(octave.info)Utility Functions
17.5 Utility Functions
======================
-- : ceil (X)
Return the smallest integer not less than X.
This is equivalent to rounding towards positive infinity.
If X is complex, return ‘ceil (real (X)) + ceil (imag (X)) * I’.
ceil ([-2.7, 2.7])
⇒ -2 3
See also: Note: floor, Note: round, Note:
fix.
-- : fix (X)
Truncate fractional portion of X and return the integer portion.
This is equivalent to rounding towards zero. If X is complex,
return ‘fix (real (X)) + fix (imag (X)) * I’.
fix ([-2.7, 2.7])
⇒ -2 2
See also: Note: ceil, Note: floor, Note:
round.
-- : floor (X)
Return the largest integer not greater than X.
This is equivalent to rounding towards negative infinity. If X is
complex, return ‘floor (real (X)) + floor (imag (X)) * I’.
floor ([-2.7, 2.7])
⇒ -3 2
See also: Note: ceil, Note: round, *note fix:
XREFfix.
-- : round (X)
Return the integer nearest to X.
If X is complex, return ‘round (real (X)) + round (imag (X)) * I’.
If there are two nearest integers, return the one further away from
zero.
round ([-2.7, 2.7])
⇒ -3 3
See also: Note: ceil, Note: floor, *note fix:
XREFfix, Note: roundb.
-- : roundb (X)
Return the integer nearest to X. If there are two nearest
integers, return the even one (banker’s rounding).
If X is complex, return ‘roundb (real (X)) + roundb (imag (X)) *
I’.
See also: Note: round.
-- : max (X)
-- : max (X, [], DIM)
-- : [W, IW] = max (X)
-- : max (X, Y)
Find maximum values in the array X.
For a vector argument, return the maximum value. For a matrix
argument, return a row vector with the maximum value of each
column. For a multi-dimensional array, ‘max’ operates along the
first non-singleton dimension.
If the optional third argument DIM is present then operate along
this dimension. In this case the second argument is ignored and
should be set to the empty matrix.
For two matrices (or a matrix and a scalar), return the pairwise
maximum.
Thus,
max (max (X))
returns the largest element of the 2-D matrix X, and
max (2:5, pi)
⇒ 3.1416 3.1416 4.0000 5.0000
compares each element of the range ‘2:5’ with ‘pi’, and returns a
row vector of the maximum values.
For complex arguments, the magnitude of the elements are used for
comparison. If the magnitudes are identical, then the results are
ordered by phase angle in the range (-pi, pi]. Hence,
max ([-1 i 1 -i])
⇒ -1
because all entries have magnitude 1, but -1 has the largest phase
angle with value pi.
If called with one input and two output arguments, ‘max’ also
returns the first index of the maximum value(s). Thus,
[x, ix] = max ([1, 3, 5, 2, 5])
⇒ x = 5
ix = 3
See also: Note: min, Note: cummax, Note:
cummin.
-- : min (X)
-- : min (X, [], DIM)
-- : [W, IW] = min (X)
-- : min (X, Y)
Find minimum values in the array X.
For a vector argument, return the minimum value. For a matrix
argument, return a row vector with the minimum value of each
column. For a multi-dimensional array, ‘min’ operates along the
first non-singleton dimension.
If the optional third argument DIM is present then operate along
this dimension. In this case the second argument is ignored and
should be set to the empty matrix.
For two matrices (or a matrix and a scalar), return the pairwise
minimum.
Thus,
min (min (X))
returns the smallest element of the 2-D matrix X, and
min (2:5, pi)
⇒ 2.0000 3.0000 3.1416 3.1416
compares each element of the range ‘2:5’ with ‘pi’, and returns a
row vector of the minimum values.
For complex arguments, the magnitude of the elements are used for
comparison. If the magnitudes are identical, then the results are
ordered by phase angle in the range (-pi, pi]. Hence,
min ([-1 i 1 -i])
⇒ -i
because all entries have magnitude 1, but -i has the smallest phase
angle with value -pi/2.
If called with one input and two output arguments, ‘min’ also
returns the first index of the minimum value(s). Thus,
[x, ix] = min ([1, 3, 0, 2, 0])
⇒ x = 0
ix = 3
See also: Note: max, Note: cummin, Note:
cummax.
-- : cummax (X)
-- : cummax (X, DIM)
-- : [W, IW] = cummax (...)
Return the cumulative maximum values along dimension DIM.
If DIM is unspecified it defaults to column-wise operation. For
example:
cummax ([1 3 2 6 4 5])
⇒ 1 3 3 6 6 6
If called with two output arguments the index of the maximum value
is also returned.
[w, iw] = cummax ([1 3 2 6 4 5])
⇒
w = 1 3 3 6 6 6
iw = 1 2 2 4 4 4
See also: Note: cummin, Note: max, *note min:
XREFmin.
-- : cummin (X)
-- : cummin (X, DIM)
-- : [W, IW] = cummin (X)
Return the cumulative minimum values along dimension DIM.
If DIM is unspecified it defaults to column-wise operation. For
example:
cummin ([5 4 6 2 3 1])
⇒ 5 4 4 2 2 1
If called with two output arguments the index of the minimum value
is also returned.
[w, iw] = cummin ([5 4 6 2 3 1])
⇒
w = 5 4 4 2 2 1
iw = 1 2 2 4 4 6
See also: Note: cummax, Note: min, *note max:
XREFmax.
-- : hypot (X, Y)
-- : hypot (X, Y, Z, ...)
Compute the element-by-element square root of the sum of the
squares of X and Y.
This is equivalent to ‘sqrt (X.^2 + Y.^2)’, but is calculated in a
manner that avoids overflows for large values of X or Y.
‘hypot’ can also be called with more than 2 arguments; in this
case, the arguments are accumulated from left to right:
hypot (hypot (X, Y), Z)
hypot (hypot (hypot (X, Y), Z), W), etc.
-- : DX = gradient (M)
-- : [DX, DY, DZ, ...] = gradient (M)
-- : [...] = gradient (M, S)
-- : [...] = gradient (M, X, Y, Z, ...)
-- : [...] = gradient (F, X0)
-- : [...] = gradient (F, X0, S)
-- : [...] = gradient (F, X0, X, Y, ...)
Calculate the gradient of sampled data or a function.
If M is a vector, calculate the one-dimensional gradient of M. If
M is a matrix the gradient is calculated for each dimension.
‘[DX, DY] = gradient (M)’ calculates the one-dimensional gradient
for X and Y direction if M is a matrix. Additional return
arguments can be use for multi-dimensional matrices.
A constant spacing between two points can be provided by the S
parameter. If S is a scalar, it is assumed to be the spacing for
all dimensions. Otherwise, separate values of the spacing can be
supplied by the X, ... arguments. Scalar values specify an
equidistant spacing. Vector values for the X, ... arguments
specify the coordinate for that dimension. The length must match
their respective dimension of M.
At boundary points a linear extrapolation is applied. Interior
points are calculated with the first approximation of the numerical
gradient
y'(i) = 1/(x(i+1)-x(i-1)) * (y(i-1)-y(i+1)).
If the first argument F is a function handle, the gradient of the
function at the points in X0 is approximated using central
difference. For example, ‘gradient (@cos, 0)’ approximates the
gradient of the cosine function in the point x0 = 0. As with
sampled data, the spacing values between the points from which the
gradient is estimated can be set via the S or DX, DY, ...
arguments. By default a spacing of 1 is used.
See also: Note: diff, Note: del2.
-- : dot (X, Y, DIM)
Compute the dot product of two vectors.
If X and Y are matrices, calculate the dot products along the first
non-singleton dimension.
If the optional argument DIM is given, calculate the dot products
along this dimension.
This is equivalent to ‘sum (conj (X) .* Y, DIM)’, but avoids
forming a temporary array and is faster. When X and Y are column
vectors, the result is equivalent to ‘X' * Y’.
See also: Note: cross, Note: divergence.
-- : cross (X, Y)
-- : cross (X, Y, DIM)
Compute the vector cross product of two 3-dimensional vectors X and
Y.
If X and Y are matrices, the cross product is applied along the
first dimension with three elements.
The optional argument DIM forces the cross product to be calculated
along the specified dimension.
Example Code:
cross ([1,1,0], [0,1,1])
⇒ [ 1; -1; 1 ]
See also: Note: dot, Note: curl, Note:
divergence.
-- : DIV = divergence (X, Y, Z, FX, FY, FZ)
-- : DIV = divergence (FX, FY, FZ)
-- : DIV = divergence (X, Y, FX, FY)
-- : DIV = divergence (FX, FY)
Calculate divergence of a vector field given by the arrays FX, FY,
and FZ or FX, FY respectively.
d d d
div F(x,y,z) = -- F(x,y,z) + -- F(x,y,z) + -- F(x,y,z)
dx dy dz
The coordinates of the vector field can be given by the arguments
X, Y, Z or X, Y respectively.
See also: Note: curl, Note: gradient, Note:
del2, Note: dot.
-- : [CX, CY, CZ, V] = curl (X, Y, Z, FX, FY, FZ)
-- : [CZ, V] = curl (X, Y, FX, FY)
-- : [...] = curl (FX, FY, FZ)
-- : [...] = curl (FX, FY)
-- : V = curl (...)
Calculate curl of vector field given by the arrays FX, FY, and FZ
or FX, FY respectively.
/ d d d d d d \
curl F(x,y,z) = | -- Fz - -- Fy, -- Fx - -- Fz, -- Fy - -- Fx |
\ dy dz dz dx dx dy /
The coordinates of the vector field can be given by the arguments
X, Y, Z or X, Y respectively. V calculates the scalar component of
the angular velocity vector in direction of the z-axis for
two-dimensional input. For three-dimensional input the scalar
rotation is calculated at each grid point in direction of the
vector field at that point.
See also: Note: divergence, *note gradient:
XREFgradient, Note: del2, Note: cross.
-- : L = del2 (M)
-- : L = del2 (M, H)
-- : L = del2 (M, DX, DY, ...)
Calculate the discrete Laplace operator.
For a 2-dimensional matrix M this is defined as
1 / d^2 d^2 \
L = --- * | --- M(x,y) + --- M(x,y) |
4 \ dx^2 dy^2 /
For N-dimensional arrays the sum in parentheses is expanded to
include second derivatives over the additional higher dimensions.
The spacing between evaluation points may be defined by H, which is
a scalar defining the equidistant spacing in all dimensions.
Alternatively, the spacing in each dimension may be defined
separately by DX, DY, etc. A scalar spacing argument defines
equidistant spacing, whereas a vector argument can be used to
specify variable spacing. The length of the spacing vectors must
match the respective dimension of M. The default spacing value is
1.
Dimensions with fewer than 3 data points are skipped. Boundary
points are calculated from the linear extrapolation of interior
points.
Example: Second derivative of 2*x^3
f = @(x) 2*x.^3;
dd = @(x) 12*x;
x = 1:6;
L = 4*del2 (f(x));
assert (L, dd (x));
See also: Note: gradient, Note: diff.
-- : factorial (N)
Return the factorial of N where N is a real non-negative integer.
If N is a scalar, this is equivalent to ‘prod (1:N)’. For vector
or matrix arguments, return the factorial of each element in the
array.
For non-integers see the generalized factorial function ‘gamma’.
Note that the factorial function grows large quite quickly, and
even with double precision values overflow will occur if N > 171.
For such cases consider ‘gammaln’.
See also: Note: prod, Note: gamma, Note:
gammaln.
-- : PF = factor (Q)
-- : [PF, N] = factor (Q)
Return the prime factorization of Q.
The prime factorization is defined as ‘prod (PF) == Q’ where every
element of PF is a prime number. If ‘Q == 1’, return 1.
With two output arguments, return the unique prime factors PF and
their multiplicities. That is, ‘prod (PF .^ N) == Q’.
Implementation Note: The input Q must be less than ‘flintmax’
(9.0072e+15) in order to factor correctly.
See also: Note: gcd, Note: lcm, *note isprime:
XREFisprime, Note: primes.
-- : G = gcd (A1, A2, ...)
-- : [G, V1, ...] = gcd (A1, A2, ...)
Compute the greatest common divisor of A1, A2, ....
If more than one argument is given then all arguments must be the
same size or scalar. In this case the greatest common divisor is
calculated for each element individually. All elements must be
ordinary or Gaussian (complex) integers. Note that for Gaussian
integers, the gcd is only unique up to a phase factor
(multiplication by 1, -1, i, or -i), so an arbitrary greatest
common divisor among the four possible is returned.
Optional return arguments V1, ..., contain integer vectors such
that,
G = V1 .* A1 + V2 .* A2 + ...
Example code:
gcd ([15, 9], [20, 18])
⇒ 5 9
See also: Note: lcm, Note: factor, Note:
isprime.
-- : lcm (X, Y)
-- : lcm (X, Y, ...)
Compute the least common multiple of X and Y, or of the list of all
arguments.
All elements must be numeric and of the same size or scalar.
See also: Note: factor, Note: gcd, Note:
isprime.
-- : rem (X, Y)
Return the remainder of the division ‘X / Y’.
The remainder is computed using the expression
x - y .* fix (x ./ y)
An error message is printed if the dimensions of the arguments do
not agree, or if either argument is complex.
Programming Notes: Floating point numbers within a few eps of an
integer will be rounded to an integer before computation for
compatibility with MATLAB.
By convention,
rem (X, 0) = NaN if X is a floating point variable
rem (X, 0) = 0 if X is an integer variable
rem (X, Y) returns a value with the signbit from X
For the opposite conventions see the ‘mod’ function. In general,
‘rem’ is best when computing the remainder after division of two
_positive_ numbers. For negative numbers, or when the values are
periodic, ‘mod’ is a better choice.
See also: Note: mod.
-- : mod (X, Y)
Compute the modulo of X and Y.
Conceptually this is given by
x - y .* floor (x ./ y)
and is written such that the correct modulus is returned for
integer types. This function handles negative values correctly.
That is, ‘mod (-1, 3)’ is 2, not -1, as ‘rem (-1, 3)’ returns.
An error results if the dimensions of the arguments do not agree,
or if either of the arguments is complex.
Programming Notes: Floating point numbers within a few eps of an
integer will be rounded to an integer before computation for
compatibility with MATLAB.
By convention,
mod (X, 0) = X
mod (X, Y) returns a value with the signbit from Y
For the opposite conventions see the ‘rem’ function. In general,
‘mod’ is a better choice than ‘rem’ when any of the inputs are
negative numbers or when the values are periodic.
See also: Note: rem.
-- : P = primes (N)
Return all primes up to N.
The output data class (double, single, uint32, etc.) is the same as
the input class of N. The algorithm used is the Sieve of
Eratosthenes.
Note: If you need a specific number of primes you can use the fact
that the distance from one prime to the next is, on average,
proportional to the logarithm of the prime. Integrating, one finds
that there are about k primes less than k*log (5*k).
See also ‘list_primes’ if you need a specific number N of primes.
See also: Note: list_primes, *note isprime:
XREFisprime.
-- : list_primes ()
-- : list_primes (N)
List the first N primes.
If N is unspecified, the first 25 primes are listed.
See also: Note: primes, Note: isprime.
-- : sign (X)
Compute the “signum” function.
This is defined as
-1, x < 0;
sign (x) = 0, x = 0;
1, x > 0.
For complex arguments, ‘sign’ returns ‘x ./ abs (X)’.
Note that ‘sign (-0.0)’ is 0. Although IEEE 754 floating point
allows zero to be signed, 0.0 and -0.0 compare equal. If you must
test whether zero is signed, use the ‘signbit’ function.
See also: Note: signbit.
-- : signbit (X)
Return logical true if the value of X has its sign bit set and
false otherwise.
This behavior is consistent with the other logical functions. See
Note: Logical Values. The behavior differs from the C language
function which returns nonzero if the sign bit is set.
This is not the same as ‘x < 0.0’, because IEEE 754 floating point
allows zero to be signed. The comparison ‘-0.0 < 0.0’ is false,
but ‘signbit (-0.0)’ will return a nonzero value.
See also: Note: sign.
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