From e5349a222d8e82418fdab3f5a91e557c78b81a34 Mon Sep 17 00:00:00 2001 From: Wolfgang Bangerth Date: Fri, 7 Jan 2022 12:19:39 -0700 Subject: [PATCH] Update step-10 output to current values. --- examples/step-10/doc/results.dox | 64 +++++++++++++++++--------------- 1 file changed, 35 insertions(+), 29 deletions(-) diff --git a/examples/step-10/doc/results.dox b/examples/step-10/doc/results.dox index a9e004936b..95c88c008b 100644 --- a/examples/step-10/doc/results.dox +++ b/examples/step-10/doc/results.dox @@ -100,7 +100,8 @@ cells. The second purpose of the program was to compute the value of pi to -good accuracy. This is the output of this part of the program: +good accuracy. This is the output of this part of the program the last +time it was updated: @code Output of grids into gnuplot files: =================================== @@ -118,10 +119,10 @@ Computation of Pi by the area: ============================== Degree = 1 cells eval.pi error - 5 1.9999999999999993 1.1416e+00 - + 5 1.9999999999999991 1.1416e+00 - 20 2.8284271247461890 3.1317e-01 1.87 80 3.0614674589207174 8.0125e-02 1.97 - 320 3.1214451522580511 2.0148e-02 1.99 + 320 3.1214451522580515 2.0148e-02 1.99 1280 3.1365484905459380 5.0442e-03 2.00 5120 3.1403311569547516 1.2615e-03 2.00 @@ -129,28 +130,28 @@ Degree = 2 cells eval.pi error 5 3.1045694996615860 3.7023e-02 - 20 3.1391475703122267 2.4451e-03 3.92 - 80 3.1414377167038290 1.5494e-04 3.98 - 320 3.1415829366419006 9.7169e-06 4.00 - 1280 3.1415920457576898 6.0783e-07 4.00 - 5120 3.1415926155921117 3.7998e-08 4.00 + 80 3.1414377167038294 1.5494e-04 3.98 + 320 3.1415829366419010 9.7169e-06 4.00 + 1280 3.1415920457576911 6.0783e-07 4.00 + 5120 3.1415926155921130 3.7998e-08 4.00 Degree = 3 cells eval.pi error - 5 3.1410033851499288 5.8927e-04 - - 20 3.1415830393583839 9.6142e-06 5.94 - 80 3.1415925017363797 1.5185e-07 5.98 - 320 3.1415926512106696 2.3791e-09 6.00 - 1280 3.1415926535525927 3.7200e-11 6.00 - 5120 3.1415926535892100 5.8302e-13 6.00 + 5 3.1410033851499310 5.8927e-04 - + 20 3.1415830393583857 9.6142e-06 5.94 + 80 3.1415925017363828 1.5185e-07 5.98 + 320 3.1415926512106704 2.3791e-09 6.00 + 1280 3.1415926535525944 3.7199e-11 6.00 + 5120 3.1415926535892122 5.8117e-13 6.00 Degree = 4 cells eval.pi error - 5 3.1415871927401131 5.4608e-06 - - 20 3.1415926314742428 2.2116e-08 7.95 - 80 3.1415926535026202 8.7173e-11 7.99 - 320 3.1415926535894498 3.4350e-13 7.99 - 1280 3.1415926535897896 3.4671e-15 6.63 - 5120 3.1415926535897909 2.4009e-15 0.53 + 5 3.1415871927401136 5.4608e-06 - + 20 3.1415926314742446 2.2116e-08 7.95 + 80 3.1415926535026215 8.7172e-11 7.99 + 320 3.1415926535894516 3.4157e-13 8.00 + 1280 3.1415926535897918 1.5339e-15 7.80 + 5120 3.1415926535897927 5.2649e-16 1.54 Computation of Pi by the perimeter: =================================== @@ -169,7 +170,7 @@ cells eval.pi error 20 3.1404050605605449 1.1876e-03 3.81 80 3.1415157631807009 7.6890e-05 3.95 320 3.1415878042798613 4.8493e-06 3.99 - 1280 3.1415923498174534 3.0377e-07 4.00 + 1280 3.1415923498174529 3.0377e-07 4.00 5120 3.1415926345931995 1.8997e-08 4.00 Degree = 3 @@ -178,23 +179,28 @@ cells eval.pi error 20 3.1415913432549156 1.3103e-06 6.23 80 3.1415926341726910 1.9417e-08 6.08 320 3.1415926532906897 2.9910e-10 6.02 - 1280 3.1415926535851355 4.6578e-12 6.00 - 5120 3.1415926535897190 7.4216e-14 5.97 + 1280 3.1415926535851364 4.6570e-12 6.01 + 5120 3.1415926535897190 7.4157e-14 5.97 Degree = 4 cells eval.pi error 5 3.1415921029432572 5.5065e-07 - 20 3.1415926513737595 2.2160e-09 7.96 - 80 3.1415926535810712 8.7222e-12 7.99 - 320 3.1415926535897576 3.5525e-14 7.94 - 1280 3.1415926535897936 4.6729e-16 6.25 - 5120 3.1415926535897918 1.4929e-15 -1.68 + 80 3.1415926535810703 8.7230e-12 7.99 + 320 3.1415926535897576 3.5705e-14 7.93 + 1280 3.1415926535897918 1.3785e-15 4.70 + 5120 3.1415926535897944 1.3798e-15 -0.00 @endcode +@note Once the error reaches a level on the + order of $10^{-14}$ or $10^{-15}$, it is essentially dominated by + round-off and consequently dominated by what precisely the library is doing + in internal computations. Since these things change, the precise values + and errors change from release to release at these round-off levels, + though the overall order of errors should of course remain the same. - -One of the immediate observations from the output is that in all cases the -values converge quickly to the true value of +One of the immediate observations from the output above is that in all cases +the values converge quickly to the true value of $\pi=3.141592653589793238462643$. Note that for the $Q_4$ mapping, we are already in the regime of roundoff errors and the convergence rate levels off, which is already quite a lot. However, also note that for the $Q_1$ mapping, -- 2.39.5