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Mathematical Functions | ![]() |
Classes | |
class | BSpline< ORDER, T > |
class | BSplineBase< ORDER, T > |
class | CatmullRomSpline< T > |
class | CoscotFunction< T > |
class | Gaussian< T > |
Functions | |
template<class Iterator > | |
Iterator | argMax (Iterator first, Iterator last) |
template<class Iterator , class UnaryFunctor > | |
Iterator | argMaxIf (Iterator first, Iterator last, UnaryFunctor condition) |
template<class Iterator > | |
Iterator | argMin (Iterator first, Iterator last) |
template<class Iterator , class UnaryFunctor > | |
Iterator | argMinIf (Iterator first, Iterator last, UnaryFunctor condition) |
UInt32 | ceilPower2 (UInt32 x) |
double | chi2 (unsigned int degreesOfFreedom, double arg, double accuracy=1e-7) |
double | chi2CDF (unsigned int degreesOfFreedom, double arg, double accuracy=1e-7) |
template<class T1 , class T2 > | |
bool | closeAtTolerance (T1 l, T2 r, typename PromoteTraits< T1, T2 >::Promote epsilon) |
double | ellipticIntegralE (double x, double k) |
double | ellipticIntegralF (double x, double k) |
double | erf (double x) |
UInt32 | floorPower2 (UInt32 x) |
template<typename IntType > | |
IntType | gcd (IntType n, IntType m) |
double | hypot (double a, double b) |
template<typename IntType > | |
IntType | lcm (IntType n, IntType m) |
Int32 | log2i (UInt32 x) |
double | noncentralChi2 (unsigned int degreesOfFreedom, double noncentrality, double arg, double accuracy=1e-7) |
double | noncentralChi2CDF (unsigned int degreesOfFreedom, double noncentrality, double arg, double accuracy=1e-7) |
double | noncentralChi2CDFApprox (unsigned int degreesOfFreedom, double noncentrality, double arg) |
template<class T > | |
NormTraits< T >::NormType | norm (T const &t) |
result_type | operator() (argument_type x) const |
float | round (float t) |
template<class T1 , class T2 > | |
T1 | sign (T1 t1, T2 t2) |
template<class T > | |
T | sign (T t) |
template<class T > | |
NumericTraits< T >::Promote | sq (T t) |
UInt32 | sqrti (UInt32 v) |
Int32 | sqrti (Int32 v) |
NormTraits< T >::SquaredNormType | squaredNorm (T const &t) |
Useful mathematical functions and functors.
float vigra::round | ( | float | t | ) |
The rounding function.
Defined for all floating point types. Rounds towards the nearest integer such that abs(round(t)) == round(abs(t))
for all t
.
#include <vigra/mathutil.hxx>
Namespace: vigra
UInt32 vigra::ceilPower2 | ( | UInt32 | x | ) |
Round up to the nearest power of 2.
Efficient algorithm for finding the smallest power of 2 which is not smaller than x (function clp2() from Henry Warren: "Hacker's Delight", Addison-Wesley, 2003, see http://www.hackersdelight.org/). If x > 2^31, the function will return 0 because integer arithmetic is defined modulo 2^32.
#include <vigra/mathutil.hxx>
Namespace: vigra
UInt32 vigra::floorPower2 | ( | UInt32 | x | ) |
Round down to the nearest power of 2.
Efficient algorithm for finding the largest power of 2 which is not greater than x (function flp2() from Henry Warren: "Hacker's Delight", Addison-Wesley, 2003, see http://www.hackersdelight.org/).
#include <vigra/mathutil.hxx>
Namespace: vigra
Int32 vigra::log2i | ( | UInt32 | x | ) |
Compute the base-2 logarithm of an integer.
Returns the position of the left-most 1-bit in the given number x, or -1 if x == 0. That is,
assert(k >= 0 && k < 32 && log2i(1 << k) == k);
The function uses Robert Harley's algorithm to determine the number of leading zeros in x (algorithm nlz10() at http://www.hackersdelight.org/). But note that the functions floorPower2() or ceilPower2() are more efficient and should be preferred when possible.
#include <vigra/mathutil.hxx>
Namespace: vigra
NumericTraits<T>::Promote vigra::sq | ( | T | t | ) |
The square function.
sq(x) = x*x
is needed so often that it makes sense to define it as a function.
#include <vigra/mathutil.hxx>
Namespace: vigra
Int32 vigra::sqrti | ( | Int32 | v | ) |
Signed integer square root.
Useful for fast fixed-point computations.
#include <vigra/mathutil.hxx>
Namespace: vigra
UInt32 vigra::sqrti | ( | UInt32 | v | ) |
Unsigned integer square root.
Useful for fast fixed-point computations.
#include <vigra/mathutil.hxx>
Namespace: vigra
double vigra::hypot | ( | double | a, | |
double | b | |||
) |
Compute the Euclidean distance (length of the hypothenuse of a right-angled triangle).
The hypot() function returns the sqrt(a*a + b*b). It is implemented in a way that minimizes round-off error.
#include <vigra/mathutil.hxx>
Namespace: vigra
T vigra::sign | ( | T | t | ) |
The sign function.
Returns 1, 0, or -1 depending on the sign of t.
#include <vigra/mathutil.hxx>
Namespace: vigra
T1 vigra::sign | ( | T1 | t1, | |
T2 | t2 | |||
) |
The binary sign function.
Transfers the sign of t2 to t1.
#include <vigra/mathutil.hxx>
Namespace: vigra
NormTraits<T>::SquaredNormType vigra::squaredNorm | ( | T const & | t | ) |
NormTraits<T>::NormType vigra::norm | ( | T const & | t | ) |
The norm of a numerical object.
For scalar types: implemented as abs(t)
otherwise: implemented as sqrt(squaredNorm(t))
.
#include <vigra/mathutil.hxx>
Namespace: vigra
Iterator vigra::argMin | ( | Iterator | first, | |
Iterator | last | |||
) |
Find the minimum element in a sequence.
The function returns the iterator refering to the minimum element.
Required Interface:
Iterator is a standard forward iterator.
bool f = *first < NumericTraits<typename std::iterator_traits<Iterator>::value_type>::max();
#include <vigra/mathutil.hxx>
Namespace: vigra
Iterator vigra::argMax | ( | Iterator | first, | |
Iterator | last | |||
) |
Find the maximum element in a sequence.
The function returns the iterator refering to the maximum element.
Required Interface:
Iterator is a standard forward iterator.
bool f = NumericTraits<typename std::iterator_traits<Iterator>::value_type>::min() < *first;
#include <vigra/mathutil.hxx>
Namespace: vigra
Iterator vigra::argMinIf | ( | Iterator | first, | |
Iterator | last, | |||
UnaryFunctor | condition | |||
) |
Find the minimum element in a sequence conforming to a condition.
The function returns the iterator refering to the minimum element, where only elements conforming to the condition (i.e. where condition(*iterator)
evaluates to true
) are considered. If no element conforms to the condition, or the sequence is empty, the end iterator last is returned.
Required Interface:
Iterator is a standard forward iterator. bool c = condition(*first); bool f = *first < NumericTraits<typename std::iterator_traits<Iterator>::value_type>::max();
#include <vigra/mathutil.hxx>
Namespace: vigra
Iterator vigra::argMaxIf | ( | Iterator | first, | |
Iterator | last, | |||
UnaryFunctor | condition | |||
) |
Find the maximum element in a sequence conforming to a condition.
The function returns the iterator refering to the maximum element, where only elements conforming to the condition (i.e. where condition(*iterator)
evaluates to true
) are considered. If no element conforms to the condition, or the sequence is empty, the end iterator last is returned.
Required Interface:
Iterator is a standard forward iterator. bool c = condition(*first); bool f = NumericTraits<typename std::iterator_traits<Iterator>::value_type>::min() < *first;
#include <vigra/mathutil.hxx>
Namespace: vigra
double vigra::ellipticIntegralF | ( | double | x, | |
double | k | |||
) |
The incomplete elliptic integral of the first kind.
Computes
according to the algorithm given in Press et al. "Numerical Recipes".
Note: In some libraries (e.g. Mathematica), the second parameter of the elliptic integral functions must be k^2 rather than k. Check the documentation when results disagree!
#include <vigra/mathutil.hxx>
Namespace: vigra
double vigra::ellipticIntegralE | ( | double | x, | |
double | k | |||
) |
The incomplete elliptic integral of the second kind.
Computes
according to the algorithm given in Press et al. "Numerical Recipes". The complete elliptic integral of the second kind is simply ellipticIntegralE(M_PI/2, k)
.
Note: In some libraries (e.g. Mathematica), the second parameter of the elliptic integral functions must be k^2 rather than k. Check the documentation when results disagree!
#include <vigra/mathutil.hxx>
Namespace: vigra
double vigra::erf | ( | double | x | ) |
The error function.
If erf()
is not provided in the C standard math library (as it should according to the new C99 standard ?), VIGRA implements erf()
as an approximation of the error function
according to the formula given in Press et al. "Numerical Recipes".
#include <vigra/mathutil.hxx>
Namespace: vigra
double vigra::chi2 | ( | unsigned int | degreesOfFreedom, | |
double | arg, | |||
double | accuracy = 1e-7 | |||
) |
Chi square distribution.
Computes the density of a chi square distribution with degreesOfFreedom and tolerance accuracy at the given argument arg by calling noncentralChi2(degreesOfFreedom, 0.0, arg, accuracy)
.
#include <vigra/mathutil.hxx>
Namespace: vigra
double vigra::chi2CDF | ( | unsigned int | degreesOfFreedom, | |
double | arg, | |||
double | accuracy = 1e-7 | |||
) |
Cumulative chi square distribution.
Computes the cumulative density of a chi square distribution with degreesOfFreedom and tolerance accuracy at the given argument arg, i.e. the probability that a random number drawn from the distribution is below arg by calling noncentralChi2CDF(degreesOfFreedom, 0.0, arg, accuracy)
.
#include <vigra/mathutil.hxx>
Namespace: vigra
double vigra::noncentralChi2 | ( | unsigned int | degreesOfFreedom, | |
double | noncentrality, | |||
double | arg, | |||
double | accuracy = 1e-7 | |||
) |
Non-central chi square distribution.
Computes the density of a non-central chi square distribution with degreesOfFreedom, noncentrality parameter noncentrality and tolerance accuracy at the given argument arg. It uses Algorithm AS 231 from Appl. Statist. (1987) Vol.36, No.3 (code ported from http://lib.stat.cmu.edu/apstat/231). The algorithm has linear complexity in the number of degrees of freedom.
#include <vigra/mathutil.hxx>
Namespace: vigra
double vigra::noncentralChi2CDF | ( | unsigned int | degreesOfFreedom, | |
double | noncentrality, | |||
double | arg, | |||
double | accuracy = 1e-7 | |||
) |
Cumulative non-central chi square distribution.
Computes the cumulative density of a chi square distribution with degreesOfFreedom, noncentrality parameter noncentrality and tolerance accuracy at the given argument arg, i.e. the probability that a random number drawn from the distribution is below arg It uses Algorithm AS 231 from Appl. Statist. (1987) Vol.36, No.3 (code ported from http://lib.stat.cmu.edu/apstat/231). The algorithm has linear complexity in the number of degrees of freedom (see noncentralChi2CDFApprox() for a constant-time algorithm).
#include <vigra/mathutil.hxx>
Namespace: vigra
double vigra::noncentralChi2CDFApprox | ( | unsigned int | degreesOfFreedom, | |
double | noncentrality, | |||
double | arg | |||
) |
Cumulative non-central chi square distribution (approximate).
Computes approximate values of the cumulative density of a chi square distribution with degreesOfFreedom, and noncentrality parameter noncentrality at the given argument arg, i.e. the probability that a random number drawn from the distribution is below arg It uses the approximate transform into a normal distribution due to Wilson and Hilferty (see Abramovitz, Stegun: "Handbook of Mathematical Functions", formula 26.3.32). The algorithm's running time is independent of the inputs, i.e. is should be used when noncentralChi2CDF() is too slow, and approximate values are sufficient. The accuracy is only about 0.1 for few degrees of freedom, but reaches about 0.001 above dof = 5.
#include <vigra/mathutil.hxx>
Namespace: vigra
bool vigra::closeAtTolerance | ( | T1 | l, | |
T2 | r, | |||
typename PromoteTraits< T1, T2 >::Promote | epsilon | |||
) |
Tolerance based floating-point comparison.
Check whether two floating point numbers are equal within the given tolerance. This is useful because floating point numbers that should be equal in theory are rarely exactly equal in practice. If the tolerance epsilon is not given, twice the machine epsilon is used.
#include <vigra/mathutil.hxx>
Namespace: vigra
IntType vigra::gcd | ( | IntType | n, | |
IntType | m | |||
) |
Calculate the greatest common divisor.
This function works for arbitrary integer types, including user-defined (e.g. infinite precision) ones.
#include <vigra/rational.hxx>
Namespace: vigra
IntType vigra::lcm | ( | IntType | n, | |
IntType | m | |||
) |
Calculate the lowest common multiple.
This function works for arbitrary integer types, including user-defined (e.g. infinite precision) ones.
#include <vigra/rational.hxx>
Namespace: vigra
CatmullRomSpline< T >::result_type operator() | ( | argument_type | x | ) | const [inherited] |
function (functor) call
© Ullrich Köthe (ullrich.koethe@iwr.uni-heidelberg.de) |
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