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Miscellaneous 128!:

128!:0 y QR. Produces the QR decomposition of a complex matrix y (in the domain of matrix inverse %.), an Hermitian matrix and a square upper triangular matrix, individually boxed.
   x=: +/ . *                  Matrix product
   A=: j./?. 2 7 4$10          A random complex matrix
   $A
7 4
   'Q R'=: 128!:0 A
   $Q
7 4
   $R
4 4
   >./|,(=i.4) - (+|:Q) x Q    Q is Hermitian
6.33846e_16
   0~:R                        R is upper triangular
1 1 1 1
0 1 1 1
0 0 1 1
0 0 0 1
   A -: Q x R
1
128!:1 y R Inv. Invert square upper triangular matrix.

x 128!:2 y Apply. x 128!:2 y applies the verb in string x to y . For example:
   '+/' 128!:2 i.2 5
5 7 9 11 13
   '+/' 128!:2"1 i.2 5
10 35
   '+/"1' 128!:2 i.2 5
10 35
   ('+/';'|.';'|."1') 128!:2&.><i.2 5
+-----------+---------+---------+
|5 7 9 11 13|5 6 7 8 9|4 3 2 1 0|
|           |0 1 2 3 4|9 8 7 6 5|
+-----------+---------+---------+

   '2 3' 128!:2 i.2 5
|syntax error
|   '2 3'    128!:2 i.2 5

   '@' 128!:2 i.2 5
|syntax error
|   '@'    128!:2 i.2 5
The ranks of 128!:2 are 1 _ , that is, apply the lists in the left argument to the right argument in toto.



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