Maxima 5.44.0 Manual: Functions and Variables for Lists

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#### 5.4.2 Functions and Variables for Lists

Operator: [
Operator: ]

  `[` and `]` mark the beginning and end, respectively, of a list.

  `[` and `]` also enclose the subscripts of a list, array, `hashed array`, or `memoizing function`. Note that other than for arrays accessing the `n`th element of a list may need an amount of time that is roughly proportional to `n`, See Performance considerations for Lists.

  Note that if an element of a subscripted variable is written to before a list or an array of this name is declared a `hashed array` (see Arrays) is created, not a list.

  Examples:

      (%i1) x: [a, b, c];
      (%o1)                       [a, b, c]

      (%i2) x[3];
      (%o2)                           c

      (%i3) array (y, fixnum, 3);
      (%o3)                           y

      (%i4) y[2]: %pi;
      (%o4)                          %pi

      (%i5) y[2];
      (%o5)                          %pi

      (%i6) z['foo]: 'bar;
      (%o6)                          bar

      (%i7) z['foo];
      (%o7)                          bar

      (%i8) g[k] := 1/(k^2+1);
                                        1
      (%o8)                     g  := ------
                                 k     2
                                      k  + 1

      (%i9) g[10];
                                      1
      (%o9)                          ---
                                     101

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  Categories: Lists ·Operators

Function: append (list_1, …, list_n)

  Returns a single list of the elements of list_1 followed by the elements of list_2, … `append` also works on general expressions, e.g. `append (f(a,b), f(c,d,e));` yields `f(a,b,c,d,e)`.

  See also `addrow`, `addcol` and `join`.

  Do `example(append);` for an example.

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  Categories: Lists ·Expressions

Function: assoc
`assoc` (key, e, default)
`assoc` (key, e)

  `assoc` searches for key as the first part of an argument of e and returns the second part of the first match, if any.

  key may be any expression. e must be a nonatomic expression, and every argument of e must have exactly two parts. `assoc` returns the second part of the first matching argument of e. Matches are determined by `is(key = first(a))` where a is an argument of e.

  If there are two or more matches, only the first is returned. If there are no matches, default is returned, if specified. Otherwise, `false` is returned.

  See also `sublist` and `member`.

  Examples:

  key may be any expression. e must be a nonatomic expression, and every argument of e must have exactly two parts. `assoc` returns the second part of the first matching argument of e.

      (%i1) assoc (f(x), foo(g(x) = y, f(x) = z + 1, h(x) = 2*u));
      (%o1)                         z + 1

  If there are two or more matches, only the first is returned.

      (%i1) assoc (yy, [xx = 111, yy = 222, yy = 333, yy = 444]);
      (%o1)                          222

  If there are no matches, default is returned, if specified. Otherwise, `false` is returned.

      (%i1) assoc (abc, [[x, 111], [y, 222], [z, 333]], none);
      (%o1)                         none
      (%i2) assoc (abc, [[x, 111], [y, 222], [z, 333]]);
      (%o2)                         false

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  Categories: Lists ·Expressions

Function: cons
`cons` (expr, list)
`cons` (expr_1, expr_2)

  `cons (expr, list)` returns a new list constructed of the element expr as its first element, followed by the elements of list. This is analogous to the Lisp language construction operation "cons".

  The Maxima function `cons` can also be used where the second argument is other than a list and this might be useful. In this case, `cons (expr_1, expr_2)` returns an expression with same operator as expr_2 but with argument `cons(expr_1, args(expr_2))`. Examples:

      (%i1) cons(a,[b,c,d]);
      (%o1)                     [a, b, c, d]

      (%i2) cons(a,f(b,c,d));
      (%o2)                     f(a, b, c, d)

  In general, `cons` applied to a nonlist doesn’t make sense. For instance, `cons(a,b^c)` results in an illegal expression, since ’^’ cannot take three arguments.

  When `inflag` is true, `cons` operates on the internal structure of an expression, otherwise `cons` operates on the displayed form. Especially when `inflag` is true, `cons` applied to a nonlist sometimes gives a surprising result; for example

      (%i1) cons(a,-a), inflag : true;
                                       2
      (%o1)                         - a

      (%i2) cons(a,-a), inflag : false;
      (%o2)                           0

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  Categories: Lists ·Expressions

Function: copylist (list)

  Returns a copy of the list list.

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  Categories: Lists

Function: create_list (form, x_1, list_1, …, x_n, list_n)

  Create a list by evaluating form with x_1 bound to each element of list_1, and for each such binding bind x_2 to each element of list_2, … The number of elements in the result will be the product of the number of elements in each list. Each variable x_i must actually be a symbol – it will not be evaluated. The list arguments will be evaluated once at the beginning of the iteration.

      (%i1) create_list (x^i, i, [1, 3, 7]);
                                      3   7
      (%o1)                      [x, x , x ]

  With a double iteration:

      (%i1) create_list ([i, j], i, [a, b], j, [e, f, h]);
      (%o1)   [[a, e], [a, f], [a, h], [b, e], [b, f], [b, h]]

  Instead of list_i two args may be supplied each of which should evaluate to a number. These will be the inclusive lower and upper bounds for the iteration.

      (%i1) create_list ([i, j], i, [1, 2, 3], j, 1, i);
      (%o1)   [[1, 1], [2, 1], [2, 2], [3, 1], [3, 2], [3, 3]]

  Note that the limits or list for the `j` variable can depend on the current value of `i`.

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  Categories: Lists

Function: delete
`delete` (expr_1, expr_2)
`delete` (expr_1, expr_2, n)

  `delete(expr_1, expr_2)` removes from expr_2 any arguments of its top-level operator which are the same (as determined by "=") as expr_1. Note that "=" tests for formal equality, not equivalence. Note also that arguments of subexpressions are not affected.

  expr_1 may be an atom or a non-atomic expression. expr_2 may be any non-atomic expression. `delete` returns a new expression; it does not modify expr_2.

  `delete(expr_1, expr_2, n)` removes from expr_2 the first n arguments of the top-level operator which are the same as expr_1. If there are fewer than n such arguments, then all such arguments are removed.

  Examples:

  Removing elements from a list.

      (%i1) delete (y, [w, x, y, z, z, y, x, w]);
      (%o1)                  [w, x, z, z, x, w]

  Removing terms from a sum.

      (%i1) delete (sin(x), x + sin(x) + y);
      (%o1)                         y + x

  Removing factors from a product.

      (%i1) delete (u - x, (u - w)*(u - x)*(u - y)*(u - z));
      (%o1)                (u - w) (u - y) (u - z)

  Removing arguments from an arbitrary expression.

      (%i1) delete (a, foo (a, b, c, d, a));
      (%o1)                     foo(b, c, d)

  Limit the number of removed arguments.

      (%i1) delete (a, foo (a, b, a, c, d, a), 2);
      (%o1)                    foo(b, c, d, a)

  Whether arguments are the same as expr_1 is determined by "=". Arguments which are `equal` but not "=" are not removed.

      (%i1) [is (equal (0, 0)), is (equal (0, 0.0)), is (equal (0, 0b0))];
      (%o1)                  [true, true, true]

      (%i2) [is (0 = 0), is (0 = 0.0), is (0 = 0b0)];
      (%o2)                 [true, false, false]

      (%i3) delete (0, [0, 0.0, 0b0]);
      (%o3)                     [0.0, 0.0b0]

      (%i4) is (equal ((x + y)*(x - y), x^2 - y^2));
      (%o4)                         true

      (%i5) is ((x + y)*(x - y) = x^2 - y^2);
      (%o5)                         false

      (%i6) delete ((x + y)*(x - y), [(x + y)*(x - y), x^2 - y^2]);
                                    2    2
      (%o6)                       [x  - y ]

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  Categories: Lists ·Expressions

Function: eighth (expr)

  Returns the 8th item of expression or list expr. See `first` for more details.

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  Categories: Lists ·Expressions

Function: endcons
`endcons` (expr, list)
`endcons` (expr_1, expr_2)

  `endcons (expr, list)` returns a new list constructed of the elements of list followed by expr. The Maxima function `endcons` can also be used where the second argument is other than a list and this might be useful. In this case, `endcons (expr_1, expr_2)` returns an expression with same operator as expr_2 but with argument `endcons(expr_1, args(expr_2))`. Examples:

      (%i1) endcons(a,[b,c,d]);
      (%o1)                     [b, c, d, a]

      (%i2) endcons(a,f(b,c,d));
      (%o2)                     f(b, c, d, a)

  In general, `endcons` applied to a nonlist doesn’t make sense. For instance, `endcons(a,b^c)` results in an illegal expression, since ’^’ cannot take three arguments.

  When `inflag` is true, `endcons` operates on the internal structure of an expression, otherwise `endcons` operates on the displayed form. Especially when `inflag` is true, `endcons` applied to a nonlist sometimes gives a surprising result; for example

      (%i1) endcons(a,-a), inflag : true;
                                       2
      (%o1)                         - a

      (%i2) endcons(a,-a), inflag : false;
      (%o2)                           0

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  Categories: Lists ·Expressions

Function: fifth (expr)

  Returns the 5th item of expression or list expr. See `first` for more details.

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  Categories: Lists ·Expressions

Function: first (expr)

  Returns the first part of expr which may result in the first element of a list, the first row of a matrix, the first term of a sum, etc.:

      (%i1) matrix([1,2],[3,4]);
                                         [ 1  2 ]
      (%o1)                              [      ]
                                         [ 3  4 ]
      (%i2) first(%);
      (%o2)                              [1,2]
      (%i3) first(%);
      (%o3)                              1
      (%i4) first(a*b/c+d+e/x);
                                         a b
      (%o4)                              ---
                                          c
      (%i5) first(a=b/c+d+e/x);
      (%o5)                              a

  Note that `first` and its related functions, `rest` and `last`, work on the form of expr which is displayed not the form which is typed on input. If the variable `inflag` is set to `true` however, these functions will look at the internal form of expr. One reason why this may make a difference is that the simplifier re-orders expressions:

      (%i1) x+y;
      (%o1)                              y+1
      (%i2) first(x+y),inflag : true;
      (%o2)                              x
      (%i3) first(x+y),inflag : false;
      (%o3)                              y

  The functions `second` … `tenth` yield the second through the tenth part of their input argument.

  See also `firstn` and `part`.

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  Categories: Lists ·Expressions

Function: firstn (expr, count)

  Returns the first count arguments of expr, if expr has at least count arguments. Returns expr if expr has less than count arguments.

  expr may be any nonatomic expression. When expr is something other than a list, `firstn` returns an expression which has the same operator as expr. count must be a nonnegative integer.

  `firstn` honors the global flag `inflag`, which governs whether the internal form of an expression is processed (when `inflag` is true) or the displayed form (when `inflag` is false).

  Note that `firstn(expr, 1)`, which returns a nonatomic expression containing the first argument, is not the same as `first(expr)`, which returns the first argument by itself.

  See also `lastn` and `rest`.

  Examples:

  `firstn` returns the first count elements of expr, if expr has at least count elements.

      (%i1) mylist : [1, a, 2, b, 3, x, 4 - y, 2*z + sin(u)];
      (%o1)        [1, a, 2, b, 3, x, 4 - y, 2 z + sin(u)]

      (%i2) firstn (mylist, 0);
      (%o2)                          []

      (%i3) firstn (mylist, 1);
      (%o3)                          [1]

      (%i4) firstn (mylist, 2);
      (%o4)                        [1, a]

      (%i5) firstn (mylist, 7);
      (%o5)               [1, a, 2, b, 3, x, 4 - y]

  `firstn` returns expr if expr has less than count elements.

      (%i1) mylist : [1, a, 2, b, 3, x, 4 - y, 2*z + sin(u)];
      (%o1)        [1, a, 2, b, 3, x, 4 - y, 2 z + sin(u)]

      (%i2) firstn (mylist, 100);
      (%o2)        [1, a, 2, b, 3, x, 4 - y, 2 z + sin(u)]

  expr may be any nonatomic expression.

      (%i1) myfoo : foo(1, a, 2, b, 3, x, 4 - y, 2*z + sin(u));
      (%o1)      foo(1, a, 2, b, 3, x, 4 - y, 2 z + sin(u))

      (%i2) firstn (myfoo, 4);
      (%o2)                    foo(1, a, 2, b)

      (%i3) mybar : bar[m, n](1, a, 2, b, 3, x, 4 - y, 2*z + sin(u));
      (%o3)    bar    (1, a, 2, b, 3, x, 4 - y, 2 z + sin(u))
                  m, n

      (%i4) firstn (mybar, 4);
      (%o4)                  bar    (1, a, 2, b)
                                m, n

      (%i5) mymatrix : genmatrix (lambda ([i, j], 10*i + j), 10, 4) $

      (%i6) firstn (mymatrix, 3);
                             [ 11  12  13  14 ]
                             [                ]
      (%o6)                  [ 21  22  23  24 ]
                             [                ]
                             [ 31  32  33  34 ]

  `firstn` honors the global flag `inflag`.

      (%i1) myexpr : a + b + c + d + e;
      (%o1)                   e + d + c + b + a

      (%i2) firstn (myexpr, 3), inflag=true;
      (%o2)                       c + b + a

      (%i3) firstn (myexpr, 3), inflag=false;
      (%o3)                       e + d + c

  Note that `firstn(expr, 1)` is not the same as `first(expr)`.

      (%i1) firstn ([w, x, y, z], 1);
      (%o1)                          [w]

      (%i2) first ([w, x, y, z]);
      (%o2)                           w

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  Categories: Lists ·Expressions

Function: fourth (expr)

  Returns the 4th item of expression or list expr. See `first` for more details.

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  Categories: Lists ·Expressions

Function: join (l, m)

  Creates a new list containing the elements of lists l and m, interspersed. The result has elements `[l[1], m[1], l[2], m[2], ...]`. The lists l and m may contain any type of elements.

  If the lists are different lengths, `join` ignores elements of the longer list.

  Maxima complains if l or m is not a list.

  See also `append`.

  Examples:

      (%i1) L1: [a, sin(b), c!, d - 1];
      (%o1)                [a, sin(b), c!, d - 1]

      (%i2) join (L1, [1, 2, 3, 4]);
      (%o2)          [a, 1, sin(b), 2, c!, 3, d - 1, 4]

      (%i3) join (L1, [aa, bb, cc, dd, ee, ff]);
      (%o3)        [a, aa, sin(b), bb, c!, cc, d - 1, dd]

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  Categories: Lists

Function: last (expr)

  Returns the last part (term, row, element, etc.) of the expr.

  See also `lastn`.

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  Categories: Lists ·Expressions

Function: lastn (expr, count)

  Returns the last count arguments of expr, if expr has at least count arguments. Returns expr if expr has less than count arguments.

  expr may be any nonatomic expression. When expr is something other than a list, `lastn` returns an expression which has the same operator as expr. count must be a nonnegative integer.

  `lastn` honors the global flag `inflag`, which governs whether the internal form of an expression is processed (when `inflag` is true) or the displayed form (when `inflag` is false).

  Note that `lastn(expr, 1)`, which returns a nonatomic expression containing the last argument, is not the same as `last(expr)`, which returns the last argument by itself.

  See also `firstn` and `rest`.

  Examples:

  `lastn` returns the last count elements of expr, if expr has at least count elements.

      (%i1) mylist : [1, a, 2, b, 3, x, 4 - y, 2*z + sin(u)];
      (%o1)        [1, a, 2, b, 3, x, 4 - y, 2 z + sin(u)]

      (%i2) lastn (mylist, 0);
      (%o2)                          []

      (%i3) lastn (mylist, 1);
      (%o3)                    [2 z + sin(u)]

      (%i4) lastn (mylist, 2);
      (%o4)                 [4 - y, 2 z + sin(u)]

      (%i5) lastn (mylist, 7);
      (%o5)         [a, 2, b, 3, x, 4 - y, 2 z + sin(u)]

  `lastn` returns expr if expr has less than count elements.

      (%i1) mylist : [1, a, 2, b, 3, x, 4 - y, 2*z + sin(u)];
      (%o1)        [1, a, 2, b, 3, x, 4 - y, 2 z + sin(u)]

      (%i2) lastn (mylist, 100);
      (%o2)        [1, a, 2, b, 3, x, 4 - y, 2 z + sin(u)]

  expr may be any nonatomic expression.

      (%i1) myfoo : foo(1, a, 2, b, 3, x, 4 - y, 2*z + sin(u));
      (%o1)      foo(1, a, 2, b, 3, x, 4 - y, 2 z + sin(u))

      (%i2) lastn (myfoo, 4);
      (%o2)            foo(3, x, 4 - y, 2 z + sin(u))

      (%i3) mybar : bar[m, n](1, a, 2, b, 3, x, 4 - y, 2*z + sin(u));
      (%o3)    bar    (1, a, 2, b, 3, x, 4 - y, 2 z + sin(u))
                  m, n

      (%i4) lastn (mybar, 4);
      (%o4)          bar    (3, x, 4 - y, 2 z + sin(u))
                        m, n

      (%i5) mymatrix : genmatrix (lambda ([i, j], 10*i + j), 10, 4) $

      (%i6) lastn (mymatrix, 3);
                           [ 81   82   83   84  ]
                           [                    ]
      (%o6)                [ 91   92   93   94  ]
                           [                    ]
                           [ 101  102  103  104 ]

  `lastn` honors the global flag `inflag`.

      (%i1) myexpr : a + b + c + d + e;
      (%o1)                   e + d + c + b + a

      (%i2) lastn (myexpr, 3), inflag=true;
      (%o2)                       e + d + c

      (%i3) lastn (myexpr, 3), inflag=false;
      (%o3)                       c + b + a

  Note that `lastn(expr, 1)` is not the same as `last(expr)`.

      (%i1) lastn ([w, x, y, z], 1);
      (%o1)                          [z]

      (%i2) last ([w, x, y, z]);
      (%o2)                           z

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  Categories: Lists ·Expressions

Function: length (expr)

  Returns (by default) the number of parts in the external (displayed) form of expr. For lists this is the number of elements, for matrices it is the number of rows, and for sums it is the number of terms (see `dispform`).

  The `length` command is affected by the `inflag` switch. So, e.g. `length(a/(b*c));` gives 2 if `inflag` is `false` (Assuming `exptdispflag` is `true`), but 3 if `inflag` is `true` (the internal representation is essentially `a*b^-1*c^-1`).

  Determining a list’s length typically needs an amount of time proportional to the number of elements in the list. If the length of a list is used inside a loop it therefore might drastically increase the performance if the length is calculated outside the loop instead.

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  Categories: Lists ·Expressions

Option variable: listarith

  Default value: `true`

  If `false` causes any arithmetic operations with lists to be suppressed; when `true`, list-matrix operations are contagious causing lists to be converted to matrices yielding a result which is always a matrix. However, list-list operations should return lists.

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  Categories: Lists ·Global flags

Function: listp (expr)

  Returns `true` if expr is a list else `false`.

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  Categories: Lists ·Predicate functions

Function: lreduce
`lreduce` (F, s)
`lreduce` (F, s, s_0)

  Extends the binary function F to an n-ary function by composition, where s is a list.

  `lreduce(F, s)` returns `F(... F(F(s_1, s_2), s_3), ... s_n)`. When the optional argument s_0 is present, the result is equivalent to `lreduce(F, cons(s_0, s))`.

  The function F is first applied to the leftmost list elements, thus the name "lreduce".

  See also `rreduce`, `xreduce`, and `tree_reduce`.

  Examples:

  `lreduce` without the optional argument.

      (%i1) lreduce (f, [1, 2, 3]);
      (%o1)                     f(f(1, 2), 3)

      (%i2) lreduce (f, [1, 2, 3, 4]);
      (%o2)                  f(f(f(1, 2), 3), 4)

  `lreduce` with the optional argument.

      (%i1) lreduce (f, [1, 2, 3], 4);
      (%o1)                  f(f(f(4, 1), 2), 3)

  `lreduce` applied to built-in binary operators. `/` is the division operator.

      (%i1) lreduce ("^", args ({a, b, c, d}));
                                     b c d
      (%o1)                       ((a ) )

      (%i2) lreduce ("/", args ({a, b, c, d}));
                                      a
      (%o2)                         -----
                                    b c d

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  Categories: Lists

Function: makelist
`makelist` ()
`makelist` (expr, n)
`makelist` (expr, i, i_max)
`makelist` (expr, i, i_0, i_max)
`makelist` (expr, i, i_0, i_max, step)
`makelist` (expr, x, list)

  The first form, `makelist ()`, creates an empty list. The second form, `makelist (expr)`, creates a list with expr as its single element. `makelist (expr, n)` creates a list of n elements generated from expr.

  The most general form, `makelist (expr, i, i_0, i_max, step)`, returns the list of elements obtained when `ev (expr, i=j)` is applied to the elements j of the sequence: i_0, i_0 + step, i_0 + 2*step, ..., with |j| less than or equal to |i_max|.

  The increment step can be a number (positive or negative) or an expression. If it is omitted, the default value 1 will be used. If both i_0 and step are omitted, they will both have a default value of 1.

  `makelist (expr, x, list)` returns a list, the `j`th element of which is equal to `ev (expr, x=list[j])` for `j` equal to 1 through `length (list)`.

  Examples:

      (%i1) makelist (concat (x,i), i, 6);
      (%o1)               [x1, x2, x3, x4, x5, x6]

      (%i2) makelist (x=y, y, [a, b, c]);
      (%o2)                 [x = a, x = b, x = c]

      (%i3) makelist (x^2, x, 3, 2*%pi, 2);
      (%o3)                        [9, 25]

      (%i4) makelist (random(6), 4);
      (%o4)                     [2, 0, 2, 5]

      (%i5) flatten (makelist (makelist (i^2, 3), i, 4));
      (%o5)        [1, 1, 1, 4, 4, 4, 9, 9, 9, 16, 16, 16]

      (%i6) flatten (makelist (makelist (i^2, i, 3), 4));
      (%o6)         [1, 4, 9, 1, 4, 9, 1, 4, 9, 1, 4, 9]

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  Categories: Lists

Function: member (expr_1, expr_2)

  Returns `true` if `is(expr_1 = a)` for some element a in `args(expr_2)`, otherwise returns `false`.

  `expr_2` is typically a list, in which case `args(expr_2) = expr_2` and `is(expr_1 = a)` for some element a in `expr_2` is the test.

  `member` does not inspect parts of the arguments of `expr_2`, so it may return `false` even if `expr_1` is a part of some argument of `expr_2`.

  See also `elementp`.

  Examples:

      (%i1) member (8, [8, 8.0, 8b0]);
      (%o1)                         true

      (%i2) member (8, [8.0, 8b0]);
      (%o2)                         false

      (%i3) member (b, [a, b, c]);
      (%o3)                         true

      (%i4) member (b, [[a, b], [b, c]]);
      (%o4)                         false

      (%i5) member ([b, c], [[a, b], [b, c]]);
      (%o5)                         true

      (%i6) F (1, 1/2, 1/4, 1/8);
                                     1  1  1
      (%o6)                     F(1, -, -, -)
                                     2  4  8

      (%i7) member (1/8, %);
      (%o7)                         true

      (%i8) member ("ab", ["aa", "ab", sin(1), a + b]);
      (%o8)                         true

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  Categories: Lists ·Expressions ·Predicate functions

Function: ninth (expr)

  Returns the 9th item of expression or list expr. See `first` for more details.

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  Categories: Lists ·Expressions

Function: pop (list)

  `pop` removes and returns the first element from the list list. The argument list must be a mapatom that is bound to a nonempty list. If the argument list is not bound to a nonempty list, Maxima signals an error. For examples, see `push`.

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  Categories: Lists ·Expressions

Function: push (item, list)

  `push` prepends the item item to the list list and returns a copy of the new list. The second argument list must be a mapatom that is bound to a list. The first argument item can be any Maxima symbol or expression. If the argument list is not bound to a list, Maxima signals an error.

  To remove the first item from a list, see `pop`.

  Examples:

      (%i1) ll: [];
      (%o1)                          []

      (%i2) push (x, ll);
      (%o2)                          [x]

      (%i3) push (x^2+y, ll);
                                       2
      (%o3)                      [y + x , x]

      (%i4) a: push ("string", ll);
                                           2
      (%o4)                  [string, y + x , x]

      (%i5) pop (ll);
      (%o5)                        string

      (%i6) pop (ll);
                                        2
      (%o6)                        y + x

      (%i7) pop (ll);
      (%o7)                           x

      (%i8) ll;
      (%o8)                          []

      (%i9) a;
                                           2
      (%o9)                  [string, y + x , x]

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  Categories: Lists ·Expressions

Function: rest
`rest` (expr, n)
`rest` (expr)

  Returns expr with its first n elements removed if n is positive and its last `- n` elements removed if n is negative. If n is 1 it may be omitted. The first argument expr may be a list, matrix, or other expression. When expr is an atom, `rest` signals an error; when expr is an empty list and `partswitch` is false, `rest` signals an error. When expr is an empty list and `partswitch` is true, `rest` returns `end`.

  Applying `rest` to expression such as `f(a,b,c)` returns `f(b,c)`. In general, applying `rest` to a nonlist doesn’t make sense. For example, because ’^’ requires two arguments, `rest(a^b)` results in an error message. The functions `args` and `op` may be useful as well, since `args(a^b)` returns `[a,b]` and `op(a^b)` returns ^.

  See also `firstn` and `lastn`.

      (%i1) rest(a+b+c);
      (%o1) b+a
      (%i2) rest(a+b+c,2);
      (%o2) a
      (%i3) rest(a+b+c,-2);
      (%o3) c

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  Categories: Lists ·Expressions

Function: reverse (list)

  Reverses the order of the members of the list (not the members themselves). `reverse` also works on general expressions, e.g. `reverse(a=b);` gives `b=a`.

  See also `sreverse`.

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  Categories: Lists ·Expressions

Function: rreduce
`rreduce` (F, s)
`rreduce` (F, s, s_{n + 1})

  Extends the binary function F to an n-ary function by composition, where s is a list.

  `rreduce(F, s)` returns `F(s_1, ... F(s_{n - 2}, F(s_{n - 1}, s_n)))`. When the optional argument s_{n + 1} is present, the result is equivalent to `rreduce(F, endcons(s_{n + 1}, s))`.

  The function F is first applied to the rightmost list elements, thus the name "rreduce".

  See also `lreduce`, `tree_reduce`, and `xreduce`.

  Examples:

  `rreduce` without the optional argument.

      (%i1) rreduce (f, [1, 2, 3]);
      (%o1)                     f(1, f(2, 3))

      (%i2) rreduce (f, [1, 2, 3, 4]);
      (%o2)                  f(1, f(2, f(3, 4)))

  `rreduce` with the optional argument.

      (%i1) rreduce (f, [1, 2, 3], 4);
      (%o1)                  f(1, f(2, f(3, 4)))

  `rreduce` applied to built-in binary operators. `/` is the division operator.

      (%i1) rreduce ("^", args ({a, b, c, d}));
                                       d
                                      c
                                     b
      (%o1)                         a

      (%i2) rreduce ("/", args ({a, b, c, d}));
                                     a c
      (%o2)                          ---
                                     b d

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  Categories: Lists

Function: second (expr)

  Returns the 2nd item of expression or list expr. See `first` for more details.

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  Categories: Lists ·Expressions

Function: seventh (expr)

  Returns the 7th item of expression or list expr. See `first` for more details.

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  Categories: Lists ·Expressions

Function: sixth (expr)

  Returns the 6th item of expression or list expr. See `first` for more details.

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  Categories: Lists ·Expressions

Function: sort
`sort` (L, P)
`sort` (L)

  `sort(L, P)` sorts a list L according to a predicate `P` of two arguments which defines a strict weak order on the elements of L. If `P(a, b)` is `true`, then `a` appears before `b` in the result. If neither `P(a, b)` nor `P(b, a)` are `true`, then `a` and `b` are equivalent, and appear in the result in the same order as in the input. That is, `sort` is a stable sort.

  If `P(a, b)` and `P(b, a)` are both `true` for some elements of L, then P is not a valid sort predicate, and the result is undefined. If `P(a, b)` is something other than `true` or `false`, `sort` signals an error.

  The predicate may be specified as the name of a function or binary infix operator, or as a `lambda` expression. If specified as the name of an operator, the name must be enclosed in double quotes.

  The sorted list is returned as a new object; the argument L is not modified.

  `sort(L)` is equivalent to `sort(L, orderlessp)`.

  The default sorting order is ascending, as determined by `orderlessp`. The predicate `ordergreatp` sorts a list in descending order.

  All Maxima atoms and expressions are comparable under `orderlessp` and `ordergreatp`.

  Operators `<` and `>` order numbers, constants, and constant expressions by magnitude. Note that `orderlessp` and `ordergreatp` do not order numbers, constants, and constant expressions by magnitude.

  `ordermagnitudep` orders numbers, constants, and constant expressions the same as `<`, and all other elements the same as `orderlessp`.

  Examples:

  `sort` sorts a list according to a predicate of two arguments which defines a strict weak order on the elements of the list.

      (%i1) sort ([1, a, b, 2, 3, c], 'orderlessp);
      (%o1)                  [1, 2, 3, a, b, c]

      (%i2) sort ([1, a, b, 2, 3, c], 'ordergreatp);
      (%o2)                  [c, b, a, 3, 2, 1]

  The predicate may be specified as the name of a function or binary infix operator, or as a `lambda` expression. If specified as the name of an operator, the name must be enclosed in double quotes.

      (%i1) L : [[1, x], [3, y], [4, w], [2, z]];
      (%o1)           [[1, x], [3, y], [4, w], [2, z]]

      (%i2) foo (a, b) := a[1] > b[1];
      (%o2)                 foo(a, b) := a  > b
                                          1    1

      (%i3) sort (L, 'foo);
      (%o3)           [[4, w], [3, y], [2, z], [1, x]]

      (%i4) infix (">>");
      (%o4)                          >>

      (%i5) a >> b := a[1] > b[1];
      (%o5)                  (a >> b) := a  > b
                                          1    1

      (%i6) sort (L, ">>");
      (%o6)           [[4, w], [3, y], [2, z], [1, x]]

      (%i7) sort (L, lambda ([a, b], a[1] > b[1]));
      (%o7)           [[4, w], [3, y], [2, z], [1, x]]

  `sort(L)` is equivalent to `sort(L, orderlessp)`.

      (%i1) L : [a, 2*b, -5, 7, 1 + %e, %pi];
      (%o1)             [a, 2 b, - 5, 7, %e + 1, %pi]

      (%i2) sort (L);
      (%o2)             [- 5, 7, %e + 1, %pi, a, 2 b]

      (%i3) sort (L, 'orderlessp);
      (%o3)             [- 5, 7, %e + 1, %pi, a, 2 b]

  The default sorting order is ascending, as determined by `orderlessp`. The predicate `ordergreatp` sorts a list in descending order.

      (%i1) L : [a, 2*b, -5, 7, 1 + %e, %pi];
      (%o1)             [a, 2 b, - 5, 7, %e + 1, %pi]

      (%i2) sort (L);
      (%o2)             [- 5, 7, %e + 1, %pi, a, 2 b]

      (%i3) sort (L, 'ordergreatp);
      (%o3)             [2 b, a, %pi, %e + 1, 7, - 5]

  All Maxima atoms and expressions are comparable under `orderlessp` and `ordergreatp`.

      (%i1) L : [11, -17, 29b0, 9*c, 7.55, foo(x, y), -5/2, b + a];
                                                       5
      (%o1)  [11, - 17, 2.9b1, 9 c, 7.55, foo(x, y), - -, b + a]
                                                       2

      (%i2) sort (L, orderlessp);
                      5
      (%o2)  [- 17, - -, 7.55, 11, 2.9b1, b + a, 9 c, foo(x, y)]
                      2

      (%i3) sort (L, ordergreatp);
                                                        5
      (%o3)  [foo(x, y), 9 c, b + a, 2.9b1, 11, 7.55, - -, - 17]
                                                        2

  Operators `<` and `>` order numbers, constants, and constant expressions by magnitude. Note that `orderlessp` and `ordergreatp` do not order numbers, constants, and constant expressions by magnitude.

      (%i1) L : [%pi, 3, 4, %e, %gamma];
      (%o1)                [%pi, 3, 4, %e, %gamma]

      (%i2) sort (L, ">");
      (%o2)                [4, %pi, 3, %e, %gamma]

      (%i3) sort (L, ordergreatp);
      (%o3)                [%pi, %gamma, %e, 4, 3]

  `ordermagnitudep` orders numbers, constants, and constant expressions the same as `<`, and all other elements the same as `orderlessp`.

      (%i1) L : [%i, 1+%i, 2*x, minf, inf, %e, sin(1), 0, 1, 2, 3, 1.0, 1.0b0];
      (%o1) [%i, %i + 1, 2 x, minf, inf, %e, sin(1), 0, 1, 2, 3, 1.0,
                                                                 1.0b0]

      (%i2) sort (L, ordermagnitudep);
      (%o2) [minf, 0, sin(1), 1, 1.0, 1.0b0, 2, %e, 3, inf, %i,
                                                           %i + 1, 2 x]

      (%i3) sort (L, orderlessp);
      (%o3) [0, 1, 1.0, 2, 3, sin(1), 1.0b0, %e, %i, %i + 1, inf,
                                                             minf, 2 x]

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  Categories: Lists

Function: sublist (list, p)

  Returns the list of elements of list for which the predicate `p` returns `true`.

  Example:

      (%i1) L: [1, 2, 3, 4, 5, 6];
      (%o1)                  [1, 2, 3, 4, 5, 6]

      (%i2) sublist (L, evenp);
      (%o2)                       [2, 4, 6]

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  Categories: Lists

Function: sublist_indices (L, P)

  Returns the indices of the elements `x` of the list L for which the predicate `maybe(P(x))` returns `true`; this excludes `unknown` as well as `false`. P may be the name of a function or a lambda expression. L must be a literal list.

  Examples:

      (%i1) sublist_indices ('[a, b, b, c, 1, 2, b, 3, b],
                             lambda ([x], x='b));
      (%o1)                     [2, 3, 7, 9]

      (%i2) sublist_indices ('[a, b, b, c, 1, 2, b, 3, b], symbolp);
      (%o2)                  [1, 2, 3, 4, 7, 9]

      (%i3) sublist_indices ([1 > 0, 1 < 0, 2 < 1, 2 > 1, 2 > 0],
                             identity);
      (%o3)                       [1, 4, 5]

      (%i4) assume (x < -1);
      (%o4)                       [x < - 1]

      (%i5) map (maybe, [x > 0, x < 0, x < -2]);
      (%o5)                [false, true, unknown]

      (%i6) sublist_indices ([x > 0, x < 0, x < -2], identity);
      (%o6)                          [2]

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  Categories: Lists

Function: tenth (expr)

  Returns the 10th item of expression or list expr. See `first` for more details.

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  Categories: Lists ·Expressions

Function: third (expr)

  Returns the 3rd item of expression or list expr. See `first` for more details.

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  Categories: Lists ·Expressions

Function: tree_reduce
`tree_reduce` (F, s)
`tree_reduce` (F, s, s_0)

  Extends the binary function F to an n-ary function by composition, where s is a set or list.

  `tree_reduce` is equivalent to the following: Apply F to successive pairs of elements to form a new list `[F(s_1, s_2), F(s_3, s_4), ...]`, carrying the final element unchanged if there are an odd number of elements. Then repeat until the list is reduced to a single element, which is the return value.

  When the optional argument s_0 is present, the result is equivalent `tree_reduce(F, cons(s_0, s))`.

  For addition of floating point numbers, `tree_reduce` may return a sum that has a smaller rounding error than either `rreduce` or `lreduce`.

  The elements of s and the partial results may be arranged in a minimum-depth binary tree, thus the name "tree_reduce".

  Examples:

  `tree_reduce` applied to a list with an even number of elements.

      (%i1) tree_reduce (f, [a, b, c, d]);
      (%o1)                  f(f(a, b), f(c, d))

  `tree_reduce` applied to a list with an odd number of elements.

      (%i1) tree_reduce (f, [a, b, c, d, e]);
      (%o1)               f(f(f(a, b), f(c, d)), e)

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  Categories: Sets ·Lists

Function: unique (L)

  Returns the unique elements of the list L.

  When all the elements of L are unique, `unique` returns a shallow copy of L, not L itself.

  If L is not a list, `unique` returns L.

  Example:

      (%i1) unique ([1, %pi, a + b, 2, 1, %e, %pi, a + b, [1]]);
      (%o1)              [1, 2, %e, %pi, [1], b + a]

Function: xreduce
`xreduce` (F, s)
`xreduce` (F, s, s_0)

  Extends the function F to an n-ary function by composition, or, if F is already n-ary, applies F to s. When F is not n-ary, `xreduce` is the same as `lreduce`. The argument s is a list.

  Functions known to be n-ary include addition `+`, multiplication `*`, `and`, `or`, `max`, `min`, and `append`. Functions may also be declared n-ary by `declare(F, nary)`. For these functions, `xreduce` is expected to be faster than either `rreduce` or `lreduce`.

  When the optional argument s_0 is present, the result is equivalent to `xreduce(s, cons(s_0, s))`.

  Floating point addition is not exactly associative; be that as it may, `xreduce` applies Maxima’s n-ary addition when s contains floating point numbers.

  Examples:

  `xreduce` applied to a function known to be n-ary. `F` is called once, with all arguments.

      (%i1) declare (F, nary);
      (%o1)                         done

      (%i2) F ([L]) := L;
      (%o2)                      F([L]) := L

      (%i3) xreduce (F, [a, b, c, d, e]);
      (%o3)                    [a, b, c, d, e]

  `xreduce` applied to a function not known to be n-ary. `G` is called several times, with two arguments each time.

      (%i1) G ([L]) := L;
      (%o1)                      G([L]) := L

      (%i2) xreduce (G, [a, b, c, d, e]);
      (%o2)                 [[[[a, b], c], d], e]

      (%i3) lreduce (G, [a, b, c, d, e]);
      (%o3)                 [[[[a, b], c], d], e]

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  Categories: Sets ·Lists

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