+
, _plus
Add expressions
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x + y + ...
_plus(x, y, …
)
x + y + ...
computes the sum of x
, y
etc.
x + y + ...
is equivalent to the function
call _plus(x, y, ...)
.
All terms that are numbers of type Type::Numeric
are automatically combined
to a single number.
Terms of a symbolic sum may be rearranged internally. Cf. Example 1. The user can control
the ordering by the preference Pref::keepOrder
. See also the documentation
for print
.
_plus
accepts an arbitrary number of arguments.
In conjunction with the sequence operator $
, this function is
the recommended tool for computing finite sums. Cf. Example 2. The function sum
may also serve for
computing such sums. However, sum
is designed for the computation of
symbolic and infinite sums. It is slower than _plus
.
x  y
is internally represented as x
+ y*(1)
= _plus(x, _mult(y, 1))
. See _subtract
for details.
For adding equalities, inequalities, and comparisons, the following rules are implemented:
Adding an arithmetical expression adds the expression to both sides.
Adding an equality adds the left hand sides and the right hand sides separately.
Adding a comparison does likewise, taking care of the correct operator. Adding a comparison to an inequality is not permitted.
Cf. Example 4.
Many library
domains overload _plus
by an appropriate slot"_plus"
. Sums involving
elements of library domains are processed as follows:
A sum x + y + ...
is searched for elements
of library domains from left to right. Let z
be
the first term that is not of one of the basic types provided by the
kernel (numbers, expressions, etc.). If the domain d
= z::dom
= domtype(z)
has
a slot"_plus"
, it
is called in the form d::_plus(x, y, ...)
. The
result returned by d::_plus
is the result of x
+ y + ...
.
Users should implement the slot d::_plus
of
their domains d
according to the following convention:
If all terms are elements of d
,
an appropriate sum of type d
should be returned.
If at least one term cannot be converted to an element
of d
, the slot should return FAIL
.
Care must be taken if there are terms that are not
of type d
, but can be converted to type d
.
Such terms should be converted only if the mathematical semantics
is obvious to any user who uses this domain as a 'black box' (e.g.,
integers may be regarded as rational numbers because of the natural
mathematical embedding). If in doubt, the "_plus"
method
should return FAIL
instead
of using implicit conversions. If implicit conversions are used, they
must be welldocumented.
Most of the library domains in the MuPAD^{®} standard installation comply with this convention.
_plus()
returns the number 0.
Polynomials of type DOM_POLY
are added by +
,
if they have the same indeterminates and the same coefficient ring.
For finite sets X
, Y
,
the sum X + Y
is the set .
Numerical terms are simplified automatically:
3 + x + y + 2*x + 5*x  1/2  sin(4) + 17/4
The ordering of the terms of a sum is not necessarily the same as on input:
x + y + z + a + b + c
1 + x + x^2 + x^10
Internally, this sum is a symbolic call of _plus
:
op(%, 0), type(%)
The functional equivalent _plus
of the operator +
is
a handy tool for computing finite sums. In the following, the terms
are generated via the sequence operator $
:
_plus(i^2 $ i = 1..100)
E.g., it is easy to add up all elements in a set:
S := {a, b, 1, 2, 27}: _plus(op(S))
The following command “zips” two lists by adding corresponding elements:
L1 := [a, b, c]: L2 := [1, 2, 3]: zip(L1, L2, _plus)
delete S, L1, L2:
Polynomials of type DOM_POLY
are added by +
,
if they have the same indeterminates and the same coefficient ring:
poly(x^2 + 1, [x]) + poly(x^2 + x  1, [x])
If the indeterminates or the coefficient rings do not match, _plus
returns
an error:
poly(x, [x]) + poly(x, [x, y])
Error: Invalid argument. [_plus]
poly(x, [x]) + poly(x, [x], Dom::Integer)
Error: Invalid argument. [_plus]
Adding a constant to an equality, an inequality, or a comparison amounts to adding it to both sides:
(a = b) + c, (a <> b) + c, (a <= b) + c, (a < b) + c
Adding an equality is performed by adding the left hand sides and the right hand sides separately:
(a = b) + (c = d), (a <> b) + (c = d), (a <= b) + (c = d), (a < b) + (c = d)
Inequalities can only be added to equalities:
(a = b) + (c <> d), (a <> b) + (c <> d), (a <= b) + (c <> d), (a < b) + (c <> d)
The addition of comparisons takes of the difference between <
and <=
into
account. Note that MuPAD uses only these two comparison operators; a > b and a ≥ b are
automatically rewritten:
(a = b) + (c <= d), (a <> b) + (c <= d), (a <= b) + (c <= d), (a < b) + (c <= d);
(a = b) + (c < d), (a <> b) + (c < d), (a <= b) + (c < d), (a < b) + (c < d);
(a = b) + (c >= d), (a <> b) + (c >= d), (a <= b) + (c >= d), (a < b) + (c >= d);
(a = b) + (c > d), (a <> b) + (c > d), (a <= b) + (c > d), (a < b) + (c > d);
For finite sets X
, Y
,
the sum X + Y
is the set :
{a, b, c} + {1, 2}
Various library domains such as matrix domains
overload _plus
:
x := Dom::Matrix(Dom::Integer)([1, 2]): y := Dom::Matrix(Dom::Rational)([2, 3]): x + y, y + x
If the terms in a sum x + y
are of different
type, the first term x
tries to convert y
to
the data type of x
. If successful, the sum is of
the same type as x
. In the previous example, x
and y
have
different types (both are matrices, but the component domains differ).
Hence the sums x + y
and y + x
differ
syntactically, because they inherit their type from the first term:
bool(x + y = y + x)
domtype(x + y), domtype(y + x)
If x
does not succeed to convert y
,
then FAIL
is
returned. In the following call, the component 2/3
cannot
be converted to an integer:
y := Dom::Matrix(Dom::Rational)([2/3, 3]): x + y
delete x, y:
This example demonstrates how to implement a slot"_plus"
for a domain.
The following domain myString
is to represent character
strings. The sum of such strings is to be the concatenation of the
strings.
The "new"
method uses expr2text
to convert any MuPAD object
to a string. This string is the internal representation of elements
of myString
. The "print"
method
turns this string into the screen output:
myString := newDomain("myString"): myString::new := proc(x) begin if args(0) = 0 then x := "": end_if; case domtype(x) of myString do return(x); of DOM_STRING do return(new(dom, x)); otherwise return(new(dom, expr2text(x))); end_case end_proc: myString::print := x > extop(x, 1):
Without a "_plus"
method, the system function _plus
handles
elements of this domain like any symbolic object:
y := myString(y): z := myString(z): 1 + x + y + z + 3/2
Now, we implement the "_plus"
method. It
checks all arguments. Arguments are converted, if they are not of
type myString
. Generally, such an implicit conversion
should be avoided. In this case, however, any object has a corresponding
string representation via expr2text
and
an implicit conversion is implemented. Finally, the sum of myString
objects
is defined as the concatenation of the internal strings:
myString::_plus := proc() local n, Arguments, i; begin print(Unquoted, "Info: myString::_plus called with the arguments:", args()): n := args(0): Arguments := [args()]; for i from 1 to n do if domtype(Arguments[i]) <> myString then // convert the ith term to myString Arguments[i] := myString::new(Arguments[i]): end_if; end_for: myString::new(_concat(extop(Arguments[i], 1) $ i = 1..n)) end_proc:
Now, myString
objects can be added:
myString("This ") + myString("is ") + myString("a string")
Info: myString::_plus called with the arguments:, This , is , a string
In the following sum, y
and z
are
elements of myString
. The term y
is
the first term that is an element of a library domain. Its "_plus"
method
is called and concatenates all terms to a string of type myString
:
1 + x + y + z + 3/2;
3 Info: myString::_plus called with the arguments:, 1, x, y, z,  2
delete myString, y, z:

arithmetical
expressions, polynomials of
type 
Arithmetical expression, a polynomial, a set, an equation, an inequality, or a comparison.
x
, y