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I want to solve a system of simple differential equations. At some point I need to solve non-linear system of algebraic equations with Newton method. I would like to efficiently create a right hand side vector using Table[] or similar command. The problem is that the system consists of two equations, so what I need is:

(*do something*)

but in fact $F$ should have $2k$ entries. For example for $k=5$ Length[F] should return $10$. How to achieve that efficiently? I know I can create different values for odd indexes and even, but then I need additional If[] or to break the whole code into two Table[] executions what I would like to avoid. I could possibly also create two element lists {a,b}, and flatten it at the end... But It still seems quite expensive.

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On what basis do you assert that Flatten is expensive? Did you time it? – Simon Woods Dec 4 '13 at 20:36
If you are looking for alternatives to Flatten you could try Last@Reap[ Do[ Sow[ .. ] , {i,k},{j,2} ] ]. Doubt its faster than Flatten@Table[ ..] though. – george2079 Dec 4 '13 at 22:12
up vote 3 down vote accepted

Posting this because I was a but surprised by the result..

n = 10^6
Last@Last@Reap[Do[ Sow[2 i + j - 2], {i, n}, {j, 2}]] == Range[2 n] // Timing
-> {3.307221, True}
Table[Unevaluated@Sequence[2 i + 1 - 2, 2 i + 2 - 2], {i, n}] == Range[2 n] // Timing
-> {1.918812, True}
Flatten@Table[ 2 i + j - 2, {i, n}, {j, 2}] == Range[2 n] // Timing
-> {0.093601, True}
Riffle[Table[ 2 i + 1 - 2, {i, n}], Table[ 2 i + 2 - 2, {i, n}]] == Range[2 n] // Timing
-> {0.078001, True}

Note the Riffle approach might really win out depending on how your expression simplifies.

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One more example: Table[Unevaluated@Sequence[2 i - 1, 2 i], {i, n}] == Range[2 n] // Timing. It is quite slow, but a bit faster then Reap/Sow. – ybeltukov Dec 4 '13 at 23:13
nice, added that to the answer for reference.. – george2079 Dec 5 '13 at 15:04
How about Flatten@Table[{2i-1,2i},{i,n}]? – Ray Koopman Dec 5 '13 at 19:58

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