I = eye () ¶I = eye (n) ¶I = eye (m, n) ¶I = eye ([m, n]) ¶I = eye (…, class) ¶Return an identity matrix.
If called with no arguments, return the scalar value 1.
If invoked with a single scalar argument n, return a square NxN identity matrix.
If supplied two scalar arguments (m, n), or a 2-element vector
[m, n], return an MxN identity matrix with
m rows and n columns.
The optional argument class specifies the return type of the matrix and
defaults to "double".
Example 1 : 1-input, square identity matrix
eye (3)
⇒ 1 0 0
0 1 0
0 0 1
Example 2 : following expressions all produce 2x2 identity matrix
eye (2) ≡ eye (2, 2) ≡ eye (size ([1, 2; 3, 4]))
⇒ 1 0
0 1
Example 3 : 2x2 uint8 identity matrix
I = eye (2, "uint8")
Programming Note: Calling eye with no arguments is equivalent to calling
it with an argument of 1. Any negative dimensions are treated as zero.
These odd definitions are for compatibility with MATLAB.
x = ones () ¶x = ones (n) ¶x = ones (m, n, …) ¶x = ones ([m, n, …]) ¶x = ones (…, class) ¶x = ones (…, "like", var) ¶Return a scalar, matrix, or N-dimensional array whose elements are all
1.
If called with no arguments, return the scalar value 1.
If invoked with a single scalar integer argument n, return a square NxN matrix.
If invoked with two or more scalar integer arguments, or a vector of integer values, return an array with the given dimensions.
The optional argument class specifies the class of the return array
and defaults to "double".
If a variable var is specified after "like", the output val
will have the same data type, complexity, and sparsity as var.
Example 1 : MxN matrix of constant value val
C = val * ones (m, n)
Example 2 : MxN matrix of uint8
C = ones (m, n, "uint8")
Programming Note: Any negative dimensions are treated as zero, and any zero dimensions will result in an empty matrix. This odd behavior is for MATLAB compatibility.
x = zeros () ¶x = zeros (n) ¶x = zeros (m, n, …) ¶x = zeros ([m, n, …]) ¶x = zeros (…, class) ¶x = zeros (…, "like", var) ¶Return a scalar, matrix, or N-dimensional array whose elements are all
0.
If called with no arguments, return the scalar value 0.
If invoked with a single scalar integer argument n, return a square NxN matrix.
If invoked with two or more scalar integer arguments, or a vector of integer values, return an array with the given dimensions.
The optional argument class specifies the class of the return array
and defaults to "double".
If a variable var is specified after "like", the output val
will have the same data type, complexity, and sparsity as var.
Example : MxN matrix of uint8
C = ones (m, n, "uint8")
Programming Note: Any negative dimensions are treated as zero, and any zero dimensions will result in an empty matrix. This odd behavior is for MATLAB compatibility.
B = repmat (A, m) ¶B = repmat (A, m, n) ¶B = repmat (A, m, n, p …) ¶B = repmat (A, [m n]) ¶B = repmat (A, [m n p …]) ¶Repeat matrix or N-D array.
Form a block matrix of size m by n, with a copy of matrix A as each element.
If n is not specified, form an m by m block matrix. For copying along more than two dimensions, specify the number of times to copy across each dimension m, n, p, …, in a vector in the second argument.
y = repelems (x, r) ¶Construct a vector of repeated elements from x.
r is a 2xN integer matrix specifying which elements to repeat and how often to repeat each element. Entries in the first row, r(1,j), select an element to repeat. The corresponding entry in the second row, r(2,j), specifies the repeat count. If x is a matrix then the columns of x are imagined to be stacked on top of each other for purposes of the selection index. A row vector is always returned.
Conceptually the result is calculated as follows:
y = []; for i = 1:columns (r) y = [y, x(r(1,i)*ones(1, r(2,i)))]; endfor
xxx = repelem (x, R) ¶xxx = repelem (x, R_1, …, R_n) ¶Construct an array of repeated elements from x and repeat instructions R_1, ...
x must be a scalar, vector, or N-dimensional array.
A repeat instruction R_j must either be a scalar or a vector. If the
instruction is a scalar then each component of x in dimension j
is repeated R_j times. If the instruction is a vector then it must
have the same number of elements as the corresponding dimension j of
x. In this case, the kth component of dimension j is
repeated R_j(k) times.
If x is a scalar or vector then repelem may be called with just
a single repeat instruction R and repelem will return a vector
with the same orientation as the input.
If x is a matrix then at least two R_js must be specified.
Note: Using repelem with a vector x and a vector for R_j
is equivalent to Run Length Decoding.
Examples:
A = [1 2 3 4 5]; B = [2 1 0 1 2]; repelem (A, B) ⇒ 1 1 2 4 5 5
A = magic (3)
⇒ A =
8 1 6
3 5 7
4 9 2
B1 = [1 2 3];
B2 = 2;
repelem (A, B1, B2)
⇒ 8 8 1 1 6 6
3 3 5 5 7 7
3 3 5 5 7 7
4 4 9 9 2 2
4 4 9 9 2 2
4 4 9 9 2 2
More R_j may be specified than the number of dimensions of x. Any excess R_j must be scalars (because x’s size in those dimensions is only 1), and x will be replicated in those dimensions accordingly.
A = [1 2 3 4 5];
B1 = 2;
B2 = [2 1 3 0 2];
B3 = 3;
repelem (A, B1, B2, B3)
⇒ ans(:,:,1) =
1 1 2 3 3 3 5 5
1 1 2 3 3 3 5 5
ans(:,:,2) =
1 1 2 3 3 3 5 5
1 1 2 3 3 3 5 5
ans(:,:,3) =
1 1 2 3 3 3 5 5
1 1 2 3 3 3 5 5
R_j must be specified in order. A placeholder of 1 may be used for dimensions which do not need replication.
repelem ([-1, 0; 0, 1], 1, 2, 1, 2)
⇒ ans(:,:,1,1) =
-1 -1 0 0
0 0 1 1
ans(:,:,1,2) =
-1 -1 0 0
0 0 1 1
If fewer R_j are given than the number of dimensions in x,
repelem will assume R_j is 1 for those dimensions.
A = cat (3, [-1 0; 0 1], [-1 0; 0 1])
⇒ ans(:,:,1) =
-1 0
0 1
ans(:,:,2) =
-1 0
0 1
repelem (A,2,3)
⇒ ans(:,:,1) =
-1 -1 -1 0 0 0
-1 -1 -1 0 0 0
0 0 0 1 1 1
0 0 0 1 1 1
ans(:,:,2) =
-1 -1 -1 0 0 0
-1 -1 -1 0 0 0
0 0 0 1 1 1
0 0 0 1 1 1
repelem preserves the class of x, and works with strings,
cell arrays, NA, and NAN inputs. If any R_j is 0 the output will
be an empty array.
repelem ("Octave", 2, 3)
⇒ OOOccctttaaavvveee
OOOccctttaaavvveee
repelem ([1 2 3; 1 2 3], 2, 0)
⇒ [](4x0)
The functions linspace and logspace make it very easy to
create vectors with evenly or logarithmically spaced elements.
See Ranges.
y = linspace (start, end) ¶y = linspace (start, end, n) ¶Return a row vector with n linearly spaced elements between start and end.
If the number of elements n is greater than one, then the endpoints start and end are always included in the range. If start is greater than end, the elements are stored in decreasing order. If the number of points n is not specified, a value of 100 is used.
The linspace function returns a row vector when both start and
end are scalars. If one, or both, inputs are vectors, then
linspace transforms them to column vectors and returns a matrix where
each row is an independent sequence between
start(row_n), end(row_n).
Programming Notes: For compatibility with MATLAB, return the second
argument (end) when a single value (n = 1) is requested. If
n is not an integer then floor (n) is used to round the
number of elements. If n is zero or negative then an empty 1x0 matrix
is returned.
y = logspace (a, b) ¶y = logspace (a, b, n) ¶y = logspace (a, pi) ¶y = logspace (a, pi, n) ¶Return a row vector with n elements logarithmically spaced from 10^a to 10^b.
If the number of elements n is unspecified it defaults to 50.
If b is equal to pi, the points are between 10^a and pi, not 10^a and 10^pi, which is useful in digital signal processing.
Programming Notes: For compatibility with MATLAB, return the right-hand
side of the range
(10^b)
when a single value (n = 1) is requested.
If n is not an integer then floor (n) is used to round
the number of elements. If n is zero or negative then an empty 1x0
matrix is returned.
See also: linspace.
x = rand () ¶x = rand (n) ¶x = rand (m, n, …) ¶x = rand ([m, n, …]) ¶x = rand (…, class) ¶v = rand ("state") ¶("state", v) ¶("state", "reset") ¶v = rand ("seed") ¶("seed", v) ¶("seed", "reset") ¶Return a scalar, matrix, or N-dimensional array whose elements are random numbers uniformly distributed on the interval (0, 1).
If called with no arguments, return a scalar random value.
If invoked with a single scalar integer argument n, return a square NxN matrix.
If invoked with two or more scalar integer arguments, or a vector of integer values, return an array with the given dimensions.
The optional argument class specifies the class of the return array.
The only valid options are "double" (default) or "single".
Programming Note: Any negative dimensions are treated as zero, and any zero dimensions will result in an empty matrix. This odd behavior is for MATLAB compatibility.
The additional calling forms provide an interface to the underlying random number generator.
You can query the state of the random number generator using the form
v = rand ("state")
This returns a column vector v of length 625. Later, you can restore the random number generator to the state v using the form
rand ("state", v)
You may also initialize the state vector from an arbitrary vector of length ≤ 625 for v. The new state will be a hash based on the value of v, not v itself.
By default, the generator is initialized by contributing entropy from the
wall clock time, the CPU time, the current fraction of a second, the process
ID and—if available—up to 1024 bits from the C++ random number source
random_device, which might be non-deterministic (implementation
specific). Note that this differs from MATLAB, which always initializes
the random number generator to the same state at startup. To obtain behavior
comparable to MATLAB, initialize with a deterministic state vector in
Octave’s startup files (see Startup Files).
Programming Notes: To compute the pseudo-random sequence, rand uses the
Mersenne Twister with a period of 2^{19937}-1 (See
M. Matsumoto and T. Nishimura, "Mersenne Twister: A
623-dimensionally equidistributed uniform pseudorandom number generator",
ACM Trans. on Modeling and Computer Simulation,
Vol. 8, No. 1, pp. 3–30, January 1998,
http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/emt.html).
Do not use for cryptography without securely hashing several returned
values together, otherwise the generator state can be learned after reading 624
consecutive values.
Older versions of Octave used a different random number generator. The new
generator is used by default as it is significantly faster than the old
generator, and produces random numbers with a significantly longer cycle time.
However, in some circumstances it might be desirable to obtain the same random
sequences as produced by the old generators. To do this the keyword
"seed" is used to specify that the old generators should be used, as in
rand ("seed", val)
which sets the seed of the generator to val. The seed of the generator can be queried with
s = rand ("seed")
However, it should be noted that querying the seed will not cause rand
to use the old generators, only setting the seed will. To cause rand to
once again use the new generators, the keyword "state" must be used
to reset rand back to using the default generator.
The state or seed of the generator can be reset to a new random value using
the "reset" keyword.
R = randi (imax) ¶R = randi (imax, n) ¶R = randi (imax, m, n, …) ¶R = randi (imax, [m, n, …]) ¶R = randi ([imin, imax], …) ¶R = randi (…, "class") ¶Return a scalar, matrix, or N-dimensional array whose elements are random integers in the range [1, imax].
If called with no size arguments, return a scalar random value.
If invoked with a single scalar size argument n, return a square NxN matrix.
If invoked with two or more scalar integer size arguments, or a vector of integer values, return an array with the given dimensions.
The integer range may optionally be described by a two-element matrix with a lower and upper bound in which case the returned integers will be on the interval [imin, imax].
The optional argument class will return a matrix of the requested type.
The default is "double". Supported classes are: "double",
"single", "int8", "int16", "int32",
"uint8", "uint16", "uint32", and "logical".
Programming Notes: randi relies internally on rand which uses
class "double" to represent numbers. This limits the maximum integer
(imax) and range (imax - imin) to the value returned by the
flintmax function. For IEEE 754 floating point numbers this value
is 2^{53} - 1.
When the output class is "logical" the function constructs an array of
random integers as specified and then applies the test x > 0
to determine which elements will be true. Because the one-input form of the
function uses 1 for imin randi will always return a matrix
of all ones.
Example: 150 integers in the range 1–10.
ri = randi (10, 150, 1)
x = randn () ¶x = randn (n) ¶x = randn (m, n, …) ¶x = randn ([m, n, …]) ¶x = randn (…, class) ¶v = randn ("state") ¶("state", v) ¶("state", "reset") ¶v = randn ("seed") ¶("seed", v) ¶("seed", "reset") ¶Return a scalar, matrix, or N-dimensional array whose elements are random
numbers from the standard normal distribution having a mean of 0 and a
variance of 1.
If called with no arguments, return a scalar random value.
If invoked with a single scalar integer argument n, return a square NxN matrix.
If invoked with two or more scalar integer arguments, or a vector of integer values, return an array with the given dimensions.
The optional argument class specifies the class of the return array.
The only valid options are "double" (default) or "single".
Programming Note: Any negative dimensions are treated as zero, and any zero dimensions will result in an empty matrix. This odd behavior is for MATLAB compatibility.
The additional calling forms provide an interface to the underlying random
number generator. See rand for documentation on
querying and controlling the random number generator.
Programming Note: By default, randn uses the
Marsaglia and Tsang “Ziggurat technique” to transform from a
uniform to a normal distribution.
Reference: G. Marsaglia and W.W. Tsang, "Ziggurat Method for Generating Random Variables", J. Statistical Software, Vol. 5, 2000, https://www.jstatsoft.org/v05/i08/
x = rande () ¶x = rande (n) ¶x = rande (m, n, …) ¶x = rande ([m, n, …]) ¶x = rande (…, class) ¶v = rande ("state") ¶("state", v) ¶("state", "reset") ¶v = rande ("seed") ¶("seed", v) ¶("seed", "reset") ¶Return a scalar, matrix, or N-dimensional array whose elements are random
numbers from the exponential distribution with rate parameter 0.
If called with no arguments, return a scalar random value.
If invoked with a single scalar integer argument n, return a square NxN matrix.
If invoked with two or more scalar integer arguments, or a vector of integer values, return an array with the given dimensions.
The optional argument class specifies the class of the return array.
The only valid options are "double" (default) or "single".
Programming Note: Any negative dimensions are treated as zero, and any zero dimensions will result in an empty matrix. This odd behavior is for MATLAB compatibility.
The additional calling forms provide an interface to the underlying random
number generator. See rand for documentation on
querying and controlling the random number generator.
Programming Note: By default, rande uses the
Marsaglia and Tsang “Ziggurat technique” to transform from a
uniform to an exponential distribution.
Reference: G. Marsaglia and W.W. Tsang, "Ziggurat Method for Generating Random Variables", J. Statistical Software, Vol 5, 2000, https://www.jstatsoft.org/v05/i08/
x = randp (l) ¶x = randp (l, n) ¶x = randp (l, m, n, …) ¶x = randp (l, [m, n, …]) ¶x = randp (…, class) ¶v = randp ("state") ¶("state", v) ¶("state", "reset") ¶v = randp ("seed") ¶("seed", v) ¶("seed", "reset") ¶Return a scalar, matrix, or N-dimensional array whose elements are random numbers from the Poisson distribution with mean value parameter l.
If called with no size arguments, return a scalar random value.
If invoked with a single scalar size argument n, return a square NxN matrix.
If invoked with two or more scalar integer size arguments, or a vector of integer values, return an array with the given dimensions.
The optional argument class specifies the class of the return array.
The only valid options are "double" (default) or "single".
Programming Note: Any negative dimensions are treated as zero, and any zero dimensions will result in an empty matrix. This odd behavior is for MATLAB compatibility.
The additional calling forms provide an interface to the underlying random
number generator. See rand for documentation on
querying and controlling the random number generator.
Programming Notes: Five different algorithms are used depending on the range of l and whether l is a scalar or a matrix.
W.H. Press, et al., Numerical Recipes in C, Cambridge University Press, 1992.
W.H. Press, et al., Numerical Recipes in C, Cambridge University Press, 1992.
E. Stadlober, et al., WinRand source code, available via FTP.
E. Stadlober, et al., WinRand source code, available via FTP, or H. Zechner, Efficient sampling from continuous and discrete unimodal distributions, Doctoral Dissertation, pp. 156, Technical University Graz, Austria, 1994.
L. Montanet, et al., "Review of Particle Properties", Physical Review D, 50, p. 1284, 1994.
x = randg (a) ¶x = randg (a, n) ¶x = randg (a, m, n, …) ¶x = randg (a, [m, n, …]) ¶x = randg (…, class) ¶v = randg ("state") ¶("state", v) ¶("state", "reset") ¶v = randg ("seed") ¶("seed", v) ¶("seed", "reset") ¶Return a scalar, matrix, or N-dimensional array whose elements are random
numbers from the gamma distribution gamma (a,1).
If called with no size arguments, return a scalar random value.
If invoked with a single scalar size argument n, return a square NxN matrix.
If invoked with two or more scalar integer size arguments, or a vector of integer values, return an array with the given dimensions.
The optional argument class specifies the class of the return array.
The only valid options are "double" (default) or "single".
Programming Note: Any negative dimensions are treated as zero, and any zero dimensions will result in an empty matrix. This odd behavior is for MATLAB compatibility.
The additional calling forms provide an interface to the underlying random
number generator. See rand for documentation on
querying and controlling the random number generator.
Programming Notes: The gamma distribution can be used to generate many other distributions:
gamma (a, b) for a > -1, b > 0r = b * randg (a)
beta (a, b) for a > -1, b > -1r1 = randg (a, 1) r = r1 / (r1 + randg (b, 1))
Erlang (a, n)r = a * randg (n)
chisq (df) for df > 0r = 2 * randg (df / 2)
t (df) for 0 < df < inf (use randn if df is infinite)r = randn () / sqrt (2 * randg (df / 2) / df)
F (n1, n2) for 0 < n1, 0 < n2## r1 equals 1 if n1 is infinite r1 = 2 * randg (n1 / 2) / n1 ## r2 equals 1 if n2 is infinite r2 = 2 * randg (n2 / 2) / n2 r = r1 / r2
binomial (n, p) for n > 0, 0 < p <= 1r = randp ((1 - p) / p * randg (n))
chisq (df, L), for df >= 0 and L > 0(use chisq if L = 0)
r = randp (L / 2) r(r > 0) = 2 * randg (r(r > 0)) r(df > 0) += 2 * randg (df(df > 0)/2)
Dirichlet (a1, … ak)r = (randg (a1), ..., randg (ak)) r = r / sum (r)
(seed) ¶(seed, "generator") ¶("shuffle") ¶("shuffle", "generator") ¶("default") ¶s = rng () ¶(s) ¶s = rng (…) ¶Set or query the seed of the random number generator used by rand and
randn.
The input seed is a scalar numeric value used to initialize the state vector of the random number generator.
The optional string generator specifies the type of random number
generator to be used. Its value can be "twister",
"v5uniform", or "v5normal". The "twister" keyword
is described below. "v5uniform" and "v5normal" refer to
older versions of Octave that used to use a different random number
generator.
The state or seed of the random number generator can be reset to a new
random value using the "shuffle" keyword.
The random number generator can be reset to default values using the
"default" keyword. The default values are to use the Mersenne
Twister generator with a seed of 0.
The optional return value s contains the state of the random number
generator at the time the function is called (i.e., before it might be
modified according to the input arguments). It is encoded as a structure
variable with three fields: "Type", "Seed", and
"State". The random number generator can be restored to the state
s using rng (s). This is useful when the identical
sequence of pseudo-random numbers is required for an algorithm.
By default, and with the "twister" option, pseudo-random sequences
are computed using the Mersenne Twister with a period of 2^{19937}-1
(See M. Matsumoto and T. Nishimura,
"Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom
number generator",
ACM Trans. on Modeling and Computer Simulation,
Vol. 8, No. 1, pp. 3–30, January 1998,
http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/emt.html).
Do not use for cryptography without securely hashing several
returned values together, otherwise the generator state can be learned after
reading 624 consecutive values.
The generators operate in the new or old style together, it is not
possible to mix the two. Initializing any generator with
"state" or "seed" causes the others to switch to the
same style for future calls.
The state of each generator is independent and calls to different generators can be interleaved without affecting the final result. For example,
rand ("state", [11, 22, 33]);
randn ("state", [44, 55, 66]);
u = rand (100, 1);
n = randn (100, 1);
and
rand ("state", [11, 22, 33]);
randn ("state", [44, 55, 66]);
u = zeros (100, 1);
n = zeros (100, 1);
for i = 1:100
u(i) = rand ();
n(i) = randn ();
end
produce equivalent results. When the generators are initialized in
the old style with "seed" only rand and randn are
independent, because the old rande, randg and
randp generators make calls to rand and randn.
The generators are initialized with random states at start-up, so that the sequences of random numbers are not the same each time you run Octave.7 If you really do need to reproduce a sequence of numbers exactly, you can set the state or seed to a specific value.
If invoked without arguments, rand and randn return a
single element of a random sequence.
The original 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.
v = randperm (n) ¶v = randperm (n, m) ¶Return a row vector containing a random permutation of 1:n.
If m is supplied, return m unique entries, sampled without
replacement from 1:n.
The complexity is O(n) in memory and O(m) in time, unless m < n/5, in which case O(m) memory is used as well. The randomization is performed using rand(). All permutations are equally likely.
See also: perms.