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6.1 Introduction to Expressions | ||
6.2 Nouns and Verbs | ||
6.3 Identifiers | ||
6.4 Inequality | ||
6.5 Functions and Variables for Expressions |
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There are a number of reserved words which should not be used as variable names. Their use would cause a possibly cryptic syntax error.
integrate next from diff in at limit sum for and elseif then else do or if unless product while thru step
Most things in Maxima are expressions. A sequence of expressions can be made into an expression by separating them by commas and putting parentheses around them. This is similar to the C comma expression.
(%i1) x: 3$ (%i2) (x: x+1, x: x^2); (%o2) 16 (%i3) (if (x > 17) then 2 else 4); (%o3) 4 (%i4) (if (x > 17) then x: 2 else y: 4, y+x); (%o4) 20
Even loops in Maxima are expressions, although the value they
return is the not too useful done
.
(%i1) y: (x: 1, for i from 1 thru 10 do (x: x*i))$ (%i2) y; (%o2) done
Whereas what you really want is probably to include a third term in the comma expression which actually gives back the value.
(%i3) y: (x: 1, for i from 1 thru 10 do (x: x*i), x)$ (%i4) y; (%o4) 3628800
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Maxima distinguishes between operators which are "nouns" and operators which are
"verbs". A verb is an operator which can be executed. A noun is an operator
which appears as a symbol in an expression, without being executed. By default,
function names are verbs. A verb can be changed into a noun by quoting the
function name or applying the nounify
function. A noun can be changed
into a verb by applying the verbify
function. The evaluation flag
nouns
causes ev
to evaluate nouns in an expression.
The verb form is distinguished by a leading dollar sign $
on the
corresponding Lisp symbol. In contrast, the noun form is distinguished by a
leading percent sign %
on the corresponding Lisp symbol. Some nouns have
special display properties, such as 'integrate
and 'derivative
(returned by diff
), but most do not. By default, the noun and verb forms
of a function are identical when displayed. The global flag noundisp
causes Maxima to display nouns with a leading quote mark '
.
See also noun
, nouns
, nounify
, and
verbify
.
Examples:
(%i1) foo (x) := x^2; 2 (%o1) foo(x) := x (%i2) foo (42); (%o2) 1764 (%i3) 'foo (42); (%o3) foo(42) (%i4) 'foo (42), nouns; (%o4) 1764 (%i5) declare (bar, noun); (%o5) done (%i6) bar (x) := x/17; x (%o6) bar(x) := -- 17 (%i7) bar (52); (%o7) bar(52) (%i8) bar (52), nouns; (%o8) bar(52) (%i9) integrate (1/x, x, 1, 42); (%o9) log(42) (%i10) 'integrate (1/x, x, 1, 42); 42 / [ 1 (%o10) I - dx ] x / 1 (%i11) ev (%, nouns); (%o11) log(42)
Categories: Evaluation · Nouns and verbs
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Maxima identifiers may comprise alphabetic characters, plus the numerals 0
through 9, plus any special character preceded by the backslash \
character.
A numeral may be the first character of an identifier if it is preceded by a backslash. Numerals which are the second or later characters need not be preceded by a backslash.
Characters may be declared alphabetic by the declare
function.
If so declared, they need not be preceded by a backslash in an identifier.
The alphabetic characters are initially A
through Z
, a
through z
, %
, and _
.
Maxima is case-sensitive. The identifiers foo
, FOO
, and
Foo
are distinct. See Lisp and Maxima for more on this point.
A Maxima identifier is a Lisp symbol which begins with a dollar sign $
.
Any other Lisp symbol is preceded by a question mark ?
when it appears
in Maxima. See Lisp and Maxima for more on this point.
Examples:
(%i1) %an_ordinary_identifier42; (%o1) %an_ordinary_identifier42 (%i2) embedded\ spaces\ in\ an\ identifier; (%o2) embedded spaces in an identifier (%i3) symbolp (%); (%o3) true (%i4) [foo+bar, foo\+bar]; (%o4) [foo + bar, foo+bar] (%i5) [1729, \1729]; (%o5) [1729, 1729] (%i6) [symbolp (foo\+bar), symbolp (\1729)]; (%o6) [true, true] (%i7) [is (foo\+bar = foo+bar), is (\1729 = 1729)]; (%o7) [false, false] (%i8) baz\~quux; (%o8) baz~quux (%i9) declare ("~", alphabetic); (%o9) done (%i10) baz~quux; (%o10) baz~quux (%i11) [is (foo = FOO), is (FOO = Foo), is (Foo = foo)]; (%o11) [false, false, false] (%i12) :lisp (defvar *my-lisp-variable* '$foo) *MY-LISP-VARIABLE* (%i12) ?\*my\-lisp\-variable\*; (%o12) foo
Categories: Syntax
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Maxima has the inequality operators <
, <=
, >=
, >
,
#
, and notequal
. See if
for a description of conditional
expressions.
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provides an alternate name for a (user or system) function, variable, array, etc. Any even number of arguments may be used.
Categories: Declarations and inferences
Default value: []
aliases
is the list of atoms which have a user defined alias (set up by
the alias
, ordergreat
, orderless
functions or by
declaring the atom a noun
with declare
.)
Categories: Declarations and inferences · Global variables
works with the part
commands (i.e. part
,
inpart
, substpart
, substinpart
,
dpart
, and lpart
).
For example,
(%i1) expr : e + d + c + b + a; (%o1) e + d + c + b + a (%i2) part (expr, [2, 5]); (%o2) d + a
while
(%i1) expr : e + d + c + b + a; (%o1) e + d + c + b + a (%i2) part (expr, allbut (2, 5)); (%o2) e + c + b
allbut
is also recognized by kill
.
(%i1) [aa : 11, bb : 22, cc : 33, dd : 44, ee : 55]; (%o1) [11, 22, 33, 44, 55] (%i2) kill (allbut (cc, dd)); (%o0) done (%i1) [aa, bb, cc, dd]; (%o1) [aa, bb, 33, 44]
kill(allbut(a_1, a_2, ...))
has the effect of
kill(all)
except that it does not kill the symbols a_1, a_2,
…
Returns the list of arguments of expr
, which may be any kind of
expression other than an atom. Only the arguments of the top-level operator
are extracted; subexpressions of expr
appear as elements or
subexpressions of elements of the list of arguments.
The order of the items in the list may depend on the global flag
inflag
.
args (expr)
is equivalent to substpart ("[", expr, 0)
.
See also substpart
, apply
, funmake
, and op
.
How to convert a matrix to a nested list:
(%i1) M:matrix([1,2],[3,4]); [ 1 2 ] (%o1) [ ] [ 3 4 ] (%i2) args(M); (%o2) [[1, 2], [3, 4]]
Since maxima internally treats a sum of n
terms as a summation command
with n
arguments args() can extract the list of terms in a sum:
(%i1) a+b+c; (%o1) c + b + a (%i2) args(%); (%o2) [c, b, a]
Categories: Expressions
Returns true
if expr is atomic (i.e. a number, name or string) else
false
. Thus atom(5)
is true
while atom(a[1])
and
atom(sin(x))
are false
(assuming a[1]
and x
are
unbound).
Categories: Expressions · Predicate functions
Returns expr enclosed in a box. The return value is an expression with
box
as the operator and expr as the argument. A box is drawn on
the display when display2d
is true
.
box (expr, a)
encloses expr in a box labelled by the
symbol a. The label is truncated if it is longer than the width of the
box.
box
evaluates its argument. However, a boxed expression does not
evaluate to its content, so boxed expressions are effectively excluded from
computations. rembox
removes the box again.
boxchar
is the character used to draw the box in box
and in the
dpart
and lpart
functions.
See also rembox
, dpart
and lpart
.
Examples:
(%i1) box (a^2 + b^2); """"""""" " 2 2" (%o1) "b + a " """"""""" (%i2) a : 1234; (%o2) 1234 (%i3) b : c - d; (%o3) c - d (%i4) box (a^2 + b^2); """""""""""""""""""" " 2 " (%o4) "(c - d) + 1522756" """""""""""""""""""" (%i5) box (a^2 + b^2, term_1); term_1"""""""""""""" " 2 " (%o5) "(c - d) + 1522756" """""""""""""""""""" (%i6) 1729 - box (1729); """""" (%o6) 1729 - "1729" """""" (%i7) boxchar: "-"; (%o7) - (%i8) box (sin(x) + cos(y)); ----------------- (%o8) -cos(y) + sin(x)- -----------------
Categories: Expressions
Default value: "
boxchar
is the character used to draw the box in the box
and in the dpart
and lpart
functions.
All boxes in an expression are drawn with the current value of boxchar
;
the drawing character is not stored with the box expression.
Categories: Expressions
Collapses expr by causing all of its common (i.e., equal) subexpressions
to share (i.e., use the same cells), thereby saving space. (collapse
is
a subroutine used by the optimize
command.) Thus, calling
collapse
may be useful after loading in a save
file. You can
collapse several expressions together by using
collapse ([expr_1, ..., expr_n])
. Similarly, you can
collapse the elements of the array A
by doing
collapse (listarray ('A))
.
Categories: Expressions
Return a copy of the Maxima expression e. Although e can be any Maxima expression, the copy function is the most useful when e is either a list or a matrix; consider:
(%i1) m : [1,[2,3]]$ (%i2) mm : m$ (%i3) mm[2][1] : x$ (%i4) m; (%o4) [1, [x, 3]] (%i5) mm; (%o5) [1, [x, 3]]
Let's try the same experiment, but this time let mm be a copy of m
(%i1) m : [1,[2,3]]$ (%i2) mm : copy(m)$ (%i3) mm[2][1] : x$ (%i4) m; (%o4) [1, [2, 3]] (%i5) mm; (%o5) [1, [x, 3]]
This time, the assignment to mm does not change the value of m.
Categories: Expressions
is similar to isolate
(expr, x)
except that it enables the
user to isolate more than one variable simultaneously. This might be useful,
for example, if one were attempting to change variables in a multiple
integration, and that variable change involved two or more of the integration
variables. This function is autoloaded from `simplification/disol.mac'.
A demo is available by demo("disol")$
.
Categories: Expressions
Returns the external representation of expr.
dispform(expr)
returns the external representation with respect to
the main (top-level) operator. dispform(expr, all)
returns the
external representation with respect to all operators in expr.
See also part
, inpart
, and inflag
.
Examples:
The internal representation of - x
is "negative one times x
"
while the external representation is "minus x
".
(%i1) - x; (%o1) - x (%i2) ?format (true, "~S~%", %); ((MTIMES SIMP) -1 $X) (%o2) false (%i3) dispform (- x); (%o3) - x (%i4) ?format (true, "~S~%", %); ((MMINUS SIMP) $X) (%o4) false
The internal representation of sqrt(x)
is "x
to the power 1/2"
while the external representation is "square root of x
".
(%i1) sqrt (x); (%o1) sqrt(x) (%i2) ?format (true, "~S~%", %); ((MEXPT SIMP) $X ((RAT SIMP) 1 2)) (%o2) false (%i3) dispform (sqrt (x)); (%o3) sqrt(x) (%i4) ?format (true, "~S~%", %); ((%SQRT SIMP) $X) (%o4) false
Use of the optional argument all
.
(%i1) expr : sin (sqrt (x)); (%o1) sin(sqrt(x)) (%i2) freeof (sqrt, expr); (%o2) true (%i3) freeof (sqrt, dispform (expr)); (%o3) true (%i4) freeof (sqrt, dispform (expr, all)); (%o4) false
Categories: Expressions
Selects the same subexpression as part
, but instead of just returning
that subexpression as its value, it returns the whole expression with the
selected subexpression displayed inside a box. The box is actually part of the
expression.
(%i1) dpart (x+y/z^2, 1, 2, 1); y (%o1) ---- + x 2 """ "z" """
Categories: Expressions
Default value: false
exptisolate
, when true
, causes isolate (expr, var)
to
examine exponents of atoms (such as %e
) which contain var
.
Categories: Expressions
Default value: false
exptsubst
, when true
, permits substitutions such as y
for %e^x
in %e^(a x)
.
(%i1) %e^(a*x); a x (%o1) %e (%i2) exptsubst; (%o2) false (%i3) subst(y, %e^x, %e^(a*x)); a x (%o3) %e (%i4) exptsubst: not exptsubst; (%o4) true (%i5) subst(y, %e^x, %e^(a*x)); a (%o5) y
Categories: Exponential and logarithm functions · Expressions
freeof (x_1, expr)
returns true
if no subexpression of
expr is equal to x_1 or if x_1 occurs only as a dummy variable
in expr, or if x_1 is neither the noun nor verb form of any operator
in expr, and returns false
otherwise.
freeof (x_1, ..., x_n, expr)
is equivalent to
freeof (x_1, expr) and ... and freeof (x_n,
expr)
.
The arguments x_1, …, x_n may be names of functions and
variables, subscripted names, operators (enclosed in double quotes), or general
expressions. freeof
evaluates its arguments.
freeof
operates only on expr as it stands (after simplification and
evaluation) and does not attempt to determine if some equivalent expression
would give a different result. In particular, simplification may yield an
equivalent but different expression which comprises some different elements than
the original form of expr.
A variable is a dummy variable in an expression if it has no binding outside of
the expression. Dummy variables recognized by freeof
are the index of a
sum or product, the limit variable in limit
, the integration variable
in the definite integral form of integrate
, the original variable in
laplace
, formal variables in at
expressions, and arguments in
lambda
expressions.
The indefinite form of integrate
is not free of its variable of
integration.
Examples:
Arguments are names of functions, variables, subscripted names, operators, and
expressions. freeof (a, b, expr)
is equivalent to
freeof (a, expr) and freeof (b, expr)
.
(%i1) expr: z^3 * cos (a[1]) * b^(c+d); d + c 3 (%o1) cos(a ) b z 1 (%i2) freeof (z, expr); (%o2) false (%i3) freeof (cos, expr); (%o3) false (%i4) freeof (a[1], expr); (%o4) false (%i5) freeof (cos (a[1]), expr); (%o5) false (%i6) freeof (b^(c+d), expr); (%o6) false (%i7) freeof ("^", expr); (%o7) false (%i8) freeof (w, sin, a[2], sin (a[2]), b*(c+d), expr); (%o8) true
freeof
evaluates its arguments.
(%i1) expr: (a+b)^5$ (%i2) c: a$ (%i3) freeof (c, expr); (%o3) false
freeof
does not consider equivalent expressions.
Simplification may yield an equivalent but different expression.
(%i1) expr: (a+b)^5$ (%i2) expand (expr); 5 4 2 3 3 2 4 5 (%o2) b + 5 a b + 10 a b + 10 a b + 5 a b + a (%i3) freeof (a+b, %); (%o3) true (%i4) freeof (a+b, expr); (%o4) false (%i5) exp (x); x (%o5) %e (%i6) freeof (exp, exp (x)); (%o6) true
A summation or definite integral is free of its dummy variable. An indefinite integral is not free of its variable of integration.
(%i1) freeof (i, 'sum (f(i), i, 0, n)); (%o1) true (%i2) freeof (x, 'integrate (x^2, x, 0, 1)); (%o2) true (%i3) freeof (x, 'integrate (x^2, x)); (%o3) false
Categories: Expressions
Default value: false
When inflag
is true
, functions for part extraction inspect the
internal form of expr
.
Note that the simplifier re-orders expressions. Thus first (x + y)
returns x
if inflag
is true
and y
if inflag
is false
. (first (y + x)
gives the same results.)
Also, setting inflag
to true
and calling part
or
substpart
is the same as calling inpart
or substinpart
.
Functions affected by the setting of inflag
are: part
,
substpart
, first
, rest
, last
,
length
, the for
… in
construct,
map
, fullmap
, maplist
, reveal
and
pickapart
.
Categories: Expressions
is similar to part
but works on the internal representation of the
expression rather than the displayed form and thus may be faster since no
formatting is done. Care should be taken with respect to the order of
subexpressions in sums and products (since the order of variables in the
internal form is often different from that in the displayed form) and in dealing
with unary minus, subtraction, and division (since these operators are removed
from the expression). part (x+y, 0)
or inpart (x+y, 0)
yield
+
, though in order to refer to the operator it must be enclosed in "s.
For example ... if inpart (%o9,0) = "+" then ...
.
Examples:
(%i1) x + y + w*z; (%o1) w z + y + x (%i2) inpart (%, 3, 2); (%o2) z (%i3) part (%th (2), 1, 2); (%o3) z (%i4) 'limit (f(x)^g(x+1), x, 0, minus); g(x + 1) (%o4) limit f(x) x -> 0- (%i5) inpart (%, 1, 2); (%o5) g(x + 1)
Categories: Expressions
Returns expr with subexpressions which are sums and which do not contain
var replaced by intermediate expression labels (these being atomic symbols
like %t1
, %t2
, …). This is often useful to avoid
unnecessary expansion of subexpressions which don't contain the variable of
interest. Since the intermediate labels are bound to the subexpressions they
can all be substituted back by evaluating the expression in which they occur.
exptisolate
(default value: false
) if true
will cause
isolate
to examine exponents of atoms (like %e
) which contain
var.
isolate_wrt_times
if true
, then isolate
will also isolate
with respect to products. See isolate_wrt_times
. See also disolate
.
Do example (isolate)
for examples.
Categories: Expressions
Default value: false
When isolate_wrt_times
is true
, isolate
will also isolate
with respect to products. E.g. compare both settings of the switch on
(%i1) isolate_wrt_times: true$ (%i2) isolate (expand ((a+b+c)^2), c); (%t2) 2 a (%t3) 2 b 2 2 (%t4) b + 2 a b + a 2 (%o4) c + %t3 c + %t2 c + %t4 (%i4) isolate_wrt_times: false$ (%i5) isolate (expand ((a+b+c)^2), c); 2 (%o5) c + 2 b c + 2 a c + %t4
Categories: Expressions
Default value: false
When listconstvars
is true
the list returned by
listofvars
contains constant variables, such as %e
,
%pi
, %i
or any variables declared as constant that
occur in expr. A variable is declared as constant
type via declare
, and constantp
returns true
for all variables declared as constant
. The default is to
omit constant variables from listofvars
return value.
Categories: Expressions
Default value: true
When listdummyvars
is false
, "dummy variables" in the expression
will not be included in the list returned by listofvars
. (The meaning
of "dummy variables" is as given in freeof
. "Dummy variables" are
mathematical things like the index of a sum or product, the limit variable,
and the definite integration variable.)
Example:
(%i1) listdummyvars: true$ (%i2) listofvars ('sum(f(i), i, 0, n)); (%o2) [i, n] (%i3) listdummyvars: false$ (%i4) listofvars ('sum(f(i), i, 0, n)); (%o4) [n]
Categories: Expressions
Returns a list of the variables in expr.
listconstvars
if true
causes listofvars
to include
%e
, %pi
, %i
, and any variables declared constant in the
list it returns if they appear in expr. The default is to omit these.
See also the option variable listdummyvars
to exclude or include
"dummy variables" in the list of variables.
(%i1) listofvars (f (x[1]+y) / g^(2+a)); (%o1) [g, a, x , y] 1
Categories: Expressions
For each member m of list, calls
freeof (m, expr)
. It returns false
if any call to
freeof
does and true
otherwise.
Example:
(%i1) lfreeof ([ a, x], x^2+b); (%o1) false (%i2) lfreeof ([ b, x], x^2+b); (%o2) false (%i3) lfreeof ([ a, y], x^2+b); (%o3) true
Categories: Expressions
is similar to dpart
but uses a labelled box. A labelled box is similar
to the one produced by dpart
but it has a name in the top line.
Categories: Expressions
You may declare variables to be mainvar
. The ordering scale for atoms is
essentially: numbers < constants (e.g., %e
, %pi
) < scalars < other
variables < mainvars. E.g., compare expand ((X+Y)^4)
with
(declare (x, mainvar), expand ((x+y)^4))
. (Note: Care should be taken if
you elect to use the above feature. E.g., if you subtract an expression in
which x
is a mainvar
from one in which x
isn't a
mainvar
, resimplification e.g. with ev (expr, simp)
may be
necessary if cancellation is to occur. Also, if you save an expression in which
x
is a mainvar
, you probably should also save x
.)
Categories: Declarations and inferences · Expressions
noun
is one of the options of the declare
command. It makes a
function so declared a "noun", meaning that it won't be evaluated
automatically.
Example:
(%i1) factor (12345678); 2 (%o1) 2 3 47 14593 (%i2) declare (factor, noun); (%o2) done (%i3) factor (12345678); (%o3) factor(12345678) (%i4) ''%, nouns; 2 (%o4) 2 3 47 14593
Categories: Nouns and verbs
Default value: false
When noundisp
is true
, nouns display with
a single quote. This switch is always true
when displaying function
definitions.
Categories: Display flags and variables · Nouns and verbs
Returns the noun form of the function name f. This is needed if one wishes to refer to the name of a verb function as if it were a noun. Note that some verb functions will return their noun forms if they can't be evaluated for certain arguments. This is also the form returned if a function call is preceded by a quote.
See also verbify
.
Categories: Nouns and verbs
Returns the number of terms that expr would have if it were fully
expanded out and no cancellations or combination of terms occurred.
Note that expressions like sin (expr)
, sqrt (expr)
,
exp (expr)
, etc. count as just one term regardless of how many
terms expr has (if it is a sum).
Categories: Expressions
Returns the main operator of the expression expr.
op (expr)
is equivalent to part (expr, 0)
.
op
returns a string if the main operator is a built-in or user-defined
prefix, binary or n-ary infix, postfix, matchfix, or nofix operator.
Otherwise, if expr is a subscripted function expression, op
returns the subscripted function; in this case the return value is not an atom.
Otherwise, expr is an array function or ordinary function expression,
and op
returns a symbol.
op
observes the value of the global flag inflag
.
op
evaluates it argument.
See also args
.
Examples:
(%i1) stringdisp: true$ (%i2) op (a * b * c); (%o2) "*" (%i3) op (a * b + c); (%o3) "+" (%i4) op ('sin (a + b)); (%o4) sin (%i5) op (a!); (%o5) "!" (%i6) op (-a); (%o6) "-" (%i7) op ([a, b, c]); (%o7) "[" (%i8) op ('(if a > b then c else d)); (%o8) "if" (%i9) op ('foo (a)); (%o9) foo (%i10) prefix (foo); (%o10) "foo" (%i11) op (foo a); (%o11) "foo" (%i12) op (F [x, y] (a, b, c)); (%o12) F x, y (%i13) op (G [u, v, w]); (%o13) G
Categories: Expressions · Operators
operatorp (expr, op)
returns true
if op is equal to the operator of expr.
operatorp (expr, [op_1, ..., op_n])
returns
true
if some element op_1, …, op_n is equal to the
operator of expr.
Categories: Operators · Predicate functions
Default value: true
When opsubst
is false
, subst
does not attempt to
substitute into the operator of an expression. E.g.,
(opsubst: false, subst (x^2, r, r+r[0]))
will work.
(%i1) r+r[0]; (%o1) r + r 0 (%i2) opsubst; (%o2) true (%i3) subst (x^2, r, r+r[0]); 2 2 (%o3) x + (x ) 0 (%i4) opsubst: not opsubst; (%o4) false (%i5) subst (x^2, r, r+r[0]); 2 (%o5) x + r 0
Categories: Expressions
Returns an expression that produces the same value and
side effects as expr but does so more efficiently by avoiding the
recomputation of common subexpressions. optimize
also has the side
effect of "collapsing" its argument so that all common subexpressions
are shared. Do example (optimize)
for examples.
Categories: Expressions
Default value: %
optimprefix
is the prefix used for generated symbols by
the optimize
command.
Categories: Expressions
ordergreat
changes the canonical ordering of Maxima expressions
such that v_1 succeeds v_2 succeeds … succeeds v_n,
and v_n succeeds any other symbol not mentioned as an argument.
orderless
changes the canonical ordering of Maxima expressions
such that v_1 precedes v_2 precedes … precedes v_n,
and v_n precedes any other variable not mentioned as an argument.
The order established by ordergreat
and orderless
is dissolved
by unorder
. ordergreat
and orderless
can be called only
once each, unless unorder
is called; only the last call to
ordergreat
and orderless
has any effect.
See also ordergreatp
.
Categories: Expressions
ordergreatp
returns true
if expr_1 succeeds expr_2 in
the canonical ordering of Maxima expressions, and false
otherwise.
orderlessp
returns true
if expr_1 precedes expr_2 in
the canonical ordering of Maxima expressions, and false
otherwise.
All Maxima atoms and expressions are comparable under ordergreatp
and
orderlessp
, although there are isolated examples of expressions for which
these predicates are not transitive; that is a bug.
The canonical ordering of atoms (symbols, literal numbers, and strings) is the following.
(integers and floats) precede (bigfloats) precede
(declared constants) precede (strings) precede (declared scalars)
precede (first argument to orderless
) precedes … precedes
(last argument to orderless
) precedes (other symbols) precede
(last argument to ordergreat
) precedes … precedes
(first argument to ordergreat
) precedes (declared main variables)
For non-atomic expressions, the canonical ordering is derived from the ordering
for atoms. For the built-in +
*
and ^
operators,
the ordering is not easily summarized. For other built-in operators and all
other functions and operators, expressions are ordered by their arguments
(beginning with the first argument), then by the name of the operator or
function. In the case of subscripted expressions, the subscripted symbol is
considered the operator and the subscript is considered an argument.
The canonical ordering of expressions is modified by the functions
ordergreat
and orderless
, and the mainvar
,
constant
, and scalar
declarations.
See also sort
.
Examples:
Ordering ordinary symbols and constants.
Note that %pi
is not ordered according to its numerical value.
(%i1) stringdisp : true; (%o1) true (%i2) sort ([%pi, 3b0, 3.0, x, X, "foo", 3, a, 4, "bar", 4.0, 4b0]); (%o2) [3, 3.0, 4, 4.0, 3.0b0, 4.0b0, %pi, "bar", "foo", X, a, x]
Effect of ordergreat
and orderless
functions.
(%i1) sort ([M, H, K, T, E, W, G, A, P, J, S]); (%o1) [A, E, G, H, J, K, M, P, S, T, W] (%i2) ordergreat (S, J); (%o2) done (%i3) orderless (M, H); (%o3) done (%i4) sort ([M, H, K, T, E, W, G, A, P, J, S]); (%o4) [M, H, A, E, G, K, P, T, W, J, S]
Effect of mainvar
, constant
, and scalar
declarations.
(%i1) sort ([aa, foo, bar, bb, baz, quux, cc, dd, A1, B1, C1]); (%o1) [A1, B1, C1, aa, bar, baz, bb, cc, dd, foo, quux] (%i2) declare (aa, mainvar); (%o2) done (%i3) declare ([baz, quux], constant); (%o3) done (%i4) declare ([A1, B1], scalar); (%o4) done (%i5) sort ([aa, foo, bar, bb, baz, quux, cc, dd, A1, B1, C1]); (%o5) [baz, quux, A1, B1, C1, bar, bb, cc, dd, foo, aa]
Ordering non-atomic expressions.
(%i1) sort ([1, 2, n, f(1), f(2), f(2, 1), g(1), g(1, 2), g(n), f(n, 1)]); (%o1) [1, 2, f(1), g(1), g(1, 2), f(2), f(2, 1), n, g(n), f(n, 1)] (%i2) sort ([foo(1), X[1], X[k], foo(k), 1, k]); (%o2) [1, X , foo(1), k, X , foo(k)] 1 k
Categories: Expressions · Predicate functions
Returns parts of the displayed form of expr
. It obtains the part of
expr
as specified by the indices n_1, …, n_k. First
part n_1 of expr
is obtained, then part n_2 of that, etc.
The result is part n_k of … part n_2 of part n_1 of
expr
. If no indices are specified expr
is returned.
part
can be used to obtain an element of a list, a row of a matrix, etc.
If the last argument to a part
function is a list of indices then
several subexpressions are picked out, each one corresponding to an
index of the list. Thus part (x + y + z, [1, 3])
is z+x
.
piece
holds the last expression selected when using the part
functions. It is set during the execution of the function and thus
may be referred to in the function itself as shown below.
If partswitch
is set to true
then end
is returned when a
selected part of an expression doesn't exist, otherwise an error message is
given.
See also inpart
, substpart
, substinpart
,
dpart
, and lpart
.
Examples:
(%i1) part(z+2*y+a,2); (%o1) 2 y (%i2) part(z+2*y+a,[1,3]); (%o2) z + a (%i3) part(z+2*y+a,2,1); (%o3) 2
example (part)
displays additional examples.
Categories: Expressions
Returns a list of two expressions. They are (1) the factors of expr (if it is a product), the terms of expr (if it is a sum), or the list (if it is a list) which don't contain x and, (2) the factors, terms, or list which do.
Examples:
(%i1) partition (2*a*x*f(x), x); (%o1) [2 a, x f(x)] (%i2) partition (a+b, x); (%o2) [b + a, 0] (%i3) partition ([a, b, f(a), c], a); (%o3) [[b, c], [a, f(a)]]
Categories: Expressions
Default value: false
When partswitch
is true
, end
is returned
when a selected part of an expression doesn't exist, otherwise an
error message is given.
Categories: Expressions
Assigns intermediate expression labels to subexpressions of expr at depth
n, an integer. Subexpressions at greater or lesser depths are not
assigned labels. pickapart
returns an expression in terms of
intermediate expressions equivalent to the original expression expr.
See also part
, dpart
, lpart
,
inpart
, and reveal
.
Examples:
(%i1) expr: (a+b)/2 + sin (x^2)/3 - log (1 + sqrt(x+1)); 2 sin(x ) b + a (%o1) - log(sqrt(x + 1) + 1) + ------- + ----- 3 2 (%i2) pickapart (expr, 0); 2 sin(x ) b + a (%t2) - log(sqrt(x + 1) + 1) + ------- + ----- 3 2 (%o2) %t2 (%i3) pickapart (expr, 1); (%t3) - log(sqrt(x + 1) + 1) 2 sin(x ) (%t4) ------- 3 b + a (%t5) ----- 2 (%o5) %t5 + %t4 + %t3 (%i5) pickapart (expr, 2); (%t6) log(sqrt(x + 1) + 1) 2 (%t7) sin(x ) (%t8) b + a %t8 %t7 (%o8) --- + --- - %t6 2 3 (%i8) pickapart (expr, 3); (%t9) sqrt(x + 1) + 1 2 (%t10) x b + a sin(%t10) (%o10) ----- - log(%t9) + --------- 2 3 (%i10) pickapart (expr, 4); (%t11) sqrt(x + 1) 2 sin(x ) b + a (%o11) ------- + ----- - log(%t11 + 1) 3 2 (%i11) pickapart (expr, 5); (%t12) x + 1 2 sin(x ) b + a (%o12) ------- + ----- - log(sqrt(%t12) + 1) 3 2 (%i12) pickapart (expr, 6); 2 sin(x ) b + a (%o12) ------- + ----- - log(sqrt(x + 1) + 1) 3 2
Categories: Expressions
Holds the last expression selected when using the part
functions.
It is set during the execution of the function and thus may be referred to in
the function itself.
Categories: Expressions
psubst(a, b, expr)
is simliar to subst
. See
subst
.
In distinction from subst
the function psubst
makes parallel
substitutions, if the first argument list is a list of equations.
See also sublis
for making parallel substitutions.
Example:
The first example shows parallel substitution with psubst
. The second
example shows the result for the function subst
, which does a serial
substitution.
(%i1) psubst ([a^2=b, b=a], sin(a^2) + sin(b)); (%o1) sin(b) + sin(a) (%i2) subst ([a^2=b, b=a], sin(a^2) + sin(b)); (%o2) 2 sin(a)
Categories: Expressions
Removes boxes from expr.
rembox (expr, unlabelled)
removes all unlabelled boxes from
expr.
rembox (expr, label)
removes only boxes bearing label.
rembox (expr)
removes all boxes, labelled and unlabelled.
Boxes are drawn by the box
, dpart
, and lpart
functions.
Examples:
(%i1) expr: (a*d - b*c)/h^2 + sin(%pi*x); a d - b c (%o1) sin(%pi x) + --------- 2 h (%i2) dpart (dpart (expr, 1, 1), 2, 2); dpart: fell off the end. -- an error. To debug this try: debugmode(true); (%i3) expr2: lpart (BAR, lpart (FOO, %, 1), 2); BAR"""""""" FOO""""""""" "a d - b c" (%o3) "sin(%pi x)" + "---------" """""""""""" " 2 " " h " """"""""""" (%i4) rembox (expr2, unlabelled); BAR"""""""" FOO""""""""" "a d - b c" (%o4) "sin(%pi x)" + "---------" """""""""""" " 2 " " h " """"""""""" (%i5) rembox (expr2, FOO); BAR"""""""" "a d - b c" (%o5) sin(%pi x) + "---------" " 2 " " h " """"""""""" (%i6) rembox (expr2, BAR); FOO""""""""" a d - b c (%o6) "sin(%pi x)" + --------- """""""""""" 2 h (%i7) rembox (expr2); a d - b c (%o7) sin(%pi x) + --------- 2 h
Categories: Expressions
Replaces parts of expr at the specified integer depth with descriptive summaries.
Sum(n)
where n is the number of operands of the sum.
Product(n)
where n is the number of operands of the product.
Expt
.
Quotient
.
Negterm
.
List(n)
where n ist the number of
elements of the list.
When depth is greater than or equal to the maximum depth of expr,
reveal (expr, depth)
returns expr unmodified.
reveal
evaluates its arguments.
reveal
returns the summarized expression.
Example:
(%i1) e: expand ((a - b)^2)/expand ((exp(a) + exp(b))^2); 2 2 b - 2 a b + a (%o1) ------------------------- b + a 2 b 2 a 2 %e + %e + %e (%i2) reveal (e, 1); (%o2) Quotient (%i3) reveal (e, 2); Sum(3) (%o3) ------ Sum(3) (%i4) reveal (e, 3); Expt + Negterm + Expt (%o4) ------------------------ Product(2) + Expt + Expt (%i5) reveal (e, 4); 2 2 b - Product(3) + a (%o5) ------------------------------------ Product(2) Product(2) 2 Expt + %e + %e (%i6) reveal (e, 5); 2 2 b - 2 a b + a (%o6) -------------------------- Sum(2) 2 b 2 a 2 %e + %e + %e (%i7) reveal (e, 6); 2 2 b - 2 a b + a (%o7) ------------------------- b + a 2 b 2 a 2 %e + %e + %e
Categories: Expressions · Display functions
Makes multiple parallel substitutions into an expression. list is a list of equations. The left hand side of the equations must be an atom.
The variable sublis_apply_lambda
controls simplification after
sublis
.
See also psubst
for making parallel substitutions.
Example:
(%i1) sublis ([a=b, b=a], sin(a) + cos(b)); (%o1) sin(b) + cos(a)
Categories: Expressions
Default value: true
Controls whether lambda
's substituted are applied in simplification after
sublis
is used or whether you have to do an ev
to get things to
apply. true
means do the application.
Categories: Expressions
Default value: false
If true
then the functions subst
and psubst
can substitute
a subscripted variable f[x]
with a number, when only the symbol f
is given.
See also subst
.
(%i1) subst(100,g,g[x]+2); subst: cannot substitute 100 for operator g in expression g x -- an error. To debug this try: debugmode(true); (%i2) subst(100,g,g[x]+2),subnumsimp:true; (%o2) 102
Categories: Expressions
Substitutes a for b in c. b must be an atom or a
complete subexpression of c. For example, x+y+z
is a complete
subexpression of 2*(x+y+z)/w
while x+y
is not. When b does
not have these characteristics, one may sometimes use substpart
or
ratsubst
(see below). Alternatively, if b is of the form
e/f
then one could use subst (a*f, e, c)
while if b is of
the form e^(1/f)
then one could use subst (a^f, e, c)
. The
subst
command also discerns the x^y
in x^-y
so that
subst (a, sqrt(x), 1/sqrt(x))
yields 1/a
. a and b
may also be operators of an expression enclosed in double-quotes "
or
they may be function names. If one wishes to substitute for the independent
variable in derivative forms then the at
function (see below) should be
used.
subst
is an alias for substitute
.
The commands subst (eq_1, expr)
or
subst ([eq_1, ..., eq_k], expr)
are other permissible
forms. The eq_i are equations indicating substitutions to be made.
For each equation, the right side will be substituted for the left in the
expression expr. The equations are substituted in serial from left to
right in expr. See the functions sublis
and psubst
for
making parallel substitutions.
exptsubst
if true
permits substitutions
like y
for %e^x
in %e^(a*x)
to take place.
When opsubst
is false
,
subst
will not attempt to substitute into the operator of an expression.
E.g. (opsubst: false, subst (x^2, r, r+r[0]))
will work.
Examples:
(%i1) subst (a, x+y, x + (x+y)^2 + y); 2 (%o1) y + x + a (%i2) subst (-%i, %i, a + b*%i); (%o2) a - %i b
The substitution is done in serial for a list of equations. Compare this with a parallel substitution:
(%i1) subst([a=b, b=c], a+b); (%o1) 2 c (%i2) sublis([a=b, b=c], a+b); (%o2) c + b
Single-character Operators like +
and -
have to be quoted in
order to be replaced by subst. It is to note, though, that a+b-c
might be expressed as a+b+(-1*c)
internally.
(%i3) subst(["+"="-"],a+b-c); (%o3) c-b+a
The difference between subst
and at
can be seen in the
following example:
(%i1) g1:y(t)=a*x(t)+b*diff(x(t),t); d (%o1) y(t) = b (-- (x(t))) + a x(t) dt (%i2) subst('diff(x(t),t)=1,g1); (%o2) y(t) = a x(t) + b (%i3) at(g1,'diff(x(t),t)=1); ! d ! (%o3) y(t) = b (-- (x(t))! ) + a x(t) dt !d !-- (x(t)) = 1 dt
For further examples, do example (subst)
.
Categories: Expressions
Similar to substpart
, but substinpart
works on the
internal representation of expr.
Examples:
(%i1) x . 'diff (f(x), x, 2); 2 d (%o1) x . (--- (f(x))) 2 dx (%i2) substinpart (d^2, %, 2); 2 (%o2) x . d (%i3) substinpart (f1, f[1](x + 1), 0); (%o3) f1(x + 1)
If the last argument to a part
function is a list of indices then
several subexpressions are picked out, each one corresponding to an
index of the list. Thus
(%i1) part (x + y + z, [1, 3]); (%o1) z + x
piece
holds the value of the last expression selected when using the
part
functions. It is set during the execution of the function and
thus may be referred to in the function itself as shown below.
If partswitch
is set to true
then end
is returned when a
selected part of an expression doesn't exist, otherwise an error
message is given.
(%i1) expr: 27*y^3 + 54*x*y^2 + 36*x^2*y + y + 8*x^3 + x + 1; 3 2 2 3 (%o1) 27 y + 54 x y + 36 x y + y + 8 x + x + 1 (%i2) part (expr, 2, [1, 3]); 2 (%o2) 54 y (%i3) sqrt (piece/54); (%o3) abs(y) (%i4) substpart (factor (piece), expr, [1, 2, 3, 5]); 3 (%o4) (3 y + 2 x) + y + x + 1 (%i5) expr: 1/x + y/x - 1/z; 1 y 1 (%o5) (- -) + - + - z x x (%i6) substpart (xthru (piece), expr, [2, 3]); y + 1 1 (%o6) ----- - - x z
Also, setting the option inflag
to true
and calling part
or substpart
is the same as calling inpart
or substinpart
.
Categories: Expressions
Substitutes x for the subexpression picked out by the rest of the
arguments as in part
. It returns the new value of expr. x
may be some operator to be substituted for an operator of expr. In some
cases x needs to be enclosed in double-quotes "
(e.g.
substpart ("+", a*b, 0)
yields b + a
).
Example:
(%i1) 1/(x^2 + 2); 1 (%o1) ------ 2 x + 2 (%i2) substpart (3/2, %, 2, 1, 2); 1 (%o2) -------- 3/2 x + 2 (%i3) a*x + f(b, y); (%o3) a x + f(b, y) (%i4) substpart ("+", %, 1, 0); (%o4) x + f(b, y) + a
Also, setting the option inflag
to true
and calling part
or substpart
is the same as calling inpart
or
substinpart
.
Categories: Expressions
Returns true
if expr is a symbol, else false
. In effect,
symbolp(x)
is equivalent to the predicate
atom(x) and not numberp(x)
.
See also Identifiers.
Categories: Predicate functions
Disables the aliasing created by the last use of the ordering commands
ordergreat
and orderless
. ordergreat
and orderless
may not be used more than one time each without calling unorder
.
unorder
does not substitute back in expressions the original symbols for
the aliases introduced by ordergreat
and orderless
. Therefore,
after execution of unorder
the aliases appear in previous expressions.
See also ordergreat
and orderless
.
Examples:
ordergreat(a)
introduces an alias for the symbol a
. Therefore,
the difference of %o2
and %o4
does not vanish. unorder
does not substitute back the symbol a
and the alias appears in the
output %o7
.
(%i1) unorder(); (%o1) [] (%i2) b*x + a^2; 2 (%o2) b x + a (%i3) ordergreat (a); (%o3) done (%i4) b*x + a^2; %th(1) - %th(3); 2 (%o4) a + b x (%i5) unorder(); 2 2 (%o5) a - a (%i6) %th(2); (%o6) [a]
Categories: Expressions
Returns the verb form of the function name f.
See also verb
, noun
, and nounify
.
Examples:
(%i1) verbify ('foo); (%o1) foo (%i2) :lisp $% $FOO (%i2) nounify (foo); (%o2) foo (%i3) :lisp $% %FOO
Categories: Nouns and verbs
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