# Solve slower in versions 9 and 10

With some systems of equations, Solve is much slower in versions 9 and 10 than in earlier versions, apparently because it is doing more simplification of the results.

With the following example linear system:

eqs = {0 == g1 - 2 g1 r[1, 1] - I oa r[1, 3] + I oa r[3, 1] + g2 r[3, 3],
0 == -I da r[1, 2] + I db r[1, 2] - 2 g1 r[1, 2] - I ob r[1, 3] + I oa r[3, 2],
0 == -I oa r[1, 1] - I ob r[1, 2] - I da r[1, 3] - 2 g1 r[1, 3] - g2 r[1, 3] + I oa r[3, 3],
0 == I da r[2, 1] - I db r[2, 1] - 2 g1 r[2, 1] - I oa r[2, 3] + I ob r[3, 1],
0 == g1 - 2 g1 r[2, 2] - I ob r[2, 3] + I ob r[3, 2] + g2 r[3, 3],
0 == -I oa r[2, 1] - I ob r[2, 2] - I db r[2, 3] - 2 g1 r[2, 3] - g2 r[2, 3] + I ob r[3, 3],
0 == I oa r[1, 1] + I ob r[2, 1] + I da r[3, 1] - 2 g1 r[3, 1] - g2 r[3, 1] - I oa r[3, 3],
0 == I oa r[1, 2] + I ob r[2, 2] + I db r[3, 2] - 2 g1 r[3, 2] - g2 r[3, 2] - I ob r[3, 3],
0 == I oa r[1, 3] + I ob r[2, 3] - I oa r[3, 1] - I ob r[3, 2] - 2 g1 r[3, 3] - 2 g2 r[3, 3]
};
vars = {r[1, 1], r[1, 2], r[1, 3], r[2, 1], r[2, 2], r[2, 3], r[3, 1], r[3, 2], r[3, 3]};

{\$VersionNumber, Round[First@Timing[sol = Solve[eqs, vars];], .01], LeafCount@sol}


I get the results:

\begin{array}{ccc} \text{Version} & \text{Timing} & \text{LeafCount} \\ 6. & 0.19 & 219384 \\ 7. & 0.08 & 227942 \\ 8. & 1.82 & 86317 \\ 9. & 63.32 & 29452 \\ 10.2 & 30.84 & 82043 \\ \end{array}

Version 10 was run on a 2.3GHz MacBook Pro, earlier versions were run on an older 2.4GHz PC. (Note that if the Solve command is executed twice, the second time is faster, presumably due to some caching of results. The timings are all for the first evaluation.)

So there is an inverse correlation between Timing and LeafCount, which makes sense if extra time is being taken for some sort of simplification. But for my purposes the ~3x reduction in LeafCount between versions 7 and 10 is not worth the ~400x slowdown.

Does anyone happen to know if there's an undocumented Method setting to get Solve to use a method from an earlier version? Or maybe some other workaround?

### Update 1:

Solve with Method -> "Legacy" (suggested by Alexey Popkov) and SystemPrivateOldSolve (suggested by Guess who) both give me a timing of 18.7 s and a LeafCount of 44087 on 10.2. So it seems that they are both doing the same thing. It's supposed to cause Solve to use the version 7 algorithm, but it's clearly not using the exact algorithm from version 7. Using Method -> "Legacy" in different versions I get

\begin{array}{ccc} \text{Version} & \text{Timing} & \text{LeafCount} \\ 8. & 1.78 & 48622 \\ 9. & 21.92 & 27358 \\ 10.2 & 18.77 & 44087 \\ \end{array}

So for this system of equations it's an improvement in both timing and leaf count in all versions, but still nowhere close to version 7's speed.

### Update 2:

As pointed out in Michael E2's answer, we can do much better with LinearSolve, and we can transfer this benefit to Solve by setting the options for RowReduce, as suggested by Daniel Lichtblau.

After setting

SetOptions[RowReduce, Method -> "CofactorExpansion"]


Solve gives the results

\begin{array}{ccc} \text{Version} & \text{Timing} & \text{LeafCount} \\ 6. & 0.42 & 64004 \\ 7. & 0.42 & 64004 \\ 8. & 0.5 & 67879 \\ 9. & 0.5 & 67879 \\ 10.2 & 0.27 & 67879 \\ \end{array}

Now it's reasonably fast, and stable across versions. It doesn't beat the default Solve in version 7 for speed, or version 9 for leaf count, but it seems like a good overall compromise.

• A good question! And very nice previous research. +10 if I could. – Dr. belisarius Jul 31 '15 at 22:10
• Very interesting! Just to add one more data point at least for LeafCount, this is the result on MMA 10.2 on Win7-64: {10.2, 24.4, 82043}. – MarcoB Jul 31 '15 at 22:16
• @belisarius Thanks. I did the research almost by accident -- it's an example from the documentation of a package I'm developing for versions 6 and later. So I was testing the docs in different versions... – Simon Rochester Jul 31 '15 at 22:26
• With Method -> "Legacy" added to Solve I get with MMa 10.2 on Win7 x64: {10.2, 20.16, 45770}. Without this option I get {10.2, 25.19, 82043}. – Alexey Popkov Aug 1 '15 at 1:59
• Changing from version 7 to version 10 I knew some things became slower, but nothing to this degree! I know that performance is a balancing act but this seems to really need some attention. The system could at least provide the old method as an option, and even attempt to evaluate it in parallel to see if it works better. – Mr.Wizard Aug 1 '15 at 2:31

This is best I can do so far. The system is linear so LinearSolve is a natural thing to try.

arrays = CoefficientArrays[eqs, vars]
(*  {SparseArray[< 2 >, {9}], SparseArray[< 35 >, {9, 9}]}  *)

vars -> LinearSolve[arrays[[2]], -arrays[[1]],
Method -> "CofactorExpansion"]]; // AbsoluteTiming
(*  {0.28347, Null}  *)


I cannot figure out how to get Solve to make this call or its equivalent.

With Daniel and Simon's help, here is how to do it with Solve:

With[{opts = Options[RowReduce]},
InternalWithLocalSettings[
SetOptions[RowReduce, Method -> "CofactorExpansion"],
Solve[eqs, vars]; // AbsoluteTiming,
SetOptions[RowReduce, opts]
]]
(*  {0.282098, Null}  *)

• SetOptions[RowReduce, Method -> "OneStepRowReduction"] will do what you have in mind; Solve goes through RowReduce at least for nonsparse linear systems. – Daniel Lichtblau Aug 5 '15 at 22:06
• @DanielLichtblau Thanks. However Solve took 7.4 sec, considerably longer than LinearSolve, with the RowReduce option set. I thought perhaps Solve is not selecting to solve as a linear system because of the symbolic parameters or because the variables have the form {r[1, 1], r[1, 2],...} -- shots in the dark, admittedly. – Michael E2 Aug 5 '15 at 22:35
• I think maybe @DanielLichtblau meant to set SetOptions[RowReduce, Method -> "CofactorExpansion"] -- that gives comparable results to your LinearSolve method on my computer. – Simon Rochester Aug 8 '15 at 20:49
• Setting the options for RowReduce` seems to work well (see the update to the question). If you add this to your answer I'll accept. – Simon Rochester Aug 8 '15 at 21:23
• @SimonRochester Thanks for figuring that out. I added my way of using the option to my answer. – Michael E2 Aug 8 '15 at 21:29