Raise a table of pre-existing data to the power of of one of the variables

Question

NOTE: I have edited this question to give a table whose first elements don't count up from 0 to 5.

I'm using a sample table here, given by Table[a/b + a^(1/2), {b, 1, 6}, {a, 1, 6}] - simply because it's small and easy. This gives me the following data:

testtable1 = {{2, 2 + Sqrt[2], 3 + Sqrt[3], 6, 5 + Sqrt[5], 
6 + Sqrt[6]}, {3/2, 1 + Sqrt[2], 3/2 + Sqrt[3], 4, 5/2 + Sqrt[5], 
3 + Sqrt[6]}, {4/3, 2/3 + Sqrt[2], 1 + Sqrt[3], 10/3, 
5/3 + Sqrt[5], 2 + Sqrt[6]}, {5/4, 1/2 + Sqrt[2], 3/4 + Sqrt[3], 3,
5/4 + Sqrt[5], 3/2 + Sqrt[6]}, {6/5, 2/5 + Sqrt[2], 3/5 + Sqrt[3],
14/5, 1 + Sqrt[5], 6/5 + Sqrt[6]}, {7/6, 1/3 + Sqrt[2], 
1/2 + Sqrt[3], 8/3, 5/6 + Sqrt[5], 1 + Sqrt[6]}}

But now I want to raise that table to the power of b - without going back to create a new table. Of course, it's easy to say Table[(a/b + a^(1/2))^b, {b, 1, 6}, {a, 1, 6}] - but that's only because this is a simple table. I'm looking for an operation that I can apply generically to any testtable1, because the calculations involved in getting to my actual testtable1 are very big and very slow, and hit the error limits of Mathematica - i.e., they become inaccurate. So, I want to crunch the data I have already generated rather than modify the original calculation.

Is this possible? Maybe I need to turn testtable1 into a dataset? Pointers on how to tackle this would be much appreciated.

Answer 1

Suppose you generate you table equivalently by

alist = Range[1, 6];
blist = Range[1, 6];
testtable1 = Table[a/b + a^(1/2), {b, blist}, {a, alist}];

Then

Table[(a/b + a^(1/2))^b, {b, blist}, {a, alist}]

can be obtained also by

testtable1^blist

and

Table[(a/b + a^(1/2))^a, {b, blist}, {a, alist}]

can be obtained by

testtable1^ConstantArray[alist, Length[blist]]

The key observation is that ^ (a.k.a. Power) has the attribute Listable.

For a preexisting table

testtable2 = RandomReal[{-1, 1}, {1000, 2000}];

the following should raise each row to the power of its row count:

poweredbyrow = testtable2^Range[1, Length[testtable2]];

The same for powering by column number:

poweredbycol = 
  testtable2^ConstantArray[
    Range[1, Dimensions[testtable2][[2]]], 
    Length[testtable2]
    ];

Answer 2

row $k$ raised to the power $k$:

MapIndexed[#^#2[[1]] &, testtable1, {1}] // MatrixForm // TeXForm

$\left( \begin{array}{cccccc} 2 & 2+\sqrt{2} & 3+\sqrt{3} & 6 & 5+\sqrt{5} & 6+\sqrt{6} \\ \frac{9}{4} & \left(1+\sqrt{2}\right)^2 & \left(\frac{3}{2}+\sqrt{3}\right)^2 & 16 & \left(\frac{5}{2}+\sqrt{5}\right)^2 & \left(3+\sqrt{6}\right)^2 \\ \frac{64}{27} & \left(\frac{2}{3}+\sqrt{2}\right)^3 & \left(1+\sqrt{3}\right)^3 & \frac{1000}{27} & \left(\frac{5}{3}+\sqrt{5}\right)^3 & \left(2+\sqrt{6}\right)^3 \\ \frac{625}{256} & \left(\frac{1}{2}+\sqrt{2}\right)^4 & \left(\frac{3}{4}+\sqrt{3}\right)^4 & 81 & \left(\frac{5}{4}+\sqrt{5}\right)^4 & \left(\frac{3}{2}+\sqrt{6}\right)^4 \\ \frac{7776}{3125} & \left(\frac{2}{5}+\sqrt{2}\right)^5 & \left(\frac{3}{5}+\sqrt{3}\right)^5 & \frac{537824}{3125} & \left(1+\sqrt{5}\right)^5 & \left(\frac{6}{5}+\sqrt{6}\right)^5 \\ \frac{117649}{46656} & \left(\frac{1}{3}+\sqrt{2}\right)^6 & \left(\frac{1}{2}+\sqrt{3}\right)^6 & \frac{262144}{729} & \left(\frac{5}{6}+\sqrt{5}\right)^6 & \left(1+\sqrt{6}\right)^6 \\ \end{array} \right)$

column $k$ raised to the power $k$:

MapIndexed[#^#2[[2]] &, testtable1, {2}] // MatrixForm // TeXForm

$\left( \begin{array}{cccccc} 2 & \left(2+\sqrt{2}\right)^2 & \left(3+\sqrt{3}\right)^3 & 1296 & \left(5+\sqrt{5}\right)^5 & \left(6+\sqrt{6}\right)^6 \\ \frac{3}{2} & \left(1+\sqrt{2}\right)^2 & \left(\frac{3}{2}+\sqrt{3}\right)^3 & 256 & \left(\frac{5}{2}+\sqrt{5}\right)^5 & \left(3+\sqrt{6}\right)^6 \\ \frac{4}{3} & \left(\frac{2}{3}+\sqrt{2}\right)^2 & \left(1+\sqrt{3}\right)^3 & \frac{10000}{81} & \left(\frac{5}{3}+\sqrt{5}\right)^5 & \left(2+\sqrt{6}\right)^6 \\ \frac{5}{4} & \left(\frac{1}{2}+\sqrt{2}\right)^2 & \left(\frac{3}{4}+\sqrt{3}\right)^3 & 81 & \left(\frac{5}{4}+\sqrt{5}\right)^5 & \left(\frac{3}{2}+\sqrt{6}\right)^6 \\ \frac{6}{5} & \left(\frac{2}{5}+\sqrt{2}\right)^2 & \left(\frac{3}{5}+\sqrt{3}\right)^3 & \frac{38416}{625} & \left(1+\sqrt{5}\right)^5 & \left(\frac{6}{5}+\sqrt{6}\right)^6 \\ \frac{7}{6} & \left(\frac{1}{3}+\sqrt{2}\right)^2 & \left(\frac{1}{2}+\sqrt{3}\right)^3 & \frac{4096}{81} & \left(\frac{5}{6}+\sqrt{5}\right)^5 & \left(1+\sqrt{6}\right)^6 \\ \end{array} \right) $

Also:

Transpose @ MapIndexed[#^#2[[1]] &, Transpose @ testtable1, {1}]

same result