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Math Objects is a math template library written in C++ using generic programming techniques. In order to use the "Math Objects" library, the user only has to include the header files he needs (e.g. Matrix.h, Polynomial.h etc.).
In order to compile the library the user needs an ISO/IEC 14882:1998 standard compliant C++ compiler (e.g. one that supports partial template specializations).
The math library has math objects like matrices, polynomials, rational functions, extended precision numbers, complex numbers etc. that can be handled in a similar way like basic numerical types (e.g. integers or floating point numbers).
One can access properties of a mathematical type through a (partial) specialization of a traits class for that type (AlgebraicTraits). Having the traits classes to expose properties of mathematical objects, one can define for example matrices of polynomials having extended precision complex coefficients and apply to them basic linear algebra algorithms using normal C++ syntax.
This library also implements two functions using two deterministic algorithms that compute the Smith form for polynomial matrices, and the Smith-McMillan form of a transfer functions matrix also keeping track of the transformation matrices.
These algorithms can be used to describe a MIMO (multi input-multi output) system by means of its zeros and poles and also give the MFD (matrix fraction description) of the system.
Enhancements:
- Recoded the LongInt class aiming for better runtime efficiency.

Math::Cephes is a Perl interface to the cephes math library.

SYNOPSIS

use Math::Cephes qw(:all);

This module provides an interface to over 150 functions of the
cephes math library of Stephen Moshier. No functions are exported
by default, but rather must be imported explicitly, as in

use Math::Cephes qw(sin cos);

There are a number of export tags defined which allow
importing groups of functions:
use Math::Cephes qw(:constants);
imports the variables

$PI : 3.14159265358979323846 # pi
$PIO2 : 1.57079632679489661923 # pi/2
$PIO4 : 0.785398163397448309616 # pi/4
$SQRT2 : 1.41421356237309504880 # sqrt(2)
$SQRTH : 0.707106781186547524401 # sqrt(2)/2
$LOG2E : 1.4426950408889634073599 # 1/log(2)
$SQ2OPI : 0.79788456080286535587989 # sqrt( 2/pi )
$LOGE2 : 0.693147180559945309417 # log(2)
$LOGSQ2 : 0.346573590279972654709 # log(2)/2
$THPIO4 : 2.35619449019234492885 # 3*pi/4
$TWOOPI : 0.636619772367581343075535 # 2/pi

As well, there are 4 machine-specific numbers available:

$MACHEP : machine roundoff error
$MAXLOG : maximum log on the machine
$MINLOG : minimum log on the machine
$MAXNUM : largest number represented
use Math::Cephes qw(:trigs);
imports

acos: Inverse circular cosine
asin: Inverse circular sine
atan: Inverse circular tangent (arctangent)
atan2: Quadrant correct inverse circular tangent
cos: Circular cosine
cosdg: Circular cosine of angle in degrees
cot: Circular cotangent
cotdg: Circular cotangent of argument in degrees
hypot: hypotenuse associated with the sides of a right triangle
radian: Degrees, minutes, seconds to radians
sin: Circular sine
sindg: Circular sine of angle in degrees
tan: Circular tangent
tandg: Circular tangent of argument in degrees
cosm1: Relative error approximations for function arguments near unity
use Math::Cephes qw(:hypers);
imports

acosh: Inverse hyperbolic cosine
asinh: Inverse hyperbolic sine
atanh: Inverse hyperbolic tangent
cosh: Hyperbolic cosine
sinh: Hyperbolic sine
tanh: Hyperbolic tangent
use Math::Cephes qw(:explog);
imports

exp: Exponential function
expxx: exp(x*x)
exp10: Base 10 exponential function (Common antilogarithm)
exp2: Base 2 exponential function
log: Natural logarithm
log10: Common logarithm
log2: Base 2 logarithm
log1p,expm1: Relative error approximations for function arguments near unity.
use Math::Cephes qw(:cmplx);
imports

new_cmplx: create a new complex number object
cabs: Complex absolute value
cacos: Complex circular arc cosine
cacosh: Complex inverse hyperbolic cosine
casin: Complex circular arc sine
casinh: Complex inverse hyperbolic sine
catan: Complex circular arc tangent
catanh: Complex inverse hyperbolic tangent
ccos: Complex circular cosine
ccosh: Complex hyperbolic cosine
ccot: Complex circular cotangent
cexp: Complex exponential function
clog: Complex natural logarithm
cadd: add two complex numbers
csub: subtract two complex numbers
cmul: multiply two complex numbers
cdiv: divide two complex numbers
cmov: copy one complex number to another
cneg: negate a complex number
cpow: Complex power function
csin: Complex circular sine
csinh: Complex hyperbolic sine
csqrt: Complex square root
ctan: Complex circular tangent
ctanh: Complex hyperbolic tangent
use Math::Cephes qw(:utils);
imports

cbrt: Cube root
ceil: ceil
drand: Pseudorandom number generator
fabs: Absolute value
fac: Factorial function
floor: floor
frexp: frexp
ldexp: multiplies x by 2**n.
lrand: Pseudorandom number generator
lsqrt: Integer square root
pow: Power function
powi: Real raised to integer power
round: Round double to nearest or even integer valued double
sqrt: Square root
use Math::Cephes qw(:bessels);
imports

i0: Modified Bessel function of order zero
i0e: Modified Bessel function of order zero, exponentially scaled
i1: Modified Bessel function of order one
i1e: Modified Bessel function of order one, exponentially scaled
iv: Modified Bessel function of noninteger order
j0: Bessel function of order zero
j1: Bessel function of order one
jn: Bessel function of integer order
jv: Bessel function of noninteger order
k0: Modified Bessel function, third kind, order zero
k0e: Modified Bessel function, third kind, order zero, exponentially scaled
k1: Modified Bessel function, third kind, order one
k1e: Modified Bessel function, third kind, order one, exponentially scaled
kn: Modified Bessel function, third kind, integer order
y0: Bessel function of the second kind, order zero
y1: Bessel function of second kind of order one
yn: Bessel function of second kind of integer order
yv: Bessel function Yv with noninteger v
use Math::Cephes qw(:dists);
imports

bdtr: Binomial distribution
bdtrc: Complemented binomial distribution
bdtri: Inverse binomial distribution
btdtr: Beta distribution
chdtr: Chi-square distribution
chdtrc: Complemented Chi-square distribution
chdtri: Inverse of complemented Chi-square distribution
fdtr: F distribution
fdtrc: Complemented F distribution
fdtri: Inverse of complemented F distribution
gdtr: Gamma distribution function
gdtrc: Complemented gamma distribution function
nbdtr: Negative binomial distribution
nbdtrc: Complemented negative binomial distribution
nbdtri: Functional inverse of negative binomial distribution
ndtr: Normal distribution function
ndtri: Inverse of Normal distribution function
pdtr: Poisson distribution
pdtrc: Complemented poisson distribution
pdtri: Inverse Poisson distribution
stdtr: Students t distribution
stdtri: Functional inverse of Students t distribution
use Math::Cephes qw(:gammas);
imports

fac: Factorial function
gamma: Gamma function
igam: Incomplete gamma integral
igamc: Complemented incomplete gamma integral
igami: Inverse of complemented imcomplete gamma integral
psi: Psi (digamma) function
rgamma: Reciprocal gamma function
use Math::Cephes qw(:betas);
imports

beta: Beta function
incbet: Incomplete beta integral
incbi: Inverse of imcomplete beta integral
lbeta: Natural logarithm of |beta|
use Math::Cephes qw(:elliptics);
imports

ellie: Incomplete elliptic integral of the second kind
ellik: Incomplete elliptic integral of the first kind
ellpe: Complete elliptic integral of the second kind
ellpj: Jacobian Elliptic Functions
ellpk: Complete elliptic integral of the first kind
use Math::Cephes qw(:hypergeometrics);
imports

hyp2f0: Gauss hypergeometric function F
hyp2f1: Gauss hypergeometric function F
hyperg: Confluent hypergeometric function
onef2: Hypergeometric function 1F2
threef0: Hypergeometric function 3F0
use Math::Cephes qw(:misc);
imports

airy: Airy function
bernum: Bernoulli numbers
dawsn: Dawsons Integral
ei: Exponential integral
erf: Error function
erfc: Complementary error function
expn: Exponential integral En
fresnl: Fresnel integral
plancki: Integral of Plancks black body radiation formula
polylog: Polylogarithm function
shichi: Hyperbolic sine and cosine integrals
sici: Sine and cosine integrals
simpson: Simpsons rule to find an integral
spence: Dilogarithm
struve: Struve function
vecang: angle between two vectors
zeta: Riemann zeta function of two arguments
zetac: Riemann zeta function
use Math::Cephes qw(:fract);
imports

new_fract: create a new fraction object
radd: add two fractions
rmul: multiply two fractions
rsub: subtracttwo fractions
rdiv: divide two fractions
euclid: finds the greatest common divisor

Math::HashSum can sum a list of key-value pairs on a per-key basis.

SYNOPSIS

use Math::HashSum qw(hashsum);

my %hash1 = (a=>.1, b=>.4);
my %hash2 = (a=>.2, b=>.5);
my %sum = hashsum(%hash1,%hash2);

print "$sum{a}n"; # Prints .3
print "$sum{b}n"; # Prints .9

This module allows you to sum a list of key-value pairs on a per-key basis. It adds up all the values associated with each key in the given list and returns a hash containing the sum associated with each key.

The example in the synopsis should explain usage of the module effectively.

Math::XOR is a package to handle XOR encryption of string buffers.

SYNOPSIS

use XOR;
print xor_buf("hello", "world"), "n";

The XOR module allows you to quickly XOR two strings together. This is the only method of encryption that (assuming the randomness of the pattern used as an encryption key) truly cannot be broken. It also has interesting, very direct mathematical properties which can be fun to play with:

XOR string 1 and string 2, you get string 3 XOR string 1 and string 3, you get string 2 XOR string 2 and string 3, you get string 1

FUNCTIONS

xor_buf($string1, $string2)

This function will return a scalar, which is the result of XORing the two strings passed to it together. The strings may contain binary data.

If $string2 is not at least as many characters long as $string1, xor_buf() will print an error and return undef. Only as many characters as there are in $string1 will be returned; excess characters in $string2 will be ignored. For this reason, when encrypting data it is good to think of $string1 as your "data" and $string2 as your "key".

Math::String::Charset is a simple charset for Math::String objects.

SYNOPSIS

use Math::String::Charset;

$a = new Math::String::Charset; # default a-z
$b = new Math::String::Charset [a..z]; # same
$c = new Math::String::Charset
{ start => [a..z], sep => }; # with between chars

print $b->length(); # a-z => 26

# construct a charset from bigram table, and an initial set (containing
# valid start-characters)
# Note: After an a, either an b, c or a can follow, in this order
# After an d only an a can follow
$bi = new Math::String::Charset ( {
start => a..d,
bi => {
a => [ b, c, a ],
b => [ c, b ],
c => [ a, c ],
d => [ a, ],
q => [ ], # q will be automatically in end
}
end => [ a, b, ],
} );
print $bi->length(); # a,b => 2 (cross of end and start)
print scalar $bi->class(2); # count of combinations with 2 letters
# will be 3+2+2+1 => 8

$d = new Math::String::Charset ( { start => [a..z],
minlen => 2, maxlen => 4, } );

print $d->first(0),"n"; # undef, too short
print $d->first(1),"n"; # undef, to short
print $d->first(2),"n"; # aa

$d = new Math::String::Charset ( { start => [a..z] } );

print $d->first(0),"n"; #
print $d->first(1),"n"; # a
print $d->last(1),"n"; # z
print $d->first(2),"n"; # aa

This module lets you create an charset object, which is used to contruct Math::String objects. This object knows how to handle simple charsets as well as complex onex consisting of bi-grams (later tri and more).

In case of more complex charsets, a reference to a Math::String::Charset::Nested or Math::String::Charset::grouped will be returned.

The default charset is the set containing "abcdefghijklmnopqrstuvwxyz" (thus producing always lower case output).

Math::Logic::Predicate is a Perl module to manage and query a predicate assertion database.

SYNOPSIS

use Math::Logic::Predicate;

$db = new Math::Logic::Predicate;

# Enter some predicates into the database
$db->add(retract( smart(_) );

# Make a query
$query = $db->parse( human(H) & name(H, X) ? );
$iter = $db->match($query, $iter);

# Get the results
$name = $db->get($iter, X);

# Store it in a rule
$db->add( human_name(H, N) := human(H) & name(H, N). );

# Use it in a query
$iter = $db->match( human_name(lister, N) ? );

# Save it to a file
use Storable;
store($db->rules, red_dwarf);

Math::BigRat package arbitrary big rational numbers.

SYNOPSIS

use Math::BigRat;

my $x = Math::BigRat->new(3/7); $x += 5/9;

print $x->bstr(),"n";
print $x ** 2,"n";

my $y = Math::BigRat->new(inf);
print "$y ", ($y->is_inf ? is : is not) , " infinityn";

my $z = Math::BigRat->new(144); $z->bsqrt();

Math::BigRat complements Math::BigInt and Math::BigFloat by providing support for arbitrary big rational numbers.

MATH LIBRARY

You can change the underlying module that does the low-level math operations by using:

use Math::BigRat try => GMP;

Note: This needs Math::BigInt::GMP installed.
The following would first try to find Math::BigInt::Foo, then Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc:

use Math::BigRat try => Foo,Math::BigInt::Bar;

If you want to get warned when the fallback occurs, replace "try" with "lib":

use Math::BigRat lib => Foo,Math::BigInt::Bar;

If you want the code to die instead, replace "try" with "only":

use Math::BigRat only => Foo,Math::BigInt::Bar;

Math::Macopt is a Perl wrapper for macopt++, which is a conjugate gradient library.

INSTALLATION

The package can be installed by the standard PERL module installation procedure:

perl Makefile.PL
make
make test
make install

Please noted that the original "macopt++" C++ source code is included in this PERL package. The static linking avoids the possible conflict to any pre-installed version of "macopt++".

SYNOPSIS

use strict;
use Math::Macopt;

&main();

sub main
{
# Some settings
my $N = 10;
my $epsilon = 0.001;

# Initialize the Macopt
my $macopt = new Math::Macopt::Base($N, 0);

# Setup the function and its gradient
my $func = sub {
my $x = shift;

my $size = $macopt->size();
my $sum = 0;
foreach my $i (0..$size-1) {
$sum += ($x->[$i]-$i)**2;
}

return $sum;
};
my $dfunc = sub {
my $x = shift;

my $size = $macopt->size();
my $g = ();
foreach my $i (0..$size-1) {
$g->[$i] = 2*($x->[$i]-$i);
}

return $g;
};
$macopt->setFunc(&$func);
$macopt->setDfunc(&$dfunc);

# Optimizer using macopt
my $x = [(1)x($N)];
$macopt->maccheckgrad($x, $N, $epsilon, 0) ;
$macopt->macoptII($x, $N);

# Display the result
printf "[%s]n", join(,, @$x);
}