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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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