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@node Numeric Data Types
@chapter Numeric Data Types
@cindex numeric constant
@cindex numeric value

A @dfn{numeric constant} may be a scalar, a vector, or a matrix, and it may
contain complex values.

The simplest form of a numeric constant, a scalar, is a single number.  Note
that by default numbers are represented within Octave by IEEE@tie{}754 double
precision (binary64) floating-point format (complex constants are stored as
pairs of binary64 values).  It is, however, possible to represent true integers
as described in @ref{Integer Data Types}.

If the numeric constant is a real integer, it can be defined in decimal,
hexadecimal, or binary notation.  An important difference, however, is that
decimal constants will be stored as binary64 values while using hexadecimal or
binary notation will result in a true integer with a storage class just large
enough to hold the specified number.  Hexadecimal notation starts with
@samp{0x} or @samp{0X}, binary notation starts with @samp{0b} or @samp{0B},
otherwise decimal notation is assumed.  As a consequence, @samp{0b} is not a
hexadecimal number, in fact, it is not a valid number at all.

For better readability, digits may be partitioned by the underscore separator
@samp{_}, which is ignored by the Octave interpreter.  Here are some examples
of real-valued integer constants, which all represent the same value:

@example
@group
42            # decimal notation, binary64
0x2A          # hexadecimal notation
0b101010      # binary notation
0b10_1010     # underscore notation
round (42.1)  # also binary64
@end group
@end example

In decimal notation, the numeric constant may be denoted as decimal fraction
or even in scientific (exponential) notation.  Note that this is not possible
for hexadecimal or binary notation.  Again, in the following example all
numeric constants represent the same value:

@example
@group
.105
1.05e-1
.00105e+2
@end group
@end example

Unlike most programming languages, complex numeric constants are denoted as
the sum of real and imaginary parts.  The imaginary part is denoted by a
real-valued numeric constant followed immediately by a complex value indicator
(@samp{i}, @samp{j}, @samp{I}, or @samp{J} which represents
@tex
  $\sqrt{-1}$).
@end tex
@ifnottex
  @code{sqrt (-1)}).
@end ifnottex
No spaces are allowed between the numeric constant and the complex value
indicator.  All complex values are stored as pairs of binary64 values and the
use of hexadecimal or binary notation does @emph{not} result in a true integer.
Some examples of complex numeric constants that all represent the same value:

@example
@group
3 + 42i
3 + 42j
3 + 42I
3 + 42J
3.0 + 42.0i
3.0 + 0x2Ai
3.0 + 0b10_1010i
0.3e1 + 420e-1i
@end group
@end example

@DOCSTRING(double)

@DOCSTRING(complex)

@menu
* Matrices::
* Ranges::
* Single Precision Data Types::
* Integer Data Types::
* Bit Manipulations::
* Logical Values::
* Automatic Conversion of Data Types::
* Predicates for Numeric Objects::
@end menu

@node Matrices
@section Matrices
@cindex matrices

@opindex [
@opindex ]
@opindex ;
@opindex ,

It is easy to define a matrix of values in Octave.  The size of the
matrix is determined automatically, so it is not necessary to explicitly
state the dimensions.  The expression

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

@noindent
results in the matrix
@tex
$$ a = \left[ \matrix{ 1 & 2 \cr 3 & 4 } \right] $$
@end tex
@ifnottex

@example
@group

        /      \
        | 1  2 |
  a  =  |      |
        | 3  4 |
        \      /

@end group
@end example

@end ifnottex

Elements of a matrix may be arbitrary expressions, provided that the
dimensions all make sense when combining the various pieces.  For
example, given the above matrix, the expression

@example
[ a, a ]
@end example

@noindent
produces the matrix

@example
@group
ans =

  1  2  1  2
  3  4  3  4
@end group
@end example

@noindent
but the expression

@example
[ a, 1 ]
@end example

@noindent
produces the error

@example
error: number of rows must match (1 != 2) near line 13, column 6
@end example

@noindent
(assuming that this expression was entered as the first thing on line
13, of course).

Inside the square brackets that delimit a matrix expression, Octave
looks at the surrounding context to determine whether spaces and newline
characters should be converted into element and row separators, or
simply ignored, so an expression like

@example
@group
a = [ 1 2
      3 4 ]
@end group
@end example

@noindent
will work.  However, some possible sources of confusion remain.  For
example, in the expression

@example
[ 1 - 1 ]
@end example

@noindent
the @samp{-} is treated as a binary operator and the result is the
scalar 0, but in the expression

@example
[ 1 -1 ]
@end example

@noindent
the @samp{-} is treated as a unary operator and the result is the
vector @code{[ 1, -1 ]}.  Similarly, the expression

@example
[ sin (pi) ]
@end example

@noindent
will be parsed as

@example
[ sin, (pi) ]
@end example

@noindent
and will result in an error since the @code{sin} function will be
called with no arguments.  To get around this, you must omit the space
between @code{sin} and the opening parenthesis, or enclose the
expression in a set of parentheses:

@example
[ (sin (pi)) ]
@end example

Whitespace surrounding the single quote character (@samp{'}, used as a
transpose operator and for delimiting character strings) can also cause
confusion.  Given @code{a = 1}, the expression

@example
[ 1 a' ]
@end example

@noindent
results in the single quote character being treated as a
transpose operator and the result is the vector @code{[ 1, 1 ]}, but the
expression

@example
[ 1 a ' ]
@end example

@noindent
produces the error message

@example
@group
parse error:

  syntax error

>>> [ 1 a ' ]
              ^
@end group
@end example

@noindent
because not doing so would cause trouble when parsing the valid expression

@example
[ a 'foo' ]
@end example

For clarity, it is probably best to always use commas and semicolons to
separate matrix elements and rows.

The maximum number of elements in a matrix is fixed when Octave is compiled.
The allowable number can be queried with the function @code{sizemax}.  Note
that other factors, such as the amount of memory available on your machine,
may limit the maximum size of matrices to something smaller.

@DOCSTRING(sizemax)

When you type a matrix or the name of a variable whose value is a
matrix, Octave responds by printing the matrix in with neatly aligned
rows and columns.  If the rows of the matrix are too large to fit on the
screen, Octave splits the matrix and displays a header before each
section to indicate which columns are being displayed.  You can use the
following variables to control the format of the output.

@DOCSTRING(output_precision)

It is possible to achieve a wide range of output styles by using
different values of @code{output_precision}.  Reasonable combinations can be
set using the @code{format} function.  @xref{Basic Input and Output}.

@DOCSTRING(split_long_rows)

Octave automatically switches to scientific notation when values become
very large or very small.  This guarantees that you will see several
significant figures for every value in a matrix.  If you would prefer to
see all values in a matrix printed in a fixed point format, you can use
the function @code{fixed_point_format}.  But doing so is not
recommended, because it can produce output that can easily be
misinterpreted.

@DOCSTRING(fixed_point_format)

@menu
* Empty Matrices::
@end menu

@node Empty Matrices
@subsection Empty Matrices

A matrix may have one or both dimensions zero, and operations on empty
matrices are handled as described by @nospell{Carl de Boor} in
@cite{An Empty Exercise}, SIGNUM, Volume 25, pages 2--6, 1990 and
@nospell{C. N. Nett and W. M. Haddad}, in
@cite{A System-Theoretic Appropriate Realization of the Empty Matrix Concept},
IEEE Transactions on Automatic Control, Volume 38, Number 5, May 1993.
@tex
Briefly, given a scalar $s$, an $m\times n$ matrix $M_{m\times n}$,
and an $m\times n$ empty matrix $[\,]_{m\times n}$ (with either one or
both dimensions equal to zero), the following are true:
$$
\eqalign{%
s \cdot [\,]_{m\times n} = [\,]_{m\times n} \cdot s &= [\,]_{m\times n}\cr
[\,]_{m\times n} + [\,]_{m\times n} &= [\,]_{m\times n}\cr
[\,]_{0\times m} \cdot M_{m\times n} &= [\,]_{0\times n}\cr
M_{m\times n} \cdot [\,]_{n\times 0} &= [\,]_{m\times 0}\cr
[\,]_{m\times 0} \cdot [\,]_{0\times n} &=  0_{m\times n}}
$$
@end tex
@ifnottex
Briefly, given a scalar @var{s}, an @var{m} by
@var{n} matrix @code{M(mxn)}, and an @var{m} by @var{n} empty matrix
@code{[](mxn)} (with either one or both dimensions equal to zero), the
following are true:

@example
@group
s * [](mxn) = [](mxn) * s = [](mxn)

    [](mxn) + [](mxn) = [](mxn)

    [](0xm) *  M(mxn) = [](0xn)

     M(mxn) * [](nx0) = [](mx0)

    [](mx0) * [](0xn) =  0(mxn)
@end group
@end example

@end ifnottex

By default, dimensions of the empty matrix are printed along with the
empty matrix symbol, @samp{[]}.  The built-in variable
@code{print_empty_dimensions} controls this behavior.

@DOCSTRING(print_empty_dimensions)

Empty matrices may also be used in assignment statements as a convenient
way to delete rows or columns of matrices.
@xref{Assignment Ops,,Assignment Expressions}.

When Octave parses a matrix expression, it examines the elements of the
list to determine whether they are all constants.  If they are, it
replaces the list with a single matrix constant.

@node Ranges
@section Ranges
@cindex range expressions
@cindex expression, range

@opindex :, range expressions

A @dfn{range} is a convenient way to write a row vector with evenly
spaced elements.  A range expression is defined by the value of the first
element in the range, an optional value for the increment between
elements, and a maximum value which the elements of the range will not
exceed.  The base, increment, and limit are separated by colons (the
@samp{:} character) and may contain any arithmetic expressions and
function calls.  If the increment is omitted, it is assumed to be 1.
For example, the range

@example
1 : 5
@end example

@noindent
defines the set of values @code{[ 1, 2, 3, 4, 5 ]}, and the range

@example
1 : 3 : 5
@end example

@noindent
defines the set of values @code{[ 1, 4 ]}.

Although a range constant specifies a row vector, Octave does @emph{not}
normally convert range constants to vectors unless it is necessary to do so.
This allows you to write a constant like @code{1 : 10000} without using
80,000 bytes of storage on a typical workstation.

A common example of when it does become necessary to convert ranges into
vectors occurs when they appear within a vector (i.e., inside square
brackets).  For instance, whereas

@example
x = 0 : 0.1 : 1;
@end example

@noindent
defines @var{x} to be a variable of type @code{double_range} and occupies 24
bytes of memory, the expression

@example
y = [ 0 : 0.1 : 1];
@end example

@noindent
defines @var{y} to be of type @code{matrix} and occupies 88 bytes of
memory.

This space saving optimization may be disabled using the function
@code{optimize_range}.

@DOCSTRING(optimize_range)

Note that the upper (or lower, if the increment is negative) bound on
the range is not always included in the set of values.  This can be useful
in some contexts.  For example:

@example
@group
## x is some predefined range or vector or matrix or array
x(1:2:end) += 1;   # increment all  odd-numbered elements
x(2:2:end) -= 1;   # decrement all even-numbered elements
@end group
@end example

The above code works correctly whether @var{x} has an odd number of elements
or not: there is no need to treat the two cases differently.

Octave uses floating point arithmetic to compute the values in the
range.  As a result, defining ranges with floating-point values can result
in pitfalls like these:

@example
@group
a = -2
b = (0.3 - 0.2 - 0.1)
x = a : b
@end group
@end example

Due to floating point rounding, @var{b} may or may not equal zero exactly,
and if it does not, it may be above zero or below zero, hence the final range
@var{x} may or may not include zero as its final value.  Similarly:

@example
@group
x = 1.80 : 0.05 : 1.90
y = 1.85 : 0.05 : 1.90
@end group
@end example

@noindent
is not as predictable as it looks.  As of Octave 8.3, the results obtained are
that @var{x} has three elements (1.80, 1.85, and 1.90), and @var{y} has only
one element (1.85 but not 1.90).  Thus, when using floating points in ranges,
changing the start of the range can easily affect the end of the range even
though the ending value was not touched in the above example.

To avoid such pitfalls with floating-points in ranges, you can use one of
the following patterns.  This change to the previous code:

@example
@group
x = (0:2) * 0.05 + 1.80
y = (0:1) * 0.05 + 1.85
@end group
@end example

@noindent
use integers to construct the range and then converts to floating point making
the overall expression safe and repeatable across platforms, compilers, and
compiler settings.  If you know the number of elements, you can use the
@code{linspace} function (@pxref{Special Utility Matrices}), which will include
the endpoints of a range.  You can also make judicious use of @code{round},
@code{floor}, @code{ceil}, @code{fix}, etc.@: to set the limits and the
increment without getting interference from floating-point rounding.  For
example, the earlier example can be made safer with one of the following:

@example
@group
a = -2
b = round ((0.3 - 0.2 - 0.1) * 1e12) / 1e12   # rounds to 12 digits
c = floor (0.3 - 0.2 - 0.1)                   # floors as integer
d = floor ((0.3 - 0.2 - 0.1) * 1e12) / 1e12   # floors at 12 digits
x = a : b
y = a : c
z = a : d
@end group
@end example

If the result of the range expression is empty then Octave returns an empty
@code{matrix}, not an empty @code{double_range}.  Similarly, if there is just
a single element in the range then Octave returns a @code{scalar}, not a
@code{double_range} with one element.

@node Single Precision Data Types
@section Single Precision Data Types

Octave includes support for single precision data types, and most of the
functions in Octave accept single precision values and return single
precision answers.  A single precision variable is created with the
@code{single} function.

@DOCSTRING(single)

for example:

@example
@group
sngl = single (rand (2, 2))
     @result{} sngl =
        0.37569   0.92982
        0.11962   0.50876
class (sngl)
    @result{} single
@end group
@end example

Many functions can also return single precision values directly.  For
example

@example
@group
ones (2, 2, "single")
zeros (2, 2, "single")
eye (2, 2,  "single")
rand (2, 2, "single")
NaN (2, 2, "single")
NA (2, 2, "single")
Inf (2, 2, "single")
@end group
@end example

@noindent
will all return single precision matrices.

@node Integer Data Types
@section Integer Data Types

Octave supports integer matrices as an alternative to using double precision.
It is possible to use both signed and unsigned integers represented by 8, 16,
32, or 64 bits.  It should be noted that most computations require floating
point data, meaning that integers will often change type when involved in
numeric computations.  For this reason integers are most often used to store
data, and not for calculations.

In general, most integer matrices are created by casting existing matrices to
integers.  The following example shows how to cast a matrix into 32-bit
integers.

@example
@group
float = rand (2, 2)
     @result{} float = 0.37569   0.92982
                0.11962   0.50876
integer = int32 (float)
     @result{} integer = 0  1
                  0  1
@end group
@end example

@noindent
As can be seen, floating point values are rounded to the nearest integer when
converted.

@DOCSTRING(isinteger)

@DOCSTRING(int8)

@DOCSTRING(uint8)

@DOCSTRING(int16)

@DOCSTRING(uint16)

@DOCSTRING(int32)

@DOCSTRING(uint32)

@DOCSTRING(int64)

@DOCSTRING(uint64)

@DOCSTRING(intmax)

@DOCSTRING(intmin)

@DOCSTRING(flintmax)

@menu
* Hexadecimal and Binary Integer Constants::
* Integer Arithmetic::
@end menu

@node Hexadecimal and Binary Integer Constants
@subsection Hexadecimal and Binary Integer Constants

The use of hexadecimal or binary notation to define a number will automatically
create an unsigned integer using a representation that is just large enough to
hold the specified value.  For example:

@example
@group
0b101        # uint8
0x100        # uint16
0xDEADBEEF   # uint32
0x1DEADBEEF  # uint64
@end group
@end example

The storage class can be specified by adding a suffix.  Use @samp{s} for signed
integers and @samp{u} for unsigned integers along with a size (@samp{8},
@samp{16}, @samp{32}, @samp{64}).  The use of the underscore separator @samp{_}
can improve readability.  For example:

@example
@group
0b101s16        # int16
0b101_s16       # int16, value and representation separated
0xDEADBEEFs32   # int32
0xDEADBEEF_u64  # uint64
@end group
@end example

Note that when defining matrices of integer constants the overall matrix will
have the storage class of its first element.  The matrix
@code{[0x1; 0x100; 0x10000]} will be of type @code{uint8} and the larger values
will be truncated because of the saturation semantics of integer values.  To
avoid this issue either: 1) declare the first integer to be of the desired size
such as @code{0x1u32; 0x100; 0x10000]}, or 2) pad constants in array
expressions with leading zeros so that they use the same number of digits for
each value such as @code{[0x00_00_01; 0x00_01_00; 0x01_00_00]}.

@node Integer Arithmetic
@subsection Integer Arithmetic

While many numerical computations can't be carried out in integers,
Octave does support basic operations like addition and multiplication
on integers.  The operators @code{+}, @code{-}, @code{.*}, and @code{./}
work on integers of the same type.  So, it is possible to add two 32-bit
integers, but not to add a 32-bit integer and a 16-bit integer.

When doing integer arithmetic one should consider the possibility of
underflow and overflow.  This happens when the result of the computation
can't be represented using the chosen integer type.  As an example it is
not possible to represent the result of @math{10 - 20} when using
unsigned integers.  Octave makes sure that the result of integer
computations is the integer that is closest to the true result.  So, the
result of @math{10 - 20} when using unsigned integers is zero.

When doing integer division Octave will round the result to the nearest
integer.  This is different from most programming languages, where the
result is often floored to the nearest integer.  So, the result of
@code{int32 (5) ./ int32 (8)} is @code{1}.

@DOCSTRING(idivide)

@node Bit Manipulations
@section Bit Manipulations

Octave provides a number of functions for the manipulation of numeric
values on a bit by bit basis.  The basic functions to set and obtain the
values of individual bits are @code{bitset} and @code{bitget}.

@DOCSTRING(bitset)

@DOCSTRING(bitget)

The arguments to all of Octave's bitwise operations can be scalar or
arrays, except for @code{bitcmp}, whose @var{k} argument must a
scalar.  In the case where more than one argument is an array, then all
arguments must have the same shape, and the bitwise operator is applied
to each of the elements of the argument individually.  If at least one
argument is a scalar and one an array, then the scalar argument is
duplicated.  Therefore

@example
bitget (100, 8:-1:1)
@end example

@noindent
is the same as

@example
bitget (100 * ones (1, 8), 8:-1:1)
@end example

It should be noted that all values passed to the bit manipulation
functions of Octave are treated as integers.  Therefore, even though the
example for @code{bitset} above passes the floating point value
@code{10}, it is treated as the bits @code{[1, 0, 1, 0]} rather than the
bits of the native floating point format representation of @code{10}.

As the maximum value that can be represented by a number is important
for bit manipulation, particularly when forming masks, Octave supplies
two utility functions: @code{flintmax} for floating point integers, and
@code{intmax} for integer objects (@code{uint8}, @code{int64}, etc.).

Octave also includes the basic bitwise 'and', 'or', and 'exclusive or'
operators.

@DOCSTRING(bitand)

@DOCSTRING(bitor)

@DOCSTRING(bitxor)

The bitwise 'not' operator is a unary operator that performs a logical
negation of each of the bits of the value.  For this to make sense, the
mask against which the value is negated must be defined.  Octave's
bitwise 'not' operator is @code{bitcmp}.

@DOCSTRING(bitcmp)

Octave also includes the ability to left-shift and right-shift values bitwise.

@DOCSTRING(bitshift)

Bits that are shifted out of either end of the value are lost.  Octave
also uses arithmetic shifts, where the sign bit of the value is kept
during a right shift.  For example:

@example
@group
bitshift (-10, -1)
@result{} -5
bitshift (int8 (-1), -1)
@result{} -1
@end group
@end example

Note that @code{bitshift (int8 (-1), -1)} is @code{-1} since the bit
representation of @code{-1} in the @code{int8} data type is @code{[1, 1,
1, 1, 1, 1, 1, 1]}.

@node Logical Values
@section Logical Values

Octave has built-in support for logical values, i.e., variables that
are either @code{true} or @code{false}.  When comparing two variables,
the result will be a logical value whose value depends on whether or
not the comparison is true.

The basic logical operations are @code{&}, @code{|}, and @code{!},
which correspond to ``Logical And'', ``Logical Or'', and ``Logical
Negation''.  These operations all follow the usual rules of logic.

It is also possible to use logical values as part of standard numerical
calculations.  In this case @code{true} is converted to @code{1}, and
@code{false} to 0, both represented using double precision floating
point numbers.  So, the result of @code{true*22 - false/6} is @code{22}.

Logical values can also be used to index matrices and cell arrays.
When indexing with a logical array the result will be a vector containing
the values corresponding to @code{true} parts of the logical array.
@xref{XREFLogicalIndexing, , Logical Indexing}.

Logical values can also be constructed by
casting numeric objects to logical values, or by using the @code{true}
or @code{false} functions.

@DOCSTRING(logical)

@DOCSTRING(true)

@DOCSTRING(false)

@node Automatic Conversion of Data Types
@section Automatic Conversion of Data Types

Many operators and functions can work with mixed data types.  For example,

@example
@group
uint8 (1) + 1
    @result{} 2
@end group

@group
single (1) + 1
    @result{} 2
@end group

@group
min (single (1), 0)
   @result{} 0
@end group
@end example

@noindent
where the results are respectively of types uint8, single, and single
respectively.  This is done for @sc{matlab} compatibility.  Valid mixed
operations are defined as follows:

@multitable @columnfractions .2 .3 .3 .2
@headitem @tab Mixed Operation @tab Result @tab
@item @tab double OP single @tab single @tab
@item @tab double OP integer @tab integer @tab
@item @tab double OP char @tab double @tab
@item @tab double OP logical @tab double @tab
@item @tab single OP integer @tab integer @tab
@item @tab single OP char @tab single @tab
@item @tab single OP logical @tab single @tab
@end multitable

When functions expect a double but are passed other types, automatic
conversion is function-dependent:

@example
@group
a = det (int8 ([1 2; 3 4]))
    @result{} a = -2
class (a)
    @result{} double
@end group

@group
a = eig (int8 ([1 2; 3 4]))
    @result{} error: eig: wrong type argument 'int8 matrix'
@end group
@end example

When two operands are both integers but of different widths, then some cases
convert them to the wider bitwidth, and other cases throw an error:

@example
@group
a = min (int8 (100), int16 (200))
    @result{} 100
class (a)
    @result{} int16
@end group

@group
int8 (100) + int16 (200)
   @result{} error: binary operator '+' not implemented
   for 'int8 scalar' by 'int16 scalar' operations
@end group
@end example

For two integer operands, they typically need to both be signed or both be
unsigned.  Mixing signed and unsigned usually causes an error, even if they
are of the same bitwidth.

@example
@group
min (int16 (100), uint16 (200))
   @result{} error: min: cannot compute min (int16 scalar, uint16 scalar)
@end group
@end example

In the case of mixed type indexed assignments, the type is not
changed.  For example,

@example
@group
x = ones (2, 2);
x(1, 1) = single (2)
   @result{} x = 2   1
          1   1
@end group
@end example

@noindent
where @code{x} remains of the double precision type.

@node Predicates for Numeric Objects
@section Predicates for Numeric Objects

Since the type of a variable may change during the execution of a
program, it can be necessary to do type checking at run-time.  Doing this
also allows you to change the behavior of a function depending on the
type of the input.  As an example, this naive implementation of @code{abs}
returns the absolute value of the input if it is a real number, and the
length of the input if it is a complex number.

@example
@group
function a = abs (x)
  if (isreal (x))
    a = sign (x) .* x;
  elseif (iscomplex (x))
    a = sqrt (real(x).^2 + imag(x).^2);
  endif
endfunction
@end group
@end example

The following functions are available for determining the type of a
variable.

@DOCSTRING(isnumeric)

@DOCSTRING(islogical)

@DOCSTRING(isfloat)

@DOCSTRING(isreal)

@DOCSTRING(iscomplex)

@DOCSTRING(ismatrix)

@DOCSTRING(isvector)

@DOCSTRING(isrow)

@DOCSTRING(iscolumn)

@DOCSTRING(isscalar)

@DOCSTRING(issquare)

@DOCSTRING(issymmetric)

@DOCSTRING(ishermitian)

@DOCSTRING(isdefinite)

@DOCSTRING(isbanded)

@DOCSTRING(isdiag)

@DOCSTRING(istril)

@DOCSTRING(istriu)

@DOCSTRING(isprime)

@DOCSTRING(isuniform)

If instead of knowing properties of variables, you wish to know which
variables are defined and to gather other information about the
workspace itself, @pxref{Status of Variables}.
