Everything in GEL is really just an expression. Expressions are stringed together with different operators. As we have seen, even the separator is simply a binary operator in GEL. Here is a list of the operators in GEL.
a;b
Разделитель, просто вычисляющий как a
, так и b
, но возвращающий только результат b
.
a=b
The assignment operator. This assigns b
to
a
(a
must be a valid lvalue) (note however that this operator
may be translated to ==
if used in a place where boolean
expression is expected)
a:=b
The assignment operator. Assigns b
to
a
(a
must be a valid lvalue). This is
different from =
because it never gets translated to a
==
.
|a|
Absolute value.
In case the expression is a complex number the result will be the modulus
(distance from the origin). For example:
|3 * e^(1i*pi)|
returns 3.
Смотрите Mathworld для дополнительной информации.
a^b
Возводит переменную a
в степень b
.
a.^b
Поэлементное возведение в степень. Возводит каждый элемент матрицы a
в степень b
. Или, если b
— матрица той же размерности, что и a
, выполняет операцию поэлементно. Если a
— число, а b
— матрица, то создаёт матрицу той же размерности, что и b
со значением a
, возведённым во все степени, содержащиеся в b
.
a+b
Addition. Adds two numbers, matrices, functions or strings. If you add a string to anything the result will just be a string. If one is a square matrix and the other a number, then the number is multiplied by the identity matrix.
a-b
Вычитание. Вычитает два числа, матрицы или функции.
a*b
Умножение. Это обычное умножение матриц.
a.*b
Поэлементное умножение, если a
и b
являются матрицами.
a/b
Division. When a
and b
are just numbers
this is the normal division. When they are matrices, then this is
equivalent to a*b^-1
.
a./b
Element by element division. Same as a/b
for
numbers, but operates element by element on matrices.
a\b
Обратное деление. Это то же самое, что b/a
.
a.\b
Поэлементное обратное деление.
a%b
The mod operator. This does not turn on the modular mode, but
just returns the remainder of integer division
a/b
.
a.%b
Element by element mod operator. Returns the remainder
after element by element integer division
a./b
.
a mod b
Modular evaluation operator. The expression a
is evaluated modulo b
. See «Modular Evaluation».
Some functions and operators behave differently modulo an integer.
a!
Факториал: 1*...*(n-2)*(n-1)*n
.
a!!
Двойной факториал: 1*...*(n-4)*(n-2)*n
.
a==b
Equality operator.
Returns true
or false
depending on a
and b
being equal or not.
a!=b
Оператор неравенства, возвращает true
, если a
не равно b
, в противном случае возвращает false
.
a<>b
Альтернативный оператор неравенства, возвращает true
, если a
не равно b
, иначе возвращает false
.
a<=b
Less than or equal operator,
returns true
if a
is
less than or equal to
b
else returns false
.
These can be chained as in a <= b <= c
(can
also be combined with the less than operator).
a>=b
Greater than or equal operator,
returns true
if a
is
greater than or equal to
b
else returns false
.
These can be chained as in a >= b >= c
(and they can also be combined with the greater than operator).
a<b
Less than operator,
returns true
if a
is
less than
b
else returns false
.
These can be chained as in a < b < c
(they can also be combined with the less than or equal to operator).
a>b
Greater than operator,
returns true
if a
is
greater than
b
else returns false
.
These can be chained as in a > b > c
(they can also be combined with the greater than or equal to operator).
a<=>b
Оператор сравнения. Если a
равно b
, возвращает 0; если a
меньше b
, возвращает -1; если a
больше b
, возвращает 1.
a and b
Logical and. Returns true if both
a
and b
are true,
else returns false. If given numbers, nonzero numbers
are treated as true.
a or b
Logical or.
Returns true if either
a
or b
is true,
else returns false. If given numbers, nonzero numbers
are treated as true.
a xor b
Logical xor.
Returns true if exactly one of
a
or b
is true,
else returns false. If given numbers, nonzero numbers
are treated as true.
not a
Logical not. Returns the logical negation of a
.
-a
Negation operator. Returns the negative of a number or a matrix (works element-wise on a matrix).
&a
Variable referencing (to pass a reference to a variable). See «References».
*a
Variable dereferencing (to access a referenced variable). See «References».
a'
Matrix conjugate transpose. That is, rows and columns get swapped and we take complex conjugate of all entries. That is
if the i,j element of a
is x+iy, then the j,i element of a'
is x-iy.
a.'
Matrix transpose, does not conjugate the entries. That is,
the i,j element of a
becomes the j,i element of a.'
.
a@(b,c)
Get element of a matrix in row b
and column
c
. If b
,
c
are vectors, then this gets the corresponding
rows, columns or submatrices.
a@(b,)
Get row of a matrix (or multiple rows if b
is a vector).
a@(b,:)
То же, что и выше.
a@(,c)
Возвращает столбец матрицы (или столбцы, если c
является вектором).
a@(:,c)
То же, что и выше.
a@(b)
Get an element from a matrix treating it as a vector. This will traverse the matrix row-wise.
a:b
Build a vector from a
to b
(or specify a row, column region for the @
operator). For example to get rows 2 to 4 of matrix A
we could do
A@(2:4,)
as 2:4
will return a vector
[2,3,4]
.
a:b:c
Build a vector from a
to c
with b
as a step. That is for example
genius> 1:2:9 = `[1, 3, 5, 7, 9]
When the numbers involved are floating point numbers, for example
1.0:0.4:3.0
, the output is what is expected
even though adding 0.4 to 1.0 five times is actually just slightly
more than 3.0 due to the way that floating point numbers are
stored in base 2 (there is no 0.4, the actual number stored is
just ever so slightly bigger). The way this is handled is the
same as in the for, sum, and prod loops. If the end is within
2^-20
times the step size of the endpoint,
the endpoint is used and we assume there were roundoff errors.
This is not perfect, but it handles the majority of the cases.
This check is done only from version 1.0.18 onwards, so execution
of your code may differ on older versions. If you want to avoid
dealing with this issue, use actual rational numbers, possibly
using the float
if you wish to get floating
point numbers in the end. For example
1:2/5:3
does the right thing and
float(1:2/5:3)
even gives you floating
point numbers and is ever so slightly more precise than
1.0:0.4:3.0
.
(a)i
Make a
into an imaginary number (multiply a
by the
imaginary). Normally the imaginary number i
is
written as 1i
. So the above is equal to
(a)*1i
`a
Quote an identifier so that it doesn't get evaluated. Or quote a matrix so that it doesn't get expanded.
a swapwith b
Меняет местами значение a
со значением b
. В настоящее время не работает с диапазонами элементов матрицы. Возвращает null
. Доступен, начиная с версии 1.0.13.
increment a
Инкремент переменной a
на 1. Если a
— матрица, то инкрементирует каждый элемент. Это эквивалентно a=a+1
, но немного быстрее. Возвращает null
. Доступен с версии 1.0.13.
increment a by b
Инкремент переменной a
на величину b
. Если a
— матрица, то инкрементирует каждый элемент. Это эквивалентно a=a+b
, но немного быстрее. Возвращает null
. Доступен с версии 1.0.13.
Оператор @() делает использование оператора : наиболее полезным. С его помощью можно указывать области матрицы. Таким образом, a@(2:4,6) — это строки 2,3,4 столбца 6. Или a@(,1:2) возвращает два первых столбца матрицы. Можно также присваивать значения оператору @(), если правое значение — это матрица, совпадающая по размеру с данной областью, или если это любой другой тип значений.
The comparison operators (except for the <=> operator, which behaves normally), are not strictly binary operators, they can in fact be grouped in the normal mathematical way, e.g.: (1<x<=y<5) is a legal boolean expression and means just what it should, that is (1<x and x≤y and y<5)
The unitary minus operates in a different fashion depending on where it
appears. If it appears before a number it binds very closely, if it appears in
front of an expression it binds less than the power and factorial operators.
So for example -1^k
is really (-1)^k
,
but -foo(1)^k
is really -(foo(1)^k)
. So
be careful how you use it and if in doubt, add parentheses.