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36.1 Introduction to Function Definition | ||
36.2 Function | ||
36.3 Macros | ||
36.4 Functions and Variables for Function Definition |
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Categories: Function definition · Programming
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To define a function in Maxima you use the :=
operator.
E.g.
f(x) := sin(x)
defines a function f
.
Anonymous functions may also be created using lambda
.
For example
lambda ([i, j], ...)
can be used instead of f
where
f(i,j) := block ([], ...); map (lambda ([i], i+1), l)
would return a list with 1 added to each term.
You may also define a function with a variable number of arguments, by having a final argument which is assigned to a list of the extra arguments:
(%i1) f ([u]) := u; (%o1) f([u]) := u (%i2) f (1, 2, 3, 4); (%o2) [1, 2, 3, 4] (%i3) f (a, b, [u]) := [a, b, u]; (%o3) f(a, b, [u]) := [a, b, u] (%i4) f (1, 2, 3, 4, 5, 6); (%o4) [1, 2, [3, 4, 5, 6]]
The right hand side of a function is an expression. Thus if you want a sequence of expressions, you do
f(x) := (expr1, expr2, ...., exprn);
and the value of exprn is what is returned by the function.
If you wish to make a return
from some expression inside the
function then you must use block
and return
.
block ([], expr1, ..., if (a > 10) then return(a), ..., exprn)
is itself an expression, and so could take the place of the right hand side of a function definition. Here it may happen that the return happens earlier than the last expression.
The first []
in the block, may contain a list of variables and
variable assignments, such as [a: 3, b, c: []]
, which would cause the
three variables a
,b
,and c
to not refer to their
global values, but rather have these special values for as long as the
code executes inside the block
, or inside functions called from
inside the block
. This is called dynamic binding, since the
variables last from the start of the block to the time it exits. Once
you return from the block
, or throw out of it, the old values (if
any) of the variables will be restored. It is certainly a good idea
to protect your variables in this way. Note that the assignments
in the block variables, are done in parallel. This means, that if
you had used c: a
in the above, the value of c
would
have been the value of a
at the time you just entered the block,
but before a
was bound. Thus doing something like
block ([a: a], expr1, ... a: a+3, ..., exprn)
will protect the external value of a
from being altered, but
would let you access what that value was. Thus the right hand
side of the assignments, is evaluated in the entering context, before
any binding occurs.
Using just block ([x], ...)
would cause the x
to have itself
as value, just as if it would have if you entered a fresh Maxima
session.
The actual arguments to a function are treated in exactly same way as the variables in a block. Thus in
f(x) := (expr1, ..., exprn);
and
f(1);
we would have a similar context for evaluation of the expressions as if we had done
block ([x: 1], expr1, ..., exprn)
Inside functions, when the right hand side of a definition,
may be computed at runtime, it is useful to use define
and
possibly buildq
.
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An array function stores the function value the first time it is called with a given argument, and returns the stored value, without recomputing it, when that same argument is given. Such a function is often called a memoizing function.
Array function names are appended to the global list arrays
(not the global list functions
). arrayinfo
returns the list of
arguments for which there are stored values, and listarray
returns the
stored values. dispfun
and fundef
return the array function
definition.
arraymake
constructs an array function call,
analogous to funmake
for ordinary functions.
arrayapply
applies an array function to its arguments,
analogous to apply
for ordinary functions.
There is nothing exactly analogous to map
for array functions,
although map(lambda([x], a[x]), L)
or
makelist(a[x], x, L)
, where L is a list,
are not too far off the mark.
remarray
removes an array function definition (including any stored
function values), analogous to remfunction
for ordinary functions.
kill(a[x])
removes the value of the array function a
stored for the argument x;
the next time a is called with argument x,
the function value is recomputed.
However, there is no way to remove all of the stored values at once,
except for kill(a)
or remarray(a)
,
which also remove the function definition.
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Substitutes variables named by the list L into the expression expr,
in parallel, without evaluating expr. The resulting expression is
simplified, but not evaluated, after buildq
carries out the substitution.
The elements of L are symbols or assignment expressions
symbol: value
, evaluated in parallel. That is, the binding
of a variable on the right-hand side of an assignment is the binding of that
variable in the context from which buildq
was called, not the binding of
that variable in the variable list L. If some variable in L is not
given an explicit assignment, its binding in buildq
is the same as in
the context from which buildq
was called.
Then the variables named by L are substituted into expr in parallel. That is, the substitution for every variable is determined before any substitution is made, so the substitution for one variable has no effect on any other.
If any variable x appears as splice (x)
in expr,
then x must be bound to a list,
and the list is spliced (interpolated) into expr instead of substituted.
Any variables in expr not appearing in L are carried into the result
verbatim, even if they have bindings in the context from which buildq
was called.
Examples
a
is explicitly bound to x
, while b
has the same binding
(namely 29) as in the calling context, and c
is carried through verbatim.
The resulting expression is not evaluated until the explicit evaluation
''%
.
(%i1) (a: 17, b: 29, c: 1729)$ (%i2) buildq ([a: x, b], a + b + c); (%o2) x + c + 29 (%i3) ''%; (%o3) x + 1758
e
is bound to a list, which appears as such in the arguments of
foo
, and interpolated into the arguments of bar
.
(%i1) buildq ([e: [a, b, c]], foo (x, e, y)); (%o1) foo(x, [a, b, c], y) (%i2) buildq ([e: [a, b, c]], bar (x, splice (e), y)); (%o2) bar(x, a, b, c, y)
The result is simplified after substitution. If simplification were applied before substitution, these two results would be the same.
(%i1) buildq ([e: [a, b, c]], splice (e) + splice (e)); (%o1) 2 c + 2 b + 2 a (%i2) buildq ([e: [a, b, c]], 2 * splice (e)); (%o2) 2 a b c
The variables in L are bound in parallel; if bound sequentially,
the first result would be foo (b, b)
.
Substitutions are carried out in parallel;
compare the second result with the result of subst
,
which carries out substitutions sequentially.
(%i1) buildq ([a: b, b: a], foo (a, b)); (%o1) foo(b, a) (%i2) buildq ([u: v, v: w, w: x, x: y, y: z, z: u], bar (u, v, w, x, y, z)); (%o2) bar(v, w, x, y, z, u) (%i3) subst ([u=v, v=w, w=x, x=y, y=z, z=u], bar (u, v, w, x, y, z)); (%o3) bar(u, u, u, u, u, u)
Construct a list of equations with some variables or expressions on the
left-hand side and their values on the right-hand side. macroexpand
shows the expression returned by show_values
.
(%i1) show_values ([L]) ::= buildq ([L], map ("=", 'L, L)); (%o1) show_values([L]) ::= buildq([L], map("=", 'L, L)) (%i2) (a: 17, b: 29, c: 1729)$ (%i3) show_values (a, b, c - a - b); (%o3) [a = 17, b = 29, c - b - a = 1683] (%i4) macroexpand (show_values (a, b, c - a - b)); (%o4) map(=, '([a, b, c - b - a]), [a, b, c - b - a])
Given a function of several arguments, create another function for which some of the arguments are fixed.
(%i1) curry (f, [a]) := buildq ([f, a], lambda ([[x]], apply (f, append (a, x))))$ (%i2) by3 : curry ("*", 3); (%o2) lambda([[x]], apply(*, append([3], x))) (%i3) by3 (a + b); (%o3) 3 (b + a)
Categories: Function definition
Returns the macro expansion of expr without evaluating it,
when expr
is a macro function call.
Otherwise, macroexpand
returns expr.
If the expansion of expr yields another macro function call, that macro function call is also expanded.
macroexpand
quotes its argument.
However, if the expansion of a macro function call has side effects,
those side effects are executed.
See also ::=
, macros
, and macroexpand1
..
Examples
(%i1) g (x) ::= x / 99; x (%o1) g(x) ::= -- 99 (%i2) h (x) ::= buildq ([x], g (x - a)); (%o2) h(x) ::= buildq([x], g(x - a)) (%i3) a: 1234; (%o3) 1234 (%i4) macroexpand (h (y)); y - a (%o4) ----- 99 (%i5) h (y); y - 1234 (%o5) -------- 99
Categories: Function application
Returns the macro expansion of expr without evaluating it,
when expr
is a macro function call.
Otherwise, macroexpand1
returns expr.
macroexpand1
quotes its argument.
However, if the expansion of a macro function call has side effects,
those side effects are executed.
If the expansion of expr yields another macro function call, that macro function call is not expanded.
See also ::=
, macros
, and macroexpand
.
Examples
(%i1) g (x) ::= x / 99; x (%o1) g(x) ::= -- 99 (%i2) h (x) ::= buildq ([x], g (x - a)); (%o2) h(x) ::= buildq([x], g(x - a)) (%i3) a: 1234; (%o3) 1234 (%i4) macroexpand1 (h (y)); (%o4) g(y - a) (%i5) h (y); y - 1234 (%o5) -------- 99
Categories: Function application
Default value: []
macros
is the list of user-defined macro functions.
The macro function definition operator ::=
puts a new macro function
onto this list, and kill
, remove
, and remfunction
remove
macro functions from the list.
See also infolists
.
Categories: Function definition · Global variables
Splices (interpolates) the list named by the atom a into an expression,
but only if splice
appears within buildq
;
otherwise, splice
is treated as an undefined function.
If appearing within buildq
as a alone (without splice
),
a is substituted (not interpolated) as a list into the result.
The argument of splice
can only be an atom;
it cannot be a literal list or an expression which yields a list.
Typically splice
supplies the arguments for a function or operator.
For a function f
, the expression f (splice (a))
within
buildq
expands to f (a[1], a[2], a[3], ...)
.
For an operator o
, the expression "o" (splice (a))
within
buildq
expands to "o" (a[1], a[2], a[3], ...)
,
where o
may be any type of operator (typically one which takes multiple
arguments). Note that the operator must be enclosed in double quotes "
.
Examples
(%i1) buildq ([x: [1, %pi, z - y]], foo (splice (x)) / length (x)); foo(1, %pi, z - y) (%o1) ----------------------- length([1, %pi, z - y]) (%i2) buildq ([x: [1, %pi]], "/" (splice (x))); 1 (%o2) --- %pi (%i3) matchfix ("<>", "<>"); (%o3) <> (%i4) buildq ([x: [1, %pi, z - y]], "<>" (splice (x))); (%o4) <>1, %pi, z - y<>
Categories: Function definition
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Constructs and evaluates an expression F(arg_1, ...,
arg_n)
.
apply
does not attempt to distinguish array functions from ordinary
functions; when F is the name of an array 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.
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
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.
Categories: Expressions · Programming
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
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
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
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:
mode_declare
or a statement like the following one:
put(x_1, bigfloat, numerical_type)
'
tells the compiler that the text is meant as an option.
Categories: Translation and compilation
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 an array 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.
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 an
array 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
Introduces a global variable into the Maxima environment.
define_variable
is useful in user-written packages, which are often
translated or compiled.
define_variable
carries out the following steps:
mode_declare (name, mode)
declares the mode of name to
the translator. See mode_declare
for a list of the possible modes.
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. -- 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'. -- an error. To debug this try: debugmode(true); (%i6) baz_quux; (%o6) baz_quux
Categories: Translation and compilation
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
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
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
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
.
Array 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
Returns the definition of the function f.
The argument may be
::=
),
:=
or define
),
:=
or define
, but enclosing arguments in square brackets [ ]
),
:=
or define
,
but enclosing some arguments in square brackets and others in parentheses
( )
),
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
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 array functions from ordinary
functions; when F is the name of an array 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.
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
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:
(%i1) f: lambda ([x], x^2); 2 (%o1) lambda([x], x ) (%i2) f(a); 2 (%o2) a
(%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 ]
''
.
(%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
(%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
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
[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
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
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
Default value: true
When mode_checkp
is true
, mode_declare
checks the modes
of bound variables.
Categories: Translation flags and variables
Default value: false
When mode_check_errorp
is true
, mode_declare
calls
error.
Categories: Translation flags and variables
Default value: true
When mode_check_warnp
is true
, mode errors are
described.
Categories: Translation flags and variables
mode_declare
is used to declare the modes of variables and functions for
subsequent translation or compilation of functions. mode_declare
is
typically placed at the beginning of a function definition, at the beginning of
a Maxima script, or executed at the interactive prompt.
The arguments of mode_declare
are pairs consisting of a variable and a
mode which is one of boolean
, fixnum
, number
,
rational
, or float
. Each variable may also be a list of variables
all of which are declared to have the same mode.
If a variable is an array, and if every element of the array which is
referenced has a value then array (yi, complete, dim1, dim2, ...)
rather than
array(yi, dim1, dim2, ...)
should be used when first declaring the bounds of the array.
If all the elements of the array are of mode fixnum
(float
), use
fixnum
(float
) instead of complete
.
Also if every element of the array is of the same mode, say m
, then
mode_declare (completearray (yi), m))
should be used for efficient translation.
Numeric code using arrays might run faster by declaring the expected size of the array, as in:
mode_declare (completearray (a [10, 10]), float)
for a floating point number array which is 10 x 10.
One may declare the mode of the result of a function by using
function (f_1, f_2, ...)
as an argument; here f_1
, f_2
,
… are the names of functions. 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
.
Categories: Translation and compilation
A special form used with mode_declare
and macros
to declare, e.g.,
a list of lists of flonums, or other compound data object. The first argument
to mode_identity
is a primitive value mode name as given to
mode_declare
(i.e., one of float
, fixnum
, number
,
list
, or any
), and the second argument is an expression which is
evaluated and returned as the value of mode_identity
. However, if the
return value is not allowed by the mode declared in the first
argument, an error or warning is signalled. The important thing is
that the mode of the expression as determined by the Maxima to Lisp
translator, will be that given as the first argument, independent of
anything that goes on in the second argument.
E.g., x: 3.3; mode_identity (fixnum, x);
yields an error.
mode_identity (flonum, x)
returns 3.3 .
This has a number of uses, e.g., if you knew that first (l)
returned a
number then you might write mode_identity (number, first (l))
. However,
a more efficient way to do it would be to define a new primitive,
firstnumb (x) ::= buildq ([x], mode_identity (number, first(x)));
and use firstnumb
every time you take the first of a list of numbers.
Categories: Translation and compilation
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 array functions or subscripted functions.
remarray
applies to those types of functions.
Categories: Function definition
Default value: true
When savedef
is true
, the Maxima version of a 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
Default value: true
When transcompile
is true
, translate
and
translate_file
generate declarations to make the translated code more
suitable for compilation.
compfile
sets transcompile: true
for the duration.
Categories: Translation flags and variables
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 a 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 a 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
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_semicompile
,
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
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
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
Default value: true
When tr_bound_function_applyp
is true
, Maxima gives a warning if a
bound variable (such as a function argument) is found being used as a function.
tr_bound_function_applyp
does not affect the code generated in such
cases.
For example, an expression such as g (f, x) := f (x+1)
will trigger
the warning message.
Categories: Translation flags and variables
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
Default value: true
Tells the Maxima-to-Lisp translator to assume that the functions
acos
, asin
, asec
, and acsc
can return complex results.
The ostensible effect of tr_float_can_branch_complex
is the following.
However, it appears that this flag has no effect on the translator output.
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
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
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
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
Default value: false
When tr_semicompile
is true
, translate_file
and
compfile
output forms which will be macroexpanded but not compiled into
machine code by the Lisp compiler.
Categories: Translation flags and variables
Default value:
[transcompile, tr_semicompile, tr_warn_undeclared, tr_warn_meval, tr_warn_fexpr, tr_warn_mode, tr_warn_undefined_variable, tr_function_call_default, tr_array_as_ref,tr_numer]
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
Prints a list of warnings which have been given by the translator during the current translation.
Categories: Translation and compilation
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
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
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
Default value: all
- Gives a warning when variables are assigned values inappropriate for their mode.
Categories: Translation flags and variables
Default value: compile
- Determines when to send warnings about undeclared variables to the TTY.
Categories: Translation flags and variables
Default value: all
- Gives a warning when undefined global variables are seen.
Categories: Translation flags and variables
Translates the Maxima file filename into Lisp, executes the Lisp compiler, and, if the translation and compilation succeed, loads the compiled code 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
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.
Categories: Translation and compilation
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