How to sum matrix elements based on finding the first (and second) non-zero elements of each row?

Question

I have a matrix:

I would like to sum all the first non-zero elements of each row so that I get a value of

$$25.5317 + 8.85471 + 6.90018 + 32.9436 + ... $$

and so on and simply ignore zero rows.

Similarly I would like to do the same again for the second non-zero elements from each row so that I get:

$$29.1235 + 11.0472 + 41.2639$$

Answer 1

 tbl = RandomChoice[{.5, .1, .05, .05, .05, .05, .05, .05, .05, .05} ->
     Range[0, 9], {10, 10}];
 tbl//TableForm

Remove all zeros from tbl and add zeros to the right to make a full array:

PadRight[tbl /. (0) -> Sequence[], Automatic] // TableForm

Then sum each column:

 Tr /@ (Transpose@PadRight[tbl /. (0) -> Sequence[], Automatic])
 (* {55, 59, 36, 28, 38, 14, 9} *)

Update: To get a matrix with dimensions {nrows, ncolumns} from PadRight change Automatic to {nrows, ncolumns}. So,

 Tr /@ (Transpose@PadRight[tbl /. (0) -> Sequence[], {10,10}])
 (* {55, 59, 36, 28, 38, 14, 9, 0 ,0, 0} *)

Answer 2

Not as nice as the method given by kguler. Nevertheless, as an alternative:

myarray = {{0., 0., 0., 0., 0}, {0., 0., 0., 0., 0},
  {0., 0., 0., 25.5317, 29.1235}, {0., 0., 0., 0., 8.85471}, {0., 0., 
   0., 0., 0}, {0., 0., 0., 0., 0}, {0., 0., 0., 0., 6.90018}, {0., 
   0., 0., 0., 32.9436}, {0., 0., 0., 0., 29.1235}, {0., 0., 0., 
   2.47854, 11.0472}, {0., 0., 16.1408, 41.2639, 45.7614}, {0., 0., 
   0., 0., 0}, {0., 0., 0., 0., 0}, {0., 0., 0., 0., 0}}

(Using Flatten to transpose a ragged array):

Flatten[Select[#, FreeQ[#, 0. | 0] &] & /@ myarray, {2}]

=>

{{25.5317, 8.85471, 6.90018, 32.9436, 29.1235, 2.47854, 16.1408}, {29.1235, 11.0472, 41.2639}, {45.7614}}

Therefore,

Total /@ Flatten[Select[#, FreeQ[#, 0. | 0] &] & /@ myarray, {2}]

=>

{121.973, 81.4346, 45.7614}

Comparing with kguler's method

tbl = RandomChoice[{.5, .1, .05, .05, .05, .05, .05, .05, .05, .05} ->
     Range[0, 9], {10, 10}];
Tr /@ (Transpose@PadRight[tbl /. (0) -> Sequence[], Automatic]) == 
 Total /@ Flatten[Select[#, FreeQ[#, 0. | 0] &] & /@ tbl, {2}]

=>

True

Edit

Better, perhaps, is

Total /@ Flatten[DeleteCases[myarray, 0 | 0., 2], {2}]

However, the best variation on the Flatten approach, I recon, is given below (as a comment) by rm -rf:

Total[Flatten[# /. 0 | 0. -> Sequence[], {2}], {2}] &@myarray

=>

{121.973, 81.4346, 45.7614}

Answer 3

One can search for the first nonzero element in each row using Cases. Generate a random table:

m = Normal@SparseArray[
      Thread@{RandomInteger[{1, 12}, 14], RandomInteger[{1, 6}, 14]} :> RandomReal[], {12, 6}];

and pick the first match in each row:

Flatten[Cases[#, _?(! PossibleZeroQ@# &), 1, 1] & /@ m]

Or extending the Sequence replacement method in kguler's answer:

m /. {0 | 0. -> Sequence[]} /. {} -> Sequence[] /. {x_?NumberQ, ___} :> x

{0.252222, 0.998427, 0.0562231, 0.849365, 0.582891, 0.0896831, 0.31189, 0.524075}

To get the second nonzero element from each row:

m /. {0 | 0. -> Sequence[]} /. {} | {_} -> Sequence[] /. {_?NumberQ, x_, ___} :> x

{0.949761, 0.841579, 0.0275406, 0.500705}

Total then does the summing.