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<H1><A NAME="SEC136" HREF="octave_toc.html#TOC136">Arithmetic</A></H1>

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

</P>



<H2><A NAME="SEC137" HREF="octave_toc.html#TOC137">Utility Functions</A></H2>

<P>
The following functions are available for working with complex numbers.
Each expects a single argument.  They are called <STRONG>mapping functions</STRONG>
because when given a matrix argument, they apply the given function to
each element of the matrix.

</P>
<P>
<DL>
<DT><U>Mapping Function:</U>  <B>ceil</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX614"></A>
Return the smallest integer not less than <VAR>x</VAR>.  If <VAR>x</VAR> is
complex, return <CODE>ceil (real (<VAR>x</VAR>)) + ceil (imag (<VAR>x</VAR>)) * I</CODE>.
</DL>

</P>
<P>
<DL>
<DT><U>Mapping Function:</U>  <B>exp</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX615"></A>
Compute the exponential of <VAR>x</VAR>.  To compute the matrix exponential,
see section <A HREF="octave_18.html#SEC143">Linear Algebra</A>.
</DL>

</P>
<P>
<DL>
<DT><U>Mapping Function:</U>  <B>fix</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX616"></A>
Truncate <VAR>x</VAR> toward zero.  If <VAR>x</VAR> is complex, return
<CODE>fix (real (<VAR>x</VAR>)) + fix (imag (<VAR>x</VAR>)) * I</CODE>.
</DL>

</P>
<P>
<DL>
<DT><U>Mapping Function:</U>  <B>floor</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX617"></A>
Return the largest integer not greater than <VAR>x</VAR>.  If <VAR>x</VAR> is
complex, return <CODE>floor (real (<VAR>x</VAR>)) + floor (imag (<VAR>x</VAR>)) * I</CODE>.
</DL>

</P>
<P>
<DL>
<DT><U>Mapping Function:</U>  <B>gcd</B> <I>(<VAR>x</VAR>, <CODE>...</CODE>)</I>
<DD><A NAME="IDX618"></A>
Compute the greatest common divisor of the elements of <VAR>x</VAR>, or the
list of all the arguments.  For example, 

</P>

<PRE>
gcd (a1, ..., ak)
</PRE>

<P>
is the same as

</P>

<PRE>
gcd ([a1, ..., ak])
</PRE>

<P>
An optional second return value, <VAR>v</VAR>
contains an integer vector such that

</P>

<PRE>
g = v(1) * a(k) + ... + v(k) * a(k)
</PRE>

</DL>

<P>
<DL>
<DT><U>Mapping Function:</U>  <B>lcm</B> <I>(<VAR>x</VAR>, <CODE>...</CODE>)</I>
<DD><A NAME="IDX619"></A>
Compute the least common multiple of the elements elements of <VAR>x</VAR>, or
the list of all the arguments.  For example, 

</P>

<PRE>
lcm (a1, ..., ak)
</PRE>

<P>
is the same as

</P>

<PRE>
lcm ([a1, ..., ak]).
</PRE>

</DL>

<P>
<DL>
<DT><U>Mapping Function:</U>  <B>log</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX620"></A>
Compute the natural logarithm of <VAR>x</VAR>.  To compute the matrix logarithm, 
see section <A HREF="octave_18.html#SEC143">Linear Algebra</A>.
</DL>

</P>
<P>
<DL>
<DT><U>Mapping Function:</U>  <B>log10</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX621"></A>
Compute the base-10 logarithm of <VAR>x</VAR>.
</DL>

</P>
<P>
<DL>
<DT><U>Mapping Function:</U> <VAR>y</VAR> = <B>log2</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX622"></A>
<DT><U>Mapping Function:</U> [<VAR>f</VAR>, <VAR>e</VAR>] <B>log2</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX623"></A>
Compute the base-2 logarithm of <VAR>x</VAR>.  With two outputs, returns
<VAR>f</VAR> and <VAR>e</VAR> such that
 1/2 &#60;= abs(f) &#60; 1 and x = f * 2^e.
</DL>

</P>
<P>
<DL>
<DT><U>Loadable Function:</U>  <B>max</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX624"></A>
For a vector argument, return the maximum value.  For a matrix argument,
return the maximum value from each column, as a row vector.  Thus,

</P>

<PRE>
max (max (<VAR>x</VAR>))
</PRE>

<P>
returns the largest element of <VAR>x</VAR>.

</P>
<P>
For complex arguments, the magnitude of the elements are used for
comparison.
</DL>

</P>
<P>
<DL>
<DT><U>Loadable Function:</U>  <B>min</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX625"></A>
Like <CODE>max</CODE>, but return the minimum value.
</DL>

</P>
<P>
<DL>
<DT><U>Function File:</U>  <B>nextpow2</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX626"></A>
If <VAR>x</VAR> is a scalar, returns the first integer <VAR>n</VAR> such that
 2^n &#62;= abs (x).

</P>
<P>
If <VAR>x</VAR> is a vector, return <CODE>nextpow2 (length (<VAR>x</VAR>))</CODE>.
</DL>

</P>
<P>
<DL>
<DT><U>Mapping Function:</U>  <B>pow2</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX627"></A>
<DT><U>Mapping Function:</U>  <B>pow2</B> <I>(<VAR>f</VAR>, <VAR>e</VAR>)</I>
<DD><A NAME="IDX628"></A>
With one argument, computes
 2 .^ x
for each element of <VAR>x</VAR>.  With two arguments, returns
 f .* (2 .^ e).
</DL>

</P>
<P>
<DL>
<DT><U>Mapping Function:</U>  <B>rem</B> <I>(<VAR>x</VAR>, <VAR>y</VAR>)</I>
<DD><A NAME="IDX629"></A>
Return the remainder of <CODE><VAR>x</VAR> / <VAR>y</VAR></CODE>, computed using the
expression

</P>

<PRE>
x - y .* fix (x ./ y)
</PRE>

<P>
An error message is printed if the dimensions of the arguments do not
agree, or if either of the arguments is complex.
</DL>

</P>
<P>
<DL>
<DT><U>Mapping Function:</U>  <B>round</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX630"></A>
Return the integer nearest to <VAR>x</VAR>.  If <VAR>x</VAR> is complex, return
<CODE>round (real (<VAR>x</VAR>)) + round (imag (<VAR>x</VAR>)) * I</CODE>.
</DL>

</P>
<P>
<DL>
<DT><U>Mapping Function:</U>  <B>sign</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX631"></A>
Compute the <STRONG>signum</STRONG> function, which is defined as

</P>

<PRE>
           -1, x &#60; 0;
sign (x) =  0, x = 0;
            1, x &#62; 0.
</PRE>

<P>
For complex arguments, <CODE>sign</CODE> returns <CODE>x ./ abs (<VAR>x</VAR>)</CODE>.
</DL>

</P>
<P>
<DL>
<DT><U>Mapping Function:</U>  <B>sqrt</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX632"></A>
Compute the square root of <VAR>x</VAR>.  If <VAR>x</VAR> is negative, a complex
result is returned.  To compute the matrix square root, see
section <A HREF="octave_18.html#SEC143">Linear Algebra</A>.
</DL>

</P>
<P>
<DL>
<DT><U>Mapping Function:</U>  <B>xor</B> <I>(<VAR>x</VAR>, <VAR>y</VAR>)</I>
<DD><A NAME="IDX633"></A>
Return the `exclusive or' of the entries of <VAR>x</VAR> and <VAR>y</VAR>.
For boolean expressions <VAR>x</VAR> and <VAR>y</VAR>,
<CODE>xor (<VAR>x</VAR>, <VAR>y</VAR>)</CODE> is true if and only if <VAR>x</VAR> or <VAR>y</VAR>
is true, but not if both <VAR>x</VAR> and <VAR>y</VAR> are true.
</DL>

</P>


<H2><A NAME="SEC138" HREF="octave_toc.html#TOC138">Complex Arithmetic</A></H2>

<P>
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,
<VAR>z</VAR> is the complex number <VAR>x</VAR> + <VAR>i</VAR><VAR>y</VAR>, where <VAR>i</VAR> is
defined as <CODE>sqrt (-1)</CODE>.

</P>
<P>
<DL>
<DT><U>Mapping Function:</U>  <B>abs</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX634"></A>
Compute the magnitude of <VAR>z</VAR>, defined as
|<VAR>z</VAR>| = <CODE>sqrt (x^2 + y^2)</CODE>.

</P>
<P>
For example,

</P>

<PRE>
abs (3 + 4i)
     => 5
</PRE>

</DL>

<P>
<DL>
<DT><U>Mapping Function:</U>  <B>arg</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX635"></A>
<DT><U>Mapping Function:</U>  <B>angle</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX636"></A>
Compute the argument of <VAR>z</VAR>, defined as
<VAR>theta</VAR> = <CODE>atan (<VAR>y</VAR>/<VAR>x</VAR>)</CODE>.

</P>
<P>
in radians. 

</P>
<P>
For example,

</P>

<PRE>
arg (3 + 4i)
     => 0.92730
</PRE>

</DL>

<P>
<DL>
<DT><U>Mapping Function:</U>  <B>conj</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX637"></A>
Return the complex conjugate of <VAR>z</VAR>, defined as
<CODE>conj (<VAR>z</VAR>)</CODE> = <VAR>x</VAR> - <VAR>i</VAR><VAR>y</VAR>.
</DL>

</P>
<P>
<DL>
<DT><U>Mapping Function:</U>  <B>imag</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX638"></A>
Return the imaginary part of <VAR>z</VAR> as a real number.
</DL>

</P>
<P>
<DL>
<DT><U>Mapping Function:</U>  <B>real</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX639"></A>
Return the real part of <VAR>z</VAR>.
</DL>

</P>


<H2><A NAME="SEC139" HREF="octave_toc.html#TOC139">Trigonometry</A></H2>

<P>
Octave provides the following trigonometric functions:

</P>
<P>
<DL>
<DT><U>Mapping Function:</U>  <B>sin</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX640"></A>
<DT><U>Mapping Function:</U>  <B>cos</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX641"></A>
<DT><U>Mapping Function:</U>  <B>tan</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX642"></A>
<DT><U>Mapping Function:</U>  <B>sec</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX643"></A>
<DT><U>Mapping Function:</U>  <B>csc</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX644"></A>
<DT><U>Mapping Function:</U>  <B>cot</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX645"></A>
The ordinary trigonometric functions.
</DL>

</P>
<P>
<DL>
<DT><U>Mapping Function:</U>  <B>asin</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX646"></A>
<DT><U>Mapping Function:</U>  <B>acos</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX647"></A>
<DT><U>Mapping Function:</U>  <B>atan</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX648"></A>
<DT><U>Mapping Function:</U>  <B>asec</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX649"></A>
<DT><U>Mapping Function:</U>  <B>acsc</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX650"></A>
<DT><U>Mapping Function:</U>  <B>acot</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX651"></A>
The ordinary inverse trigonometric functions.
</DL>

</P>
<P>
<DL>
<DT><U>Mapping Function:</U>  <B>sinh</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX652"></A>
<DT><U>Mapping Function:</U>  <B>cosh</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX653"></A>
<DT><U>Mapping Function:</U>  <B>tanh</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX654"></A>
<DT><U>Mapping Function:</U>  <B>sech</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX655"></A>
<DT><U>Mapping Function:</U>  <B>csch</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX656"></A>
<DT><U>Mapping Function:</U>  <B>coth</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX657"></A>
Hyperbolic trigonometric functions.
</DL>

</P>
<P>
<DL>
<DT><U>Mapping Function:</U>  <B>asinh</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX658"></A>
<DT><U>Mapping Function:</U>  <B>acosh</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX659"></A>
<DT><U>Mapping Function:</U>  <B>atanh</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX660"></A>
<DT><U>Mapping Function:</U>  <B>asech</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX661"></A>
<DT><U>Mapping Function:</U>  <B>acsch</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX662"></A>
<DT><U>Mapping Function:</U>  <B>acoth</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX663"></A>
Inverse hyperbolic trigonometric functions.
</DL>

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

</P>

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

<P>
<DL>
<DT><U>Mapping Function:</U>  <B>atan2</B> <I>(<VAR>y</VAR>, <VAR>x</VAR>)</I>
<DD><A NAME="IDX664"></A>
Return the arctangent of <VAR>y</VAR>/<VAR>x</VAR>.  The signs of the arguments
are used to determine the quadrant of the result, which is in the range
<CODE>pi</CODE> to -<CODE>pi</CODE>.
</DL>

</P>


<H2><A NAME="SEC140" HREF="octave_toc.html#TOC140">Sums and Products</A></H2>

<P>
<DL>
<DT><U>Built-in Function:</U>  <B>sum</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX665"></A>
For a vector argument, return the sum of all the elements.  For a matrix
argument, return the sum of the elements in each column, as a row
vector.  The sum of an empty matrix is 0 if it has no columns, or a
vector of zeros if it has no rows (see section <A HREF="octave_5.html#SEC50">Empty Matrices</A>).
</DL>

</P>
<P>
<DL>
<DT><U>Built-in Function:</U>  <B>prod</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX666"></A>
For a vector argument, return the product of all the elements.  For a
matrix argument, return the product of the elements in each column, as a
row vector.  The product of an empty matrix is 1 if it has no columns,
or a vector of ones if it has no rows (see section <A HREF="octave_5.html#SEC50">Empty Matrices</A>).
</DL>

</P>
<P>
<DL>
<DT><U>Built-in Function:</U>  <B>cumsum</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX667"></A>
Return the cumulative sum of each column of <VAR>x</VAR>.  For example,

</P>

<PRE>
cumsum ([1, 2; 3, 4])
     =>  1  2
         4  6
</PRE>

</DL>

<P>
<DL>
<DT><U>Built-in Function:</U>  <B>cumprod</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX668"></A>
Return the cumulative product of each column of <VAR>x</VAR>.  For example,

</P>

<PRE>
cumprod ([1, 2; 3, 4])
     =>  1  2
         3  8
</PRE>

</DL>

<P>
<DL>
<DT><U>Built-in Function:</U>  <B>sumsq</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX669"></A>
For a vector argument, return the sum of the squares of all the
elements.  For a matrix argument, return the sum of the squares of the
elements in each column, as a row vector.
</DL>

</P>


<H2><A NAME="SEC141" HREF="octave_toc.html#TOC141">Special Functions</A></H2>

<P>
<DL>
<DT><U>Mapping Function:</U>  <B>besseli</B> <I>(<VAR>alpha</VAR>, <VAR>x</VAR>)</I>
<DD><A NAME="IDX670"></A>
<DT><U>Mapping Function:</U>  <B>besselj</B> <I>(<VAR>alpha</VAR>, <VAR>x</VAR>)</I>
<DD><A NAME="IDX671"></A>
<DT><U>Mapping Function:</U>  <B>besselk</B> <I>(<VAR>alpha</VAR>, <VAR>x</VAR>)</I>
<DD><A NAME="IDX672"></A>
<DT><U>Mapping Function:</U>  <B>bessely</B> <I>(<VAR>alpha</VAR>, <VAR>x</VAR>)</I>
<DD><A NAME="IDX673"></A>
Compute Bessel functions of the following types:

</P>
<DL COMPACT>

<DT><CODE>besselj</CODE>
<DD>
Bessel functions of the first kind.

<DT><CODE>bessely</CODE>
<DD>
Bessel functions of the second kind.

<DT><CODE>besseli</CODE>
<DD>
Modified Bessel functions of the first kind.

<DT><CODE>besselk</CODE>
<DD>
Modified Bessel functions of the second kind.
</DL>

<P>
The second argument, <VAR>x</VAR>, must be a real matrix, vector, or scalar.

</P>
<P>
The first argument, <VAR>alpha</VAR>, must be greater than or equal to zero.
If <VAR>alpha</VAR> is a range, it must have an increment equal to one.

</P>
<P>
If <VAR>alpha</VAR> is a scalar, the result is the same size as <VAR>x</VAR>.

</P>
<P>
If <VAR>alpha</VAR> is a range, <VAR>x</VAR> must be a vector or scalar, and the
result is a matrix with <CODE>length(<VAR>x</VAR>)</CODE> rows and
<CODE>length(<VAR>alpha</VAR>)</CODE> columns.
</DL>

</P>
<P>
<DL>
<DT><U>Mapping Function:</U>  <B>beta</B> <I>(<VAR>a</VAR>, <VAR>b</VAR>)</I>
<DD><A NAME="IDX674"></A>
Return the Beta function,

</P>

<PRE>
beta (a, b) = gamma (a) * gamma (b) / gamma (a + b).
</PRE>

</DL>

<P>
<DL>
<DT><U>Mapping Function:</U>  <B>betai</B> <I>(<VAR>a</VAR>, <VAR>b</VAR>, <VAR>x</VAR>)</I>
<DD><A NAME="IDX675"></A>
Return the incomplete Beta function,

</P>

<PRE>
                                    x
                                   /
betai (a, b, x) = beta (a, b)^(-1) | t^(a-1) (1-t)^(b-1) dt.
                                   /
                                t=0
</PRE>

<P>
If x has more than one component, both <VAR>a</VAR> and <VAR>b</VAR> must be
scalars.  If <VAR>x</VAR> is a scalar, <VAR>a</VAR> and <VAR>b</VAR> must be of
compatible dimensions.
</DL>

</P>
<P>
<DL>
<DT><U>Mapping Function:</U>  <B>bincoeff</B> <I>(<VAR>n</VAR>, <VAR>k</VAR>)</I>
<DD><A NAME="IDX676"></A>
Return the binomial coefficient of <VAR>n</VAR> and <VAR>k</VAR>, defined as

</P>

<PRE>
 /   \
 | n |    n (n-1) (n-2) ... (n-k+1)
 |   |  = -------------------------
 | k |               k!
 \   /
</PRE>

<P>
For example,

</P>

<PRE>
bincoeff (5, 2)
     => 10
</PRE>

</DL>

<P>
<DL>
<DT><U>Mapping Function:</U>  <B>erf</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX677"></A>
Computes the error function,

</P>

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

</DL>

<P>
<DL>
<DT><U>Mapping Function:</U>  <B>erfc</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX678"></A>
Computes the complementary error function,
<CODE>1 - erf (<VAR>z</VAR>)</CODE>.
</DL>

</P>
<P>
<DL>
<DT><U>Mapping Function:</U>  <B>erfinv</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX679"></A>
Computes the inverse of the error function,
</DL>

</P>
<P>
<DL>
<DT><U>Mapping Function:</U>  <B>gamma</B> <I>(<VAR>z</VAR>)</I>
<DD><A NAME="IDX680"></A>
Computes the Gamma function,

</P>

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

</DL>

<P>
<DL>
<DT><U>Mapping Function:</U>  <B>gammai</B> <I>(<VAR>a</VAR>, <VAR>x</VAR>)</I>
<DD><A NAME="IDX681"></A>
Computes the incomplete gamma function,

</P>

<PRE>
                              x
                    1        /
gammai (a, x) = ---------    | exp (-t) t^(a-1) dt
                gamma (a)    /
                          t=0
</PRE>

<P>
If <VAR>a</VAR> is scalar, then <CODE>gammai (<VAR>a</VAR>, <VAR>x</VAR>)</CODE> is returned
for each element of <VAR>x</VAR> and vice versa.

</P>
<P>
If neither <VAR>a</VAR> nor <VAR>x</VAR> is scalar, the sizes of <VAR>a</VAR> and
<VAR>x</VAR> must agree, and <VAR>gammai</VAR> is applied element-by-element.
</DL>

</P>
<P>
<DL>
<DT><U>Mapping Function:</U>  <B>lgamma</B> <I>(<VAR>a</VAR>, <VAR>x</VAR>)</I>
<DD><A NAME="IDX682"></A>
<DT><U>Mapping Function:</U>  <B>gammaln</B> <I>(<VAR>a</VAR>, <VAR>x</VAR>)</I>
<DD><A NAME="IDX683"></A>
Return the natural logarithm of the gamma function.
</DL>

</P>
<P>
<DL>
<DT><U>Function File:</U>  <B>cross</B> <I>(<VAR>x</VAR>, <VAR>y</VAR>)</I>
<DD><A NAME="IDX684"></A>
Computes the vector cross product of the two 3-dimensional vectors
<VAR>x</VAR> and <VAR>y</VAR>.  For example,

</P>

<PRE>
cross ([1,1,0], [0,1,1])
     => [ 1; -1; 1 ]
</PRE>

</DL>

<P>
<DL>
<DT><U>Function File:</U>  <B>commutation_matrix</B> <I>(<VAR>m</VAR>, <VAR>n</VAR>)</I>
<DD><A NAME="IDX685"></A>
Return the commutation matrix
 K(m,n)
 which is the unique
 <VAR>m</VAR>*<VAR>n</VAR> by <VAR>m</VAR>*<VAR>n</VAR>
 matrix such that
 <VAR>K</VAR>(<VAR>m</VAR>,<VAR>n</VAR>) * vec (<VAR>A</VAR>) = vec (<VAR>A</VAR>')
 for all
 <VAR>m</VAR> by <VAR>n</VAR>
 matrices
 <VAR>A</VAR>.

</P>
<P>
If only one argument <VAR>m</VAR> is given,
 K(m,m)
 is returned.

</P>
<P>
See Magnus and Neudecker (1988), Matrix differential calculus with
applications in statistics and econometrics.
</DL>

</P>
<P>
<DL>
<DT><U>Function File:</U>  <B>duplication_matrix</B> <I>(<VAR>n</VAR>)</I>
<DD><A NAME="IDX686"></A>
Return the duplication matrix
 <VAR>D</VAR>_<VAR>n</VAR>
 which is the unique
 <VAR>n</VAR>^2 by <VAR>n</VAR>*(<VAR>n</VAR>+1)/2
 matrix such that
 <VAR>D</VAR>_<VAR>n</VAR> \cdot vech (<VAR>A</VAR>) = vec (<VAR>A</VAR>)
 for all symmetric
 <VAR>n</VAR> by <VAR>n</VAR>
 matrices
 <VAR>A</VAR>.

</P>
<P>
See Magnus and Neudecker (1988), Matrix differential calculus with
applications in statistics and econometrics.
</DL>

</P>


<H2><A NAME="SEC142" HREF="octave_toc.html#TOC142">Mathematical Constants</A></H2>

<P>
<DL>
<DT><U>Built-in Variable:</U> <B>I</B>
<DD><A NAME="IDX687"></A>
<DT><U>Built-in Variable:</U> <B>J</B>
<DD><A NAME="IDX688"></A>
<DT><U>Built-in Variable:</U> <B>i</B>
<DD><A NAME="IDX689"></A>
<DT><U>Built-in Variable:</U> <B>j</B>
<DD><A NAME="IDX690"></A>
A pure imaginary number, defined as
  <CODE>sqrt (-1)</CODE>.
The <CODE>I</CODE> and <CODE>J</CODE> forms are true constants, and cannot be
modified.  The <CODE>i</CODE> and <CODE>j</CODE> 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 <A HREF="octave_8.html#SEC61">Status of Variables</A>.
</DL>

</P>
<P>
<DL>
<DT><U>Built-in Variable:</U> <B>Inf</B>
<DD><A NAME="IDX691"></A>
<DT><U>Built-in Variable:</U> <B>inf</B>
<DD><A NAME="IDX692"></A>
Infinity.  This is the result of an operation like 1/0, or an operation
that results in a floating point overflow.
</DL>

</P>
<P>
<DL>
<DT><U>Built-in Variable:</U> <B>NaN</B>
<DD><A NAME="IDX693"></A>
<DT><U>Built-in Variable:</U> <B>nan</B>
<DD><A NAME="IDX694"></A>
Not a number.  This is the result of an operation like
0/0, or <SAMP>`Inf - Inf'</SAMP>,
or any operation with a NaN.

</P>
<P>
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 <CODE>isnan</CODE> function.
</DL>

</P>
<P>
<DL>
<DT><U>Built-in Variable:</U> <B>pi</B>
<DD><A NAME="IDX695"></A>
The ratio of the circumference of a circle to its diameter.
Internally, <CODE>pi</CODE> is computed as <SAMP>`4.0 * atan (1.0)'</SAMP>.
</DL>

</P>
<P>
<DL>
<DT><U>Built-in Variable:</U> <B>e</B>
<DD><A NAME="IDX696"></A>
The base of natural logarithms.  The constant
 <VAR>e</VAR>
 satisfies the equation
 <CODE>log</CODE> (<VAR>e</VAR>) = 1.
</DL>

</P>
<P>
<DL>
<DT><U>Built-in Variable:</U> <B>eps</B>
<DD><A NAME="IDX697"></A>
The machine precision.  More precisely, <CODE>eps</CODE> 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, <CODE>eps</CODE>
is approximately
 2.2204e-16.
</DL>

</P>
<P>
<DL>
<DT><U>Built-in Variable:</U> <B>realmax</B>
<DD><A NAME="IDX698"></A>
The largest floating point number that is representable.  The actual
value is system-dependent.  On machines that support 64 bit IEEE
floating point arithmetic, <CODE>realmax</CODE> is approximately
 1.7977e+308
</DL>

</P>
<P>
<DL>
<DT><U>Built-in Variable:</U> <B>realmin</B>
<DD><A NAME="IDX699"></A>
The smallest floating point number that is representable.  The actual
value is system-dependent.  On machines that support 64 bit IEEE
floating point arithmetic, <CODE>realmin</CODE> is approximately
 2.2251e-308
</DL>

</P>
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