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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

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

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

Mapping Function: 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

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.

Built-in Function: max (x)
Built-in Function: max (x, [], dim)
Built-in Function: [w, iw] = max (x)
Built-in Function: 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

Built-in Function: min (x)
Built-in Function: min (x, [], dim)
Built-in Function: [w, iw] = min (x)
Built-in Function: 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

Built-in Function: cummax (x)
Built-in Function: cummax (x, dim)
Built-in Function: [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

Built-in Function: cummin (x)
Built-in Function: cummin (x, dim)
Built-in 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

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

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 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.
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.

Built-in Function: 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.

Function File: cross (x, y)
Function File: 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 ]

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.

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.

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.

Function File: 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.

Function File: pf = factor (q)
Function File: [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 bitmax (9.0072e+15) in order to factor correctly.

Built-in Function: g = gcd (a1, a2, …)
Built-in Function: [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

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 numeric and of the same size or scalar.

Function File: chop (x, ndigits, base)

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
Mapping Function: 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 of the arguments is complex.

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.

Function File: 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.

Notes: 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.

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

List the first n primes.

If n is unspecified, the first 25 primes are listed.

Mapping Function: 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.