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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.
   mp=: +/ . *                  NB. Matrix product
   A=: j./?. 2 7 4$10           NB. A random complex matrix
   $A
7 4
   'Q R'=: 128!:0 A
   $Q
7 4
   $R
4 4
   >./|,(=i.4) - (+|:Q) mp Q    NB. Q is Hermitian
2.70972e_16
   0~:R                         NB. R is upper triangular
1 1 1 1
0 1 1 1
0 0 1 1
0 0 0 1
   A -: Q mp 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.

[x] 128!:3 y CRC. A CRC polynomial is a boolean list or an integer. The following CRC polynomials p and q are equivalent:

   p=: 1 1 1 0  1 1 0 1  1 0 1 1  1 0 0 0  1 0 0 0  0 0 1 1  0 0 1 0  0 0 0 0
   q=: _306674912

   _8]\p
1 1 1 0 1 1 0 1
1 0 1 1 1 0 0 0
1 0 0 0 0 0 1 1
0 0 1 0 0 0 0 0

   q -: (_2 _2,30$2) #. p
1
   p -: (32$2) #: q
1

x 128!:3 y computes the CRC encoding of string y according to x , which may be p or p;i , where p is a CRC polynomial and i is the initial CRC value. If i is elided then _1 is used. 128!:3 y is equivalent to _306674912 (128!:3) y .

The monad x&(128!:3) is supported by special code which pre-computes a look-up table of CRC values for each byte.

Examples:

   f=: 128!:3
   f '123456789'
_873187034
   f 'assiduously avoid any and all asinine alliterations'
1439575093
128!:4 y RNG Raw. See Roll/Deal (?).

128!:5 y

Is NaN. If z=: 128!:5 y , then z has the same shape as y and an atom of z is 1 if and only if the corresponding atom of y is _. (NaN) or contains _. .

Examples:

   (128!:5) 3.45 6 _. 7
0 0 1 0
   (128!:5) t=. 2 2$1 2;3 4 5;(<<_. 7);_.
0 0 
1 1
   (128!:5) < t
1
   (128!:5) ;:'Cogito, ergo sum. _.'
0 0 0 0 0


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