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### 17.5 Utility Functions

— Mapping Function: 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: floor, round, fix.

— Function File: [cx, cy, cz, v] = curl (x, y, z, fx, fy, fz)
— Function File: [cz, v] = curl (x, y, fx, fy)
— Function File: [...] = curl (fx, fy, fz)
— Function File: [...] = curl (fx, fy)
— Function File: 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: divergence, gradient, del2, cross, dot.

— Function File: cross (x, y)
— Function File: cross (x, y, dim)

> 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.

— Function File: d = del2 (M)
— Function File: d = del2 (M, h)
— Function File: d = del2 (M, dx, dy, ...)

> 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.

> See also: gradient, diff.

— Function File: div = divergence (x, y, z, fx, fy, fz)
— Function File: div = divergence (fx, fy, fz)
— Function File: div = divergence (x, y, fx, fy)
— Function File: 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: curl, gradient, del2, cross, dot.

— Function File: p = factor (q)
— Function File: [p, n] = factor (q)

> Return 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.

> See also: gcd, lcm.

— Function File: factorial (n)

> 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`.

> See also: prod, gamma.

— Mapping Function: 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: ceil, floor, round.

— Mapping Function: 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: ceil, round, fix.

— Loadable Function: g = gcd (a1, a2, ...)
— Loadable Function: [g, v1, ...] = gcd (a1, a2, ...)

> 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. For example,

> and

>               gcd ([15, 9], [20, 18])
>                   ⇒  5  9

> Optional return arguments v1, etc., contain integer vectors such that,

>               g = v1 .* a1 + v2 .* a2 + ...

> See also: lcm, factor.

— Function File: dx = gradient (m)
— Function File: [dx, dy, dz, ...] = gradient (m)
— Function File: [...] = gradient (m, s)
— Function File: [...] = gradient (m, x, y, z, ...)
— Function File: [...] = gradient (f, x0)
— Function File: [...] = gradient (f, x0, s)
— Function File: [...] = 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: diff, del2.

— Built-in Function: hypot (x, y)
— Built-in Function: 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 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.

— Mapping Function: lcm (x, y)
— Mapping Function: lcm (x, y, ...)

> 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.

> See also: factor, gcd.

— Function File: list_primes ()
— Function File: list_primes (n)

> 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.

> See also: primes, isprime.

— Loadable Function: max (x)
— Loadable Function: max (x, y)
— Loadable Function: max (x, y, dim)
— Loadable Function: [w, iw] = max (x)

> 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 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

> See also: min, cummax, cummin.

— Loadable Function: min (x)
— Loadable Function: min (x, y)
— Loadable Function: min (x, y, dim)
— Loadable Function: [w, iw] = min (x)

> 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, 0])
>                   ⇒  x = 0
>                       ix = 3

> See also: max, cummin, cummax.

— Loadable Function: cummax (x)
— Loadable Function: cummax (x, dim)
— Loadable Function: [w, iw] = cummax (x)

> 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

> The call

>               [w, iw] = cummax (x, dim)

> with `x` a vector, is equivalent to the following code:

>               w = iw = zeros (size (x));
>               for i = 1:length (x)
>                 [w(i), iw(i)] = max (x(1:i));
>               endfor

> but computed in a much faster manner.

> See also: cummin, max, min.

— Loadable Function: cummin (x)
— Loadable Function: cummin (x, dim)
— Loadable Function: [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

> The call

>                 [w, iw] = cummin (x)

> with `x` a vector, is equivalent to the following code:

>               w = iw = zeros (size (x));
>               for i = 1:length (x)
>                 [w(i), iw(i)] = max (x(1:i));
>               endfor

> but computed in a much faster manner.

> See also: cummax, min, max.

— Mapping Function: 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. `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.

— Function File: primes (n)

> 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.

— Mapping Function: rem (x, y)
— Mapping Function: fmod (x, y)

> 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.

— Mapping Function: round (x)

> Return the integer nearest to x. If x is complex, return `round (real (`x`)) + round (imag (`x`)) * I`.

>               round ([-2.7, 2.7])
>                    ⇒ -3   3

> See also: ceil, floor, fix.

— Mapping Function: 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: round.

— Mapping Function: sign (x)

> 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`)`.