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@node Expressions
@chapter Expressions
@cindex expressions

Expressions are the basic building block of statements in Octave.  An
expression evaluates to a value, which you can print, test, store in a
variable, pass to a function, or assign a new value to a variable with
an assignment operator.

An expression can serve as a statement on its own.  Most other kinds of
statements contain one or more expressions which specify data to be
operated on.  As in other languages, expressions in Octave include
variables, array references, constants, and function calls, as well as
combinations of these with various operators.

@menu
* Index Expressions::
* Calling Functions::
* Arithmetic Ops::
* Comparison Ops::
* Boolean Expressions::
* Assignment Ops::
* Increment Ops::
* Operator Precedence::
@end menu

@node Index Expressions
@section Index Expressions

@opindex (
@opindex )
@opindex :

An @dfn{index expression} allows you to reference or extract selected
elements of a matrix or vector.

Indices may be scalars, vectors, ranges, or the special operator
@samp{:}, which may be used to select entire rows or columns.

Vectors are indexed using a single index expression.  Matrices (2-D)
and higher multi-dimensional arrays are indexed using either one index
or @math{N} indices where @math{N} is the dimension of the array.
When using a single index expression to index 2-D or higher data the
elements of the array are taken in column-first order (like Fortran).

The output from indexing assumes the dimensions of the index
expression.  For example:

@example
@group
a(2)       # result is a scalar
a(1:2)     # result is a row vector
a([1; 2])  # result is a column vector
@end group
@end example

As a special case, when a colon is used as a single index, the output
is a column vector containing all the elements of the vector or
matrix.  For example:

@example
@group
a(:)       # result is a column vector
a(:)'      # result is a row vector
@end group
@end example

The above two code idioms are often used in place of @code{reshape}
when a simple vector, rather than an arbitrarily sized array, is
needed.

Given the matrix

@example
a = [1, 2; 3, 4]
@end example

@noindent
all of the following expressions are equivalent and select the first
row of the matrix.

@example
@group
a(1, [1, 2])  # row 1, columns 1 and 2
a(1, 1:2)     # row 1, columns in range 1-2
a(1, :)       # row 1, all columns
@end group
@end example

@cindex @code{end}, indexing
@cindex :end

In index expressions the keyword @code{end} automatically refers to
the last entry for a particular dimension.  This magic index can also
be used in ranges and typically eliminates the needs to call
@code{size} or @code{length} to gather array bounds before indexing.
For example:

@example
@group
a = [1, 2, 3, 4];

a(1:end/2)        # first half of a => [1, 2]
a(end + 1) = 5;   # append element
a(end) = [];      # delete element
a(1:2:end)        # odd elements of a => [1, 3]
a(2:2:end)        # even elements of a => [2, 4]
a(end:-1:1)       # reversal of a => [4, 3, 2 , 1]
@end group
@end example

@menu
* Advanced Indexing::
@end menu

@node Advanced Indexing
@subsection Advanced Indexing

An array with @samp{n} dimensions can be indexed using @samp{m}
indices.  More generally, the set of index tuples determining the
result is formed by the Cartesian product of the index vectors (or
ranges or scalars).

For the ordinary and most common case, @w{@code{m == n}}, and each
index corresponds to its respective dimension.  If @w{@code{m < n}}
and every index is less than the size of the array in the
@math{i^{th}} dimension, @code{m(i) < n(i)}, then the index expression
is padded with trailing singleton dimensions (@code{[ones (m-n, 1)]}).
If @w{@code{m < n}} but one of the indices @code{m(i)} is outside the
size of the current array, then the last @w{@code{n-m+1}} dimensions
are folded into a single dimension with an extent equal to the product
of extents of the original dimensions.  This is easiest to understand
with an example.

@example
a = reshape (1:8, 2, 2, 2)  # Create 3-D array
a =

ans(:,:,1) =

   1   3
   2   4

ans(:,:,2) =

   5   7
   6   8

a(2,1,2);   # Case (m == n): ans = 6
a(2,1);     # Case (m < n), idx within array:
            # equivalent to a(2,1,1), ans = 2
a(2,4);     # Case (m < n), idx outside array:
            # Dimension 2 & 3 folded into new dimension of size 2x2 = 4
            # Select 2nd row, 4th element of [2, 4, 6, 8], ans = 8
@end example

One advanced use of indexing is to create arrays filled with a single
value.  This can be done by using an index of ones on a scalar value.
The result is an object with the dimensions of the index expression
and every element equal to the original scalar.  For example, the
following statements

@example
@group
a = 13;
a(ones (1, 4))
@end group
@end example

@noindent
produce a vector whose four elements are all equal to 13.

Similarly, by indexing a scalar with two vectors of ones it is
possible to create a matrix.  The following statements

@example
@group
a = 13;
a(ones (1, 2), ones (1, 3))
@end group
@end example

@noindent
create a 2x3 matrix with all elements equal to 13.

The last example could also be written as

@example
@group
13(ones (2, 3))
@end group
@end example

It is more efficient to use indexing rather than the code construction
@code{scalar * ones (N, M, @dots{})} because it avoids the unnecessary
multiplication operation.  Moreover, multiplication may not be
defined for the object to be replicated whereas indexing an array is
always defined.  The following code shows how to create a 2x3 cell
array from a base unit which is not itself a scalar.

@example
@group
@{"Hello"@}(ones (2, 3))
@end group
@end example

It should be, noted that @code{ones (1, n)} (a row vector of ones)
results in a range (with zero increment).  A range is stored
internally as a starting value, increment, end value, and total number
of values; hence, it is more efficient for storage than a vector or
matrix of ones whenever the number of elements is greater than 4.  In
particular, when @samp{r} is a row vector, the expressions

@example
  r(ones (1, n), :)
@end example

@example
  r(ones (n, 1), :)
@end example

@noindent
will produce identical results, but the first one will be
significantly faster, at least for @samp{r} and @samp{n} large enough.
In the first case the index is held in compressed form as a range
which allows Octave to choose a more efficient algorithm to handle the
expression.

A general recommendation, for a user unaware of these subtleties, is
to use the function @code{repmat} for replicating smaller arrays into
bigger ones.

A second use of indexing is to speed up code.  Indexing is a fast
operation and judicious use of it can reduce the requirement for
looping over individual array elements which is a slow operation.

Consider the following example which creates a 10-element row vector
@math{a} containing the values
@tex
$a_i = \sqrt{i}$.
@end tex
@ifnottex
a(i) = sqrt (i).
@end ifnottex

@example
@group
for i = 1:10
  a(i) = sqrt (i);
endfor
@end group
@end example

@noindent
It is quite inefficient to create a vector using a loop like this.  In
this case, it would have been much more efficient to use the
expression

@example
a = sqrt (1:10);
@end example

@noindent
which avoids the loop entirely.

In cases where a loop cannot be avoided, or a number of values must be
combined to form a larger matrix, it is generally faster to set the
size of the matrix first (pre-allocate storage), and then insert
elements using indexing commands.  For example, given a matrix
@code{a},

@example
@group
[nr, nc] = size (a);
x = zeros (nr, n * nc);
for i = 1:n
  x(:,(i-1)*nc+1:i*nc) = a;
endfor
@end group
@end example

@noindent
is considerably faster than

@example
@group
x = a;
for i = 1:n-1
  x = [x, a];
endfor
@end group
@end example

@noindent
because Octave does not have to repeatedly resize the intermediate
result.

@c sub2ind libinterp/corefcn/sub2ind.cc
@anchor{XREFsub2ind}
@deftypefn  {Function File} {@var{ind} =} sub2ind (@var{dims}, @var{i}, @var{j})
@deftypefnx {Function File} {@var{ind} =} sub2ind (@var{dims}, @var{s1}, @var{s2}, @dots{}, @var{sN})
Convert subscripts to a linear index.

The following example shows how to convert the two-dimensional index
@code{(2,3)} of a 3-by-3 matrix to a linear index.  The matrix is linearly
indexed moving from one column to next, filling up all rows in each column.

@example
@group
linear_index = sub2ind ([3, 3], 2, 3)
@result{} 8
@end group
@end example
@seealso{@ref{XREFind2sub,,ind2sub}}
@end deftypefn


@c ind2sub libinterp/corefcn/sub2ind.cc
@anchor{XREFind2sub}
@deftypefn {Function File} {[@var{s1}, @var{s2}, @dots{}, @var{sN}] =} ind2sub (@var{dims}, @var{ind})
Convert a linear index to subscripts.

The following example shows how to convert the linear index @code{8}
in a 3-by-3 matrix into a subscript.  The matrix is linearly indexed
moving from one column to next, filling up all rows in each column.

@example
@group
[r, c] = ind2sub ([3, 3], 8)
    @result{} r =  2
    @result{} c =  3
@end group
@end example
@seealso{@ref{XREFsub2ind,,sub2ind}}
@end deftypefn


@c isindex libinterp/corefcn/utils.cc
@anchor{XREFisindex}
@deftypefn  {Built-in Function} {} isindex (@var{ind})
@deftypefnx {Built-in Function} {} isindex (@var{ind}, @var{n})
Return true if @var{ind} is a valid index.

Valid indices are either positive integers (although possibly of real data
type), or logical arrays.

If present, @var{n} specifies the maximum extent of the dimension to be
indexed.  When possible the internal result is cached so that subsequent
indexing using @var{ind} will not perform the check again.

Implementation Note: Strings are first converted to double values before the
checks for valid indices are made.  Unless a string contains the NULL
character @nospell{"\0"}, it will always be a valid index.
@end deftypefn


@c allow_noninteger_range_as_index libinterp/octave-value/ov-range.cc
@anchor{XREFallow_noninteger_range_as_index}
@deftypefn  {Built-in Function} {@var{val} =} allow_noninteger_range_as_index ()
@deftypefnx {Built-in Function} {@var{old_val} =} allow_noninteger_range_as_index (@var{new_val})
@deftypefnx {Built-in Function} {} allow_noninteger_range_as_index (@var{new_val}, "local")
Query or set the internal variable that controls whether non-integer
ranges are allowed as indices.

This might be useful for @sc{matlab} compatibility; however, it is still not
entirely compatible because @sc{matlab} treats the range expression
differently in different contexts.

When called from inside a function with the @qcode{"local"} option, the
variable is changed locally for the function and any subroutines it calls.
The original variable value is restored when exiting the function.
@end deftypefn


@node Calling Functions
@section Calling Functions

A @dfn{function} is a name for a particular calculation.  Because it has
a name, you can ask for it by name at any point in the program.  For
example, the function @code{sqrt} computes the square root of a number.

A fixed set of functions are @dfn{built-in}, which means they are
available in every Octave program.  The @code{sqrt} function is one of
these.  In addition, you can define your own functions.
@xref{Functions and Scripts}, for information about how to do this.

@cindex arguments in function call
The way to use a function is with a @dfn{function call} expression,
which consists of the function name followed by a list of
@dfn{arguments} in parentheses.  The arguments are expressions which give
the raw materials for the calculation that the function will do.  When
there is more than one argument, they are separated by commas.  If there
are no arguments, you can omit the parentheses, but it is a good idea to
include them anyway, to clearly indicate that a function call was
intended.  Here are some examples:

@example
@group
sqrt (x^2 + y^2)      # @r{One argument}
ones (n, m)           # @r{Two arguments}
rand ()               # @r{No arguments}
@end group
@end example

Each function expects a particular number of arguments.  For example, the
@code{sqrt} function must be called with a single argument, the number
to take the square root of:

@example
sqrt (@var{argument})
@end example

Some of the built-in functions take a variable number of arguments,
depending on the particular usage, and their behavior is different
depending on the number of arguments supplied.

Like every other expression, the function call has a value, which is
computed by the function based on the arguments you give it.  In this
example, the value of @code{sqrt (@var{argument})} is the square root of
the argument.  A function can also have side effects, such as assigning
the values of certain variables or doing input or output operations.

Unlike most languages, functions in Octave may return multiple values.
For example, the following statement

@example
[u, s, v] = svd (a)
@end example

@noindent
computes the singular value decomposition of the matrix @code{a} and
assigns the three result matrices to @code{u}, @code{s}, and @code{v}.

The left side of a multiple assignment expression is itself a list of
expressions, and is allowed to be a list of variable names or index
expressions.  See also @ref{Index Expressions}, and @ref{Assignment Ops}.

@menu
* Call by Value::
* Recursion::
@end menu

@node Call by Value
@subsection Call by Value

In Octave, unlike Fortran, function arguments are passed by value, which
means that each argument in a function call is evaluated and assigned to
a temporary location in memory before being passed to the function.
There is currently no way to specify that a function parameter should be
passed by reference instead of by value.  This means that it is
impossible to directly alter the value of a function parameter in the
calling function.  It can only change the local copy within the function
body.  For example, the function

@example
@group
function f (x, n)
  while (n-- > 0)
    disp (x);
  endwhile
endfunction
@end group
@end example

@noindent
displays the value of the first argument @var{n} times.  In this
function, the variable @var{n} is used as a temporary variable without
having to worry that its value might also change in the calling
function.  Call by value is also useful because it is always possible to
pass constants for any function parameter without first having to
determine that the function will not attempt to modify the parameter.

The caller may use a variable as the expression for the argument, but
the called function does not know this: it only knows what value the
argument had.  For example, given a function called as

@example
@group
foo = "bar";
fcn (foo)
@end group
@end example

@noindent
you should not think of the argument as being ``the variable
@code{foo}.''  Instead, think of the argument as the string value,
@qcode{"bar"}.

Even though Octave uses pass-by-value semantics for function arguments,
values are not copied unnecessarily.  For example,

@example
@group
x = rand (1000);
f (x);
@end group
@end example

@noindent
does not actually force two 1000 by 1000 element matrices to exist
@emph{unless} the function @code{f} modifies the value of its
argument.  Then Octave must create a copy to avoid changing the
value outside the scope of the function @code{f}, or attempting (and
probably failing!) to modify the value of a constant or the value of a
temporary result.

@node Recursion
@subsection Recursion
@cindex factorial function

With some restrictions@footnote{Some of Octave's functions are
implemented in terms of functions that cannot be called recursively.
For example, the ODE solver @code{lsode} is ultimately implemented in a
Fortran subroutine that cannot be called recursively, so @code{lsode}
should not be called either directly or indirectly from within the
user-supplied function that @code{lsode} requires.  Doing so will result
in an error.}, recursive function calls are allowed.  A
@dfn{recursive function} is one which calls itself, either directly or
indirectly.  For example, here is an inefficient@footnote{It would be
much better to use @code{prod (1:n)}, or @code{gamma (n+1)} instead,
after first checking to ensure that the value @code{n} is actually a
positive integer.} way to compute the factorial of a given integer:

@example
@group
function retval = fact (n)
  if (n > 0)
    retval = n * fact (n-1);
  else
    retval = 1;
  endif
endfunction
@end group
@end example

This function is recursive because it calls itself directly.  It
eventually terminates because each time it calls itself, it uses an
argument that is one less than was used for the previous call.  Once the
argument is no longer greater than zero, it does not call itself, and
the recursion ends.

The built-in variable @code{max_recursion_depth} specifies a limit to
the recursion depth and prevents Octave from recursing infinitely.

@c max_recursion_depth libinterp/parse-tree/pt-eval.cc
@anchor{XREFmax_recursion_depth}
@deftypefn  {Built-in Function} {@var{val} =} max_recursion_depth ()
@deftypefnx {Built-in Function} {@var{old_val} =} max_recursion_depth (@var{new_val})
@deftypefnx {Built-in Function} {} max_recursion_depth (@var{new_val}, "local")
Query or set the internal limit on the number of times a function may
be called recursively.

If the limit is exceeded, an error message is printed and control returns to
the top level.

When called from inside a function with the @qcode{"local"} option, the
variable is changed locally for the function and any subroutines it calls.
The original variable value is restored when exiting the function.
@end deftypefn


@node Arithmetic Ops
@section Arithmetic Operators
@cindex arithmetic operators
@cindex operators, arithmetic
@cindex addition
@cindex subtraction
@cindex multiplication
@cindex matrix multiplication
@cindex division
@cindex quotient
@cindex negation
@cindex unary minus
@cindex exponentiation
@cindex transpose
@cindex Hermitian operator
@cindex transpose, complex-conjugate
@cindex complex-conjugate transpose

The following arithmetic operators are available, and work on scalars
and matrices.  The element-by-element operators and functions broadcast
(@pxref{Broadcasting}).

@table @asis
@item @var{x} + @var{y}
@opindex +
Addition.  If both operands are matrices, the number of rows and columns
must both agree, or they must be broadcastable to the same shape.

@item @var{x} .+ @var{y}
@opindex .+
Element-by-element addition.  This operator is equivalent to @code{+}.

@item @var{x} - @var{y}
@opindex -
Subtraction.  If both operands are matrices, the number of rows and
columns of both must agree, or they must be broadcastable to the same
shape.

@item @var{x} .- @var{y}
Element-by-element subtraction.  This operator is equivalent to @code{-}.

@item @var{x} * @var{y}
@opindex *
Matrix multiplication.  The number of columns of @var{x} must agree with
the number of rows of @var{y}, or they must be broadcastable to the same
shape.

@item @var{x} .* @var{y}
@opindex .*
Element-by-element multiplication.  If both operands are matrices, the
number of rows and columns must both agree, or they must be
broadcastable to the same shape.

@item @var{x} / @var{y}
@opindex /
Right division.  This is conceptually equivalent to the expression

@example
(inverse (y') * x')'
@end example

@noindent
but it is computed without forming the inverse of @var{y'}.

If the system is not square, or if the coefficient matrix is singular,
a minimum norm solution is computed.

@item @var{x} ./ @var{y}
@opindex ./
Element-by-element right division.

@item @var{x} \ @var{y}
@opindex \
Left division.  This is conceptually equivalent to the expression

@example
inverse (x) * y
@end example

@noindent
but it is computed without forming the inverse of @var{x}.

If the system is not square, or if the coefficient matrix is singular,
a minimum norm solution is computed.

@item @var{x} .\ @var{y}
@opindex .\
Element-by-element left division.  Each element of @var{y} is divided
by each corresponding element of @var{x}.

@item  @var{x} ^ @var{y}
@itemx @var{x} ** @var{y}
@opindex **
@opindex ^
Power operator.  If @var{x} and @var{y} are both scalars, this operator
returns @var{x} raised to the power @var{y}.  If @var{x} is a scalar and
@var{y} is a square matrix, the result is computed using an eigenvalue
expansion.  If @var{x} is a square matrix, the result is computed by
repeated multiplication if @var{y} is an integer, and by an eigenvalue
expansion if @var{y} is not an integer.  An error results if both
@var{x} and @var{y} are matrices.

The implementation of this operator needs to be improved.

@item  @var{x} .^ @var{y}
@itemx @var{x} .** @var{y}
@opindex .**
@opindex .^
Element-by-element power operator.  If both operands are matrices, the
number of rows and columns must both agree, or they must be
broadcastable to the same shape.  If several complex results are
possible, the one with smallest non-negative argument (angle) is taken.
This rule may return a complex root even when a real root is also possible.
Use @code{realpow}, @code{realsqrt}, @code{cbrt}, or @code{nthroot} if a
real result is preferred.

@item -@var{x}
@opindex -
Negation.

@item +@var{x}
@opindex +
Unary plus.  This operator has no effect on the operand.

@item @var{x}'
@opindex '
Complex conjugate transpose.  For real arguments, this operator is the
same as the transpose operator.  For complex arguments, this operator is
equivalent to the expression

@example
conj (x.')
@end example

@item @var{x}.'
@opindex .'
Transpose.
@end table

Note that because Octave's element-by-element operators begin with a
@samp{.}, there is a possible ambiguity for statements like

@example
1./m
@end example

@noindent
because the period could be interpreted either as part of the constant
or as part of the operator.  To resolve this conflict, Octave treats the
expression as if you had typed

@example
(1) ./ m
@end example

@noindent
and not

@example
(1.) / m
@end example

@noindent
Although this is inconsistent with the normal behavior of Octave's
lexer, which usually prefers to break the input into tokens by
preferring the longest possible match at any given point, it is more
useful in this case.

@opindex '
@c ctranspose libinterp/corefcn/data.cc
@anchor{XREFctranspose}
@deftypefn {Built-in Function} {} ctranspose (@var{x})
Return the complex conjugate transpose of @var{x}.

This function and @tcode{x'} are equivalent.
@seealso{@ref{XREFtranspose,,transpose}}
@end deftypefn


@opindex .\
@c ldivide libinterp/corefcn/data.cc
@anchor{XREFldivide}
@deftypefn {Built-in Function} {} ldivide (@var{x}, @var{y})
Return the element-by-element left division of @var{x} and @var{y}.

This function and @w{@tcode{x .@xbackslashchar{} y}} are equivalent.
@seealso{@ref{XREFrdivide,,rdivide}, @ref{XREFmldivide,,mldivide}, @ref{XREFtimes,,times}, @ref{XREFplus,,plus}}
@end deftypefn


@opindex -
@c minus libinterp/corefcn/data.cc
@anchor{XREFminus}
@deftypefn {Built-in Function} {} minus (@var{x}, @var{y})
This function and @w{@tcode{x - y}} are equivalent.
@seealso{@ref{XREFplus,,plus}, @ref{XREFuminus,,uminus}}
@end deftypefn


@opindex \
@c mldivide libinterp/corefcn/data.cc
@anchor{XREFmldivide}
@deftypefn {Built-in Function} {} mldivide (@var{x}, @var{y})
Return the matrix left division of @var{x} and @var{y}.

This function and @w{@tcode{x @xbackslashchar{} y}} are equivalent.
@seealso{@ref{XREFmrdivide,,mrdivide}, @ref{XREFldivide,,ldivide}, @ref{XREFrdivide,,rdivide}}
@end deftypefn


@opindex **
@opindex ^
@c mpower libinterp/corefcn/data.cc
@anchor{XREFmpower}
@deftypefn {Built-in Function} {} mpower (@var{x}, @var{y})
Return the matrix power operation of @var{x} raised to the @var{y} power.

This function and @w{@tcode{x ^ y}} are equivalent.
@seealso{@ref{XREFpower,,power}, @ref{XREFmtimes,,mtimes}, @ref{XREFplus,,plus}, @ref{XREFminus,,minus}}
@end deftypefn


@opindex /
@c mrdivide libinterp/corefcn/data.cc
@anchor{XREFmrdivide}
@deftypefn {Built-in Function} {} mrdivide (@var{x}, @var{y})
Return the matrix right division of @var{x} and @var{y}.

This function and @w{@tcode{x / y}} are equivalent.
@seealso{@ref{XREFmldivide,,mldivide}, @ref{XREFrdivide,,rdivide}, @ref{XREFplus,,plus}, @ref{XREFminus,,minus}}
@end deftypefn


@opindex *
@c mtimes libinterp/corefcn/data.cc
@anchor{XREFmtimes}
@deftypefn  {Built-in Function} {} mtimes (@var{x}, @var{y})
@deftypefnx {Built-in Function} {} mtimes (@var{x1}, @var{x2}, @dots{})
Return the matrix multiplication product of inputs.

This function and @w{@tcode{x * y}} are equivalent.
If more arguments are given, the multiplication is applied
cumulatively from left to right:

@example
(@dots{}((x1 * x2) * x3) * @dots{})
@end example

At least one argument is required.
@seealso{@ref{XREFtimes,,times}, @ref{XREFplus,,plus}, @ref{XREFminus,,minus}, @ref{XREFrdivide,,rdivide}, @ref{XREFmrdivide,,mrdivide}, @ref{XREFmldivide,,mldivide}, @ref{XREFmpower,,mpower}}
@end deftypefn


@opindex +
@c plus libinterp/corefcn/data.cc
@anchor{XREFplus}
@deftypefn  {Built-in Function} {} plus (@var{x}, @var{y})
@deftypefnx {Built-in Function} {} plus (@var{x1}, @var{x2}, @dots{})
This function and @w{@tcode{x + y}} are equivalent.

If more arguments are given, the summation is applied
cumulatively from left to right:

@example
(@dots{}((x1 + x2) + x3) + @dots{})
@end example

At least one argument is required.
@seealso{@ref{XREFminus,,minus}, @ref{XREFuplus,,uplus}}
@end deftypefn


@opindex .**
@opindex .^
@c power libinterp/corefcn/data.cc
@anchor{XREFpower}
@deftypefn {Built-in Function} {} power (@var{x}, @var{y})
Return the element-by-element operation of @var{x} raised to the
@var{y} power.

This function and @w{@tcode{x .^ y}} are equivalent.

If several complex results are possible, returns the one with smallest
non-negative argument (angle).  Use @code{realpow}, @code{realsqrt},
@code{cbrt}, or @code{nthroot} if a real result is preferred.

@seealso{@ref{XREFmpower,,mpower}, @ref{XREFrealpow,,realpow}, @ref{XREFrealsqrt,,realsqrt}, @ref{XREFcbrt,,cbrt}, @ref{XREFnthroot,,nthroot}}
@end deftypefn


@opindex ./
@c rdivide libinterp/corefcn/data.cc
@anchor{XREFrdivide}
@deftypefn {Built-in Function} {} rdivide (@var{x}, @var{y})
Return the element-by-element right division of @var{x} and @var{y}.

This function and @w{@tcode{x ./ y}} are equivalent.
@seealso{@ref{XREFldivide,,ldivide}, @ref{XREFmrdivide,,mrdivide}, @ref{XREFtimes,,times}, @ref{XREFplus,,plus}}
@end deftypefn


@opindex .*
@c times libinterp/corefcn/data.cc
@anchor{XREFtimes}
@deftypefn  {Built-in Function} {} times (@var{x}, @var{y})
@deftypefnx {Built-in Function} {} times (@var{x1}, @var{x2}, @dots{})
Return the element-by-element multiplication product of inputs.

This function and @w{@tcode{x .* y}} are equivalent.
If more arguments are given, the multiplication is applied
cumulatively from left to right:

@example
(@dots{}((x1 .* x2) .* x3) .* @dots{})
@end example

At least one argument is required.
@seealso{@ref{XREFmtimes,,mtimes}, @ref{XREFrdivide,,rdivide}}
@end deftypefn


@opindex .'
@c transpose libinterp/corefcn/data.cc
@anchor{XREFtranspose}
@deftypefn {Built-in Function} {} transpose (@var{x})
Return the transpose of @var{x}.

This function and @tcode{x.'} are equivalent.
@seealso{@ref{XREFctranspose,,ctranspose}}
@end deftypefn


@opindex -
@c uminus libinterp/corefcn/data.cc
@anchor{XREFuminus}
@deftypefn {Built-in Function} {} uminus (@var{x})
This function and @w{@tcode{- x}} are equivalent.
@seealso{@ref{XREFuplus,,uplus}, @ref{XREFminus,,minus}}
@end deftypefn


@opindex +
@c uplus libinterp/corefcn/data.cc
@anchor{XREFuplus}
@deftypefn {Built-in Function} {} uplus (@var{x})
This function and @w{@tcode{+ x}} are equivalent.
@seealso{@ref{XREFuminus,,uminus}, @ref{XREFplus,,plus}, @ref{XREFminus,,minus}}
@end deftypefn


@node Comparison Ops
@section Comparison Operators
@cindex comparison expressions
@cindex expressions, comparison
@cindex relational operators
@cindex operators, relational
@cindex less than operator
@cindex greater than operator
@cindex equality operator
@cindex tests for equality
@cindex equality, tests for

@dfn{Comparison operators} compare numeric values for relationships
such as equality.  They are written using
@emph{relational operators}.

All of Octave's comparison operators return a value of 1 if the
comparison is true, or 0 if it is false.  For matrix values, they all
work on an element-by-element basis.  Broadcasting rules apply.
@xref{Broadcasting}.  For example:

@example
@group
[1, 2; 3, 4] == [1, 3; 2, 4]
     @result{}  1  0
         0  1
@end group
@end example

According to broadcasting rules, if one operand is a scalar and the
other is a matrix, the scalar is compared to each element of the matrix
in turn, and the result is the same size as the matrix.

@table @code
@item @var{x} < @var{y}
@opindex <
True if @var{x} is less than @var{y}.

@item @var{x} <= @var{y}
@opindex <=
True if @var{x} is less than or equal to @var{y}.

@item @var{x} == @var{y}
@opindex ==
True if @var{x} is equal to @var{y}.

@item @var{x} >= @var{y}
@opindex >=
True if @var{x} is greater than or equal to @var{y}.

@item @var{x} > @var{y}
@opindex >
True if @var{x} is greater than @var{y}.

@item  @var{x} != @var{y}
@itemx @var{x} ~= @var{y}
@opindex !=
@opindex ~=
True if @var{x} is not equal to @var{y}.
@end table

For complex numbers, the following ordering is defined:
@var{z1} < @var{z2}
if and only if

@example
@group
  abs (@var{z1}) < abs (@var{z2})
  || (abs (@var{z1}) == abs (@var{z2}) && arg (@var{z1}) < arg (@var{z2}))
@end group
@end example

This is consistent with the ordering used by @dfn{max}, @dfn{min} and
@dfn{sort}, but is not consistent with @sc{matlab}, which only compares the real
parts.

String comparisons may also be performed with the @code{strcmp}
function, not with the comparison operators listed above.
@xref{Strings}.

@opindex ==
@c eq libinterp/corefcn/data.cc
@anchor{XREFeq}
@deftypefn {Built-in Function} {} eq (@var{x}, @var{y})
Return true if the two inputs are equal.

This function is equivalent to @w{@code{x == y}}.
@seealso{@ref{XREFne,,ne}, @ref{XREFisequal,,isequal}, @ref{XREFle,,le}, @ref{XREFge,,ge}, @ref{XREFgt,,gt}, @ref{XREFne,,ne}, @ref{XREFlt,,lt}}
@end deftypefn


@opindex >=
@c ge libinterp/corefcn/data.cc
@anchor{XREFge}
@deftypefn {Built-in Function} {} ge (@var{x}, @var{y})
This function is equivalent to @w{@code{x >= y}}.
@seealso{@ref{XREFle,,le}, @ref{XREFeq,,eq}, @ref{XREFgt,,gt}, @ref{XREFne,,ne}, @ref{XREFlt,,lt}}
@end deftypefn


@opindex >
@c gt libinterp/corefcn/data.cc
@anchor{XREFgt}
@deftypefn {Built-in Function} {} gt (@var{x}, @var{y})
This function is equivalent to @w{@code{x > y}}.
@seealso{@ref{XREFle,,le}, @ref{XREFeq,,eq}, @ref{XREFge,,ge}, @ref{XREFne,,ne}, @ref{XREFlt,,lt}}
@end deftypefn


@c isequal scripts/general/isequal.m
@anchor{XREFisequal}
@deftypefn {Function File} {} isequal (@var{x1}, @var{x2}, @dots{})
Return true if all of @var{x1}, @var{x2}, @dots{} are equal.
@seealso{@ref{XREFisequaln,,isequaln}}
@end deftypefn


@c isequaln scripts/general/isequaln.m
@anchor{XREFisequaln}
@deftypefn {Function File} {} isequaln (@var{x1}, @var{x2}, @dots{})
Return true if all of @var{x1}, @var{x2}, @dots{} are equal under the
additional assumption that NaN == NaN (no comparison of NaN placeholders
in dataset).
@seealso{@ref{XREFisequal,,isequal}}
@end deftypefn


@opindex <=
@c le libinterp/corefcn/data.cc
@anchor{XREFle}
@deftypefn {Built-in Function} {} le (@var{x}, @var{y})
This function is equivalent to @w{@code{x <= y}}.
@seealso{@ref{XREFeq,,eq}, @ref{XREFge,,ge}, @ref{XREFgt,,gt}, @ref{XREFne,,ne}, @ref{XREFlt,,lt}}
@end deftypefn


@opindex <
@c lt libinterp/corefcn/data.cc
@anchor{XREFlt}
@deftypefn {Built-in Function} {} lt (@var{x}, @var{y})
This function is equivalent to @w{@code{x < y}}.
@seealso{@ref{XREFle,,le}, @ref{XREFeq,,eq}, @ref{XREFge,,ge}, @ref{XREFgt,,gt}, @ref{XREFne,,ne}}
@end deftypefn


@opindex !=
@opindex ~=
@c ne libinterp/corefcn/data.cc
@anchor{XREFne}
@deftypefn {Built-in Function} {} ne (@var{x}, @var{y})
Return true if the two inputs are not equal.

This function is equivalent to @w{@code{x != y}}.
@seealso{@ref{XREFeq,,eq}, @ref{XREFisequal,,isequal}, @ref{XREFle,,le}, @ref{XREFge,,ge}, @ref{XREFlt,,lt}}
@end deftypefn


@node Boolean Expressions
@section Boolean Expressions
@cindex expressions, boolean
@cindex boolean expressions
@cindex expressions, logical
@cindex logical expressions
@cindex operators, boolean
@cindex boolean operators
@cindex logical operators
@cindex operators, logical
@cindex and operator
@cindex or operator
@cindex not operator

@menu
* Element-by-element Boolean Operators::
* Short-circuit Boolean Operators::
@end menu

@node Element-by-element Boolean Operators
@subsection Element-by-element Boolean Operators
@cindex element-by-element evaluation

An @dfn{element-by-element boolean expression} is a combination of
comparison expressions using the boolean
operators ``or'' (@samp{|}), ``and'' (@samp{&}), and ``not'' (@samp{!}),
along with parentheses to control nesting.  The truth of the boolean
expression is computed by combining the truth values of the
corresponding elements of the component expressions.  A value is
considered to be false if it is zero, and true otherwise.

Element-by-element boolean expressions can be used wherever comparison
expressions can be used.  They can be used in @code{if} and @code{while}
statements.  However, a matrix value used as the condition in an
@code{if} or @code{while} statement is only true if @emph{all} of its
elements are nonzero.

Like comparison operations, each element of an element-by-element
boolean expression also has a numeric value (1 if true, 0 if false) that
comes into play if the result of the boolean expression is stored in a
variable, or used in arithmetic.

Here are descriptions of the three element-by-element boolean operators.

@table @code
@item @var{boolean1} & @var{boolean2}
@opindex &
Elements of the result are true if both corresponding elements of
@var{boolean1} and @var{boolean2} are true.

@item @var{boolean1} | @var{boolean2}
@opindex |
Elements of the result are true if either of the corresponding elements
of @var{boolean1} or @var{boolean2} is true.

@item  ! @var{boolean}
@itemx ~ @var{boolean}
@opindex ~
@opindex !
Each element of the result is true if the corresponding element of
@var{boolean} is false.
@end table

These operators work on an element-by-element basis.  For example, the
expression

@example
[1, 0; 0, 1] & [1, 0; 2, 3]
@end example

@noindent
returns a two by two identity matrix.

For the binary operators, broadcasting rules apply.  @xref{Broadcasting}.
In particular, if one of the operands is a scalar and the other a
matrix, the operator is applied to the scalar and each element of the
matrix.

For the binary element-by-element boolean operators, both subexpressions
@var{boolean1} and @var{boolean2} are evaluated before computing the
result.  This can make a difference when the expressions have side
effects.  For example, in the expression

@example
a & b++
@end example

@noindent
the value of the variable @var{b} is incremented even if the variable
@var{a} is zero.

This behavior is necessary for the boolean operators to work as
described for matrix-valued operands.

@opindex &
@c and libinterp/corefcn/data.cc
@anchor{XREFand}
@deftypefn  {Built-in Function} {@var{z} =} and (@var{x}, @var{y})
@deftypefnx {Built-in Function} {@var{z} =} and (@var{x1}, @var{x2}, @dots{})
Return the logical AND of @var{x} and @var{y}.

This function is equivalent to the operator syntax @w{@code{x & y}}.  If
more than two arguments are given, the logical AND is applied cumulatively
from left to right:

@example
(@dots{}((x1 & x2) & x3) & @dots{})
@end example

At least one argument is required.
@seealso{@ref{XREFor,,or}, @ref{XREFnot,,not}, @ref{XREFxor,,xor}}
@end deftypefn


@opindex ~
@opindex !
@c not libinterp/corefcn/data.cc
@anchor{XREFnot}
@deftypefn {Built-in Function} {@var{z} =} not (@var{x})
Return the logical NOT of @var{x}.

This function is equivalent to the operator syntax @w{@code{! x}}.
@seealso{@ref{XREFand,,and}, @ref{XREFor,,or}, @ref{XREFxor,,xor}}
@end deftypefn


@opindex |
@c or libinterp/corefcn/data.cc
@anchor{XREFor}
@deftypefn  {Built-in Function} {@var{z} =} or (@var{x}, @var{y})
@deftypefnx {Built-in Function} {@var{z} =} or (@var{x1}, @var{x2}, @dots{})
Return the logical OR of @var{x} and @var{y}.

This function is equivalent to the operator syntax @w{@code{x | y}}.  If
more than two arguments are given, the logical OR is applied cumulatively
from left to right:

@example
(@dots{}((x1 | x2) | x3) | @dots{})
@end example

At least one argument is required.
@seealso{@ref{XREFand,,and}, @ref{XREFnot,,not}, @ref{XREFxor,,xor}}
@end deftypefn


@node Short-circuit Boolean Operators
@subsection Short-circuit Boolean Operators
@cindex short-circuit evaluation

Combined with the implicit conversion to scalar values in @code{if} and
@code{while} conditions, Octave's element-by-element boolean operators
are often sufficient for performing most logical operations.  However,
it is sometimes desirable to stop evaluating a boolean expression as
soon as the overall truth value can be determined.  Octave's
@dfn{short-circuit} boolean operators work this way.

@table @code
@item @var{boolean1} && @var{boolean2}
@opindex &&
The expression @var{boolean1} is evaluated and converted to a scalar
using the equivalent of the operation @code{all (@var{boolean1}(:))}.
If it is false, the result of the overall expression is 0.  If it is
true, the expression @var{boolean2} is evaluated and converted to a
scalar using the equivalent of the operation @code{all
(@var{boolean1}(:))}.  If it is true, the result of the overall expression
is 1.  Otherwise, the result of the overall expression is 0.

@strong{Warning:} there is one exception to the rule of evaluating
@code{all (@var{boolean1}(:))}, which is when @code{boolean1} is the
empty matrix.  The truth value of an empty matrix is always @code{false}
so @code{[] && true} evaluates to @code{false} even though
@code{all ([])} is @code{true}.

@item @var{boolean1} || @var{boolean2}
@opindex ||
The expression @var{boolean1} is evaluated and converted to a scalar
using the equivalent of the operation @code{all (@var{boolean1}(:))}.
If it is true, the result of the overall expression is 1.  If it is
false, the expression @var{boolean2} is evaluated and converted to a
scalar using the equivalent of the operation @code{all
(@var{boolean1}(:))}.  If it is true, the result of the overall expression
is 1.  Otherwise, the result of the overall expression is 0.

@strong{Warning:} the truth value of an empty matrix is always @code{false},
see the previous list item for details.
@end table

The fact that both operands may not be evaluated before determining the
overall truth value of the expression can be important.  For example, in
the expression

@example
a && b++
@end example

@noindent
the value of the variable @var{b} is only incremented if the variable
@var{a} is nonzero.

This can be used to write somewhat more concise code.  For example, it
is possible write

@example
@group
function f (a, b, c)
  if (nargin > 2 && ischar (c))
    @dots{}
@end group
@end example

@noindent
instead of having to use two @code{if} statements to avoid attempting to
evaluate an argument that doesn't exist.  For example, without the
short-circuit feature, it would be necessary to write

@example
@group
function f (a, b, c)
  if (nargin > 2)
    if (ischar (c))
      @dots{}
@end group
@end example

@noindent
Writing

@example
@group
function f (a, b, c)
  if (nargin > 2 & ischar (c))
    @dots{}
@end group
@end example

@noindent
would result in an error if @code{f} were called with one or two
arguments because Octave would be forced to try to evaluate both of the
operands for the operator @samp{&}.

@sc{matlab} has special behavior that allows the operators @samp{&} and
@samp{|} to short-circuit when used in the truth expression for @code{if} and
@code{while} statements.  Octave also behaves the same way by default,
though the use of the @samp{&} and @samp{|} operators in this way is
strongly discouraged.  Instead, you should use the @samp{&&} and @samp{||}
operators that always have short-circuit behavior.

Finally, the ternary operator (?:) is not supported in Octave.  If
short-circuiting is not important, it can be replaced by the @code{ifelse}
function.

@c merge libinterp/corefcn/data.cc
@anchor{XREFmerge}
@deftypefn  {Built-in Function} {} merge (@var{mask}, @var{tval}, @var{fval})
@deftypefnx {Built-in Function} {} ifelse (@var{mask}, @var{tval}, @var{fval})
Merge elements of @var{true_val} and @var{false_val}, depending on the
value of @var{mask}.

If @var{mask} is a logical scalar, the other two arguments can be arbitrary
values.  Otherwise, @var{mask} must be a logical array, and @var{tval},
@var{fval} should be arrays of matching class, or cell arrays.  In the
scalar mask case, @var{tval} is returned if @var{mask} is true, otherwise
@var{fval} is returned.

In the array mask case, both @var{tval} and @var{fval} must be either
scalars or arrays with dimensions equal to @var{mask}.  The result is
constructed as follows:

@example
@group
result(mask) = tval(mask);
result(! mask) = fval(! mask);
@end group
@end example

@var{mask} can also be arbitrary numeric type, in which case it is first
converted to logical.
@seealso{@ref{XREFlogical,,logical}, @ref{XREFdiff,,diff}}
@end deftypefn


@node Assignment Ops
@section Assignment Expressions
@cindex assignment expressions
@cindex assignment operators
@cindex operators, assignment
@cindex expressions, assignment

@opindex =

An @dfn{assignment} is an expression that stores a new value into a
variable.  For example, the following expression assigns the value 1 to
the variable @code{z}:

@example
z = 1
@end example

@noindent
After this expression is executed, the variable @code{z} has the value 1.
Whatever old value @code{z} had before the assignment is forgotten.
The @samp{=} sign is called an @dfn{assignment operator}.

Assignments can store string values also.  For example, the following
expression would store the value @qcode{"this food is good"} in the
variable @code{message}:

@example
@group
thing = "food"
predicate = "good"
message = [ "this " , thing , " is " , predicate ]
@end group
@end example

@noindent
(This also illustrates concatenation of strings.)

@cindex side effect
Most operators (addition, concatenation, and so on) have no effect
except to compute a value.  If you ignore the value, you might as well
not use the operator.  An assignment operator is different.  It does
produce a value, but even if you ignore the value, the assignment still
makes itself felt through the alteration of the variable.  We call this
a @dfn{side effect}.

@cindex lvalue
The left-hand operand of an assignment need not be a variable
(@pxref{Variables}).  It can also be an element of a matrix
(@pxref{Index Expressions}) or a list of return values
(@pxref{Calling Functions}).  These are all called @dfn{lvalues}, which
means they can appear on the left-hand side of an assignment operator.
The right-hand operand may be any expression.  It produces the new value
which the assignment stores in the specified variable, matrix element,
or list of return values.

It is important to note that variables do @emph{not} have permanent types.
The type of a variable is simply the type of whatever value it happens
to hold at the moment.  In the following program fragment, the variable
@code{foo} has a numeric value at first, and a string value later on:

@example
@group
octave:13> foo = 1
foo = 1
octave:13> foo = "bar"
foo = bar
@end group
@end example

@noindent
When the second assignment gives @code{foo} a string value, the fact that
it previously had a numeric value is forgotten.

Assignment of a scalar to an indexed matrix sets all of the elements
that are referenced by the indices to the scalar value.  For example, if
@code{a} is a matrix with at least two columns,

@example
@group
a(:, 2) = 5
@end group
@end example

@noindent
sets all the elements in the second column of @code{a} to 5.

Assigning an empty matrix @samp{[]} works in most cases to allow you to
delete rows or columns of matrices and vectors.  @xref{Empty Matrices}.
For example, given a 4 by 5 matrix @var{A}, the assignment

@example
A (3, :) = []
@end example

@noindent
deletes the third row of @var{A}, and the assignment

@example
A (:, 1:2:5) = []
@end example

@noindent
deletes the first, third, and fifth columns.

An assignment is an expression, so it has a value.  Thus, @code{z = 1}
as an expression has the value 1.  One consequence of this is that you
can write multiple assignments together:

@example
x = y = z = 0
@end example

@noindent
stores the value 0 in all three variables.  It does this because the
value of @code{z = 0}, which is 0, is stored into @code{y}, and then
the value of @code{y = z = 0}, which is 0, is stored into @code{x}.

This is also true of assignments to lists of values, so the following is
a valid expression

@example
[a, b, c] = [u, s, v] = svd (a)
@end example

@noindent
that is exactly equivalent to

@example
@group
[u, s, v] = svd (a)
a = u
b = s
c = v
@end group
@end example

In expressions like this, the number of values in each part of the
expression need not match.  For example, the expression

@example
[a, b] = [u, s, v] = svd (a)
@end example

@noindent
is equivalent to

@example
@group
[u, s, v] = svd (a)
a = u
b = s
@end group
@end example

@noindent
The number of values on the left side of the expression can, however,
not exceed the number of values on the right side.  For example, the
following will produce an error.

@example
@group
[a, b, c, d] = [u, s, v] = svd (a);
@print{} error: element number 4 undefined in return list
@end group
@end example

The symbol @code{~} may be used as a placeholder in the list of lvalues,
indicating that the corresponding return value should be ignored and not stored
anywhere:

@example
@group
[~, s, v] = svd (a);
@end group
@end example

This is cleaner and more memory efficient than using a dummy variable.
The @code{nargout} value for the right-hand side expression is not affected.
If the assignment is used as an expression, the return value is a
comma-separated list with the ignored values dropped.

@opindex +=
A very common programming pattern is to increment an existing variable
with a given value, like this

@example
a = a + 2;
@end example

@noindent
This can be written in a clearer and more condensed form using the
@code{+=} operator

@example
a += 2;
@end example

@noindent
@opindex -=
@opindex *=
@opindex /=
Similar operators also exist for subtraction (@code{-=}),
multiplication (@code{*=}), and division (@code{/=}).  An expression
of the form

@example
@var{expr1} @var{op}= @var{expr2}
@end example

@noindent
is evaluated as

@example
@var{expr1} = (@var{expr1}) @var{op} (@var{expr2})
@end example

@noindent
where @var{op} can be either @code{+}, @code{-}, @code{*}, or @code{/},
as long as @var{expr2} is a simple expression with no side effects.  If
@var{expr2} also contains an assignment operator, then this expression
is evaluated as

@example
@group
@var{temp} = @var{expr2}
@var{expr1} = (@var{expr1}) @var{op} @var{temp}
@end group
@end example

@noindent
where @var{temp} is a placeholder temporary value storing the computed
result of evaluating @var{expr2}.  So, the expression

@example
a *= b+1
@end example

@noindent
is evaluated as

@example
a = a * (b+1)
@end example

@noindent
and @emph{not}

@example
a = a * b + 1
@end example

You can use an assignment anywhere an expression is called for.  For
example, it is valid to write @code{x != (y = 1)} to set @code{y} to 1
and then test whether @code{x} equals 1.  But this style tends to make
programs hard to read.  Except in a one-shot program, you should rewrite
it to get rid of such nesting of assignments.  This is never very hard.

@cindex increment operator
@cindex decrement operator
@cindex operators, increment
@cindex operators, decrement

@node Increment Ops
@section Increment Operators

@emph{Increment operators} increase or decrease the value of a variable
by 1.  The operator to increment a variable is written as @samp{++}.  It
may be used to increment a variable either before or after taking its
value.

For example, to pre-increment the variable @var{x}, you would write
@code{++@var{x}}.  This would add one to @var{x} and then return the new
value of @var{x} as the result of the expression.  It is exactly the
same as the expression @code{@var{x} = @var{x} + 1}.

To post-increment a variable @var{x}, you would write @code{@var{x}++}.
This adds one to the variable @var{x}, but returns the value that
@var{x} had prior to incrementing it.  For example, if @var{x} is equal
to 2, the result of the expression @code{@var{x}++} is 2, and the new
value of @var{x} is 3.

For matrix and vector arguments, the increment and decrement operators
work on each element of the operand.

Here is a list of all the increment and decrement expressions.

@table @code
@item ++@var{x}
@opindex ++
This expression increments the variable @var{x}.  The value of the
expression is the @emph{new} value of @var{x}.  It is equivalent to the
expression @code{@var{x} = @var{x} + 1}.

@item --@var{x}
@opindex @code{--}
This expression decrements the variable @var{x}.  The value of the
expression is the @emph{new} value of @var{x}.  It is equivalent to the
expression @code{@var{x} = @var{x} - 1}.

@item @var{x}++
@opindex ++
This expression causes the variable @var{x} to be incremented.  The
value of the expression is the @emph{old} value of @var{x}.

@item @var{x}--
@opindex @code{--}
This expression causes the variable @var{x} to be decremented.  The
value of the expression is the @emph{old} value of @var{x}.
@end table

@node Operator Precedence
@section Operator Precedence
@cindex operator precedence

@dfn{Operator precedence} determines how operators are grouped, when
different operators appear close by in one expression.  For example,
@samp{*} has higher precedence than @samp{+}.  Thus, the expression
@code{a + b * c} means to multiply @code{b} and @code{c}, and then add
@code{a} to the product (i.e., @code{a + (b * c)}).

You can overrule the precedence of the operators by using parentheses.
You can think of the precedence rules as saying where the parentheses
are assumed if you do not write parentheses yourself.  In fact, it is
wise to use parentheses whenever you have an unusual combination of
operators, because other people who read the program may not remember
what the precedence is in this case.  You might forget as well, and then
you too could make a mistake.  Explicit parentheses will help prevent
any such mistake.

When operators of equal precedence are used together, the leftmost
operator groups first, except for the assignment operators, which group
in the opposite order.  Thus, the expression @code{a - b + c} groups as
@code{(a - b) + c}, but the expression @code{a = b = c} groups as
@code{a = (b = c)}.

The precedence of prefix unary operators is important when another
operator follows the operand.  For example, @code{-x^2} means
@code{-(x^2)}, because @samp{-} has lower precedence than @samp{^}.

Here is a table of the operators in Octave, in order of decreasing
precedence.  Unless noted, all operators group left to right.

@table @code
@item function call and array indexing, cell array indexing, and structure element indexing
@samp{()}  @samp{@{@}} @samp{.}

@item postfix increment, and postfix decrement
@samp{++}  @samp{--}

These operators group right to left.

@item transpose and exponentiation
@samp{'} @samp{.'} @samp{^} @samp{**} @samp{.^} @samp{.**}

@item unary plus, unary minus, prefix increment, prefix decrement, and logical "not"
@samp{+} @samp{-} @samp{++}  @samp{--} @samp{~} @samp{!}

@item multiply and divide
@samp{*} @samp{/} @samp{\} @samp{.\} @samp{.*} @samp{./}

@item add, subtract
@samp{+} @samp{-}

@item colon
@samp{:}

@item relational
@samp{<} @samp{<=} @samp{==} @samp{>=} @samp{>} @samp{!=}
@samp{~=}

@item element-wise "and"
@samp{&}

@item element-wise "or"
@samp{|}

@item logical "and"
@samp{&&}

@item logical "or"
@samp{||}

@item assignment
@samp{=} @samp{+=} @samp{-=} @samp{*=} @samp{/=} @samp{\=}
@samp{^=} @samp{.*=} @samp{./=} @samp{.\=} @samp{.^=} @samp{|=}
@samp{&=}

These operators group right to left.
@end table
