 0.893886 + 0.i
> >> 0.907204 + 0.i
> >> 0.907405 + 0.i
> >> 0.908615 + 0.i
> >> 0.909567 + 0.i
> >> 0.909714 + 0.i
> >> 0.916938 + 0.i
> >> 0.920716 + 0.i
> >> 0.926377 + 0.i
> >> 0.926776 + 0.i
> >> 0.928557 + 0.i
> >> 0.928572 + 0.i
> >> 0.939078 + 0.i
> >> 0.939392 + 0.i
> >> 0.940033 + 0.i
> >> 0.941661 + 0.i
> >> 0.942081 + 0.i
> >> 0.942419 + 0.i
> >> 0.942522 + 0.i
> >> 0.951621 + 0.i
> >> 0.952499 + 0.i
> >> 0.959928 + 0.i
> >> 0.96003 + 0.i
> >> 0.960158 + 0.i
> >> 0.96049 + 0.i
> >> 0.974053 + 0.i
> >> 0.974116 + 0.i
> >> 0.988145 + 0.i
> >> 0.988274 + 0.i
> >> 0.988988 + 0.i
> >> 0.989061 + 0.i
> >> 0.98914 + 0.i
> >> 0.991796 + 0.i
> >> 0.991902 + 0.i
> >> 0.994804 + 0.i
> >> 0.994863 + 0.i
> >> 1.0045 + 0.i
> >> 1.00453 + 0.i
> >> 1.00987 + 0.i
> >> 1.02901 + 0.i
> >> 1.03109 + 0.i
> >> 1.0311 + 0.i
> >> 1.03117 + 0.i
> >> 1.03133 + 0.i
> >> 1.03275 + 0.i
> >> 1.0329 + 0.i
> >> 1.03517 + 0.i
> >> 1.03527 + 0.i
> >> 1.03931 + 0.i
> >> 1.03945 + 0.i
> >> 1.07673 + 0.i
> >> 1.07703 + 0.i
> >> 1.07789 + 0.i
> >> 1.07789 + 0.i
> >> 1.08547 + 0.i
> >> 1.08549 + 0.i
> >> 1.09246 + 0.i
> >> 1.09269 + 0.i
> >> 1.09886 + 0.i
> >> 1.09886 + 0.i
> >> 1.10179 + 0.i
> >> 1.10188 + 0.i
> >> 1.10626 + 0.i
> >> 1.10653 + 0.i
> >> 1.10814 + 0.i
> >> 1.10819 + 0.i
> >> 1.10901 + 0.i
> >> 1.10911 + 0.i
> >> 1.10923 + 0.i
> >> 1.10954 + 0.i
> >> 1.11153 + 0.i
> >> 1.11368 + 0.i
> >> 1.11412 + 0.i
> >> 1.11424 + 0.i
> >> 1.11729 + 0.i
> >> 1.1174 + 0.i
> >> 1.14549 + 0.i
> >> 1.14715 + 0.i
> >> 1.14717 + 0.i
> >> 1.14717 + 0.i
> >> 1.14746 + 0.i
> >> 1.14758 + 0.i
> >> 1.16574 + 0.i
> >> 1.16597 + 0.i
> >> 1.17815 + 0.i
> >> 1.17845 + 0.i
> >> 1.18206 + 0.i
> >> 1.18208 + 0.i
> >> 1.20113 + 0.i
> >> 1.20116 + 0.i
> >> 1.20318 + 0.i
> >> 1.2033 + 0.i
> >> 1.20947 + 0.i
> >> 1.20955 + 0.i
> >> 1.21223 + 0.i
> >> 1.21596 + 0.i
> >> 1.21596 + 0.i
> >> 1.22727 + 0.i
> >> 1.2273 + 0.i
> >> 1.23049 + 0.i
> >> 1.23086 + 0.i
> >> 1.23474 + 0.i
> >> 1.23636 + 0.i
> >> 1.2396 + 0.i
> >> 1.23962 + 0.i
> >> 1.25889 + 0.i
> >> 1.25919 + 0.i
> >> 1.25948 + 0.i
> >> 1.25961 + 0.i
> >> 1.26135 + 0.i
> >> 1.26146 + 0.i
> >> 1.27463 + 0.i
> >> 1.27486 + 0.i
> >> 1.30179 + 0.i
> >> 1.30211 + 0.i
> >> 1.30257 + 0.i
> >> 1.30268 + 0.i
> >> 1.34742 + 0.i
> >> 1.34745 + 0.i
> >> 1.34827 + 0.i
> >> 1.34845 + 0.i
> >> 1.35084 + 0.i
> >> 1.35085 + 0.i
> >> 1.36396 + 0.i
> >> 1.36397 + 0.i
> >> 1.36462 + 0.i
> >> 1.36466 + 0.i
> >> 1.36931 + 0.i
> >> 1.36973 + 0.i
> >> 1.37666 + 0.i
> >> 1.37668 + 0.i
> >> 1.3813 + 0.i
> >> 1.38144 + 0.i
> >> 1.40369 + 0.i
> >> 1.4043 + 0.i
> >> 1.41065 + 0.i
> >> 1.41108 + 0.i
> >> 1.41788 + 0.i
> >> 1.42358 + 0.i
> >> 1.4236 + 0.i
> >> 1.42376 + 0.i
> >> 1.42376 + 0.i
> >> 1.42417 + 0.i
> >> 1.42649 + 0.i
> >> 1.4265 + 0.i
> >> 1.45085 + 0.i
> >> 1.45403 + 0.i
> >> 1.45407 + 0.i
> >> 1.45408 + 0.i
> >> 1.48153 + 0.i
> >> 1.48178 + 0.i
> >> 1.48257 + 0.i
> >> 1.48285 + 0.i
> >> 1.49974 + 0.i
> >> 1.50306 + 0.i
> >> 1.50322 + 0.i
> >> 1.50531 + 0.i
> >> 1.50533 + 0.i
> >> 1.52388 + 0.i
> >> 1.52389 + 0.i
> >> 1.52647 + 0.i
> >> 1.52651 + 0.i
> >> 1.53664 + 0.i
> >> 1.53809 + 0.i
> >> 1.53809 + 0.i
> >> 1.53813 + 0.i
> >> 1.5383 + 0.i
> >> 1.53846 + 0.i
> >> 1.54051 + 0.i
> >> 1.54722 + 0.i
> >> 1.54724 + 0.i
> >> 1.54724 + 0.i
> >> 1.54841 + 0.i
> >> 1.54889 + 0.i
> >> 1.57043 + 0.i
> >> 1.5721 + 0.i
> >> 1.58658 + 0.i
> >> 1.58678 + 0.i
> >> 1.6039 + 0.i
> >> 1.60416 + 0.i
> >> 1.60877 + 0.i
> >> 1.60896 + 0.i
> >> 1.60942 + 0.i
> >> 1.60954 + 0.i
> >> 1.63187 + 0.i
> >> 1.63223 + 0.i
> >> 1.6468 + 0.i
> >> 1.64683 + 0.i
> >> 1.66346 + 0.i
> >> 1.66348 + 0.i
> >> 1.67861 + 0.i
> >> 1.68121 + 0.i
> >> 1.68151 + 0.i
> >> 1.70128 + 0.i
> >> 1.70146 + 0.i
> >> 1.71076 + 0.i
> >> 1.71132 + 0.i
> >> 1.71362 + 0.i
> >> 1.71366 + 0.i
> >> 1.73403 + 0.i
> >> 1.73465 + 0.i
> >> 1.73479 + 0.i
> >> 1.74324 + 0.i
> >> 1.74327 + 0.i
> >> 1.74354 + 0.i
> >> 1.74354 + 0.i
> >> 1.74419 + 0.i
> >> 1.7508 + 0.i
> >> 1.75082 + 0.i
> >> 1.76674 + 0.i
> >> 1.76702 + 0.i
> >> 1.78801 + 0.i
> >> 1.78817 + 0.i
> >> 1.80169 + 0.i
> >> 1.80507 + 0.i
> >> 1.80564 + 0.i
> >> 1.80578 + 0.i
> >> 1.81023 + 0.i
> >> 1.81038 + 0.i
> >> 1.83173 + 0.i
> >> 1.83187 + 0.i
> >> 1.83281 + 0.i
> >> 1.83288 + 0.i
> >> 1.84212 + 0.i
> >> 1.84215 + 0.i
> >> 1.85367 + 0.i
> >> 1.8537 + 0.i
> >> 1.87126 + 0.i
> >> 1.87299 + 0.i
> >> 1.87655 + 0.i
> >> 1.87656 + 0.i
> >> 1.89401 + 0.i
> >> 1.89404 + 0.i
> >> 1.93557 + 0.i
> >> 1.93584 + 0.i
> >> 1.93787 + 0.i
> >> 1.9379 + 0.i
> >> 1.95212 + 0.i
> >> 1.96311 + 0.i
> >> 1.97136 + 0.i
> >> 1.97138 + 0.i
> >> 1.97138 + 0.i
> >> 1.97192 + 0.i
> >> 1.97218 + 0.i
> >> 1.98881 + 0.i
> >> 1.98909 + 0.i
> >> 2.00743 + 0.i
> >> 2.00744 + 0.i
> >> 2.01614 + 0.i
> >> 2.01641 + 0.i
> >> 2.01714 + 0.i
> >> 2.01731 + 0.i
> >> 2.04198 + 0.i
> >> 2.04202 + 0.i
> >> 2.04887 + 0.i
> >> 2.04913 + 0.i
> >> 2.05054 + 0.i
> >> 2.05063 + 0.i
> >> 2.06708 + 0.i
> >> 2.06709 + 0.i
> >> 2.08633 + 0.i
> >> 2.08636 + 0.i
> >> 2.08982 + 0.i
> >> 2.09 + 0.i
> >> 2.104 + 0.i
> >> 2.10422 + 0.i
> >> 2.15289 + 0.i
> >> 2.15334 + 0.i
> >> 2.16771 + 0.i
> >> 2.16774 + 0.i
> >> 2.23282 + 0.i
> >> 2.233 + 0.i
> >> 2.23696 + 0.i
> >> 2.237 + 0.i
> >> 2.2372 + 0.i
> >> 2.2615 + 0.i
> >> 2.26457 + 0.i
> >> 2.2646 + 0.i
> >> 2.26937 + 0.i
> >> 2.26981 + 0.i
> >> 2.27639 + 0.i
> >> 2.27642 + 0.i
> >> 2.27643 + 0.i
> >> 2.27643 + 0.i
> >> 2.27649 + 0.i
> >> 2.27795 + 0.i
> >> 2.27799 + 0.i
> >> 2.28052 + 0.i
> >> 2.28057 + 0.i
> >> 2.28567 + 0.i
> >> 2.28568 + 0.i
> >> 2.29304 + 0.i
> >> 2.29558 + 0.i
> >> 2.29618 + 0.i
> >> 2.29631 + 0.i
> >> 2.33763 + 0.i
> >> 2.33788 + 0.i
> >> 2.36223 + 0.i
> >> 2.36224 + 0.i
> >> 2.4134 + 0.i
> >> 2.41361 + 0.i
> >> 2.42906 + 0.i
> >> 2.42923 + 0.i
> >> 2.4563 + 0.i
> >> 2.45677 + 0.i
> >> 2.48269 + 0.i
> >> 2.48306 + 0.i
> >> 2.48393 + 0.i
> >> 2.48416 + 0.i
> >> 2.48829 + 0.i
> >> 2.48835 + 0.i
> >> 2.50243 + 0.i
> >> 2.50255 + 0.i
> >> 2.50619 + 0.i
> >> 2.51896 + 0.i
> >> 2.51899 + 0.i
> >> 2.51899 + 0.i
> >> 2.51902 + 0.i
> >> 2.51902 + 0.i
> >> 2.51964 + 0.i
> >> 2.51996 + 0.i
> >> 2.54248 + 0.i
> >> 2.5425 + 0.i
> >> 2.54795 + 0.i
> >> 2.54806 + 0.i
> >> 2.60176 + 0.i
> >> 2.60315 + 0.i
> >> 2.62963 + 0.i
> >> 2.62983 + 0.i
> >> 2.63234 + 0.i
> >> 2.63238 + 0.i
> >> 2.63751 + 0.i
> >> 2.66482 + 0.i
> >> 2.66483 + 0.i
> >> 2.68963 + 0.i
> >> 2.68966 + 0.i
> >> 2.71695 + 0.i
> >> 2.71733 + 0.i
> >> 2.72705 + 0.i
> >> 2.7271 + 0.i
> >> 2.72955 + 0.i
> >> 2.73537 + 0.i
> >> 2.73583 + 0.i
> >> 2.73595 + 0.i
> >> 2.78551 + 0.i
> >> 2.78567 + 0.i
> >> 2.79934 + 0.i
> >> 2.81368 + 0.i
> >> 2.82558 + 0.i
> >> 2.82562 + 0.i
> >> 2.82586 + 0.i
> >> 2.82586 + 0.i
> >> 2.82649 + 0.i
> >> 2.83904 + 0.i
> >> 2.83906 + 0.i
> >> 2.85523 + 0.i
> >> 2.85553 + 0.i
> >> 2.87397 + 0.i
> >> 2.87398 + 0.i
> >> 2.90074 + 0.i
> >> 2.90076 + 0.i
> >> 2.94142 + 0.i
> >> 2.94174 + 0.i
> >> 2.96105 + 0.i
> >> 2.96115 + 0.i
> >> 2.99596 + 0.i
> >> 2.99649 + 0.i
> >> 2.99656 + 0.i
> >> 2.99662 + 0.i
> >> 3.02402 + 0.i
> >> 3.02424 + 0.i
> >> 3.03894 + 0.i
> >> 3.03896 + 0.i
> >> 3.05052 + 0.i
> >> 3.05054 + 0.i
> >> 3.07641 + 0.i
> >> 3.07642 + 0.i
> >> 3.09111 + 0.i
> >> 3.10507 + 0.i
> >> 3.10509 + 0.i
> >> 3.11683 + 0.i
> >> 3.11704 + 0.i
> >> 3.14499 + 0.i
> >> 3.15685 + 0.i
> >> 3.15689 + 0.i
> >> 3.15689 + 0.i
> >> 3.15747 + 0.i
> >> 3.15777 + 0.i
> >> 3.20644 + 0.i
> >> 3.20655 + 0.i
> >> 3.2738 + 0.i
> >> 3.27616 + 0.i
> >> 3.29973 + 0.i
> >> 3.29978 + 0.i
> >> 3.32425 + 0.i
> >> 3.32745 + 0.i
> >> 3.33317 + 0.i
> >> 3.33317 + 0.i
> >> 3.33317 + 0.i
> >> 3.34044 + 0.i
> >> 3.34045 + 0.i
> >> 3.37155 + 0.i
> >> 3.37158 + 0.i
> >> 3.39934 + 0.i
> >> 3.39938 + 0.i
> >> 3.44127 + 0.i
> >> 3.44128 + 0.i
> >> 3.47332 + 0.i
> >> 3.47336 + 0.i
> >> 3.52504 + 0.i
> >> 3.52537 + 0.i
> >> 3.52593 + 0.i
> >> 3.52611 + 0.i
> >> 3.57033 + 0.i
> >> 3.57053 + 0.i
> >> 3.57644 + 0.i
> >> 3.5895 + 0.i
> >> 3.5895 + 0.i
> >> 3.59299 + 0.i
> >> 3.59346 + 0.i
> >> 3.59352 + 0.i
> >> 3.60413 + 0.i
> >> 3.60415 + 0.i
> >> 3.61629 + 0.i
> >> 3.6163 + 0.i
> >> 3.64999 + 0.i
> >> 3.65001 + 0.i
> >> 3.70438 + 0.i
> >> 3.7044 + 0.i
> >> 3.75027 + 0.i
> >> 3.75027 + 0.i
> >> 3.78498 + 0.i
> >> 3.78501 + 0.i
> >> 3.79402 + 0.i
> >> 3.80135 + 0.i
> >> 3.80244 + 0.i
> >> 3.80273 + 0.i
> >> 3.8261 + 0.i
> >> 3.82995 + 0.i
> >> 3.86128 + 0.i
> >> 3.86376 + 0.i
> >> 3.86632 + 0.i
> >> 3.86861 + 0.i
> >> 3.87662 + 0.i
> >> 3.88743 + 0.i
> >> 3.88747 + 0.i
> >> 3.88747 + 0.i
> >> 3.90168 + 0.i
> >> 3.90383 + 0.i
> >> 3.94272 + 0.i
> >> 4.07034 + 0.i
> >> 4.07073 + 0.i
> >> 4.07106 + 0.i
> >> 4.0712 + 0.i
> >> 4.12802 + 0.i
> >> 4.179 + 0.i
> >> 4.179 + 0.i
> >> 4.21785 + 0.i
> >> 4.21785 + 0.i
> >> 4.23041 + 0.i
> >> 4.26551 + 0.i
> >> 4.26984 + 0.i
> >> 4.26984 + 0.i
> >> 4.28215 + 0.i
> >> 4.28222 + 0.i
> >> 4.28253 + 0.i
> >> 4.28253 + 0.i
> >> 4.28439 + 0.i
> >> 4.30964 + 0.i
> >> 4.41513 + 0.i
> >> 4.42009 + 0.i
> >> 4.4201 + 0.i
> >> 4.4201 + 0.i
> >> 4.55339 + 0.i
> >> 4.55351 + 0.i
> >> 4.5543 + 0.i
> >> 4.5544 + 0.i
> >> 4.6759 + 0.i
> >> 4.69125 + 0.i
> >> 4.6913 + 0.i
> >> 4.6913 + 0.i
> >> 4.69141 + 0.i
> >> 4.69152 + 0.i
> >> 4.779 + 0.i
> >> 4.78551 + 0.i
> >> 4.78649 + 0.i
> >> 4.78681 + 0.i
> >> 4.96201 + 0.i
> >> 4.96209 + 0.i
> >> 4.981 + 0.i
> >> 4.98151 + 0.i
> >> 5.06651 + 0.i
> >> 5.19733 + 0.i
> >> 5.21834 + 0.i
> >> 5.2184 + 0.i
> >> 5.2184 + 0.i
> >> 5.2184 + 0.i
> >> 5.21846 + 0.i
> >> 5.49277 + 0.i
> >> 5.50119 + 0.i
> >> 5.50517 + 0.i
> >> 5.5204 + 0.i
> >> 5.52046 + 0.i
> >> 5.52046 + 0.i
> >> 5.52081 + 0.i
> >> 5.52102 + 0.i
> >> 5.5294 + 0.i
> >> 5.52954 + 0.i
> >> 5.5956 + 0.i
> >> 5.59687 + 0.i
> >> 5.59694 + 0.i
> >> 5.59766 + 0.i
> >> 5.6491 + 0.i
> >> 5.64982 + 0.i
> >> 5.64984 + 0.i
> >> 5.64987 + 0.i
> >> 5.8325 + 0.i
> >> 5.93872 + 0.i
> >> 5.95213 + 0.i
> >> 5.95213 + 0.i
> >> 5.95367 + 0.i
> >> 5.95377 + 0.i
> >> 5.96576 + 0.i
> >> 6.13806 + 0.i
> >> 6.13941 + 0.i
> >> 6.36023 + 0.i
> >> 6.37305 + 0.i
> >> 6.37312 + 0.i
> >> 6.37312 + 0.i
> >> 6.37498 + 0.i
> >> 6.37574 + 0.i
> >> 6.8939 + 0.i
> >> 6.92001 + 0.i
> >> 6.92009 + 0.i
> >> 6.92009 + 0.i
> >> 6.92009 + 0.i
> >> 6.92017 + 0.i
> >> 7.10562 + 0.i
> >> 7.12889 + 0.i
> >> 7.12897 + 0.i
> >> 7.12897 + 0.i
> >> 7.13024 + 0.i
> >> 7.13089 + 0.i
> >> 7.40688 + 0.i
> >> 7.46155 + 0.i
> >> 7.47877 + 0.i
> >> 7.49077 + 0.i
> >> 7.49077 + 0.i
> >> 7.49145 + 0.i
> >> 7.49157 + 0.i
> >> 7.49954 + 0.i
> >> 7.56572 + 0.i
> >> 7.56602 + 0.i
> >> 7.76145 + 0.i
> >> 7.76855 + 0.i
> >> 7.76864 + 0.i
> >> 7.76864 + 0.i
> >> 7.8675 + 0.i
> >> 7.86924 + 0.i
> >> 7.93069 + 0.i
> >> 7.93406 + 0.i
> >> 7.93419 + 0.i
> >> 8.58745 + 0.i
> >> 8.58793 + 0.i
> >> 8.65657 + 0.i
> >> 8.65693 + 0.i
> >> 8.65703 + 0.i
> >> 8.65703 + 0.i
> >> 8.76218 + 0.i
> >> 8.76323 + 0.i
> >> 8.76344 + 0.i
> >> 8.76345 + 0.i
> >> 8.76364 + 0.i
> >> 8.76379 + 0.i
> >> 8.84516 + 0.i
> >> 8.84531 + 0.i
> >> 8.84538 + 0.i
> >> 8.84542 + 0.i
> >> 8.84542 + 0.i
> >> 8.84542 + 0.i
> >> 9.96128 + 0.i
> >> 10.0422 + 0.i
> >> 10.3582 + 0.i
> >> 11.1981 + 0.i
> >> 11.5961 + 0.i
> >> 11.5961 + 0.i
> >> 11.5961 + 0.i
> >> 11.5962 + 0.i
> >> 11.5962 + 0.i
> >> 11.5962 + 0.i
> >> 11.5962 + 0.i
> >> 11.5963 + 0.i
> >> 16.048 + 0.i
> >> 17.1999 + 0.i
> >> 17.2057 + 0.i
> >> 18.1546 + 0.i
> >> 18.1548 + 0.i
> >> 18.1548 + 0.i
> >> 18.155 + 0.i
> >> 18.155 + 0.i
> >> 18.155 + 0.i
> >> 53.4484 + 0.i
> >> 285.167 + 0.i
> >> 285.353 + 0.i
> >
>

-- 
What most experimenters take for granted before they begin their
experiments is infinitely more interesting than any results to which their
experiments lead.
-- Norbert Wiener
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