### 36.4 Functions and Variables for Function Definition

Function: apply (F, [x_1, …, x_n])

  Constructs and evaluates an expression `F(arg_1, ..., arg_n)`.

  `apply` does not attempt to distinguish a `memoizing function` from an ordinary function; when F is the name of a memoizing function, `apply` evaluates `F(...)` (that is, a function call with parentheses instead of square brackets). `arrayapply` evaluates a function call with square brackets in this case.

  See also `funmake` and `args`.

  Examples:

  `apply` evaluates its arguments. In this example, `min` is applied to the value of `L`.

      (%i1) L : [1, 5, -10.2, 4, 3];
      (%o1)                 [1, 5, - 10.2, 4, 3]

      (%i2) apply (min, L);
      (%o2)                        - 10.2

  `apply` evaluates arguments, even if the function F quotes them.

      (%i1) F (x) := x / 1729;
                                         x
      (%o1)                     F(x) := ----
                                        1729

      (%i2) fname : F;
      (%o2)                           F

      (%i3) dispfun (F);
                                         x
      (%t3)                     F(x) := ----
                                        1729

      (%o3)                         [%t3]

      (%i4) dispfun (fname);
      fundef: no such function: fname
       -- an error. To debug this try: debugmode(true);

      (%i5) apply (dispfun, [fname]);
                                         x
      (%t5)                     F(x) := ----
                                        1729

      (%o5)                         [%t5]

  `apply` evaluates the function name F. Single quote `'` defeats evaluation. `demoivre` is the name of a global variable and also a function.

      (%i1) demoivre;
      (%o1)                         false

      (%i2) demoivre (exp (%i * x));
      (%o2)                  %i sin(x) + cos(x)

      (%i3) apply (demoivre, [exp (%i * x)]);
      apply: found false where a function was expected.
       -- an error. To debug this try: debugmode(true);

      (%i4) apply ('demoivre, [exp (%i * x)]);
      (%o4)                  %i sin(x) + cos(x)

  How to convert a nested list into a matrix:

      (%i1) a:[[1,2],[3,4]];
      (%o1)                   [[1, 2], [3, 4]]

      (%i2) apply(matrix,a);
                                  [ 1  2 ]
      (%o2)                       [      ]
                                  [ 3  4 ]

  ·

  Categories: Function application

Function: block
`block` ([v_1, …, v_m], expr_1, …, expr_n)
`block` (expr_1, …, expr_n)

  The function `block` allows to make the variables v_1, …, v_m to be local for a sequence of commands. If these variables are already bound `block` saves the current values of the variables v_1, …, v_m (if any) upon entry to the block, then unbinds the variables so that they evaluate to themselves; The local variables may be bound to arbitrary values within the block but when the block is exited the saved values are restored, and the values assigned within the block are lost.

  If there is no need to define local variables then the list at the beginning of the `block` command may be omitted. In this case if neither `return` nor `go` are used `block` behaves similar to the following construct:

      ( expr_1, expr_2,... , expr_n );

  expr_1, …, expr_n will be evaluated in sequence and the value of the last expression will be returned. The sequence can be modified by the `go`, `throw`, and `return` functions. The last expression is expr_n unless `return` or an expression containing `throw` is evaluated.

  The declaration `local(v_1, ..., v_m)` within `block` saves the properties associated with the symbols v_1, …, v_m, removes any properties before evaluating other expressions, and restores any saved properties on exit from the block. Some declarations are implemented as properties of a symbol, including `:=`, `array`, `dependencies`, `atvalue`, `matchdeclare`, `atomgrad`, `constant`, `nonscalar`, `assume`, and some others. The effect of `local` is to make such declarations effective only within the block; otherwise declarations within a block are actually global declarations.

  `block` may appear within another `block`. Local variables are established each time a new `block` is evaluated. Local variables appear to be global to any enclosed blocks. If a variable is non-local in a block, its value is the value most recently assigned by an enclosing block, if any, otherwise, it is the value of the variable in the global environment. This policy may coincide with the usual understanding of "dynamic scope".

  The value of the block is the value of the last statement or the value of the argument to the function `return` which may be used to exit explicitly from the block. The function `go` may be used to transfer control to the statement of the block that is tagged with the argument to `go`. To tag a statement, precede it by an atomic argument as another statement in the block. For example: `block ([x], x:1, loop, x: x+1, ..., go(loop), ...)`. The argument to `go` must be the name of a tag appearing within the block. One cannot use `go` to transfer to a tag in a block other than the one containing the `go`.

  Blocks typically appear on the right side of a function definition but can be used in other places as well.

  See also `return` and `go`.

  ·

  Categories: Expressions ·Programming

Function: break (expr_1, …, expr_n)

  Evaluates and prints expr_1, …, expr_n and then causes a Maxima break at which point the user can examine and change his environment. Upon typing `exit;` the computation resumes.

  ·

  Categories: Debugging

Function: catch (expr_1, …, expr_n)

  Evaluates expr_1, …, expr_n one by one; if any leads to the evaluation of an expression of the form `throw (arg)`, then the value of the `catch` is the value of `throw (arg)`, and no further expressions are evaluated. This "non-local return" thus goes through any depth of nesting to the nearest enclosing `catch`. If there is no `catch` enclosing a `throw`, an error message is printed.

  If the evaluation of the arguments does not lead to the evaluation of any `throw` then the value of `catch` is the value of expr_n.

      (%i1) lambda ([x], if x < 0 then throw(x) else f(x))$
      (%i2) g(l) := catch (map (''%, l))$
      (%i3) g ([1, 2, 3, 7]);
      (%o3)               [f(1), f(2), f(3), f(7)]
      (%i4) g ([1, 2, -3, 7]);
      (%o4)                          - 3

  The function `g` returns a list of `f` of each element of `l` if `l` consists only of non-negative numbers; otherwise, `g` "catches" the first negative element of `l` and "throws" it up.

  ·

  Categories: Programming

Function: compfile
`compfile` (filename, f_1, …, f_n)
`compfile` (filename, functions)
`compfile` (filename, all)

  Translates Maxima functions into Lisp and writes the translated code into the file filename.

  `compfile(filename, f_1, ..., f_n)` translates the specified functions. `compfile (filename, functions)` and `compfile (filename, all)` translate all user-defined functions.

  The Lisp translations are not evaluated, nor is the output file processed by the Lisp compiler. `translate` creates and evaluates Lisp translations. `compile_file` translates Maxima into Lisp, and then executes the Lisp compiler.

  See also `translate`, `translate_file`, and `compile_file`.

  ·

  Categories: Translation and compilation

Function: compile
`compile` (f_1, …, f_n)
`compile` (functions)
`compile` (all)

  Translates Maxima functions f_1, …, f_n into Lisp, evaluates the Lisp translations, and calls the Lisp function `COMPILE` on each translated function. `compile` returns a list of the names of the compiled functions.

  `compile (all)` or `compile (functions)` compiles all user-defined functions.

  `compile` quotes its arguments; the quote-quote operator `''` defeats quotation.

  Compiling a function to native code can mean a big increase in speed and might cause the memory footprint to reduce drastically. Code tends to be especially effective when the flexibility it needs to provide is limited. If compilation doesn’t provide the speed that is needed a few ways to limit the code’s functionality are the following:

  - If the function accesses global variables the complexity of the function can be drastically be reduced by limiting these variables to one data type, for example using `mode_declare` or a statement like the following one: `put(x_1, bigfloat, numerical_type)`
  - The compiler might warn about undeclared variables if text could either be a named option to a command or (if they are assigned a value to) the name of a variable. Prepending the option with a single quote `'` tells the compiler that the text is meant as an option.
  ·

  Categories: Translation and compilation

Function: define
`define` (f(x_1, …, x_n), expr)
`define` (f[x_1, …, x_n], expr)
`define` (f[x_1, …, x_n](y_1, …, y_m), expr)
`define` (funmake (f, [x_1, …, x_n]), expr)
`define` (arraymake (f, [x_1, …, x_n]), expr)
`define` (ev (expr_1), expr_2)

  Defines a function named f with arguments x_1, …, x_n and function body expr. `define` always evaluates its second argument (unless explicitly quoted). The function so defined may be an ordinary Maxima function (with arguments enclosed in parentheses) or a `memoizing function` (with arguments enclosed in square brackets).

  When the last or only function argument x_n is a list of one element, the function defined by `define` accepts a variable number of arguments. Actual arguments are assigned one-to-one to formal arguments x_1, …, x_(n - 1), and any further actual arguments, if present, are assigned to x_n as a list.

  When the first argument of `define` is an expression of the form `f(x_1, ..., x_n)` or `f[x_1, ..., x_n]`, the function arguments are evaluated but f is not evaluated, even if there is already a function or variable by that name.

  When the first argument is an expression with operator `funmake`, `arraymake`, or `ev`, the first argument is evaluated; this allows for the function name to be computed, as well as the body.

  All function definitions appear in the same namespace; defining a function `f` within another function `g` does not automatically limit the scope of `f` to `g`. However, `local(f)` makes the definition of function `f` effective only within the block or other compound expression in which `local` appears.

  If some formal argument x_k is a quoted symbol (after evaluation), the function defined by `define` does not evaluate the corresponding actual argument. Otherwise all actual arguments are evaluated.

  See also `:=` and `::=`.

  Examples:

  `define` always evaluates its second argument (unless explicitly quoted).

      (%i1) expr : cos(y) - sin(x);
      (%o1)                    cos(y) - sin(x)

      (%i2) define (F1 (x, y), expr);
      (%o2)              F1(x, y) := cos(y) - sin(x)

      (%i3) F1 (a, b);
      (%o3)                    cos(b) - sin(a)

      (%i4) F2 (x, y) := expr;
      (%o4)                   F2(x, y) := expr

      (%i5) F2 (a, b);
      (%o5)                    cos(y) - sin(x)

  The function defined by `define` may be an ordinary Maxima function or a `memoizing function`.

      (%i1) define (G1 (x, y), x.y - y.x);
      (%o1)               G1(x, y) := x . y - y . x

      (%i2) define (G2 [x, y], x.y - y.x);
      (%o2)                G2     := x . y - y . x
                             x, y

  When the last or only function argument x_n is a list of one element, the function defined by `define` accepts a variable number of arguments.

      (%i1) define (H ([L]), '(apply ("+", L)));
      (%o1)                H([L]) := apply("+", L)

      (%i2) H (a, b, c);
      (%o2)                       c + b + a

  When the first argument is an expression with operator `funmake`, `arraymake`, or `ev`, the first argument is evaluated.

      (%i1) [F : I, u : x];
      (%o1)                        [I, x]

      (%i2) funmake (F, [u]);
      (%o2)                         I(x)

      (%i3) define (funmake (F, [u]), cos(u) + 1);
      (%o3)                  I(x) := cos(x) + 1

      (%i4) define (arraymake (F, [u]), cos(u) + 1);
      (%o4)                   I  := cos(x) + 1
                               x

      (%i5) define (foo (x, y), bar (y, x));
      (%o5)                foo(x, y) := bar(y, x)

      (%i6) define (ev (foo (x, y)), sin(x) - cos(y));
      (%o6)             bar(y, x) := sin(x) - cos(y)

  ·

  Categories: Function definition

Function: define_variable (name, default_value, mode)

  Introduces a global variable into the Maxima environment. `define_variable` is useful in user-written packages, which are often translated or compiled as it gives the compiler hints of the type (“mode”) of a variable and therefore avoids requiring it to generate generic code that can deal with every variable being an integer, float, maxima object, array etc.

  `define_variable` carries out the following steps:

  1. `mode_declare (name, mode)` declares the mode (“type”) of name to the translator which can considerably speed up compiled code as it allows having to create generic code. See `mode_declare` for a list of the possible modes.
  2. If the variable is unbound, default_value is assigned to name.
  3. Associates name with a test function to ensure that name is only assigned values of the declared mode.

  The `value_check` property can be assigned to any variable which has been defined via `define_variable` with a mode other than `any`. The `value_check` property is a lambda expression or the name of a function of one variable, which is called when an attempt is made to assign a value to the variable. The argument of the `value_check` function is the would-be assigned value.

  `define_variable` evaluates `default_value`, and quotes `name` and `mode`. `define_variable` returns the current value of `name`, which is `default_value` if `name` was unbound before, and otherwise it is the previous value of `name`.

  Examples:

  `foo` is a Boolean variable, with the initial value `true`.

      (%i1) define_variable (foo, true, boolean);
      (%o1)                         true

      (%i2) foo;
      (%o2)                         true

      (%i3) foo: false;
      (%o3)                         false

      (%i4) foo: %pi;
      translator: foo was declared with mode boolean
                                                , but it has value: %pi
       -- an error. To debug this try: debugmode(true);

      (%i5) foo;
      (%o5)                         false

  `bar` is an integer variable, which must be prime.

      (%i1) define_variable (bar, 2, integer);
      (%o1)                           2

      (%i2) qput (bar, prime_test, value_check);
      (%o2)                      prime_test

      (%i3) prime_test (y) := if not primep(y) then
                                 error (y, "is not prime.");
      (%o3) prime_test(y) := if not primep(y)
                                         then error(y, "is not prime.")

      (%i4) bar: 1439;
      (%o4)                         1439

      (%i5) bar: 1440;
      1440 is not prime.
      #0: prime_test(y=1440)
       -- an error. To debug this try: debugmode(true);

      (%i6) bar;
      (%o6)                         1439

  `baz_quux` is a variable which cannot be assigned a value. The mode `any_check` is like `any`, but `any_check` enables the `value_check` mechanism, and `any` does not.

      (%i1) define_variable (baz_quux, 'baz_quux, any_check);
      (%o1)                       baz_quux

      (%i2) F: lambda ([y], if y # 'baz_quux then
                       error ("Cannot assign to `baz_quux'."));
      (%o2) lambda([y], if y # 'baz_quux
                              then error(Cannot assign to `baz_quux'.))

      (%i3) qput (baz_quux, ''F, value_check);
      (%o3) lambda([y], if y # 'baz_quux
                              then error(Cannot assign to `baz_quux'.))

      (%i4) baz_quux: 'baz_quux;
      (%o4)                       baz_quux

      (%i5) baz_quux: sqrt(2);
      Cannot assign to `baz_quux'.
      #0: lambda([y],if y # 'baz_quux then
                  error("Cannot assign to `baz_quux'."))(y=sqrt(2))
       -- an error. To debug this try: debugmode(true);

      (%i6) baz_quux;
      (%o6)                       baz_quux

  ·

  Categories: Translation and compilation

Function: dispfun
`dispfun` (f_1, …, f_n)
`dispfun` (all)

  Displays the definition of the user-defined functions f_1, …, f_n. Each argument may be the name of a macro (defined with `::=`), an ordinary function (defined with `:=` or `define`), an array function (defined with `:=` or `define`, but enclosing arguments in square brackets `[ ]`), a subscripted function (defined with `:=` or `define`, but enclosing some arguments in square brackets and others in parentheses `( )`), one of a family of subscripted functions selected by a particular subscript value, or a subscripted function defined with a constant subscript.

  `dispfun (all)` displays all user-defined functions as given by the `functions`, `arrays`, and `macros` lists, omitting subscripted functions defined with constant subscripts.

  `dispfun` creates an intermediate expression label (`%t1`, `%t2`, etc.) for each displayed function, and assigns the function definition to the label. In contrast, `fundef` returns the function definition.

  `dispfun` quotes its arguments; the quote-quote operator `''` defeats quotation. `dispfun` returns the list of intermediate expression labels corresponding to the displayed functions.

  Examples:

      (%i1) m(x, y) ::= x^(-y);
                                           - y
      (%o1)                   m(x, y) ::= x

      (%i2) f(x, y) :=  x^(-y);
                                           - y
      (%o2)                    f(x, y) := x

      (%i3) g[x, y] :=  x^(-y);
                                          - y
      (%o3)                     g     := x
                                 x, y

      (%i4) h[x](y) :=  x^(-y);
                                          - y
      (%o4)                     h (y) := x
                                 x

      (%i5) i[8](y) :=  8^(-y);
                                          - y
      (%o5)                     i (y) := 8
                                 8

      (%i6) dispfun (m, f, g, h, h[5], h[10], i[8]);
                                           - y
      (%t6)                   m(x, y) ::= x

                                           - y
      (%t7)                    f(x, y) := x

                                          - y
      (%t8)                     g     := x
                                 x, y

                                          - y
      (%t9)                     h (y) := x
                                 x

                                          1
      (%t10)                     h (y) := --
                                  5        y
                                          5

                                           1
      (%t11)                    h  (y) := ---
                                 10         y
                                          10

                                          - y
      (%t12)                    i (y) := 8
                                 8

      (%o12)       [%t6, %t7, %t8, %t9, %t10, %t11, %t12]

      (%i13) ''%;
                           - y              - y            - y
      (%o13) [m(x, y) ::= x   , f(x, y) := x   , g     := x   ,
                                                  x, y
                        - y           1              1             - y
              h (y) := x   , h (y) := --, h  (y) := ---, i (y) := 8   ]
               x              5        y   10         y   8
                                      5             10

  ·

  Categories: Function definition ·Display functions

Function: fullmap (f, expr_1, …)

  Similar to `map`, but `fullmap` keeps mapping down all subexpressions until the main operators are no longer the same.

  `fullmap` is used by the Maxima simplifier for certain matrix manipulations; thus, Maxima sometimes generates an error message concerning `fullmap` even though `fullmap` was not explicitly called by the user.

  Examples:

      (%i1) a + b * c;
      (%o1)                        b c + a

      (%i2) fullmap (g, %);
      (%o2)                   g(b) g(c) + g(a)

      (%i3) map (g, %th(2));
      (%o3)                     g(b c) + g(a)

  ·

  Categories: Function application ·Expressions

Function: fullmapl (f, list_1, …)

  Similar to `fullmap`, but `fullmapl` only maps onto lists and matrices.

  Example:

      (%i1) fullmapl ("+", [3, [4, 5]], [[a, 1], [0, -1.5]]);
      (%o1)                [[a + 3, 4], [4, 3.5]]

  ·

  Categories: Function application ·Expressions

System variable: functions

  Default value: `[]`

  `functions` is the list of ordinary Maxima functions in the current session. An ordinary function is a function constructed by `define` or `:=` and called with parentheses `()`. A function may be defined at the Maxima prompt or in a Maxima file loaded by `load` or `batch`.

  `Memoizing functions` (called with square brackets, e.g., `F[x]`) and subscripted functions (called with square brackets and parentheses, e.g., `F[x](y)`) are listed by the global variable `arrays`, and not by `functions`.

  Lisp functions are not kept on any list.

  Examples:

      (%i1) F_1 (x) := x - 100;
      (%o1)                   F_1(x) := x - 100

      (%i2) F_2 (x, y) := x / y;
                                            x
      (%o2)                    F_2(x, y) := -
                                            y

      (%i3) define (F_3 (x), sqrt (x));
      (%o3)                   F_3(x) := sqrt(x)

      (%i4) G_1 [x] := x - 100;
      (%o4)                    G_1  := x - 100
                                  x

      (%i5) G_2 [x, y] := x / y;
                                           x
      (%o5)                     G_2     := -
                                   x, y    y

      (%i6) define (G_3 [x], sqrt (x));
      (%o6)                    G_3  := sqrt(x)
                                  x

      (%i7) H_1 [x] (y) := x^y;
                                            y
      (%o7)                     H_1 (y) := x
                                   x

      (%i8) functions;
      (%o8)              [F_1(x), F_2(x, y), F_3(x)]

      (%i9) arrays;
      (%o9)                 [G_1, G_2, G_3, H_1]

  ·

  Categories: Function definition ·Global variables

Function: fundef (f)

  Returns the definition of the function f.

  The argument may be

  - the name of a macro (defined with `::=`),
  - an ordinary function (defined with `:=` or `define`),
  - a `memoizing function` (defined with `:=` or `define`, but enclosing arguments in square brackets `[ ]`),
  - a subscripted function (defined with `:=` or `define`, but enclosing some arguments in square brackets and others in parentheses `( )`),
  - one of a family of subscripted functions selected by a particular subscript value,
  - or a subscripted function defined with a constant subscript.

  `fundef` quotes its argument; the quote-quote operator `''` defeats quotation.

  `fundef (f)` returns the definition of f. In contrast, `dispfun (f)` creates an intermediate expression label and assigns the definition to the label.

  ·

  Categories: Function definition

Function: funmake (F, [arg_1, …, arg_n])

  Returns an expression `F(arg_1, ..., arg_n)`. The return value is simplified, but not evaluated, so the function F is not called, even if it exists.

  `funmake` does not attempt to distinguish `memoizing functions` from ordinary functions; when F is the name of a memoizing function, `funmake` returns `F(...)` (that is, a function call with parentheses instead of square brackets). `arraymake` returns a function call with square brackets in this case.

  `funmake` evaluates its arguments.

  See also `apply` and `args`.

  Examples:

  `funmake` applied to an ordinary Maxima function.

      (%i1) F (x, y) := y^2 - x^2;
                                         2    2
      (%o1)                  F(x, y) := y  - x

      (%i2) funmake (F, [a + 1, b + 1]);
      (%o2)                    F(a + 1, b + 1)

      (%i3) ''%;
                                    2          2
      (%o3)                  (b + 1)  - (a + 1)

  `funmake` applied to a macro.

      (%i1) G (x) ::= (x - 1)/2;
                                        x - 1
      (%o1)                    G(x) ::= -----
                                          2

      (%i2) funmake (G, [u]);
      (%o2)                         G(u)

      (%i3) ''%;
                                    u - 1
      (%o3)                         -----
                                      2

  `funmake` applied to a subscripted function.

      (%i1) H [a] (x) := (x - 1)^a;
                                              a
      (%o1)                   H (x) := (x - 1)
                               a

      (%i2) funmake (H [n], [%e]);
                                             n
      (%o2)               lambda([x], (x - 1) )(%e)

      (%i3) ''%;
                                          n
      (%o3)                       (%e - 1)

      (%i4) funmake ('(H [n]), [%e]);
      (%o4)                        H (%e)
                                    n

      (%i5) ''%;
                                          n
      (%o5)                       (%e - 1)

  `funmake` applied to a symbol which is not a defined function of any kind.

      (%i1) funmake (A, [u]);
      (%o1)                         A(u)

      (%i2) ''%;
      (%o2)                         A(u)

  `funmake` evaluates its arguments, but not the return value.

      (%i1) det(a,b,c) := b^2 -4*a*c;
                                          2
      (%o1)              det(a, b, c) := b  - 4 a c

      (%i2) (x : 8, y : 10, z : 12);
      (%o2)                          12

      (%i3) f : det;
      (%o3)                          det

      (%i4) funmake (f, [x, y, z]);
      (%o4)                    det(8, 10, 12)

      (%i5) ''%;
      (%o5)                         - 284

  Maxima simplifies `funmake`’s return value.

      (%i1) funmake (sin, [%pi / 2]);
      (%o1)                           1

  ·

  Categories: Function application ·Expressions

Function: lambda
`lambda` ([x_1, …, x_m], expr_1, …, expr_n)
`lambda` ([[L]], expr_1, …, expr_n)
`lambda` ([x_1, …, x_m, [L]], expr_1, …, expr_n)

  Defines and returns a lambda expression (that is, an anonymous function). The function may have required arguments x_1, …, x_m and/or optional arguments L, which appear within the function body as a list. The return value of the function is expr_n. A lambda expression can be assigned to a variable and evaluated like an ordinary function. A lambda expression may appear in some contexts in which a function name is expected.

  When the function is evaluated, unbound local variables x_1, …, x_m are created. `lambda` may appear within `block` or another `lambda`; local variables are established each time another `block` or `lambda` is evaluated. Local variables appear to be global to any enclosed `block` or `lambda`. If a variable is not local, its value is the value most recently assigned in an enclosing `block` or `lambda`, if any, otherwise, it is the value of the variable in the global environment. This policy may coincide with the usual understanding of "dynamic scope".

  After local variables are established, expr_1 through expr_n are evaluated in turn. The special variable `%%`, representing the value of the preceding expression, is recognized. `throw` and `catch` may also appear in the list of expressions.

  `return` cannot appear in a lambda expression unless enclosed by `block`, in which case `return` defines the return value of the block and not of the lambda expression, unless the block happens to be expr_n. Likewise, `go` cannot appear in a lambda expression unless enclosed by `block`.

  `lambda` quotes its arguments; the quote-quote operator `''` defeats quotation.

  Examples:

  - A lambda expression can be assigned to a variable and evaluated like an ordinary function.

      (%i1) f: lambda ([x], x^2);
                                            2
      (%o1)                    lambda([x], x )

      (%i2) f(a);
                                      2
      (%o2)                          a

  - A lambda expression may appear in contexts in which a function evaluation is expected.

      (%i1) lambda ([x], x^2) (a);
                                      2
      (%o1)                          a

      (%i2) apply (lambda ([x], x^2), [a]);
                                      2
      (%o2)                          a

      (%i3) map (lambda ([x], x^2), [a, b, c, d, e]);
                              2   2   2   2   2
      (%o3)                 [a , b , c , d , e ]

  - Argument variables are local variables. Other variables appear to be global variables. Global variables are evaluated at the time the lambda expression is evaluated, unless some special evaluation is forced by some means, such as `''`.

      (%i1) a: %pi$
      (%i2) b: %e$

      (%i3) g: lambda ([a], a*b);
      (%o3)                   lambda([a], a b)

      (%i4) b: %gamma$

      (%i5) g(1/2);
                                   %gamma
      (%o5)                        ------
                                     2

      (%i6) g2: lambda ([a], a*''b);
      (%o6)                 lambda([a], a %gamma)

      (%i7) b: %e$

      (%i8) g2(1/2);
                                   %gamma
      (%o8)                        ------
                                     2

  - Lambda expressions may be nested. Local variables within the outer lambda expression appear to be global to the inner expression unless masked by local variables of the same names.

      (%i1) h: lambda ([a, b], h2: lambda ([a], a*b), h2(1/2));
                                                         1
      (%o1)     lambda([a, b], h2 : lambda([a], a b), h2(-))
                                                         2

      (%i2) h(%pi, %gamma);
                                   %gamma
      (%o2)                        ------
                                     2

  - Since `lambda` quotes its arguments, lambda expression `i` below does not define a "multiply by `a`" function. Such a function can be defined via `buildq`, as in lambda expression `i2` below.

      (%i1) i: lambda ([a], lambda ([x], a*x));
      (%o1)             lambda([a], lambda([x], a x))

      (%i2) i(1/2);
      (%o2)                   lambda([x], a x)

      (%i3) i2: lambda([a], buildq([a: a], lambda([x], a*x)));
      (%o3)    lambda([a], buildq([a : a], lambda([x], a x)))

      (%i4) i2(1/2);
                                          1
      (%o4)                  lambda([x], (-) x)
                                          2

      (%i5) i2(1/2)(%pi);
                                     %pi
      (%o5)                          ---
                                      2

  - A lambda expression may take a variable number of arguments, which are indicated by `[L]` as the sole or final argument. The arguments appear within the function body as a list.

      (%i1) f : lambda ([aa, bb, [cc]], aa * cc + bb);
      (%o1)          lambda([aa, bb, [cc]], aa cc + bb)

      (%i2) f (foo, %i, 17, 29, 256);
      (%o2)       [17 foo + %i, 29 foo + %i, 256 foo + %i]

      (%i3) g : lambda ([[aa]], apply ("+", aa));
      (%o3)             lambda([[aa]], apply(+, aa))

      (%i4) g (17, 29, x, y, z, %e);
      (%o4)                  z + y + x + %e + 46

  ·

  Categories: Function definition

Function: local (v_1, …, v_n)

  Saves the properties associated with the symbols v_1, …, v_n, removes any properties before evaluating other expressions, and restores any saved properties on exit from the block or other compound expression in which `local` appears.

  Some declarations are implemented as properties of a symbol, including `:=`, `array`, `dependencies`, `atvalue`, `matchdeclare`, `atomgrad`, `constant`, `nonscalar`, `assume`, and some others. The effect of `local` is to make such declarations effective only within the block or other compound expression in which `local` appears; otherwise such declarations are global declarations.

  `local` can only appear in `block` or in the body of a function definition or `lambda` expression, and only one occurrence is permitted in each.

  `local` quotes its arguments. `local` returns `done`.

  Example:

  A local function definition.

      (%i1) foo (x) := 1 - x;
      (%o1)                    foo(x) := 1 - x

      (%i2) foo (100);
      (%o2)                         - 99

      (%i3) block (local (foo), foo (x) := 2 * x, foo (100));
      (%o3)                          200

      (%i4) foo (100);
      (%o4)                         - 99

  ·

  Categories: Function definition ·Programming

Option variable: macroexpansion

  Default value: `false`

  `macroexpansion` controls whether the expansion (that is, the return value) of a macro function is substituted for the macro function call. A substitution may speed up subsequent expression evaluations, at the cost of storing the expansion.

  `false`

    The expansion of a macro function is not substituted for the macro function call.

  `expand`

    The first time a macro function call is evaluated, the expansion is stored. The expansion is not recomputed on subsequent calls; any side effects (such as `print` or assignment to global variables) happen only when the macro function call is first evaluated. Expansion in an expression does not affect other expressions which have the same macro function call.

  `displace`

    The first time a macro function call is evaluated, the expansion is substituted for the call, thus modifying the expression from which the macro function was called. The expansion is not recomputed on subsequent calls; any side effects happen only when the macro function call is first evaluated. Expansion in an expression does not affect other expressions which have the same macro function call.

  Examples

  When `macroexpansion` is `false`, a macro function is called every time the calling expression is evaluated, and the calling expression is not modified.

      (%i1) f (x) := h (x) / g (x);
                                        h(x)
      (%o1)                     f(x) := ----
                                        g(x)

      (%i2) g (x) ::= block (print ("x + 99 is equal to", x),
                             return (x + 99));
      (%o2) g(x) ::= block(print("x + 99 is equal to", x),
                                                        return(x + 99))

      (%i3) h (x) ::= block (print ("x - 99 is equal to", x),
                             return (x - 99));
      (%o3) h(x) ::= block(print("x - 99 is equal to", x),
                                                        return(x - 99))

      (%i4) macroexpansion: false;
      (%o4)                         false

      (%i5) f (a * b);
      x - 99 is equal to x
      x + 99 is equal to x
                                  a b - 99
      (%o5)                       --------
                                  a b + 99

      (%i6) dispfun (f);
                                        h(x)
      (%t6)                     f(x) := ----
                                        g(x)

      (%o6)                         [%t6]

      (%i7) f (a * b);
      x - 99 is equal to x
      x + 99 is equal to x
                                  a b - 99
      (%o7)                       --------
                                  a b + 99

  When `macroexpansion` is `expand`, a macro function is called once, and the calling expression is not modified.

      (%i1) f (x) := h (x) / g (x);
                                        h(x)
      (%o1)                     f(x) := ----
                                        g(x)

      (%i2) g (x) ::= block (print ("x + 99 is equal to", x),
                             return (x + 99));
      (%o2) g(x) ::= block(print("x + 99 is equal to", x),
                                                        return(x + 99))

      (%i3) h (x) ::= block (print ("x - 99 is equal to", x),
                             return (x - 99));
      (%o3) h(x) ::= block(print("x - 99 is equal to", x),
                                                        return(x - 99))

      (%i4) macroexpansion: expand;
      (%o4)                        expand

      (%i5) f (a * b);
      x - 99 is equal to x
      x + 99 is equal to x
                                  a b - 99
      (%o5)                       --------
                                  a b + 99

      (%i6) dispfun (f);
                            mmacroexpanded(x - 99, h(x))
      (%t6)         f(x) := ----------------------------
                            mmacroexpanded(x + 99, g(x))

      (%o6)                         [%t6]

      (%i7) f (a * b);
                                  a b - 99
      (%o7)                       --------
                                  a b + 99

  When `macroexpansion` is `displace`, a macro function is called once, and the calling expression is modified.

      (%i1) f (x) := h (x) / g (x);
                                        h(x)
      (%o1)                     f(x) := ----
                                        g(x)

      (%i2) g (x) ::= block (print ("x + 99 is equal to", x),
                             return (x + 99));
      (%o2) g(x) ::= block(print("x + 99 is equal to", x),
                                                        return(x + 99))

      (%i3) h (x) ::= block (print ("x - 99 is equal to", x),
                             return (x - 99));
      (%o3) h(x) ::= block(print("x - 99 is equal to", x),
                                                        return(x - 99))

      (%i4) macroexpansion: displace;
      (%o4)                       displace

      (%i5) f (a * b);
      x - 99 is equal to x
      x + 99 is equal to x
                                  a b - 99
      (%o5)                       --------
                                  a b + 99

      (%i6) dispfun (f);
                                       x - 99
      (%t6)                    f(x) := ------
                                       x + 99

      (%o6)                         [%t6]

      (%i7) f (a * b);
                                  a b - 99
      (%o7)                       --------
                                  a b + 99

  ·

  Categories: Function application ·Global flags

Function: mode_declare (y_1, mode_1, …, y_n, mode_n)
Function: modedeclare (y_1, mode_1, …, y_n, mode_n)

  A `mode_declare` informs the compiler which type (lisp programmers name the type: “mode”) a function parameter or its return value will be of. This can greatly boost the efficiency of the code the compiler generates: Without knowing the type of all variables and knowing the return value of all functions a function uses in advance very generic (and thus potentially slow) code needs to be generated.

  The arguments of `mode_declare` are pairs consisting of a variable (or a list of variables all having the same mode) and a mode. Available modes (“types”) are:

      array            an declared array (see the detailed description below)
      boolean          true or false
      integer          integers (including arbitrary-size integers)
      fixnum           integers (excluding arbitrary-size integers)
      float            machine-size floating-point numbers
      real             machine-size floating-point or integer
      number           Numbers
      any              any kind of object (useful for arrays of any)

  A function parameter named `a` can be declared as an array filled with elements of the type `t` the following way:

      mode_declare (a, array(t, dim1, dim2, ...))

  If none of the elements of the array `a` needs to be checked if it still doesn’t contain a value additional code can be omitted by declaring this fact, too:

      mode_declare (a, array (t, complete, dim1, dim2, ...))

  The `complete` has no effect if all array elements are of the type `fixnum` or `float`: Machine-sized numbers inevitably contain a value (and will automatically be initialized to 0 in most lisp implementations).

  Another way to tell that all entries of the array `a` are of the type (“mode”) `m` and have been assigned a value to would be:

      mode_declare (completearray (a), m))

  Numeric code using arrays might run faster still if the size of the array is known at compile time, as well, as in:

      mode_declare (completearray (a [10, 10]), float)

  for a floating point number array named `a` which is 10 x 10.

  `mode_declare` also can be used in order to declare the type of the result of a function. In this case the function compilation needs to be preceded by another `mode_declare` statement. For example the expression,

      mode_declare ([function (f_1, f_2, ...)], fixnum)

  declares that the values returned by `f_1`, `f_2`, … are single-word integers.

  `modedeclare` is a synonym for `mode_declare`.

  If the type of function parameters and results doesn’t match the declaration by `mode_declare` the function may misbehave or a warning or an error might occur, see `mode_checkp`, `mode_check_errorp` and `mode_check_warnp`.

  See `mode_identity` for declaring the type of lists and `define_variable` for declaring the type of all global variables compiled code uses, as well.

  Example:

      (%i1) square_float(f):=(
           mode_declare(f,float),
           f*f
       );
      (%o1)   square_float(f) := (mode_declare(f, float), f f)

      (%i2) mode_declare([function(f)],float);
      (%o2)                    [[function(f)]]

      (%i3) compile(square_float);
      (%o3)                    [square_float]

      (%i4) square_float(100.0);
      (%o4)                        10000.0

  ·

  Categories: Translation and compilation

Option variable: mode_checkp

  Default value: `true`

  When `mode_checkp` is `true`, `mode_declare` does not only define which type a variable will be of so the compiler can generate more efficient code, but will also create a runtime warning if the variable isn’t of the variable type the code was compiled to deal with.

      (%i1) mode_checkp:true;
      (%o1)                         true

      (%i2) square(f):=(
          mode_declare(f,float),
          f^2);
                                                         2
      (%o2)       square(f) := (mode_declare(f, float), f )

      (%i3) compile(square);
      (%o3)                       [square]

      (%i4) square(2.3);
      (%o4)                   5.289999999999999

      (%i5) square(4);
      Maxima encountered a Lisp error:

       The value
         4
       is not of type
         DOUBLE-FLOAT
       when binding $F

      Automatically continuing.
      To enable the Lisp debugger set *debugger-hook* to nil.

  ·

  Categories: Translation flags and variables

Option variable: mode_check_errorp

  Default value: `false`

  When `mode_check_errorp` is `true`, `mode_declare` calls error.

  ·

  Categories: Translation flags and variables

Option variable: mode_check_warnp

  Default value: `true`

  When `mode_check_warnp` is `true`, mode errors are described.

  ·

  Categories: Translation flags and variables

Function: mode_identity (arg_1, arg_2)

  `mode_identity` works similar to `mode_declare`, but is used for informing the compiler that a thing like a `macro` or a list operation will only return a specific type of object. The purpose of doing so is that maxima supports many objects: Machine integers, arbitrary length integers, equations, machine floats, big floats, which means that for everything that deals with return values of operations that can result in any object the compiler needs to output generic (and therefore potentially slow) code.

  The first argument to `mode_identity` is the type of return value something will return (for possible types see `mode_declare`). (i.e., one of `float`, `fixnum`, `number`, The second argument is the expression that will return an object of this type.

  If the the return value of this expression is of a type the code was not compiled for error or warning is signalled.

  If you knew that `first (l)` returned a number then you could write

      mode_identity (number, first (l)).

  However, if you need this construct more often it would be more efficient to define a function that returns a number fist:

      firstnumb (x) ::= buildq ([x], mode_identity (number, first(x)));
      compile(firstnumb)

  `firstnumb` now can be used every time you need the first element of a list that is guaranteed to be filled with numbers.

  ·Translation and compilation

Function: remfunction
`remfunction` (f_1, …, f_n)
`remfunction` (all)

  Unbinds the function definitions of the symbols f_1, …, f_n. The arguments may be the names of ordinary functions (created by `:=` or `define`) or macro functions (created by `::=`).

  `remfunction (all)` unbinds all function definitions.

  `remfunction` quotes its arguments.

  `remfunction` returns a list of the symbols for which the function definition was unbound. `false` is returned in place of any symbol for which there is no function definition.

  `remfunction` does not apply to `memoizing functions` or subscripted functions. `remarray` applies to those types of functions.

  ·

  Categories: Function definition

Option variable: savedef

  Default value: `true`

  When `savedef` is `true`, the Maxima version of an user function is preserved when the function is translated. This permits the definition to be displayed by `dispfun` and allows the function to be edited.

  When `savedef` is `false`, the names of translated functions are removed from the `functions` list.

  ·

  Categories: Translation flags and variables

Function: translate
`translate` (f_1, …, f_n)
`translate` (functions)
`translate` (all)

  Translates the user-defined functions f_1, …, f_n from the Maxima language into Lisp and evaluates the Lisp translations. Typically the translated functions run faster than the originals.

  `translate (all)` or `translate (functions)` translates all user-defined functions.

  Functions to be translated should include a call to `mode_declare` at the beginning when possible in order to produce more efficient code. For example:

      f (x_1, x_2, ...) := block ([v_1, v_2, ...],
          mode_declare (v_1, mode_1, v_2, mode_2, ...), ...)

  where the x_1, x_2, … are the parameters to the function and the v_1, v_2, … are the local variables.

  The names of translated functions are removed from the `functions` list if `savedef` is `false` (see below) and are added to the `props` lists.

  Functions should not be translated unless they are fully debugged.

  Expressions are assumed simplified; if they are not, correct but non-optimal code gets generated. Thus, the user should not set the `simp` switch to `false` which inhibits simplification of the expressions to be translated.

  The switch `translate`, if `true`, causes automatic translation of an user’s function to Lisp.

  Note that translated functions may not run identically to the way they did before translation as certain incompatibilities may exist between the Lisp and Maxima versions. Principally, the `rat` function with more than one argument and the `ratvars` function should not be used if any variables are `mode_declare`’d canonical rational expressions (CRE). Also the `prederror: false` setting will not translate.

  `savedef` - if `true` will cause the Maxima version of an user function to remain when the function is `translate`’d. This permits the definition to be displayed by `dispfun` and allows the function to be edited.

  `transrun` - if `false` will cause the interpreted version of all functions to be run (provided they are still around) rather than the translated version.

  The result returned by `translate` is a list of the names of the functions translated.

  ·

  Categories: Translation and compilation

Function: translate_file
`translate_file` (maxima_filename)
`translate_file` (maxima_filename, lisp_filename)

  Translates a file of Maxima code into a file of Lisp code. `translate_file` returns a list of three filenames: the name of the Maxima file, the name of the Lisp file, and the name of file containing additional information about the translation. `translate_file` evaluates its arguments.

  `translate_file ("foo.mac"); load("foo.LISP")` is the same as the command `batch ("foo.mac")` except for certain restrictions, the use of `''` and `%`, for example.

  `translate_file (maxima_filename)` translates a Maxima file maxima_filename into a similarly-named Lisp file. For example, `foo.mac` is translated into `foo.LISP`. The Maxima filename may include a directory name or names, in which case the Lisp output file is written to the same directory from which the Maxima input comes.

  `translate_file (maxima_filename, lisp_filename)` translates a Maxima file maxima_filename into a Lisp file lisp_filename. `translate_file` ignores the filename extension, if any, of `lisp_filename`; the filename extension of the Lisp output file is always `LISP`. The Lisp filename may include a directory name or names, in which case the Lisp output file is written to the specified directory.

  `translate_file` also writes a file of translator warning messages of various degrees of severity. The filename extension of this file is `UNLISP`. This file may contain valuable information, though possibly obscure, for tracking down bugs in translated code. The `UNLISP` file is always written to the same directory from which the Maxima input comes.

  `translate_file` emits Lisp code which causes some declarations and definitions to take effect as soon as the Lisp code is compiled. See `compile_file` for more on this topic.

  See also

  `tr_array_as_ref` `tr_bound_function_applyp`, `tr_exponent` `tr_file_tty_messagesp`, `tr_float_can_branch_complex`, `tr_function_call_default`, `tr_numer`, `tr_optimize_max_loop`, `tr_state_vars`, `tr_warnings_get`, `tr_warn_bad_function_calls` `tr_warn_fexpr`, `tr_warn_meval`, `tr_warn_mode`, `tr_warn_undeclared`, and `tr_warn_undefined_variable`.

  ·

  Categories: Translation and compilation

Option variable: transrun

  Default value: `true`

  When `transrun` is `false` will cause the interpreted version of all functions to be run (provided they are still around) rather than the translated version.

  ·

  Categories: Translation flags and variables

Option variable: tr_array_as_ref

  Default value: `true`

  If `translate_fast_arrays` is `false`, array references in Lisp code emitted by `translate_file` are affected by `tr_array_as_ref`. When `tr_array_as_ref` is `true`, array names are evaluated, otherwise array names appear as literal symbols in translated code.

  `tr_array_as_ref` has no effect if `translate_fast_arrays` is `true`.

  ·

  Categories: Translation flags and variables

Option variable: tr_bound_function_applyp

  Default value: `true`

  When `tr_bound_function_applyp` is `true` and `tr_function_call_default` is `general`, if a bound variable (such as a function argument) is found being used as a function then Maxima will rewrite that function call using `apply` and print a warning message.

  For example, if `g` is defined by `g(f,x) := f(x+1)` then translating `g` will cause Maxima to print a warning and rewrite `f(x+1)` as `apply(f,[x+1])`.

      (%i1) f (x) := x^2$
      (%i2) g (f) := f (3)$
      (%i3) tr_bound_function_applyp : true$

      (%i4) translate (g)$
      warning: f is a bound variable in f(3), but it is used as a function.
      note: instead I'll translate it as: apply(f,[3])

      (%i5) g (lambda ([x], x));
      (%o5)                           3

      (%i6) tr_bound_function_applyp : false$
      (%i7) translate (g)$

      (%i8) g (lambda ([x], x));
      (%o8)                           9

  ·

  Categories: Translation flags and variables

Option variable: tr_file_tty_messagesp

  Default value: `false`

  When `tr_file_tty_messagesp` is `true`, messages generated by `translate_file` during translation of a file are displayed on the console and inserted into the UNLISP file. When `false`, messages about translation of the file are only inserted into the UNLISP file.

  ·

  Categories: Translation flags and variables

Option variable: tr_float_can_branch_complex

  Default value: `true`

  Tells the Maxima-to-Lisp translator to assume that the functions `acos`, `asin`, `asec`, `acsc`, `acosh`, `asech`, `atanh`, `acoth`, `log` and `sqrt` can return complex results.

  When it is `true` then `acos(x)` is of mode `any` even if `x` is of mode `float` (as set by `mode_declare`). When `false` then `acos(x)` is of mode `float` if and only if `x` is of mode `float`.

  ·

  Categories: Translation flags and variables

Option variable: tr_function_call_default

  Default value: `general`

  `false` means give up and call `meval`, `expr` means assume Lisp fixed arg function. `general`, the default gives code good for `mexprs` and `mlexprs` but not `macros`. `general` assures variable bindings are correct in compiled code. In `general` mode, when translating F(X), if F is a bound variable, then it assumes that `apply (f, [x])` is meant, and translates a such, with appropriate warning. There is no need to turn this off. With the default settings, no warning messages implies full compatibility of translated and compiled code with the Maxima interpreter.

  ·

  Categories: Translation flags and variables

Option variable: tr_numer

  Default value: `false`

  When `tr_numer` is `true`, `numer` properties are used for atoms which have them, e.g. `%pi`.

  ·

  Categories: Translation flags and variables

Option variable: tr_optimize_max_loop

  Default value: 100

  `tr_optimize_max_loop` is the maximum number of times the macro-expansion and optimization pass of the translator will loop in considering a form. This is to catch macro expansion errors, and non-terminating optimization properties.

  ·

  Categories: Translation flags and variables

System variable: tr_state_vars

  Default value:

      [translate_fast_arrays, tr_function_call_default, tr_bound_function_applyp,
      tr_array_as_ref, tr_numer, tr_float_can_branch_complex, define_variable]

  The list of the switches that affect the form of the translated output. This information is useful to system people when trying to debug the translator. By comparing the translated product to what should have been produced for a given state, it is possible to track down bugs.

  ·

  Categories: Translation flags and variables

Function: tr_warnings_get ()

  Prints a list of warnings which have been given by the translator during the current translation.

  ·

  Categories: Translation and compilation

Option variable: tr_warn_bad_function_calls

  Default value: `true`

  - Gives a warning when when function calls are being made which may not be correct due to improper declarations that were made at translate time.

  ·

  Categories: Translation flags and variables

Option variable: tr_warn_fexpr

  Default value: `compfile`

  - Gives a warning if any FEXPRs are encountered. FEXPRs should not normally be output in translated code, all legitimate special program forms are translated.

  ·

  Categories: Translation flags and variables

Option variable: tr_warn_meval

  Default value: `compfile`

  - Gives a warning if the function `meval` gets called. If `meval` is called that indicates problems in the translation.

  ·

  Categories: Translation flags and variables

Option variable: tr_warn_mode

  Default value: `all`

  - Gives a warning when variables are assigned values inappropriate for their mode.

  ·

  Categories: Translation flags and variables

Option variable: tr_warn_undeclared

  Default value: `compile`

  - Determines when to send warnings about undeclared variables to the TTY.

  ·

  Categories: Translation flags and variables

Option variable: tr_warn_undefined_variable

  Default value: `all`

  - Gives a warning when undefined global variables are seen.

  ·

  Categories: Translation flags and variables

Function: compile_file
`compile_file` (filename)
`compile_file` (filename, compiled_filename)
`compile_file` (filename, compiled_filename, lisp_filename)

  Translates the Maxima file filename into Lisp, and executes the Lisp compiler. The compiled code is not loaded into Maxima.

  `compile_file` returns a list of the names of four files: the original Maxima file, the Lisp translation, notes on translation, and the compiled code. If the compilation fails, the fourth item is `false`.

  Some declarations and definitions take effect as soon as the Lisp code is compiled (without loading the compiled code). These include functions defined with the `:=` operator, macros define with the `::=` operator, `alias`, `declare`, `define_variable`, `mode_declare`, and `infix`, `matchfix`, `nofix`, `postfix`, `prefix`, and `compfile`.

  Assignments and function calls are not evaluated until the compiled code is loaded. In particular, within the Maxima file, assignments to the translation flags (`tr_numer`, etc.) have no effect on the translation.

  filename may not contain `:lisp` statements.

  `compile_file` evaluates its arguments.

  ·

  Categories: Translation and compilation

Function: declare_translated (f_1, f_2, …)

  When translating a file of Maxima code to Lisp, it is important for the translator to know which functions it sees in the file are to be called as translated or compiled functions, and which ones are just Maxima functions or undefined. Putting this declaration at the top of the file, lets it know that although a symbol does which does not yet have a Lisp function value, will have one at call time. `(MFUNCTION-CALL fn arg1 arg2 ...)` is generated when the translator does not know `fn` is going to be a Lisp function.

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  Categories: Translation and compilation