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The following functions are available for working with complex numbers. Each expects a single argument. They are called mapping functions because when given a matrix argument, they apply the given function to each element of the matrix.
Return the smallest integer not less than x. If x is
complex, return ceil (real (x)) + ceil (imag (x)) * I.
Sort the numbers z into complex conjugate pairs ordered by
increasing real part. With identical real parts, order by increasing
imaginary magnitude. Place the negative imaginary complex number
first within each pair. Place all the real numbers after all the
complex pairs (those with abs (imag (z) / z) <
tol), where the default value of tol is 100 *
eps.
By default the complex pairs are sorted along the first non-singleton dimension of z. If dim is specified, then the complex pairs are sorted along this dimension.
Signal an error if some complex numbers could not be paired. Requires all complex numbers to be exact conjugates within tol, or signals an error. Note that there are no guarantees on the order of the returned pairs with identical real parts but differing imaginary parts.
cplxpair (exp(2i*pi*[0:4]'/5)) == exp(2i*pi*[3; 2; 4; 1; 0]/5) |
Calculates the discrete Laplace operator. If m is a matrix this is defined as
1 / d^2 d^2 \
D = --- * | --- M(x,y) + --- M(x,y) |
4 \ dx^2 dy^2 /
|
The above to continued to N-dimensional arrays calculating the second derivative over the higher dimensions.
The spacing between evaluation points may be defined by h, which is a scalar defining the spacing in all dimensions. Or alternative, the spacing in each dimension may be defined separately by dx, dy, etc. Scalar spacing values give equidistant spacing, whereas vector spacing values can be used to specify variable spacing. The length of the vectors must match the respective dimension of m. The default spacing value is 1.
You need at least 3 data points for each dimension. Boundary points are calculated as the linear extrapolation of the interior points.
See also: gradient, diff.
Compute the exponential of x. To compute the matrix exponential, see Linear Algebra.
Return prime factorization of q. That is prod (p)
== q. If q == 1, returns 1.
With two output arguments, returns the unique primes p and
their multiplicities. That is prod (p .^ n) ==
q.
Return the factorial of n. If n is scalar, this is
equivalent to prod (1:n). If n is an array,
the factorial of the elements of the array are returned.
Truncate x toward zero. If x is complex, return
fix (real (x)) + fix (imag (x)) * I.
Return the largest integer not greater than x. If x is
complex, return floor (real (x)) + floor (imag (x)) * I.
Compute the floating point remainder of dividing x by y
using the C library function fmod. The result has the same
sign as x. If y is zero, the result implementation-defined.
If a single argument is given then compute the greatest common divisor of the elements of this argument. Otherwise 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 element individually. All elements must be integers. For example,
gcd ([15, 20])
⇒ 5
|
and
gcd ([15, 9], [20 18])
⇒ 5 9
|
Optional return arguments v1, etc, contain integer vectors such that,
For backward compatibility with previous versions of this function, when
all arguments are scalar, a single return argument v1 containing
all of the values of v1, … is acceptable.
See also: lcm, min, max, ceil, floor.
Calculates the gradient. x = gradient (M)
calculates the one dimensional gradient if M is a vector. If
M is a matrix the gradient is calculated for each row.
[x, y] = gradient (M) calculates the one
dimensional gradient for each direction if M if M is a
matrix. Additional return arguments can be use for multi-dimensional
matrices.
Spacing values between two points can be provided by the dx, dy or h parameters. If h is supplied it is assumed to be the spacing in all directions. Otherwise, separate values of the spacing can be supplied by the dx, etc variables. A scalar value specifies an equidistant spacing, while a vector value can be used to specify a variable spacing. 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)). |
Compute the least common multiple of the elements of x, or the list of all the arguments. For example,
lcm (a1, ..., ak) |
is the same as
lcm ([a1, ..., ak]). |
All elements must be the same size or scalar.
See also: gcd, min, max, ceil, floor.
Compute the natural logarithm for each element of x. To compute the
matrix logarithm, see Linear Algebra.
See also: log2, log10, logspace, exp.
Compute the base-10 logarithm for each element of x.
See also: log, log2, logspace, exp.
Compute the base-2 logarithm of x. With two outputs, returns
f and e such that
See also: log, log10, logspace, exp.
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. For two matrices (or a matrix and scalar), return the pair-wise maximum. Thus,
max (max (x)) |
returns the largest element of 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. 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, 5])
⇒ x = 0
ix = 3
|
Compute modulo function, using
x - y .* floor (x ./ y) |
Note that this handles negative numbers correctly:
mod (-1, 3) is 2, not -1 as rem (-1, 3) returns.
Also, mod (x, 0) returns x.
An error message is printed if the dimensions of the arguments do not
agree, or if either of the arguments is complex.
See also: rem, round.
If x is a scalar, returns the first integer n such that
If x is a vector, return nextpow2 (length (x)).
See also: pow2.
Compute the nth root of x, returning real results for real components of x. For example
nthroot (-1, 3) ⇒ -1 (-1) ^ (1 / 3) ⇒ 0.50000 - 0.86603i |
With one argument, computes
for each element of x. With two arguments, returns
See also: nextpow2.
Return all primes up to n.
Note that if you need a specific number of primes, you can use the fact the distance from one prime to the next is on average proportional to the logarithm of the prime. Integrating, you find that there are about k primes less than k \log ( 5 k ).
The algorithm used is called the Sieve of Erastothenes.
Return the remainder of 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, round.
Return the integer nearest to x. If x is complex, return
round (real (x)) + round (imag (x)) * I.
See also: rem.
Compute the signum function, which is defined as
For complex arguments, sign returns x ./ abs (x).
Compute the square root of x. If x is negative, a complex result is returned. To compute the matrix square root, see Linear Algebra.
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