extraction matrix and list entry extraction

## Syntax

                x(i)
                x(i,j)
                x(i,j,k,..)
                [...] = l(i)
                [...] = l(k1)...(kn)(i) or [...] = l(list(k1,...,kn,i))
                l(k1)...(kn)(i,j) or l(list(k1,...,kn,list(i,j))

## Arguments

x

  matrix of any possible types

l

  list variable

i,j, k

  indices

k1,...kn

  indices

## Description

### Matrix case

`i`, `j`, `k`,.. can be:

a real scalar or a vector or a matrix with positive elements.

  - `r=x(i,j)` builds the matrix `r` such as `r(l,k)=x(int(i(l)),int(j(k)))` for `l` from 1 to `size(i,'*')` and `k` from 1 to `size(j,'*')`.

    `i` (`j`) Maximum value must be less or equal to `size(x,1)` (`size(x,2)`).

  - `r=x(i)` with `x` a 1x1 matrix builds the matrix `r` such as `r(l,k)=x(int(i(l)),int(i(k)))` for `l` from 1 to `size(i,1)` and `k` from 1 to `size(i,2)`.

    Note that in this case index `i` is valid only if all its entries are equal to one.

  - `r=x(i)` with `x` a row vector builds the row vector `r` such as `r(l)=x(int(i(l)))` for `l` from 1 to `size(i,'*')``i` Maximum value must be less or equal to `size(x,'*')`.

  - `r=x(i)` with `x` a matrix with one or more columns builds the column vector `r` such as `r(l)` (`l` from 1 to `size(i,'*')`) contains the `int(i(l))` entry of the column vector formed by the concatenation of the `x`'s columns.

    `i` Maximum value must be less or equal to `size(x,'*')`.

the symbol `:`

  `` which stands for "all elements".

  - `r=x(i,:)` builds the matrix `r` such as `r(l,k)=x(int(i(l)),k))` for `l` from 1 to `size(i,'*')` and `k` from 1 to `size(x,2)`

  - `r=x(:,j)` builds the matrix `r` such as `r(l,k)=x(l,int(j(k)))` for `l` from 1 to `size(r,1)` and `k` from 1 to `size(j,'*')`.

  - `r=x(:)` builds the column vector `r` formed by the column concatenations of `x` columns. It is equivalent to `matrix(x,size(x,'*'),1)`.

a vector of boolean

  If an index (`i` or `j`) is a vector of booleans it is interpreted as `find(i)` or respectively `find(j)`

a polynomial

  If an index (`i` or `j`) is a vector of polynomials or implicit polynomial vector it is interpreted as `horner(i,m)` or respectively `horner(j,n)` where `m` and `n` are associated `x` dimensions. Even if this feature works for all polynomials, it is recommended to use polynomials in `$` for readability.

For matrices with more than 2 dimensions (see:hypermatrices) the dimensionality is automatically reduced when right-most dimensions are equal to 1.

### List or Tlist case

If they are present the `ki` give the path to a sub-list entry of `l` data structure. They allow a recursive extraction without intermediate copies. The instructions

`[...]=l(k1)...(kn)(i)`

and

`[...]=l(list(k1,...,kn,i))`

are interpreted as:

`lk1 = l(k1)`

`.. = ..`

`lkn = lkn-1(kn)`

`[...] = lkn(i)`.

And the `l(k1)...(kn)(i,j)` and

`l(list(k1,...,kn,list(i,j))` instructions are interpreted as:

`lk1 = l(k1)`

`.. = ..`

`lkn = lkn-1(kn)`

`lkn(i,j)`

When path points to more than one list component, the instruction must have as many left-hand side arguments as selected components. But if the extraction syntax is used as function parameters, each returned list component is added to the function parameters.

Note that `l(list())` is the same as `l`.

i and j may be:
  real scalar or vector or matrix with positive elements.

    `[r1,...rn]=l(i)` extracts the `i(k)` elements from the list `l` and store them in `rk` variables for `k` from 1 to `size(i,'*')`

  the symbol `:`

    which stands for "all elements".

  a vector of booleans.

    If `i` is a vector of booleans it is interpreted as `find(i)`.

  a polynomial

    If `i` is a vector of polynomials or implicit polynomial vector it is interpreted as `horner(i,m)` where `m=size(l)`. Even if this feature works for all polynomials, it is recommended to use polynomials in `$` for readability.

k1 ... kn may be :
  real positive scalar

  a polynomial

    interpreted as `horner(ki,m)` where `m` is the corresponding sub-list size.

  a character string

    associated with a sub-list entry name.

### Remarks

For soft coded matrix types such as rational functions and state space linear systems, `x(i)` syntax may not be used for vector element extraction due to confusion with list element extraction. `x(1,j)` or `x(i,1)` syntax must be used.

## Examples

Matrix case:

    a = [1 2 3 ; 4 5 6]
    a(1,2)
    a([1 1], 2)
    a(:, 1)
    a(:, 3:-1:1)
    a(1)
    a(6)
    a(:)
    a([%t %f %f %t])
    a([%t %f], [2 3])
    a(1:2, $-1)
    a($:-1:1, 2)
    a($)
    //
    x = 'test'
    x([1 1 ; 1 1 ; 1 1])
    //
    b = [1/%s, (%s+1)/(%s-1)]
    b(1,1)
    b(1,$)
    b(2) // the numerator

List ot Tlist case:

    l = list(1,'qwerw',%s)
    l(1)
    [a,b] = l([3 2])
    l($)
    x = tlist(l(2:3)) // forms a tlist with the last 2 components of l
    //
    dts = list(1, tlist(['x';'a';'b'],10,[2 3]));
    dts(2)('a')
    dts(2)('b')(1,2)
    [a,b] = dts(2)(['a','b'])

## See also

- insertion
- colon
- find
- horner
- parentheses