A 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 that can be an integer, a decimal fraction, a number in scientific (exponential) notation, or a complex number. Note that all numeric constants are represented within Octave in double-precision floating point format (complex constants are stored as pairs of double-precision floating point values). Here are some examples of real-valued numeric constants, which all have the same value:
105 1.05e+2 1050e-1
To specify complex constants, you can write an expression of the form
3 + 4i 3.0 + 4.0i 0.3e1 + 40e-1i
all of which are equivalent. The letter `i' in the previous example
stands for the pure imaginary constant, defined as
sqrt (-1).
For Octave to recognize a value as the imaginary part of a complex constant, a space must not appear between the number and the `i'. If it does, Octave will print an error message, like this:
octave:13> 3 + 4 i
parse error:
3 + 4 i
^
You may also use `j', `I', or `J' in place of the `i' above. All four forms are equivalent.
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
a = [1, 2; 3, 4]
results in the matrix
/ \
| 1 2 |
a = | |
| 3 4 |
\ /
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
[ a, a ]
produces the matrix
ans = 1 2 1 2 3 4 3 4
but the expression
[ a, 1 ]
produces the error
error: number of rows must match near line 13, column 6
(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 commands like
[ linspace (1, 2) ]
and
a = [ 1 2
3 4 ]
will work. However, some possible sources of confusion remain. For example, in the expression
[ 1 - 1 ]
the `-' is treated as a binary operator and the result is the scalar 0, but in the expression
[ 1 -1 ]
the `-' is treated as a unary operator and the result is the
vector [ 1, -1 ].
Given a = 1, the expression
[ 1 a' ]
results in the single quote character `'' being treated as a
transpose operator and the result is the vector [ 1, 1 ], but the
expression
[ 1 a ' ]
produces the error message
error: unterminated string constant
because to not do so would make it impossible to correctly parse the valid expression
[ a 'foo' ]
For clarity, it is probably best to always use commas and semicolons to
separate matrix elements and rows. It is possible to enforce this style
by setting the built-in variable whitespace_in_literal_matrix to
"ignore".
[m (1)] or
[ 1, 2, 3, 4 ]
If the value of whitespace_in_literal_matrix is "ignore",
Octave will never insert a comma or a semicolon in a literal matrix
list. For example, the expression [1 2] will result in an error
instead of being treated the same as [1, 2], and the expression
[ 1, 2, 3, 4 ]
will result in the vector [ 1, 2, 3, 4 ] instead of a matrix.
If the value of whitespace_in_literal_matrix is "traditional",
Octave will convert spaces to a comma between identifiers and `('. For
example, given the matrix
m = [3 2]
the expression
[m (1)]
will be parsed as
[m, (1)]
and will result in
[3 2 1]
and the expression
[ 1, 2, 3, 4 ]
will result in a matrix because the newline character is converted to a semicolon (row separator) even though there is a comma at the end of the first line (trailing commas or semicolons are ignored). This is apparently how MATLAB behaves.
Any other value for whitespace_in_literal_matrix results in behavior
that is the same as traditional, except that Octave does not
convert spaces to a comma between identifiers and `('. For
example, the expression
[m (1)]
will produce `3'. This is the way Octave has always behaved.
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.
It is possible to achieve a wide range of output styles by using
different values of output_precision and
output_max_field_width. Reasonable combinations can be set using
the format function. See section Basic Input and Output.
If the value of split_long_rows is nonzero, Octave will display
the matrix in a series of smaller pieces, each of which can fit within
the limits of your terminal width. Each set of rows is labeled so that
you can easily see which columns are currently being displayed.
For example:
octave:13> rand (2,10) ans = Columns 1 through 6: 0.75883 0.93290 0.40064 0.43818 0.94958 0.16467 0.75697 0.51942 0.40031 0.61784 0.92309 0.40201 Columns 7 through 10: 0.90174 0.11854 0.72313 0.73326 0.44672 0.94303 0.56564 0.82150
The default value of split_long_rows is nonzero.
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 set
the built-in variable fixed_point_format to a nonzero value. But
doing so is not recommended, because it can produce output that can
easily be misinterpreted.
octave:1> logspace (1, 7, 5)' ans = 1.0e+07 * 0.00000 0.00003 0.00100 0.03162 1.00000
Notice that first value appears to be zero when it is actually 1. For
this reason, you should be careful when setting
fixed_point_format to a nonzero value.
The default value of fixed_point_format is 0.
A matrix may have one or both dimensions zero, and operations on empty
matrices are handled as described by Carl de Boor in An Empty
Exercise, SIGNUM, Volume 25, pages 2--6, 1990 and C. N. Nett and W. M.
Haddad, in A System-Theoretic Appropriate Realization of the Empty
Matrix Concept, IEEE Transactions on Automatic Control, Volume 38,
Number 5, May 1993.
Briefly, given a scalar s, an m by
n matrix M(mxn), and an m by n empty matrix
[](mxn) (with either one or both dimensions equal to zero), the
following are true:
s * [](mxn) = [](mxn) * s = [](mxn)
[](mxn) + [](mxn) = [](mxn)
[](0xm) * M(mxn) = [](0xn)
M(mxn) * [](nx0) = [](mx0)
[](mx0) * [](0xn) = 0(mxn)
By default, dimensions of the empty matrix are printed along with the
empty matrix symbol, `[]'. The built-in variable
print_empty_dimensions controls this behavior.
print_empty_dimensions is nonzero, the
dimensions of empty matrices are printed along with the empty matrix
symbol, `[]'. For example, the expression
zeros (3, 0)
will print
ans = [](3x0)
Empty matrices may also be used in assignment statements as a convenient way to delete rows or columns of matrices. See section Assignment Expressions.
Octave will normally issue a warning if it finds an empty matrix in the
list of elements that make up another matrix. You can use the variable
empty_list_elements_ok to suppress the warning or to treat it as
an error.
For example, if the value of empty_list_elements_ok is
nonzero, Octave will ignore the empty matrices in the expression
a = [1, [], 3, [], 5]
and the variable a will be assigned the value [ 1, 3, 5 ].
The default value is "warn".
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.
propagate_empty_matrices is nonzero,
functions like inverse and svd will return an empty matrix
if they are given one as an argument. The default value is 1.
A 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 `:' 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
1 : 5
defines the set of values `[ 1, 2, 3, 4, 5 ]', and the range
1 : 3 : 5
defines the set of values `[ 1, 4 ]'.
Although a range constant specifies a row vector, Octave does not convert range constants to vectors unless it is necessary to do so. This allows you to write a constant like `1 : 10000' without using 80,000 bytes of storage on a typical 32-bit workstation.
Note that the upper (or lower, if the increment is negative) bound on
the range is not always included in the set of values, and that ranges
defined by floating point values can produce surprising results because
Octave uses floating point arithmetic to compute the values in the
range. If it is important to include the endpoints of a range and the
number of elements is known, you should use the linspace function
instead (see section Special Utility Matrices).
When Octave parses a range expression, it examines the elements of the expression to determine whether they are all constants. If they are, it replaces the range expression with a single range constant.
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