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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
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
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
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
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: round.
For a vector argument, return the maximum value. For a matrix argument, return the maximum value from each column, as a row vector, or over the dimension dim if defined, in which case y should be set to the empty matrix (it’s ignored otherwise). For two matrices (or a matrix and scalar), return the pair-wise maximum. Thus,
max (max (x))
returns the largest element of the 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 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
For a vector argument, return the minimum value. For a matrix argument, return the minimum value from each column, as a row vector, or over the dimension dim if defined, in which case y should be set to the empty matrix (it’s ignored otherwise). For two matrices (or a matrix and scalar), return the pair-wise minimum. Thus,
min (min (x))
returns the smallest element of 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 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
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
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
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 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.
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.
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: cross, divergence.
Compute the vector cross product of two 3-dimensional vectors x and y.
cross ([1,1,0], [0,1,1])
⇒ [ 1; -1; 1 ]
If x and y are matrices, the cross product is applied along the first dimension with 3 elements. The optional argument dim forces the cross product to be calculated along the specified dimension.
See also: dot, curl, divergence.
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.
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: divergence, gradient, del2, cross.
Calculate the discrete Laplace operator. For a 2-dimensional matrix M this is defined as
1 / d^2 d^2 \
D = --- * | --- 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.
At least 3 data points are needed for each dimension. Boundary points are calculated from the linear extrapolation of interior points.
Return the factorial of n where n is a positive 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.
Return the prime factorization of q. That is,
prod (p) == q and every element of p is a prime
number. If q == 1, return 1.
With two output arguments, return the unique primes p and
their multiplicities. That is, prod (p .^ n) ==
q.
Implementation Note: The input q must not be greater than
bitmax (9.0072e+15) in order to factor correctly.
Compute the greatest common divisor of a1, a2, …. If more than one argument is given 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 not unique up to units (multiplication by 1, -1, i or -i), so an arbitrary greatest common divisor amongst four possible is returned.
Example code:
gcd ([15, 9], [20, 18]) ⇒ 5 9
Optional return arguments v1, etc., contain integer vectors such that,
g = v1 .* a1 + v2 .* a2 + …
Compute the least common multiple of x and y, or of the list of all arguments. All elements must be the same size or scalar.
Truncate elements of x to a length of ndigits such that the resulting numbers are exactly divisible by base. If base is not specified it defaults to 10.
chop (-pi, 5, 10) ⇒ -3.14200000000000 chop (-pi, 5, 5) ⇒ -3.14150000000000
Return the remainder of the division x / y, 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 of the arguments is complex.
See also: mod.
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.
mod (x, 0) returns x.
An error results if the dimensions of the arguments do not agree, or if either of the arguments is complex.
See also: rem.
Return all primes up to n.
The algorithm used is the Sieve of Eratosthenes.
Note that 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, isprime.
List the first n primes. If n is unspecified, the first 25 primes are listed.
The algorithm used is from page 218 of the TeXbook.
Compute the signum function, which 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: signbit.
Return logical true if the value of x has its sign bit set. Otherwise return logical false. This behavior is consistent with the other logical functions. SeeLogical Values. The behavior differs from the C language function which returns non-zero 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: sign.
Next: Special Functions, Previous: Sums and Products, Up: Arithmetic [Contents][Index]