Go to the first, previous, next, last section, table of contents.


Arithmetic

Unless otherwise noted, all of the functions described in this chapter will work for real and complex scalar or matrix arguments.

Utility Functions

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.

Mapping Function: ceil (x)
Return the smallest integer not less than x. If x is complex, return ceil (real (x)) + ceil (imag (x)) * I.

Mapping Function: exp (x)
Compute the exponential of x. To compute the matrix exponential, see section Linear Algebra.

Mapping Function: fix (x)
Truncate x toward zero. If x is complex, return fix (real (x)) + fix (imag (x)) * I.

Mapping Function: floor (x)
Return the largest integer not greater than x. If x is complex, return floor (real (x)) + floor (imag (x)) * I.

Mapping Function: gcd (x, ...)
Compute the greatest common divisor of the elements of x, or the list of all the arguments. For example,
 gcd (a1, ..., ak)

is the same as

 gcd ([a1, ..., ak])

An optional second return value, v contains an integer vector such that

 g = v(1) * a(k) + ... + v(k) * a(k)

Mapping Function: lcm (x, ...)
Compute the least common multiple of the elements elements of x, or the list of all the arguments. For example,
 lcm (a1, ..., ak)

is the same as

 lcm ([a1, ..., ak]).

Mapping Function: log (x)
Compute the natural logarithm for each element of x. To compute the matrix logarithm, see section Linear Algebra.

See also: log2, log10, logspace, exp

Mapping Function: log10 (x)
Compute the base-10 logarithm for each element of x.

See also: log, log2, logspace, exp

Mapping Function: y = log2 (x)
Mapping Function: [f, e] log2 (x)
Compute the base-2 logarithm of x. With two outputs, returns f and e such that 1/2 <= abs(f) < 1 and x = f * 2^e.

max (X): maximum value(s) of a vector (matrix)

min (X): minimum value(s) of a vector (matrix)

Function File: nextpow2 (x)
If x is a scalar, returns the first integer n such that 2^n >= abs (x). If x is a vector, return nextpow2 (length (x)).

Mapping Function: pow2 (x)
Mapping Function: pow2 (f, e)
With one argument, computes 2 .^ x for each element of x. With two arguments, returns f .* (2 .^ e).

Mapping Function: rem (x, y)
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.

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

See also: rem

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).

Mapping Function: sqrt (x)
Compute the square root of x. If x is negative, a complex result is returned. To compute the matrix square root, see section Linear Algebra.

Complex Arithmetic

The following functions are available for working with complex numbers. Each expects a single argument. Given a matrix they work on an element by element basis. In the descriptions of the following functions, z is the complex number x + iy, where i is defined as sqrt (-1).

Mapping Function: abs (z)
Compute the magnitude of z, defined as |z| = sqrt (x^2 + y^2).

For example,

abs (3 + 4i)
     => 5

Mapping Function: angle (z)
Compute the argument of z, defined as theta = atan (y/x).

in radians.

For example,

arg (3 + 4i)
     => 0.92730

Mapping Function: conj (z)
Return the complex conjugate of z, defined as conj (z) = x - iy.

See also: real, imag

Mapping Function: imag (z)
Return the imaginary part of z as a real number.

See also: real, conj

Mapping Function: real (z)
Return the real part of z.

See also: imag, conj

Trigonometry

Octave provides the following trigonometric functions. Angles are specified in radians. To convert from degrees to radians multipy by pi/180 (e.g. sin (30 * pi/180) returns the sine of 30 degrees).

Mapping Function: sin (X)
sin (X): compute the sin of X for each element of X

Mapping Function: cos (X)
cos (X): compute the cosine of X for each element of X

Mapping Function: tan (z)
tan (X): compute tanget of X for each element of X

Mapping Function: sec (X)
sec (X): compute the secant of X for each element of X

Mapping Function: csc (X)
csc (X): compute the cosecant of X for each element of X

Mapping Function: cot (X)
cot (X): compute the cotangent of X for each element of X

Mapping Function: asin (X)
asin (X): compute inverse sin (X) for each element of X

Mapping Function: acos (X)
acos (X): compute the inverse cosine of X for each element of X

Mapping Function: atan (X)
atan (X): compute the inverse tangent of (X) for each element of X

Mapping Function: asec (X)
asec (X): compute the inverse secant of X for each element of X

Mapping Function: acsc (X)
acsc (X): compute the inverse cosecant of X for each element of X

Mapping Function: acot (X)
acot (X): compute the inverse cotangent of X for each element of X

Mapping Function: sinh (X)
sinh (X): compute the inverse hyperbolic sin of X for each element of X

Mapping Function: acosh (X)
acosh (X): compute the inverse hyperbolic cosine of X for each element of X

Mapping Function: tanh (X)
tanh (X): compute hyperbolic tangent of X for each element of X

Mapping Function: sech (X)
sech (X): compute the hyperbolic secant of X for each element of X

Mapping Function: coth (X)
coth (X): compute the hyperbolic cotangent of X for each element of X

Mapping Function: asinh (X)
asinh (X): compute the inverse hyperbolic sin (X) for each element of X

Mapping Function: acosh (X)
acosh (X): compute the inverse hyperbolic cosine of X for each element of X.

Mapping Function: atanh (X)
atanh (X): compute the inverse hyperbolic tanget of X for each element of X

Mapping Function: asech (X)
asech (X): compute the inverse hyperbolic secant of X for each element of X

Mapping Function: acsch (X)
acsch (X): compute the inverse hyperbolic for each element of X

acoth (z): compute the inverse hyperbolic cotangent for each element of z.

Each of these functions expect a single argument. For matrix arguments, they work on an element by element basis. For example,

sin ([1, 2; 3, 4])
     =>  0.84147   0.90930
         0.14112  -0.75680

atan2 (Y, X): atan (Y / X) in range -pi to pi

Sums and Products

sum (X): sum of elements

prod (X): products

cumsum (X): cumulative sums

cumprod (X): cumulative products

sumsq (X): sum of squares of elements.

This function is equivalent to computing

sum (X .* conj (X))

but it uses less memory and avoids calling conj if X is real.

Special Functions

Mapping Function: besseli (alpha, x)
Mapping Function: besselj (alpha, x)
Mapping Function: besselk (alpha, x)
Mapping Function: bessely (alpha, x)
Compute Bessel functions of the following types:
besselj
Bessel functions of the first kind.
bessely
Bessel functions of the second kind.
besseli
Modified Bessel functions of the first kind.
besselk
Modified Bessel functions of the second kind.

The second argument, x, must be a real matrix, vector, or scalar. The first argument, alpha, must be greater than or equal to zero. If alpha is a range, it must have an increment equal to one. If alpha is a scalar, the result is the same size as x. If alpha is a range, x must be a vector or scalar, and the result is a matrix with length(x) rows and length(alpha) columns.

Mapping Function: beta (a, b)
Return the Beta function,
 beta (a, b) = gamma (a) * gamma (b) / gamma (a + b).

Mapping Function: betai (a, b, x)
Return the incomplete Beta function,
                                     x
                                    /
 betai (a, b, x) = beta (a, b)^(-1) | t^(a-1) (1-t)^(b-1) dt.
                                    /
                                 t=0

If x has more than one component, both a and b must be scalars. If x is a scalar, a and b must be of compatible dimensions.

Mapping Function: bincoeff (n, k)
Return the binomial coefficient of n and k, defined as
  /   \
  | n |    n (n-1) (n-2) ... (n-k+1)
  |   |  = -------------------------
  | k |               k!
  \   /

For example,

 bincoeff (5, 2)
=> 10

Mapping Function: erf (z)
Computes the error function,

                         z
                        /
erf (z) = (2/sqrt (pi)) | e^(-t^2) dt
                        /
                     t=0

See also: erfc, erfinv

Mapping Function: erfc (z)
Computes the complementary error function, 1 - erf (z).

See also: erf, erfinv

Mapping Function: erfinv (z)
Computes the inverse of the error function,

Mapping Function: gamma (z)
Computes the Gamma function,

            infinity
            /
gamma (z) = | t^(z-1) exp (-t) dt.
            /
         t=0

See also: gammai, lgamma

Mapping Function: gammai (a, x)
Computes the incomplete gamma function,
                               x
                     1        /
 gammai (a, x) = ---------    | exp (-t) t^(a-1) dt
                 gamma (a)    /
                           t=0

If a is scalar, then gammai (a, x) is returned for each element of x and vice versa. If neither a nor x is scalar, the sizes of a and x must agree, and gammai is applied element-by-element.

Mapping Function: lgamma (a, x)
Mapping Function: gammaln (a, x)
Return the natural logarithm of the gamma function.

See also: gamma, gammai

Function File: cross (x, y)
Computes the vector cross product of the two 3-dimensional vectors x and y. For example,
 cross ([1,1,0], [0,1,1])
=> [ 1; -1; 1 ]

Function File: commutation_matrix (m, n)
Return the commutation matrix K(m,n) which is the unique m*n by m*n matrix such that K(m,n) * vec (A) = vec (A') for all m by n matrices A. If only one argument m is given, K(m,m) is returned. See Magnus and Neudecker (1988), Matrix differential calculus with applications in statistics and econometrics.

Function File: duplication_matrix (n)
Return the duplication matrix D_n which is the unique n^2 by n*(n+1)/2 matrix such that D_n \cdot vech (A) = vec (A) for all symmetric n by n matrices A. See Magnus and Neudecker (1988), Matrix differential calculus with applications in statistics and econometrics.

Mathematical Constants

Built-in Variable: I
Built-in Variable: J
Built-in Variable: i
Built-in Variable: j
A pure imaginary number, defined as sqrt (-1). The I and J forms are true constants, and cannot be modified. The i and j forms are like ordinary variables, and may be used for other purposes. However, unlike other variables, they once again assume their special predefined values if they are cleared See section Status of Variables.

Built-in Variable: Inf
Built-in Variable: inf
Infinity. This is the result of an operation like 1/0, or an operation that results in a floating point overflow.

Built-in Variable: NaN
Built-in Variable: nan
Not a number. This is the result of an operation like 0/0, or `Inf - Inf', or any operation with a NaN.

Note that NaN always compares not equal to NaN. This behavior is specified by the IEEE standard for floating point arithmetic. To find NaN values, you must use the isnan function.

Built-in Variable: pi
The ratio of the circumference of a circle to its diameter. Internally, pi is computed as `4.0 * atan (1.0)'.

Built-in Variable: e
The base of natural logarithms. The constant e satisfies the equation log (e) = 1.

Built-in Variable: eps
The machine precision. More precisely, eps is the largest relative spacing between any two adjacent numbers in the machine's floating point system. This number is obviously system-dependent. On machines that support 64 bit IEEE floating point arithmetic, eps is approximately 2.2204e-16.

Built-in Variable: realmax
The largest floating point number that is representable. The actual value is system-dependent. On machines that support 64 bit IEEE floating point arithmetic, realmax is approximately 1.7977e+308

Built-in Variable: realmin
The smallest floating point number that is representable. The actual value is system-dependent. On machines that support 64 bit IEEE floating point arithmetic, realmin is approximately 2.2251e-308

% DO NOT EDIT! Generated automatically by munge-texi.


Go to the first, previous, next, last section, table of contents.