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Re: Setting tolerances for multigrid

To: petsc-users@xxxxxxxxxxx
Subject: Re: Setting tolerances for multigrid
From: Knut Erik Teigen <knutert@xxxxxxxxxxxx>
Date: Sun, 17 Jun 2007 10:47:39 +0200
In-reply-to: <Pine.OSX.4.64.0706161652260.227@bsmith.local>
References: <200706141301.37757.timothy.stitt@ichec.ie> <a9f269830706140532g326e7461vc78418869932b05d@mail.gmail.com> <200706141409.08369.timothy.stitt@ichec.ie> <1181835970.3606.62.camel@iept0415.ivt.ntnu.no> <a9f269830706140932p3ef131f8rce0d2bf600339071@mail.gmail.com> <1181894796.3606.67.camel@iept0415.ivt.ntnu.no> <a9f269830706150121o642dac96s461ead71dc5b1c99@mail.gmail.com> <1181901771.3606.78.camel@iept0415.ivt.ntnu.no> <Pine.OSX.4.64.0706151009460.227@bsmith.local> <1181927340.3606.93.camel@iept0415.ivt.ntnu.no> <Pine.OSX.4.64.0706151415330.227@bsmith.local> <1181983629.3606.109.camel@iept0415.ivt.ntnu.no> <Pine.OSX.4.64.0706161652260.227@bsmith.local>
Reply-to: petsc-users@xxxxxxxxxxx
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Ok, I've attached two files, one with
./run -dmmg_galerkin -dmmg_nlevels 1 -ksp_type gmres -pc_type lu
-ksp_truemonitor -ksp_converged_reason > lev1 

and one with

./run -dmmg_galerkin -dmmg_nlevels 2 -ksp_type gmres -pc_type lu
-mg_levels_1_ksp_type gmres -mg_levels_1_pc_type lu
-ksp_converged_reason -ksp_truemonitor > lev2

The relative residual is 1.e-12, and I am using
KSPSetInitialGuessNonzero.
I've also printed the solution, and if you diff the two files, you can
see that there is a large difference in the third step, e.g. 71.7852 vs.
70.9845. This shouldn't happen, even with a true residual of 1.e-5?

-Knut Erik-

On Sat, 2007-06-16 at 16:56 -0500, Barry Smith wrote:
> 
> On Sat, 16 Jun 2007, Knut Erik Teigen wrote:
> 
> > On Fri, 2007-06-15 at 14:30 -0500, Barry Smith wrote:
> > >   What happens if you run the two level with -ksp_rtol 1.e-12
> > 
> > The exact same thing happens. At the third time step, the solution with
> > two levels begin to deviate from the one with one level.
> > 
> > > 
> > >   BTW: when you just change -dmmg_nlevels from 1 to 2 you are now
> > > solving a NEW problem that has twice as many points in each direction,
> > > this can also effect the time-stepping. You can use -dmmg_nlevels 2
> > > -pc_type lu to compare with the -dmmg_nlevels 2
> > 
> > I forgot to mention that I adjust the coarsest level according to the
> > number of levels, so that the finest level always contains the same
> > amount of points. I.e. with one level it is 65x129, with two 33x65 and
> > so on.
> > 
> > > 
> > >   Always verify things with a direct solver before using multigrid
> > > or any other iterative linear solver. Multigrid is ONLY an optimization
> > > (faster, less memory), debug BEFORE turning on the optimizations.
> > 
> > Yes, I've already done this. With a direct solver, I get the same
> > solution as with one level. Since multigrid is only an optimization as
> > you mention, I don't understand why the solution should be different
> > when running with two or more levels as opposed to only one.
> > Here's the output when running with -pc_type lu. As you can see, with
> > two levels, suddenly more iterations are needed for the third step.
> > 
> > -dmmg_nlevels 1
> >  Start of time loop:
> >  L-s reinit converged after   10 steps with norm  1.716E-09
> >   0 KSP preconditioned resid norm 1.366287028032e+04 true resid norm 3.168341703825e+08 ||Ae||/||Ax|| 1.000000000000e+00
> >   1 KSP preconditioned resid norm 3.325862387561e+02 true resid norm 8.783238319618e-07 ||Ae||/||Ax|| 2.772187832207e-15
> >   2 KSP preconditioned resid norm 4.237324220037e-10 true resid norm 9.365968491320e-07 ||Ae||/||Ax|| 2.956110598807e-15
> > Linear solve converged due to CONVERGED_RTOL
>     0 KSP preconditioned resid norm 1.366743784848e+04 true resid norm 3.168385551957e+08 ||Ae||/||Ax|| 1.000000000000e+00
> >   1 KSP preconditioned resid norm 3.578541063222e+02 true resid norm 3.047639475331e-06 ||Ae||/||Ax|| 9.618903461571e-15
> >   2 KSP preconditioned resid norm 2.490853654289e-10 true resid norm 2.047397542219e-06 ||Ae||/||Ax|| 6.461958333808e-15
> > Linear solve converged due to CONVERGED_RTOL
> >   0 KSP preconditioned resid norm 1.373641039770e+04 true resid norm 4.083933364397e+08 ||Ae||/||Ax|| 1.000000000000e+00
> >   1 KSP preconditioned resid norm 2.756300642422e+02 true resid norm 1.333942531534e-06 ||Ae||/||Ax|| 3.266318062786e-15
> >   2 KSP preconditioned resid norm 1.967864218276e-09 true resid norm 4.708778143639e-06 ||Ae||/||Ax|| 1.153000728339e-14
> > Linear solve converged due to CONVERGED_RTOL
> > 
> > -dmmg_nlevels 2
> >  Start of time loop:
> >  L-s reinit converged after   10 steps with norm  1.716E-09
> >   0 KSP preconditioned resid norm 1.366287028032e+04 true resid norm 3.168341703825e+08 ||Ae||/||Ax|| 1.000000000000e+00
> >   1 KSP preconditioned resid norm 3.325862387561e+02 true resid norm 8.783238319618e-07 ||Ae||/||Ax|| 2.772187832207e-15
> >   2 KSP preconditioned resid norm 4.237324220037e-10 true resid norm 9.365968491320e-07 ||Ae||/||Ax|| 2.956110598807e-15
> > Linear solve converged due to CONVERGED_RTOL
> >   0 KSP preconditioned resid norm 1.366743784848e+04 true resid norm 3.168385551957e+08 ||Ae||/||Ax|| 1.000000000000e+00
> >   1 KSP preconditioned resid norm 3.578541063222e+02 true resid norm 3.047639475331e-06 ||Ae||/||Ax|| 9.618903461571e-15
> >   2 KSP preconditioned resid norm 2.490853654289e-10 true resid norm 2.047397542219e-06 ||Ae||/||Ax|| 6.461958333808e-15
> > Linear solve converged due to CONVERGED_RTOL
> >   0 KSP preconditioned resid norm 1.373641039770e+04 true resid norm 4.083933364397e+08 ||Ae||/||Ax|| 1.000000000000e+00
> >   1 KSP preconditioned resid norm 2.826990181167e+02 true resid norm 3.621921471554e+07 ||Ae||/||Ax|| 8.868708542429e-02
> >   2 KSP preconditioned resid norm 2.151728459050e+02 true resid norm 5.126312558070e+05 ||Ae||/||Ax|| 1.255239030774e-03
> >   3 KSP preconditioned resid norm 9.420491033127e+01 true resid norm 7.221320422298e+02 ||Ae||/||Ax|| 1.768226799500e-06
> > (...)
> > 118 KSP preconditioned resid norm 2.011348229632e-04 true resid norm 1.933530019765e-05 ||Ae||/||Ax|| 4.734479843920e-14
> > 119 KSP preconditioned resid norm 1.002113384828e-04 true resid norm 1.933774400466e-05 ||Ae||/||Ax|| 4.735078239337e-14
> > 120 KSP preconditioned resid norm 3.976002405523e-07 true resid norm 1.947236185243e-05 ||Ae||/||Ax|| 4.768041031762e-14
> > Linear solve converged due to CONVERGED_RTOL
> > 
> > 
> 
>   The true residuals both me, they are very big at "convergence" 1.e-5? Are you using KSPSetInitialGuessNonzero() and
> using the previous solution as initial guess?
> 
>   Run with -ksp_type gmres -pc_type lu -ksp_monitor_true_residual. I bet the converged true residual is
> much better. Send the output.
> 
>    Barry
> 
> 
> > 
> > > 
> > > 
> > >    Barry
> > > 
> > > On Fri, 15 Jun 2007, Knut Erik Teigen wrote:
> > > 
> > > > When running with those options I get this after the first timestep for
> > > > both one and two levels:
> > > > 159 KSP preconditioned resid norm 1.397494864058e-07 true resid norm
> > > > 4.996893774970e-02 ||Ae||/||Ax|| 1.577132216812e-10
> > > > Linear solve converged due to CONVERGED_RTOL
> > > > 
> > > > However, after the second timestep they start to get different.
> > > > With two levels the norm is actually lower, but still, the computation
> > > > diverges after about 6 time steps. With one level the computation runs
> > > > smoothly. Here are the numbers:
> > > > 
> > > > -dmmg_nlevels 1
> > > >   0 KSP preconditioned resid norm 1.674785901465e+02 
> > > > true resid norm 4.083933362528e+08 ||Ae||/||Ax|| 1.000000000000e+00
> > > >   1 KSP preconditioned resid norm 6.256839347623e+01 
> > > > true resid norm 4.220478752135e+07 ||Ae||/||Ax|| 1.033434774147e-01
> > > > (...)
> > > > 159 KSP preconditioned resid norm 1.318896741223e-07 
> > > > true resid norm 5.104451971417e-02 ||Ae||/||Ax|| 1.249886204866e-10
> > > > Linear solve converged due to CONVERGED_RTOL
> > > > 
> > > > -dmmg_nlevels 2
> > > >   0 KSP preconditioned resid norm 1.674785901465e+02 
> > > > true resid norm 4.083933362528e+08 ||Ae||/||Ax|| 1.000000000000e+00
> > > >   1 KSP preconditioned resid norm 6.252455641723e+01 
> > > > true resid norm 5.561545924132e+07 ||Ae||/||Ax|| 1.361811134129e-01
> > > > (...) 
> > > > 159 KSP preconditioned resid norm 1.305901659826e-07 
> > > > true resid norm 4.297164043386e-02 ||Ae||/||Ax|| 1.052212086224e-10
> > > > Linear solve converged due to CONVERGED_RTOL
> > > > 
> > > > -Knut Erik-
> > > > 
> > > > On Fri, 2007-06-15 at 10:10 -0500, Barry Smith wrote:
> > > > >   You need to make sure the linear solvers are converging properly
> > > > > before even worrying about time-stepping. Run with -ksp_monitor_true_residual
> > > > > (-ksp_truemonitor on older versions of PETSc) and -ksp_converged_reason
> > > > > 
> > > > >    Barry
> > > > > 
> > > > > 
> > > > > On Fri, 15 Jun 2007, Knut Erik Teigen wrote:
> > > > > 
> > > > > > The computation runs just fine with -dmmg_nlevels 1.
> > > > > > I use a grid of 65x129, with three levels, so that the coarsest grid is
> > > > > > 17x33. 
> > > > > > I've diffed the solutions, and they are equal for the first two time
> > > > > > steps, but then they start to differ somewhat. Here is an excerpt:
> > > > > > -dmmg_nlevels 1:
> > > > > > 65.0427492820557        65.0216231814457      64.9793945321386        
> > > > > > -dmmg_nlevels 3:
> > > > > > 65.0920864319067        65.0711253401585      65.0292265582817
> > > > > > 
> > > > > > Using 2 or 4 levels produce the same result as 3 levels.
> > > > > > 
> > > > > > -Knut Erik-
> > > > > > 
> > > > > > 
> > > > > > 
> > > > > > On Fri, 2007-06-15 at 10:21 +0200, Matthew Knepley wrote:
> > > > > > > This does not sound right to me. I think something must be
> > > > > > > configured incorrectly. All MG does is accelerate the solution
> > > > > > > of the fine grid problem. I think something must be wrong with
> > > > > > > the specification here. It is easy to check. Run with -dmmg_nlevels 1
> > > > > > > which just solves on the fine grid.
> > > > > > > 
> > > > > > >   Matt
> > > > > > > 
> > > > > > > On 6/15/07, Knut Erik Teigen <knutert@xxxxxxxxxxxx> wrote:
> > > > > > > > Thanks, Matthew. It seems like the tolerances weren't my problem,
> > > > > > > > though. Even with a tolerance of 1e-10, the computation diverges after
> > > > > > > > only a few time steps when using multigrid. With ordinary linear
> > > > > > > > solvers, I can use a tolerance of 1e-4 and still get a satisfactory
> > > > > > > > solution.
> > > > > > > > Could it be that the Galerkin approximation for the coarser levels is
> > > > > > > > too inaccurate for my problem?
> > > > > > > >
> > > > > > > > -Knut Erik-
> > > > > > > >
> > > > > > > > On Thu, 2007-06-14 at 11:32 -0500, Matthew Knepley wrote:
> > > > > > > > > On 6/14/07, Knut Erik Teigen <knutert@xxxxxxxxxxxx> wrote:
> > > > > > > > > > Hello,
> > > > > > > > > >
> > > > > > > > > > How do I set the tolerances for the KSP solvers when using the
> > > > > > > > > > DMMG routines without using command line arguments? It seems that
> > > > > > > > > > I can use the argument -ksp_rtol on the command line, but in the code I
> > > > > > > > > > can't use KSPSetTolerances since I don't have access to the KSP solver
> > > > > > > > > > context?
> > > > > > > > >
> > > > > > > > > You can use DMMGGetKSP() for the finest level, and dmmg[level]->ksp for
> > > > > > > > > any others.
> > > > > > > > >
> > > > > > > > > > Also, I see there is a reference to the function DMMGSetUseGalerkin
> > > > > > > > > > in the source code, but I can't find the documentation for it.
> > > > > > > > > > Does this function not exist yet? Again I would like to use
> > > > > > > > > > -dmmg_galerkin, but setting it in the code instead of on the command
> > > > > > > > > > line.
> > > > > > > > >
> > > > > > > > > I think you want this:
> > > > > > > > >
> > > > > > > > > http://www-unix.mcs.anl.gov/petsc/petsc-as/snapshots/petsc-current/docs/manualpages/DA/DMMGSetUseGalerkinCoarse.html
> > > > > > > > >
> > > > > > > > >   Matt
> > > > > > > > >
> > > > > > > > > > Regards,
> > > > > > > > > > Knut Erik Teigen
> > > > > > > > > >
> > > > > > > > > >
> > > > > > > > >
> > > > > > > > >
> > > > > > > >
> > > > > > > >
> > > > > > > 
> > > > > > > 
> > > > > > 
> > > > > > 
> > > > > 
> > > > > 
> > > > 
> > > > 
> > > 
> > > 
> > 
> > 
> 
>

  nlevels           1
 coarsest_x,coarsest_y:          65         129
 Using SSP-RK3()4 (SSP) scheme for system           1
 Using SSP-RK2()4 (SSP) scheme for system           2
 Using SSP-RK3()4 (SSP) scheme for system           3
 u                   : is f-variable number:    1
 v                   : is f-variable number:    2
 p                   : is f-variable number:    3
 fi                  : is f-variable number:    4
 Initial configuration: Rising bubble                           
 Using CFL-number=  0.3000
 Exact area, phase 1: 3.491E-05
 Numerical area, phase 1: 3.499E-05
 Using WENO boundary trick!
 Writing inputfile to file: out.inp
 c_conv= 0.00E+00 c_visc= 6.07E+02 c_grav= 1.78E+02 c_st= 1.58E+03
 -------------------------------------------
 Start of time loop:
 L-s reinit converged after   10 steps with norm  1.716E-09
  0 KSP preconditioned resid norm 3.168341703825e+08 true resid norm 3.168341703825e+08 ||Ae||/||Ax|| 1.000000000000e+00
  0 KSP preconditioned resid norm 3.168341703825e+08 true resid norm 3.168341703825e+08 ||Ae||/||Ax|| 1.000000000000e+00
  1 KSP preconditioned resid norm 7.830661072700e-07 true resid norm 7.337644742621e-07 ||Ae||/||Ax|| 2.315925941246e-15
  1 KSP preconditioned resid norm 7.830661072700e-07 true resid norm 7.337644742621e-07 ||Ae||/||Ax|| 2.315925941246e-15
Linear solve converged due to CONVERGED_RTOL
Process [0]
66.5366
66.5153
66.473
66.4098
66.326
66.222
66.0984
65.9559
65.7952
65.6174
65.4234
65.2146
64.9923
64.7582
64.514
64.2616
64.0031
63.7408
63.4772
63.2148
62.9562
62.7043
62.4619
62.2317
62.0167
61.8195
61.6427
61.4887
61.3596
61.2572
61.1829
61.138
61.1229
61.138
61.1829
61.2572
61.3596
61.4887
61.6427
61.8195
62.0167
62.2317
62.4619
62.7043
62.9562
63.2148
63.4772
63.7408
64.0031
64.2616
64.514
64.7582
64.9923
65.2146
65.4234
65.6174
65.7952
65.9559
66.0984
66.222
66.326
66.4098
66.473
66.5153
66.5366
63.5162
63.4949
63.4524
63.3888
63.3046
63.2001
63.0758
62.9324
62.7708
62.5918
62.3966
62.1864
61.9625
61.7266
61.4804
61.2259
60.9651
60.7004
60.4341
60.169
59.9076
59.6528
59.4075
59.1745
58.9568
58.757
58.5778
58.4216
58.2906
58.1867
58.1114
58.0657
58.0504
58.0657
58.1114
58.1867
58.2906
58.4216
58.5778
58.757
58.9568
59.1745
59.4075
59.6528
59.9076
60.169
60.4341
60.7004
60.9651
61.2259
61.4804
61.7266
61.9625
62.1864
62.3966
62.5918
62.7708
62.9324
63.0758
63.2001
63.3046
63.3888
63.4524
63.4949
63.5162
60.5176
60.4961
60.4531
60.3889
60.3038
60.1982
60.0725
59.9276
59.764
59.5829
59.3851
59.172
58.9449
58.7054
58.4553
58.1965
57.9311
57.6614
57.39
57.1193
56.8523
56.5917
56.3406
56.1019
55.8785
55.6735
55.4894
55.3288
55.1941
55.0872
55.0097
54.9627
54.9469
54.9627
55.0097
55.0872
55.1941
55.3288
55.4894
55.6735
55.8785
56.1019
56.3406
56.5917
56.8523
57.1193
57.39
57.6614
57.9311
58.1965
58.4553
58.7054
58.9449
59.172
59.3851
59.5829
59.764
59.9276
60.0725
60.1982
60.3038
60.3889
60.4531
60.4961
60.5176
57.5408
57.5189
57.4754
57.4102
57.3238
57.2166
57.0889
56.9415
56.7752
56.5907
56.3892
56.1718
55.9399
55.6951
55.4391
55.1738
54.9015
54.6243
54.3449
54.066
53.7904
53.521
53.2611
53.0136
52.7818
52.5686
52.3771
52.2099
52.0694
51.9579
51.8769
51.8278
51.8114
51.8278
51.8769
51.9579
52.0694
52.2099
52.3771
52.5686
52.7818
53.0136
53.2611
53.521
53.7904
54.066
54.3449
54.6243
54.9015
55.1738
55.4391
55.6951
55.9399
56.1718
56.3892
56.5907
56.7752
56.9415
57.0889
57.2166
57.3238
57.4102
57.4754
57.5189
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54.5862
54.5639
54.5195
54.4531
54.3651
54.2556
54.1254
53.9748
53.8047
53.6159
53.4093
53.1863
52.948
52.6961
52.4323
52.1584
51.8767
51.5895
51.2995
51.0093
50.7221
50.4408
50.1689
49.9095
49.6661
49.4419
49.2401
49.0637
48.9154
48.7975
48.7118
48.6599
48.6425
48.6599
48.7118
48.7975
48.9154
49.0637
49.2401
49.4419
49.6661
49.9095
50.1689
50.4408
50.7221
51.0093
51.2995
51.5895
51.8767
52.1584
52.4323
52.6961
52.948
53.1863
53.4093
53.6159
53.8047
53.9748
54.1254
54.2556
54.3651
54.4531
54.5195
54.5639
54.5862
51.6541
51.6314
51.586
51.5181
51.428
51.3159
51.1824
51.0279
50.8532
50.659
50.4463
50.2162
49.97
49.7093
49.4356
49.1509
48.8575
48.5576
48.2541
47.9497
47.6477
47.3512
47.0638
46.7892
46.5308
46.2924
46.0774
45.8891
45.7305
45.6043
45.5125
45.4568
45.4382
45.4568
45.5125
45.6043
45.7305
45.8891
46.0774
46.2924
46.5308
46.7892
47.0638
47.3512
47.6477
47.9497
48.2541
48.5576
48.8575
49.1509
49.4356
49.7093
49.97
50.2162
50.4463
50.659
50.8532
51.0279
51.1824
51.3159
51.428
51.5181
51.586
51.6314
51.6541
48.7452
48.7219
48.6753
48.6057
48.5131
48.398
48.2607
48.1016
47.9215
47.721
47.501
47.2625
47.0069
46.7355
46.45
46.1524
45.8447
45.5295
45.2096
44.8878
44.5675
44.2524
43.946
43.6523
43.3753
43.1191
42.8875
42.6842
42.5128
42.3761
42.2766
42.2162
42.1959
42.2162
42.2766
42.3761
42.5128
42.6842
42.8875
43.1191
43.3753
43.6523
43.946
44.2524
44.5675
44.8878
45.2096
45.5295
45.8447
46.1524
46.45
46.7355
47.0069
47.2625
47.501
47.721
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-285.071
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-285.073
-285.073
-285.073
-285.073
-285.074
-285.074
-285.074
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-288.116
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-288.113
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-288.111
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-288.11
-288.11
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-288.108
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-288.107
-288.107
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-288.106
-288.106
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-288.105
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-288.111
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-288.113
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-288.114
-288.114
-288.114
-288.115
-288.115
-288.115
-288.115
-288.115
-288.116
-288.116
-291.157
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-291.157
-291.157
-291.156
-291.156
-291.156
-291.155
-291.155
-291.155
-291.154
-291.154
-291.153
-291.153
-291.152
-291.152
-291.151
-291.151
-291.15
-291.15
-291.149
-291.149
-291.149
-291.148
-291.148
-291.148
-291.148
-291.147
-291.147
-291.147
-291.147
-291.147
-291.147
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-291.148
-291.148
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-291.148
-291.149
-291.149
-291.149
-291.15
-291.15
-291.151
-291.151
-291.152
-291.152
-291.153
-291.153
-291.154
-291.154
-291.155
-291.155
-291.155
-291.156
-291.156
-291.156
-291.157
-291.157
-291.157
-291.157
-291.157
-291.157
-294.199
-294.199
-294.199
-294.199
-294.199
-294.199
-294.198
-294.198
-294.198
-294.197
-294.197
-294.196
-294.196
-294.195
-294.195
-294.194
-294.194
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-294.192
-294.191
-294.191
-294.19
-294.19
-294.19
-294.19
-294.189
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-294.199
  0 KSP preconditioned resid norm 3.005729294791e+04 true resid norm 3.005729294791e+04 ||Ae||/||Ax|| 9.486627323288e-05
  0 KSP preconditioned resid norm 3.005729294791e+04 true resid norm 3.005729294791e+04 ||Ae||/||Ax|| 9.486627323288e-05
  0 KSP preconditioned resid norm 3.005729294791e+04 true resid norm 3.005729294791e+04 ||Ae||/||Ax|| 9.486627323288e-05
  1 KSP preconditioned resid norm 5.352387062140e-07 true resid norm 8.438016085313e-07 ||Ae||/||Ax|| 2.663191062748e-15
  1 KSP preconditioned resid norm 5.352387062140e-07 true resid norm 8.438016085313e-07 ||Ae||/||Ax|| 2.663191062748e-15
  1 KSP preconditioned resid norm 5.352387062140e-07 true resid norm 8.438016085313e-07 ||Ae||/||Ax|| 2.663191062748e-15
Linear solve converged due to CONVERGED_RTOL
Process [0]
71.0932
71.0721
71.0298
70.9666
70.8829
70.779
70.6555
70.5131
70.3525
70.1748
69.9809
69.7722
69.5501
69.3161
69.072
68.8197
68.5614
68.2993
68.0357
67.7734
67.515
67.2632
67.0208
66.7908
66.5759
66.3787
66.202
66.048
65.919
65.8166
65.7424
65.6974
65.6823
65.6974
65.7424
65.8166
65.919
66.048
66.202
66.3787
66.5759
66.7908
67.0208
67.2632
67.515
67.7734
68.0357
68.2993
68.5614
68.8197
69.072
69.3161
69.5501
69.7722
69.9809
70.1748
70.3525
70.5131
70.6555
70.779
70.8829
70.9666
71.0298
71.0721
71.0932
68.0729
68.0516
68.0091
67.9456
67.8615
67.757
67.6328
67.4896
67.3281
67.1492
66.9541
66.744
66.5203
66.2845
66.0384
65.784
65.5234
65.2588
64.9927
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64.4664
64.2117
63.9665
63.7336
63.5159
63.3162
63.1371
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62.85
62.7462
62.6709
62.6252
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62.6252
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62.85
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63.1371
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65.0742
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61.678
61.4111
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60.2328
60.0488
59.8883
59.7536
59.6467
59.5692
59.5222
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59.5222
59.5692
59.6467
59.7536
59.8883
60.0488
60.2328
60.4378
60.661
60.8996
61.1506
61.4111
61.678
61.9485
62.2199
62.4894
62.7547
63.0133
63.2633
63.5026
63.7296
63.9426
64.1402
64.3213
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64.7551
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65.0098
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61.9669
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57.128
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60.2529
60.4976
60.7293
60.9466
61.148
61.3323
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62.032
62.0755
62.0973
59.1426
59.1204
59.0761
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58.8124
58.6822
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58.3618
58.1731
57.9667
57.7438
57.5057
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56.9902
56.7165
56.435
56.148
55.8581
55.5681
55.2809
54.9998
54.728
54.4687
54.2254
54.0013
53.7997
53.6233
53.4751
53.3572
53.2716
53.2196
53.2022
53.2196
53.2716
53.3572
53.4751
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53.7997
54.0013
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54.4687
54.728
54.9998
55.2809
55.5681
55.8581
56.148
56.435
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56.2105
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55.9846
55.8726
55.7392
55.5849
55.4103
55.2162
55.0036
54.7737
54.5277
54.267
53.9935
53.709
53.4157
53.1161
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52.5085
52.2066
51.9103
51.6231
51.3485
51.0903
50.8519
50.637
50.4488
50.2903
50.1641
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49.998
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50.1641
50.2903
50.4488
50.637
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51.0903
51.3485
51.6231
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52.5085
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53.4157
53.709
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54.267
54.5277
54.7737
55.0036
55.2162
55.4103
55.5849
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55.8726
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56.0746
56.1425
56.1878
56.2105
53.3014
53.2782
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53.1621
53.0696
52.9546
52.8174
52.6585
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52.2781
52.0582
51.8199
51.5645
51.2932
51.0079
50.7105
50.403
50.088
49.7682
49.4466
49.1265
48.8115
48.5052
48.2117
47.9349
47.6787
47.4472
47.2441
47.0727
46.936
46.8366
46.7762
46.7559
46.7762
46.8366
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47.0727
47.2441
47.4472
47.6787
47.9349
48.2117
48.5052
48.8115
49.1265
49.4466
49.7682
50.088
50.403
50.7105
51.0079
51.2932
51.5645
51.8199
52.0582
52.2781
52.4785
52.6585
52.8174
52.9546
53.0696
53.1621
53.2317
53.2782
53.3014
50.416
50.3921
50.3444
50.2728
50.1776
50.0591
49.9176
49.7535
49.5673
49.3597
49.1315
48.8836
48.6173
48.3337
48.0347
47.722
47.3979
47.0648
46.7255
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46.0413
45.7038
45.3745
45.0579
44.7584
44.4805
44.2286
44.0071
43.8198
43.6702
43.5612
43.4949
43.4726
43.4949
43.5612
43.6702
43.8198
44.0071
44.2286
44.4805
44.7584
45.0579
45.3745
45.7038
46.0413
46.3832
46.7255
47.0648
47.3979
47.722
48.0347
48.3337
48.6173
48.8836
49.1315
49.3597
49.5673
49.7535
49.9176
50.0591
50.1776
50.2728
50.3444
50.3921
50.416
47.555
47.5304
47.4812
47.4075
47.3094
47.1871
47.0408
46.871
46.678
46.4623
46.2248
45.9661
45.6874
45.39
45.0753
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44.4019
44.0478
43.6858
43.3193
42.9518
42.5876
42.231
41.8868
41.5601
41.2558
40.9793
40.7353
40.5286
40.3631
40.2424
40.1688
40.1442
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41.2558
41.5601
41.8868
42.231
42.5876
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45.0753
45.39
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47.3094
47.4075
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41.782
41.4169
41.0389
40.6509
40.2563
39.8591
39.4636
39.0748
38.698
38.3388
38.0031
37.6968
37.4258
37.1954
37.0107
36.8756
36.7932
36.7656
36.7932
36.8756
37.0107
37.1954
37.4258
37.6968
38.0031
38.3388
38.698
39.0748
39.4636
39.8591
40.2563
40.6509
41.0389
41.4169
41.782
42.1316
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42.7767
43.069
43.3394
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44.012
44.188
44.3393
44.4656
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44.6936
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41.9087
41.8826
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41.6474
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40.9695
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40.4771
40.1939
39.8869
39.5571
39.2057
38.8343
38.4449
38.04
37.6225
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36.3329
35.9065
35.4913
35.0937
34.7204
34.3785
34.0749
33.816
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33.3621
33.3308
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33.6078
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34.0749
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35.0937
35.4913
35.9065
36.3329
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37.196
37.6225
38.04
38.4449
38.8343
39.2057
39.5571
39.8869
40.1939
40.4771
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40.9695
41.1777
41.3602
41.5168
41.6474
41.7519
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41.9087
39.125
39.0981
39.0443
38.9635
38.8557
38.7208
38.5588
38.3696
38.1533
37.9099
37.6398
37.343
37.0203
36.6723
36.3
35.9049
35.4888
35.054
34.6034
34.1407
33.6702
33.1969
32.7269
32.2667
31.8238
31.406
31.0214
30.6785
30.385
30.1483
29.9745
29.8682
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29.9745
30.1483
30.385
30.6785
31.0214
31.406
31.8238
32.2667
32.7269
33.1969
33.6702
34.1407
34.6034
35.054
35.4888
35.9049
36.3
36.6723
37.0203
37.343
37.6398
37.9099
38.1533
38.3696
38.5588
38.7208
38.8557
38.9635
39.0443
39.0981
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36.3686
36.341
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35.9523
35.7849
35.589
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33.4176
32.9969
32.5516
32.0839
31.5966
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30.5785
30.0577
29.5372
29.0246
28.5284
28.0576
27.6222
27.2319
26.8965
26.625
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26.3027
26.4251
26.625
26.8965
27.2319
27.6222
28.0576
28.5284
29.0246
29.5372
30.0577
30.5785
31.0933
31.5966
32.0839
32.5516
32.9969
33.4176
33.8121
34.1794
34.5186
34.8293
35.1113
35.3645
35.589
35.7849
35.9523
36.0915
36.2025
36.2857
36.341
36.3686
33.6403
33.612
33.5553
33.4699
33.3558
33.2125
33.0399
32.8375
32.6049
32.3419
32.048
31.723
31.3669
30.9797
30.5617
30.1138
29.6372
29.1338
28.6061
28.0576
27.493
26.918
26.3395
25.766
25.2071
24.6735
24.1769
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23.0291
22.7971
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25.2071
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26.3395
26.918
27.493
28.0576
28.6061
29.1338
29.6372
30.1138
30.5617
30.9797
31.3669
31.723
32.048
32.3419
32.6049
32.8375
33.0399
33.2125
33.3558
33.4699
33.5553
33.612
33.6403
30.9409
30.9119
30.8539
30.7666
30.6496
30.5026
30.3251
30.1166
29.8764
29.6038
29.2983
28.9592
28.586
28.1784
27.7363
27.2599
26.7501
26.2083
25.6366
25.0384
24.4181
23.7818
23.1368
22.4926
21.86
21.2518
20.6819
20.165
19.7163
19.3498
19.078
18.9108
18.8544
18.9108
19.078
19.3498
19.7163
20.165
20.6819
21.2518
21.86
22.4926
23.1368
23.7818
24.4181
25.0384
25.6366
26.2083
26.7501
27.2599
27.7363
28.1784
28.586
28.9592
29.2983
29.6038
29.8764
30.1166
30.3251
30.5026
30.6496
30.7666
30.8539
30.9119
30.9409
28.2709
28.2414
28.1823
28.0933
27.9739
27.8237
27.6419
27.4278
27.1806
26.8992
26.5826
26.2299
25.84
25.4122
24.9456
24.44
23.8956
23.313
22.6941
22.0415
21.3596
20.6543
19.9334
19.2072
18.4882
17.7912
17.1329
16.5315
16.0059
15.5743
15.2529
15.0545
14.9874
15.0545
15.2529
15.5743
16.0059
16.5315
17.1329
17.7912
18.4882
19.2072
19.9334
20.6543
21.3596
22.0415
22.6941
23.313
23.8956
24.44
24.9456
25.4122
25.84
26.2299
26.5826
26.8992
27.1806
27.4278
27.6419
27.8237
27.9739
28.0933
28.1823
28.2414
28.2709
25.6308
25.601
25.5411
25.4508
25.3296
25.1768
24.9915
24.7728
24.5195
24.2303
23.9037
23.5384
23.1327
22.6852
22.1945
21.6596
21.0797
20.4548
19.7857
19.0745
18.3249
17.5425
16.7354
15.9147
15.0942
14.2912
13.526
12.821
12.2001
11.6868
11.3027
11.0648
10.9842
11.0648
11.3027
11.6868
12.2001
12.821
13.526
14.2912
15.0942
15.9147
16.7354
17.5425
18.3249
19.0745
19.7857
20.4548
21.0797
21.6596
22.1945
22.6852
23.1327
23.5384
23.9037
24.2303
24.5195
24.7728
24.9915
25.1768
25.3296
25.4508
25.5411
25.601
25.6308
23.0212
22.9911
22.9308
22.8397
22.7173
22.5628
22.3751
22.1529
21.8949
21.5993
21.2642
20.8878
20.4678
20.0021
19.4885
18.9248
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