There are a number of functions available for checking to see if the elements of a matrix meet some condition, and for rearranging the elements of a matrix. For example, Octave can easily tell you if all the elements of a matrix are finite, or are less than some specified value. Octave can also rotate the elements, extract the upper- or lower-triangular parts, or sort the columns of a matrix.
The functions any
and all
are useful for determining
whether any or all of the elements of a matrix satisfy some condition.
The find
function is also useful in determining which elements of
a matrix meet a specified condition.
For a matrix argument, return a row vector of ones and zeros with each element indicating whether any of the elements of the corresponding column of the matrix are nonzero. For example,
any (eye (2, 4)) => [ 1, 1, 0, 0 ]
To see if any of the elements of a matrix are nonzero, you can use a statement like
any (any (a))
all
behaves like the function any
, except
that it returns true only if all the elements of a vector, or all the
elements in a column of a matrix, are nonzero.
Since the comparison operators (see section Comparison Operators) return matrices of ones and zeros, it is easy to test a matrix for many things, not just whether the elements are nonzero. For example,
all (all (rand (5) < 0.9)) => 0
tests a random 5 by 5 matrix to see if all of it's elements are less than 0.9.
Note that in conditional contexts (like the test clause of if
and
while
statements) Octave treats the test as if you had typed
all (all (condition))
.
[errorcode, a, b] = common_size ([1 2; 3 4], 5) => errorcode = 0 => a = [ 1, 2; 3, 4 ] => b = [ 5, 5; 5, 5 ]
This is useful for implementing functions where arguments can either be scalars or of common size.
diff (x)
is the
vector of first differences
If x is a matrix, diff (x)
is the matrix of column
differences.
The second argument is optional. If supplied, diff (x,
k)
, where k is a nonnegative integer, returns the
k-th differences.
isinf ([13, Inf, NaN]) => [ 0, 1, 0 ]
isnan ([13, Inf, NaN]) => [ 0, 0, 1 ]
finite ([13, Inf, NaN]) => [ 1, 0, 0 ]
find (eye (2)) => [ 1; 4 ]
If two outputs are requested, find
returns the row and column
indices of nonzero elements of a matrix. For example,
[i, j] = find (2 * eye (2)) => i = [ 1; 2 ] => j = [ 1; 2 ]
If three outputs are requested, find
also returns a vector
containing the nonzero values. For example,
[i, j, v] = find (3 * eye (2)) => i = [ 1; 2 ] => j = [ 1; 2 ] => v = [ 3; 3 ]
fliplr ([1, 2; 3, 4]) => 2 1 4 3
flipud ([1, 2; 3, 4]) => 3 4 1 2
rot90 ([1, 2; 3, 4], -1) => 3 1 4 2
rotates the given matrix clockwise by 90 degrees. The following are all equivalent statements:
rot90 ([1, 2; 3, 4], -1) == rot90 ([1, 2; 3, 4], 3) == rot90 ([1, 2; 3, 4], 7)
For example,
reshape ([1, 2, 3, 4], 2, 2) => 1 3 2 4
If the variable do_fortran_indexing
is nonzero, the
reshape
function is equivalent to
retval = zeros (m, n); retval (:) = a;
but it is somewhat less cryptic to use reshape
instead of the
colon operator. Note that the total number of elements in the original
matrix must match the total number of elements in the new matrix.
If x is a matrix, do the same for each column of x.
sort
orders the elements in each
column.
For example,
sort ([1, 2; 2, 3; 3, 1]) => 1 1 2 2 3 3
The sort
function may also be used to produce a matrix
containing the original row indices of the elements in the sorted
matrix. For example,
[s, i] = sort ([1, 2; 2, 3; 3, 1]) => s = 1 1 2 2 3 3 => i = 1 3 2 1 3 2
Since the sort
function does not allow sort keys to be specified,
it can't be used to order the rows of a matrix according to the values
of the elements in various columns(6)
in a single call. Using the second output, however, it is possible to
sort all rows based on the values in a given column. Here's an example
that sorts the rows of a matrix based on the values in the second
column.
a = [1, 2; 2, 3; 3, 1]; [s, i] = sort (a (:, 2)); a (i, :) => 3 1 1 2 2 3
tril
)
or upper (triu
) triangular part of the matrix a, and
setting all other elements to zero. The second argument is optional,
and specifies how many diagonals above or below the main diagonal should
also be set to zero.
The default value of k is zero, so that triu
and
tril
normally include the main diagonal as part of the result
matrix.
If the value of k is negative, additional elements above (for
tril
) or below (for triu
) the main diagonal are also
selected.
The absolute value of k must not be greater than the number of sub- or super-diagonals.
For example,
tril (ones (3), -1) => 0 0 0 1 0 0 1 1 0
and
tril (ones (3), 1) => 1 1 0 1 1 1 1 1 1
eye
returns a square matrix with the dimension specified. If you
supply two scalar arguments, eye
takes them to be the number of
rows and columns. If given a vector with two elements, eye
uses
the values of the elements as the number of rows and columns,
respectively. For example,
eye (3) => 1 0 0 0 1 0 0 0 1
The following expressions all produce the same result:
eye (2) == eye (2, 2) == eye (size ([1, 2; 3, 4])
For compatibility with MATLAB, calling eye
with no arguments
is equivalent to calling it with an argument of 1.
eye
.
If you need to create a matrix whose values are all the same, you should use an expression like
val_matrix = val * ones (n, m)
eye
.
"seed"
, x)
eye
. In
addition, you can set the seed for the random number generator using the
form
rand ("seed", x)
where x is a scalar value. If called as
rand ("seed")
rand
returns the current value of the seed.
"seed"
, x)
eye
. In
addition, you can set the seed for the random number generator using the
form
randn ("seed", x)
where x is a scalar value. If called as
randn ("seed")
randn
returns the current value of the seed.
The rand
and randn
functions use separate generators.
This ensures that
rand ("seed", 13); randn ("seed", 13); u = rand (100, 1); n = randn (100, 1);
and
rand ("seed", 13); randn ("seed", 13); u = zeros (100, 1); n = zeros (100, 1); for i = 1:100 u(i) = rand (); n(i) = randn (); end
produce equivalent results.
Normally, rand
and randn
obtain their initial
seeds from the system clock, so that the sequence of random numbers is
not the same each time you run Octave. If you really do need for to
reproduce a sequence of numbers exactly, you can set the seed to a
specific value.
If it is invoked without arguments, rand
and randn
return a
single element of a random sequence.
The rand
and randn
functions use Fortran code from
RANLIB, a library of fortran routines for random number generation,
compiled by Barry W. Brown and James Lovato of the Department of
Biomathematics at The University of Texas, M.D. Anderson Cancer Center,
Houston, TX 77030.
diag ([1, 2, 3], 1) => 0 1 0 0 0 0 2 0 0 0 0 3 0 0 0 0
The functions linspace
and logspace
make it very easy to
create vectors with evenly or logarithmically spaced elements.
See section Ranges.
The linspace
function always returns a row vector, regardless of
the value of prefer_column_vectors
.
linspace
except that the values are logarithmically
spaced from
If limit is equal to the points are between not in order to be compatible with the corresponding MATLAB function.
treat_neg_dim_as_zero
is nonzero, expressions
like
eye (-1)
produce an empty matrix (i.e., row and column dimensions are zero). Otherwise, an error message is printed and control is returned to the top level. The default value is 0.
The following functions return famous matrix forms.
A Hankel matrix formed from an m-vector c, and an n-vector r, has the elements
inverse (hilb (n))
,
which suffers from the ill-conditioning of the Hilbert matrix, and the
finite precision of your computer's floating point arithmetic.
A square Toeplitz matrix has the form
A Vandermonde matrix has the form
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