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# 37. Program Flow

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## 37.1 Lisp and Maxima

Maxima is written in Lisp, and it is easy to access Lisp functions and variables from Maxima and vice versa. Lisp and Maxima symbols are distinguished by a naming convention. A Lisp symbol which begins with a dollar sign `$` corresponds to a Maxima symbol without the dollar sign.

A Maxima symbol which begins with a question mark `?` corresponds to a Lisp symbol without the question mark. For example, the Maxima symbol `foo` corresponds to the Lisp symbol `$FOO`, while the Maxima symbol `?foo` corresponds to the Lisp symbol `FOO`. Note that `?foo` is written without a space between `?` and `foo`; otherwise it might be mistaken for `describe ("foo")`.

Hyphen `-`, asterisk `*`, or other special characters in Lisp symbols must be escaped by backslash `\` where they appear in Maxima code. For example, the Lisp identifier `*foo-bar*` is written `?\*foo\-bar\*` in Maxima.

Lisp code may be executed from within a Maxima session. A single line of Lisp (containing one or more forms) may be executed by the special command `:lisp`. For example,

    (%i1) :lisp (foo $x $y)

calls the Lisp function `foo` with Maxima variables `x` and `y` as arguments. The `:lisp` construct can appear at the interactive prompt or in a file processed by `batch` or `demo`, but not in a file processed by `load`, `batchload`, `translate_file`, or `compile_file`.

The function `to_lisp` opens an interactive Lisp session. Entering `(to-maxima)` closes the Lisp session and returns to Maxima.

Lisp functions and variables which are to be visible in Maxima as functions and variables with ordinary names (no special punctuation) must have Lisp names beginning with the dollar sign `$`.

Maxima is case-sensitive, distinguishing between lowercase and uppercase letters in identifiers. There are some rules governing the translation of names between Lisp and Maxima.

1. A Lisp identifier not enclosed in vertical bars corresponds to a Maxima identifier in lowercase. Whether the Lisp identifier is uppercase, lowercase, or mixed case, is ignored. E.g., Lisp `$foo`, `$FOO`, and `$Foo` all correspond to Maxima `foo`. But this is because `$foo`, `$FOO` and `$Foo` are converted by the Lisp reader by default to the Lisp symbol `$FOO`.
2. A Lisp identifier which is all uppercase or all lowercase and enclosed in vertical bars corresponds to a Maxima identifier with case reversed. That is, uppercase is changed to lowercase and lowercase to uppercase. E.g., Lisp `|$FOO|` and `|$foo|` correspond to Maxima `foo` and `FOO`, respectively.
3. A Lisp identifier which is mixed uppercase and lowercase and enclosed in vertical bars corresponds to a Maxima identifier with the same case. E.g., Lisp `|$Foo|` corresponds to Maxima `Foo`.

The `#$` Lisp macro allows the use of Maxima expressions in Lisp code. `#$expr$` expands to a Lisp expression equivalent to the Maxima expression expr.

    (msetq $foo #$[x, y]$)

This has the same effect as entering

    (%i1) foo: [x, y];

The Lisp function `displa` prints an expression in Maxima format.

    (%i1) :lisp #$[x, y, z]$
    ((MLIST SIMP) $X $Y $Z)
    (%i1) :lisp (displa '((MLIST SIMP) $X $Y $Z))
    [x, y, z]
    NIL

Functions defined in Maxima are not ordinary Lisp functions. The Lisp function `mfuncall` calls a Maxima function. For example:

    (%i1) foo(x,y) := x*y$
    (%i2) :lisp (mfuncall '$foo 'a 'b)
    ((MTIMES SIMP) A B)

Some Lisp functions are shadowed in the Maxima package, namely the following.

       complement     continue      //
       float          functionp     array
       exp            listen        signum
       atan           asin          acos
       asinh          acosh         atanh
       tanh           cosh          sinh
       tan            break         gcd

·

@ref{Category: Programming}

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## 37.2 Garbage Collection

Symbolic computation tends to create a good deal of garbage (temporary or intermediate results that are eventually not used), and effective handling of this can be crucial to successful completion of some programs.

Under GCL, on UNIX systems where the mprotect system call is available (including SUN OS 4.0 and some variants of BSD) a stratified garbage collection is available. This limits the collection to pages which have been recently written to. See the GCL documentation under ALLOCATE and GBC. At the Lisp level doing (setq si::*notify-gbc* t) will help you determine which areas might need more space.

For other Lisps that run Maxima, we refer the reader to the documentation for that Lisp on how to control GC.

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## 37.3 Introduction to Program Flow

Maxima provides a `do` loop for iteration, as well as more primitive constructs such as `go`.

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## 37.4 Functions and Variables for Program Flow

Function: backtrace ()
Function: backtrace (n)

  Prints the call stack, that is, the list of functions which called the currently active function.

  `backtrace()` prints the entire call stack.

  `backtrace (n)` prints the n most recent functions, including the currently active function.

  `backtrace` can be called from a script, a function, or the interactive prompt (not only in a debugging context).

  Examples:

  - `backtrace()` prints the entire call stack.

        (%i1) h(x) := g(x/7)$
        (%i2) g(x) := f(x-11)$
        (%i3) f(x) := e(x^2)$
        (%i4) e(x) := (backtrace(), 2*x + 13)$
        (%i5) h(10);
        #0: e(x=4489/49)
        #1: f(x=-67/7)
        #2: g(x=10/7)
        #3: h(x=10)
                                      9615
        (%o5)                         ----
                                       49

  - `backtrace (n)` prints the n most recent functions, including the currently active function.

        (%i1) h(x) := (backtrace(1), g(x/7))$
        (%i2) g(x) := (backtrace(1), f(x-11))$
        (%i3) f(x) := (backtrace(1), e(x^2))$
        (%i4) e(x) := (backtrace(1), 2*x + 13)$
        (%i5) h(10);
        #0: h(x=10)
        #0: g(x=10/7)
        #0: f(x=-67/7)
        #0: e(x=4489/49)
                                      9615
        (%o5)                         ----
                                       49

  ·

  @ref{Category: Debugging}

Special operator: do

  The `do` statement is used for performing iteration. Due to its great generality the `do` statement will be described in two parts. First the usual form will be given which is analogous to that used in several other programming languages (Fortran, Algol, PL/I, etc.); then the other features will be mentioned.

  There are three variants of this form that differ only in their terminating conditions. They are:

  - `for variable: initial_value step increment thru limit do body`
  - `for variable: initial_value step increment while condition do body`
  - `for variable: initial_value step increment unless condition do body`

  (Alternatively, the `step` may be given after the termination condition or limit.)

  initial_value, increment, limit, and body can be any expressions. If the increment is 1 then "`step 1`" may be omitted.

  The execution of the `do` statement proceeds by first assigning the initial_value to the variable (henceforth called the control-variable). Then: (1) If the control-variable has exceeded the limit of a `thru` specification, or if the condition of the `unless` is `true`, or if the condition of the `while` is `false` then the `do` terminates. (2) The body is evaluated. (3) The increment is added to the control-variable. The process from (1) to (3) is performed repeatedly until the termination condition is satisfied. One may also give several termination conditions in which case the `do` terminates when any of them is satisfied.

  In general the `thru` test is satisfied when the control-variable is greater than the limit if the increment was non-negative, or when the control-variable is less than the limit if the increment was negative. The increment and limit may be non-numeric expressions as long as this inequality can be determined. However, unless the increment is syntactically negative (e.g. is a negative number) at the time the `do` statement is input, Maxima assumes it will be positive when the `do` is executed. If it is not positive, then the `do` may not terminate properly.

  Note that the limit, increment, and termination condition are evaluated each time through the loop. Thus if any of these involve much computation, and yield a result that does not change during all the executions of the body, then it is more efficient to set a variable to their value prior to the `do` and use this variable in the `do` form.

  The value normally returned by a `do` statement is the atom `done`. However, the function `return` may be used inside the body to exit the `do` prematurely and give it any desired value. Note however that a `return` within a `do` that occurs in a `block` will exit only the `do` and not the `block`. Note also that the `go` function may not be used to exit from a `do` into a surrounding `block`.

  The control-variable is always local to the `do` and thus any variable may be used without affecting the value of a variable with the same name outside of the `do`. The control-variable is unbound after the `do` terminates.

      (%i1) for a:-3 thru 26 step 7 do display(a)$
                                   a = - 3

                                    a = 4

                                   a = 11

                                   a = 18

                                   a = 25

      (%i1) s: 0$
      (%i2) for i: 1 while i <= 10 do s: s+i;
      (%o2)                         done
      (%i3) s;
      (%o3)                          55

  Note that the condition `while i <= 10` is equivalent to `unless i > 10` and also `thru 10`.

      (%i1) series: 1$
      (%i2) term: exp (sin (x))$
      (%i3) for p: 1 unless p > 7 do
                (term: diff (term, x)/p,
                 series: series + subst (x=0, term)*x^p)$
      (%i4) series;
                        7    6     5    4    2
                       x    x     x    x    x
      (%o4)            -- - --- - -- - -- + -- + x + 1
                       90   240   15   8    2

  which gives 8 terms of the Taylor series for `e^sin(x)`.

      (%i1) poly: 0$
      (%i2) for i: 1 thru 5 do
                for j: i step -1 thru 1 do
                    poly: poly + i*x^j$
      (%i3) poly;
                        5      4       3       2
      (%o3)          5 x  + 9 x  + 12 x  + 14 x  + 15 x
      (%i4) guess: -3.0$
      (%i5) for i: 1 thru 10 do
                (guess: subst (guess, x, 0.5*(x + 10/x)),
                 if abs (guess^2 - 10) < 0.00005 then return (guess));
      (%o5)                  - 3.162280701754386

  This example computes the negative square root of 10 using the Newton- Raphson iteration a maximum of 10 times. Had the convergence criterion not been met the value returned would have been `done`.

  Instead of always adding a quantity to the control-variable one may sometimes wish to change it in some other way for each iteration. In this case one may use `next expression` instead of `step increment`. This will cause the control-variable to be set to the result of evaluating expression each time through the loop.

      (%i6) for count: 2 next 3*count thru 20 do display (count)$
                                  count = 2

                                  count = 6

                                 count = 18

  As an alternative to `for variable: value ...do...` the syntax `for variable from value ...do...` may be used. This permits the `from value` to be placed after the `step` or `next` value or after the termination condition. If `from value` is omitted then 1 is used as the initial value.

  Sometimes one may be interested in performing an iteration where the control-variable is never actually used. It is thus permissible to give only the termination conditions omitting the initialization and updating information as in the following example to compute the square-root of 5 using a poor initial guess.

      (%i1) x: 1000$
      (%i2) thru 20 do x: 0.5*(x + 5.0/x)$
      (%i3) x;
      (%o3)                   2.23606797749979
      (%i4) sqrt(5), numer;
      (%o4)                   2.23606797749979

  If it is desired one may even omit the termination conditions entirely and just give `do body` which will continue to evaluate the body indefinitely. In this case the function `return` should be used to terminate execution of the `do`.

      (%i1) newton (f, x):= ([y, df, dfx], df: diff (f ('x), 'x),
                do (y: ev(df), x: x - f(x)/y,
                    if abs (f (x)) < 5e-6 then return (x)))$
      (%i2) sqr (x) := x^2 - 5.0$
      (%i3) newton (sqr, 1000);
      (%o3)                   2.236068027062195

  (Note that `return`, when executed, causes the current value of `x` to be returned as the value of the `do`. The `block` is exited and this value of the `do` is returned as the value of the `block` because the `do` is the last statement in the block.)

  One other form of the `do` is available in Maxima. The syntax is:

      for variable in list end_tests do body

  The elements of list are any expressions which will successively be assigned to the `variable` on each iteration of the body. The optional termination tests end_tests can be used to terminate execution of the `do`; otherwise it will terminate when the list is exhausted or when a `return` is executed in the body. (In fact, `list` may be any non-atomic expression, and successive parts are taken.)

      (%i1)  for f in [log, rho, atan] do ldisp(f(1))$
      (%t1)                                  0
      (%t2)                                rho(1)
                                           %pi
      (%t3)                                 ---
                                            4
      (%i4) ev(%t3,numer);
      (%o4)                             0.78539816

  ·

  @ref{Category: Programming}

Function: errcatch (expr_1, …, expr_n)

  Evaluates expr_1, …, expr_n one by one and returns `[expr_n]` (a list) if no error occurs. If an error occurs in the evaluation of any argument, `errcatch` prevents the error from propagating and returns the empty list `[]` without evaluating any more arguments.

  `errcatch` is useful in `batch` files where one suspects an error might occur which would terminate the `batch` if the error weren't caught.

  ·

  @ref{Category: Programming}

Function: error (expr_1, …, expr_n)
System variable: error

  Evaluates and prints expr_1, …, expr_n, and then causes an error return to top level Maxima or to the nearest enclosing `errcatch`.

  The variable `error` is set to a list describing the error. The first element of `error` is a format string, which merges all the strings among the arguments expr_1, …, expr_n, and the remaining elements are the values of any non-string arguments.

  `errormsg()` formats and prints `error`. This is effectively reprinting the most recent error message.

  ·

  @ref{Category: Programming}

Option variable: error_size

  Default value: 10

  `error_size` modifies error messages according to the size of expressions which appear in them. If the size of an expression (as determined by the Lisp function `ERROR-SIZE`) is greater than `error_size`, the expression is replaced in the message by a symbol, and the symbol is assigned the expression. The symbols are taken from the list `error_syms`.

  Otherwise, the expression is smaller than `error_size`, and the expression is displayed in the message.

  See also `error` and `error_syms`.

  Example:

  The size of `U`, as determined by `ERROR-SIZE`, is 24.

      (%i1) U: (C^D^E + B + A)/(cos(X-1) + 1)$

      (%i2) error_size: 20$

      (%i3) error ("Example expression is", U);

      Example expression is errexp1
       -- an error.  Quitting.  To debug this try debugmode(true);
      (%i4) errexp1;
                                  E
                                 D
                                C   + B + A
      (%o4)                    --------------
                               cos(X - 1) + 1
      (%i5) error_size: 30$

      (%i6) error ("Example expression is", U);

                               E
                              D
                             C   + B + A
      Example expression is --------------
                            cos(X - 1) + 1
       -- an error.  Quitting.  To debug this try debugmode(true);

  ·

  @ref{Category: Debugging} · @ref{Category: Display flags and variables}

Option variable: error_syms

  Default value: `[errexp1, errexp2, errexp3]`

  In error messages, expressions larger than `error_size` are replaced by symbols, and the symbols are set to the expressions. The symbols are taken from the list `error_syms`. The first too-large expression is replaced by `error_syms[1]`, the second by `error_syms[2]`, and so on.

  If there are more too-large expressions than there are elements of `error_syms`, symbols are constructed automatically, with the n-th symbol equivalent to `concat ('errexp, n)`.

  See also `error` and `error_size`.

  ·

  @ref{Category: Debugging} · @ref{Category: Display flags and variables}

Function: errormsg ()

  Reprints the most recent error message. The variable `error` holds the message, and `errormsg` formats and prints it.

  ·

  @ref{Category: Programming}

Option variable: errormsg

  Default value: `true`

  When `false` the output of error messages is suppressed.

  The option variable `errormsg` can not be set in a block to a local value. The global value of `errormsg` is always present.

      (%i1) errormsg;
      (%o1)                                true
      (%i2) sin(a,b);
      Wrong number of arguments to sin
       -- an error. To debug this try: debugmode(true);
      (%i3) errormsg:false;
      (%o3)                                false
      (%i4) sin(a,b);

       -- an error. To debug this try: debugmode(true);

  The option variable `errormsg` can not be set in a block to a local value.

      (%i1) f(bool):=block([errormsg:bool],
                           print ("value of errormsg is",errormsg))$
      (%i2) errormsg:true;
      (%o2)                                true
      (%i3) f(false);
      value of errormsg is true
      (%o3)                                true
      (%i4) errormsg:false;
      (%o4)                                false
      (%i5) f(true);
      value of errormsg is false
      (%o5)                                false

  ·

  @ref{Category: Programming}

Special operator: for

  Used in iterations. See `do` for a description of Maxima's iteration facilities.

  ·

  @ref{Category: Programming}

Function: go (tag)

  is used within a `block` to transfer control to the statement of the block which 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+1, ..., go(loop), ...)

  The argument to `go` must be the name of a tag appearing in the same `block`. One cannot use `go` to transfer to tag in a `block` other than the one containing the `go`.

  ·

  @ref{Category: Programming}

Special operator: if

  Represents conditional evaluation. Various forms of `if` expressions are recognized.

  `if cond_1 then expr_1 else expr_0` evaluates to expr_1 if cond_1 evaluates to `true`, otherwise the expression evaluates to expr_0.

  The command `if cond_1 then expr_1 elseif cond_2 then expr_2 elseif ... else expr_0` evaluates to expr_k if cond_k is `true` and all preceding conditions are `false`. If none of the conditions are `true`, the expression evaluates to `expr_0`.

  A trailing `else false` is assumed if `else` is missing. That is, the command `if cond_1 then expr_1` is equivalent to `if cond_1 then expr_1 else false`, and the command `if cond_1 then expr_1 elseif ... elseif cond_n then expr_n` is equivalent to `if cond_1 then expr_1 elseif ... elseif cond_n then expr_n else false`.

  The alternatives expr_0, …, expr_n may be any Maxima expressions, including nested `if` expressions. The alternatives are neither simplified nor evaluated unless the corresponding condition is `true`.

  The conditions cond_1, …, cond_n are expressions which potentially or actually evaluate to `true` or `false`. When a condition does not actually evaluate to `true` or `false`, the behavior of `if` is governed by the global flag `prederror`. When `prederror` is `true`, it is an error if any evaluated condition does not evaluate to `true` or `false`. Otherwise, conditions which do not evaluate to `true` or `false` are accepted, and the result is a conditional expression.

  Among other elements, conditions may comprise relational and logical operators as follows.

      Operation            Symbol      Type

      less than            <           relational infix
      less than            <=
        or equal to                    relational infix
      equality (syntactic) =           relational infix
      negation of =        #           relational infix
      equality (value)     equal       relational function
      negation of equal    notequal    relational function
      greater than         >=
        or equal to                    relational infix
      greater than         >           relational infix
      and                  and         logical infix
      or                   or          logical infix
      not                  not         logical prefix

  ·

  @ref{Category: Programming} · @ref{Category: Predicate functions}

Function: map (f, expr_1, …, expr_n)

  Returns an expression whose leading operator is the same as that of the expressions expr_1, …, expr_n but whose subparts are the results of applying f to the corresponding subparts of the expressions. f is either the name of a function of n arguments or is a `lambda` form of n arguments.

  `maperror` - if `false` will cause all of the mapping functions to (1) stop when they finish going down the shortest expr_i if not all of the expr_i are of the same length and (2) apply f to [expr_1, expr_2, …] if the expr_i are not all the same type of object. If `maperror` is `true` then an error message will be given in the above two instances.

  One of the uses of this function is to `map` a function (e.g. `partfrac`) onto each term of a very large expression where it ordinarily wouldn't be possible to use the function on the entire expression due to an exhaustion of list storage space in the course of the computation.

      (%i1) map(f,x+a*y+b*z);
      (%o1)                        f(b z) + f(a y) + f(x)
      (%i2) map(lambda([u],partfrac(u,x)),x+1/(x^3+4*x^2+5*x+2));
                                 1       1        1
      (%o2)                     ----- - ----- + -------- + x
                               x + 2   x + 1          2
                                               (x + 1)
      (%i3) map(ratsimp, x/(x^2+x)+(y^2+y)/y);
                                            1
      (%o3)                            y + ----- + 1
                                          x + 1
      (%i4) map("=",[a,b],[-0.5,3]);
      (%o4)                          [a = - 0.5, b = 3]

  ·

  @ref{Category: Function application}

Function: mapatom (expr)

  Returns `true` if and only if expr is treated by the mapping routines as an atom. "Mapatoms" are atoms, numbers (including rational numbers), and subscripted variables.

  ·

  @ref{Category: Predicate functions}

Option variable: maperror

  Default value: `true`

  When `maperror` is `false`, causes all of the mapping functions, for example

      map (f, expr_1, expr_2, ...)

  to (1) stop when they finish going down the shortest expr_i if not all of the expr_i are of the same length and (2) apply f to [expr_1, expr_2, …] if the expr_i are not all the same type of object.

  If `maperror` is `true` then an error message is displayed in the above two instances.

  ·

  @ref{Category: Function application}

Option variable: mapprint

  Default value: `true`

  When `mapprint` is `true`, various information messages from `map`, `mapl`, and `fullmap` are produced in certain situations. These include situations where `map` would use `apply`, or `map` is truncating on the shortest list.

  If `mapprint` is `false`, these messages are suppressed.

  ·

  @ref{Category: Function application}

Function: maplist (f, expr_1, …, expr_n)

  Returns a list of the applications of f to the parts of the expressions expr_1, …, expr_n. f is the name of a function, or a lambda expression.

  `maplist` differs from `map(f, expr_1, ..., expr_n)` which returns an expression with the same main operator as expr_i has (except for simplifications and the case where `map` does an `apply`).

  ·

  @ref{Category: Function application}

Option variable: prederror

  Default value: `false`

  When `prederror` is `true`, an error message is displayed whenever the predicate of an `if` statement or an `is` function fails to evaluate to either `true` or `false`.

  If `false`, `unknown` is returned instead in this case. The `prederror: false` mode is not supported in translated code; however, `maybe` is supported in translated code.

  See also `is` and `maybe`.

  ·

  @ref{Category: Programming} · @ref{Category: Predicate functions}

Function: return (value)

  May be used to exit explicitly from a block, bringing its argument. See `block` for more information.

  ·

  @ref{Category: Programming}

Function: scanmap (f, expr)
Function: scanmap (f, expr, bottomup)

  Recursively applies f to expr, in a top down manner. This is most useful when complete factorization is desired, for example:

      (%i1) exp:(a^2+2*a+1)*y + x^2$
      (%i2) scanmap(factor,exp);
                                          2      2
      (%o2)                         (a + 1)  y + x

  Note the way in which `scanmap` applies the given function `factor` to the constituent subexpressions of expr; if another form of expr is presented to `scanmap` then the result may be different. Thus, `%o2` is not recovered when `scanmap` is applied to the expanded form of `exp`:

      (%i3) scanmap(factor,expand(exp));
                                 2                  2
      (%o3)                      a  y + 2 a y + y + x

  Here is another example of the way in which `scanmap` recursively applies a given function to all subexpressions, including exponents:

      (%i4) expr : u*v^(a*x+b) + c$
      (%i5) scanmap('f, expr);
                          f(f(f(a) f(x)) + f(b))
      (%o5) f(f(f(u) f(f(v)                      )) + f(c))

  `scanmap (f, expr, bottomup)` applies f to expr in a bottom-up manner. E.g., for undefined `f`,

      scanmap(f,a*x+b) ->
         f(a*x+b) -> f(f(a*x)+f(b)) -> f(f(f(a)*f(x))+f(b))
      scanmap(f,a*x+b,bottomup) -> f(a)*f(x)+f(b)
          -> f(f(a)*f(x))+f(b) ->
           f(f(f(a)*f(x))+f(b))

  In this case, you get the same answer both ways.

  ·

  @ref{Category: Function application}

Function: throw (expr)

  Evaluates expr and throws the value back to the most recent `catch`. `throw` is used with `catch` as a nonlocal return mechanism.

  ·

  @ref{Category: Programming}

Special operator: while
Special operator: unless

  See `do`.

  ·

  @ref{Category: Programming}

Function: outermap (f, a_1, …, a_n)

  Applies the function f to each one of the elements of the outer product a_1 cross a_2 … cross a_n.

  f is the name of a function of n arguments or a lambda expression of n arguments. Each argument a_k may be a list or nested list, or a matrix, or any other kind of expression.

  The `outermap` return value is a nested structure. Let x be the return value. Then x has the same structure as the first list, nested list, or matrix argument, `x[i_1]...[i_m]` has the same structure as the second list, nested list, or matrix argument, `x[i_1]...[i_m][j_1]...[j_n]` has the same structure as the third list, nested list, or matrix argument, and so on, where m, n, … are the numbers of indices required to access the elements of each argument (one for a list, two for a matrix, one or more for a nested list). Arguments which are not lists or matrices have no effect on the structure of the return value.

  Note that the effect of `outermap` is different from that of applying f to each one of the elements of the outer product returned by `cartesian_product`. `outermap` preserves the structure of the arguments in the return value, while `cartesian_product` does not.

  `outermap` evaluates its arguments.

  See also `map`, `maplist`, and `apply`.

  Examples:

  Elementary examples of `outermap`. To show the argument combinations more clearly, `F` is left undefined.

      (%i1) outermap(F, [a, b, c], [1, 2, 3]);
      (%o1) [[F(a, 1), F(a, 2), F(a, 3)], [F(b, 1), F(b, 2), F(b, 3)],
                                           [F(c, 1), F(c, 2), F(c, 3)]]
      (%i2) outermap(F, matrix([a, b],[c, d]), matrix([1, 2],[3, 4]));
               [ [ F(a, 1)  F(a, 2) ]  [ F(b, 1)  F(b, 2) ] ]
               [ [                  ]  [                  ] ]
               [ [ F(a, 3)  F(a, 4) ]  [ F(b, 3)  F(b, 4) ] ]
      (%o2)    [                                            ]
               [ [ F(c, 1)  F(c, 2) ]  [ F(d, 1)  F(d, 2) ] ]
               [ [                  ]  [                  ] ]
               [ [ F(c, 3)  F(c, 4) ]  [ F(d, 3)  F(d, 4) ] ]
      (%i3) outermap (F, [a, b], x, matrix ([1, 2], [3, 4]));
             [ F(a, x, 1)  F(a, x, 2) ]  [ F(b, x, 1)  F(b, x, 2) ]
      (%o3) [[                        ], [                        ]]
             [ F(a, x, 3)  F(a, x, 4) ]  [ F(b, x, 3)  F(b, x, 4) ]
      (%i4) outermap (F, [a, b], matrix ([1, 2]), matrix ([x], [y]));
             [ [ F(a, 1, x) ]  [ F(a, 2, x) ] ]
      (%o4) [[ [            ]  [            ] ],
             [ [ F(a, 1, y) ]  [ F(a, 2, y) ] ]
                                    [ [ F(b, 1, x) ]  [ F(b, 2, x) ] ]
                                    [ [            ]  [            ] ]]
                                    [ [ F(b, 1, y) ]  [ F(b, 2, y) ] ]
      (%i5) outermap ("+", [a, b, c], [1, 2, 3]);
      (%o5) [[a + 1, a + 2, a + 3], [b + 1, b + 2, b + 3],
                                                 [c + 1, c + 2, c + 3]]

  A closer examination of the `outermap` return value. The first, second, and third arguments are a matrix, a list, and a matrix, respectively. The return value is a matrix. Each element of that matrix is a list, and each element of each list is a matrix.

      (%i1) arg_1 :  matrix ([a, b], [c, d]);
                                  [ a  b ]
      (%o1)                       [      ]
                                  [ c  d ]
      (%i2) arg_2 : [11, 22];
      (%o2)                       [11, 22]
      (%i3) arg_3 : matrix ([xx, yy]);
      (%o3)                      [ xx  yy ]
      (%i4) xx_0 : outermap(lambda([x, y, z], x / y + z), arg_1,
                                                         arg_2, arg_3);
                     [  [      a        a  ]  [      a        a  ]  ]
                     [ [[ xx + --  yy + -- ], [ xx + --  yy + -- ]] ]
                     [  [      11       11 ]  [      22       22 ]  ]
      (%o4)  Col 1 = [                                              ]
                     [  [      c        c  ]  [      c        c  ]  ]
                     [ [[ xx + --  yy + -- ], [ xx + --  yy + -- ]] ]
                     [  [      11       11 ]  [      22       22 ]  ]
                       [  [      b        b  ]  [      b        b  ]  ]
                       [ [[ xx + --  yy + -- ], [ xx + --  yy + -- ]] ]
                       [  [      11       11 ]  [      22       22 ]  ]
               Col 2 = [                                              ]
                       [  [      d        d  ]  [      d        d  ]  ]
                       [ [[ xx + --  yy + -- ], [ xx + --  yy + -- ]] ]
                       [  [      11       11 ]  [      22       22 ]  ]
      (%i5) xx_1 : xx_0 [1][1];
                 [      a        a  ]  [      a        a  ]
      (%o5)     [[ xx + --  yy + -- ], [ xx + --  yy + -- ]]
                 [      11       11 ]  [      22       22 ]
      (%i6) xx_2 : xx_0 [1][1] [1];
                            [      a        a  ]
      (%o6)                 [ xx + --  yy + -- ]
                            [      11       11 ]
      (%i7) xx_3 : xx_0 [1][1] [1] [1][1];
                                        a
      (%o7)                        xx + --
                                        11
      (%i8) [op (arg_1), op (arg_2), op (arg_3)];
      (%o8)                  [matrix, [, matrix]
      (%i9) [op (xx_0), op (xx_1), op (xx_2)];
      (%o9)                  [matrix, [, matrix]

  `outermap` preserves the structure of the arguments in the return value, while `cartesian_product` does not.

      (%i1) outermap (F, [a, b, c], [1, 2, 3]);
      (%o1) [[F(a, 1), F(a, 2), F(a, 3)], [F(b, 1), F(b, 2), F(b, 3)],
                                           [F(c, 1), F(c, 2), F(c, 3)]]
      (%i2) setify (flatten (%));
      (%o2) {F(a, 1), F(a, 2), F(a, 3), F(b, 1), F(b, 2), F(b, 3),
                                             F(c, 1), F(c, 2), F(c, 3)}
      (%i3) map(lambda([L], apply(F, L)),
                           cartesian_product({a, b, c}, {1, 2, 3}));
      (%o3) {F(a, 1), F(a, 2), F(a, 3), F(b, 1), F(b, 2), F(b, 3),
                                             F(c, 1), F(c, 2), F(c, 3)}
      (%i4) is (equal (%, %th (2)));
      (%o4)                         true

  ·

  @ref{Category: Function application}

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