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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 IndexIterator , class InIterator , class OutIterator >
void | applyPermutation (IndexIterator index_first, IndexIterator index_last, InIterator in, OutIterator out)
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)
double | besselJ (int n, double x)
double | besselY (int n, double x)
UInt32 | ceilPower2 (UInt32 x)
UInt32 | checksum (const char *data, unsigned int size)
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)
UInt32 | concatenateChecksum (UInt32 checksum, const char *data, unsigned int size)
template<class PointArray1 , class PointArray2 >
void | convexHull (const PointArray1 &points, PointArray2 &convex_hull)
| Compute convex hull of a 2D polygon.
template<class REAL >
REAL | cos_pi (REAL x)
double | ellipticIntegralE (double x, double k)
double | ellipticIntegralF (double x, double k)
bool | even (int t)
UInt32 | floorPower2 (UInt32 x)
double | gamma (double x)
template<typename IntType >
IntType | gcd (IntType n, IntType m)
template<class Iterator , class IndexIterator , class Compare >
void | indexSort (Iterator first, Iterator last, IndexIterator index_first, Compare c)
template<... >
void | inspectSequence (...)
| Call an analyzing functor at every element of a sequence.
template<class InIterator , class OutIterator >
void | inversePermutation (InIterator first, InIterator last, OutIterator out)
template<typename IntType >
IntType | lcm (IntType n, IntType m)
template<class REAL >
REAL | legendre (unsigned int l, int m, REAL x)
template<class REAL >
REAL | legendre (unsigned int l, REAL x)
template<class Iterator , class Value >
void | linearSequence (Iterator first, Iterator last, Value start, Value step)
Int32 | log2i (UInt32 x)
double | loggamma (double 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)
bool | odd (int t)
result_type | operator() (argument_type x) const
template<unsigned n, class V >
V | power (const V &x)
REAL | round (REAL v)
int | roundi (double t)
template<class T >
T | sign (T t)
template<class T1 , class T2 >
T1 | sign (T1 t1, T2 t2)
template<class T >
int | signi (T t)
template<class REAL >
REAL | sin_pi (REAL x)
template<class T >
NumericTraits< T >::Promote | sq (T t)
Int32 | sqrti (Int32 v)
NormTraits< T >::SquaredNormType | squaredNorm (T const &t)
template<class T >
void | symmetric2x2Eigenvalues (T a00, T a01, T a11, T *r0, T *r1)
template<class T >
void | symmetric3x3Eigenvalues (T a00, T a01, T a02, T a11, T a12, T a22, T *r0, T *r1, T *r2)

## Detailed Description

Useful mathematical functions and functors.

## Function Documentation

Iterator vigra::argMin | ( | Iterator | first,
|  | Iterator | last
| ) |  |

Find the minimum element in a sequence.

The function returns the iterator referring to the minimum element. This is identical to the function `std::min_element()`.

Required Interface:

            Iterator is a standard forward iterator.

            bool f = *first < NumericTraits<typename std::iterator_traits<Iterator>::value_type>::max();

#include <vigra/algorithm.hxx>
Namespace: vigra

Iterator vigra::argMax | ( | Iterator | first,
|  | Iterator | last
| ) |  |

Find the maximum element in a sequence.

The function returns the iterator referring to the maximum element. This is identical to the function `std::max_element()`.

Required Interface:

            Iterator is a standard forward iterator.

            bool f = NumericTraits<typename std::iterator_traits<Iterator>::value_type>::min() < *first;

#include <vigra/algorithm.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 referring 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/algorithm.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 referring 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/algorithm.hxx>
Namespace: vigra

void vigra::linearSequence | ( | Iterator | first,
|  | Iterator | last,
|  | Value | start,
|  | Value | step
| ) |  |

Fill an array with a sequence of numbers.

The sequence starts at start and is incremented with step. Default start and stepsize are 0 and 1 respectively.

Declaration:

            namespace vigra {
                template <class Iterator, class Value>
                void linearSequence(Iterator first, Iterator last,
                              Value const & start = 0, Value const & step = 1);
            }

Required Interface:

            Iterator is a standard forward iterator.

            *first = start;
            start += step;

#include <vigra/algorithm.hxx>
Namespace: vigra

void vigra::inspectSequence | ( |  | ... | ) |

Call an analyzing functor at every element of a sequence.

This function can be used to collect statistics of the sequence `[first, last)` defined by these two input interators. The results must be stored in the functor, which serves as a return value.

Declarations:

        namespace vigra {
            template <class InputIterator, class Functor>
            void
            inspectSequence(InputIterator first, InputIterator last, Functor & f);
        }

Usage:

#include <vigra/algorithm.hxx>
Namespace: vigra

        std::vector array(100);

        // init functor
        vigra::FindMinMax<int> minmax;

        vigra::inspectSequence(array.begin(), array.end(), minmax);

        cout << "Min: " << minmax.min << " Max: " << minmax.max;

void vigra::indexSort | ( | Iterator | first,
|  | Iterator | last,
|  | IndexIterator | index_first,
|  | Compare | c
| ) |  |

Return the index permutation that would sort the input array.

To actually sort an array according to the ordering thus determined, use applyPermutation().

Declarations:

            namespace vigra {
                // compare using std::less
                template <class Iterator, class IndexIterator>
                void indexSort(Iterator first, Iterator last, IndexIterator index_first);

                // compare using functor Compare
                template <class Iterator, class IndexIterator, class Compare>
                void indexSort(Iterator first, Iterator last, IndexIterator index_first, Compare compare);
            }

Required Interface:

            Iterator and IndexIterators are random access iterators.

            bool res = compare(first[*index_first], first[*index_first]);

#include <vigra/algorithm.hxx>
Namespace: vigra

void vigra::applyPermutation | ( | IndexIterator | index_first,
|  | IndexIterator | index_last,
|  | InIterator | in,
|  | OutIterator | out
| ) |  |

Sort an array according to the given index permutation.

The iterators in and out may not refer to the same array, as this would overwrite the input prematurely.

Declaration:

            namespace vigra {
                template <class IndexIterator, class InIterator, class OutIterator>
                void applyPermutation(IndexIterator index_first, IndexIterator index_last,
                                      InIterator in, OutIterator out);
            }

Required Interface:

            OutIterator and IndexIterators are forward iterators.
            InIterator is a random access iterator.

            *out = in[*index_first];

#include <vigra/algorithm.hxx>
Namespace: vigra

void vigra::inversePermutation | ( | InIterator | first,
|  | InIterator | last,
|  | OutIterator | out
| ) |  |

Compute the inverse of a given permutation.

This is just another name for indexSort(), referring to another semantics.

Declaration:

            namespace vigra {
                template <class InIterator, class OutIterator>
                void inversePermutation(InIterator first, InIterator last,
                                        OutIterator out);
            }

Required Interface:

            InIterator and OutIterator are random access iterators.

            *out = in[*index_first];

#include <vigra/algorithm.hxx>
Namespace: vigra

UInt32 vigra::checksum | ( | const char * | data,
|  | unsigned int | size
| ) |  |

Compute the CRC-32 checksum of a byte array.

Implementation note: This function is slower on big-endian machines because the "4 bytes at a time" optimization is only implemented for little-endian.

UInt32 vigra::concatenateChecksum | ( | UInt32 | checksum,
|  | const char * | data,
|  | unsigned int | size
| ) |  |

Concatenate a byte array to an existing CRC-32 checksum.

double vigra::besselJ | ( | int | n,
|  | double | x
| ) |  |

Bessel function of the first kind.

Computes the value of BesselJ of integer order `n` and argument `x`. Negative `x` are unsupported and will result in a `std::domain_error`.

This function wraps a number of existing implementations and falls back to a rather slow algorithm if none of them is available. In particular, it uses boost::math when `HasBoostMath` is #defined, or native implementations on gcc and MSVC otherwise.

#include <vigra/bessel.hxx>
Namespace: vigra

double vigra::besselY | ( | int | n,
|  | double | x
| ) |  |

Bessel function of the second kind.

Computes the value of BesselY of integer order `n` and argument `x`. Negative `x` are unsupported and will result in a `std::domain_error`.

This function wraps a number of existing implementations and falls back to a rather slow algorithm if none of them is available. In particular, it uses boost::math when `HasBoostMath` is #defined, or native implementations on gcc and MSVC otherwise.

#include <vigra/bessel.hxx>
Namespace: vigra

REAL vigra::round | ( | REAL | v | ) |

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

int vigra::roundi | ( | double | t | ) |

Round and cast to integer.

Rounds to the nearest integer like round(), but casts the result to `int` (this will be faster and is usually needed anyway).

#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

V vigra::power | ( | const V & | x | ) |

Exponentiation to a positive integer power by squaring.

#include <vigra/mathutil.hxx>
Namespace: vigra

UInt32 sqrti | ( | Int32 | v | ) |

Signed integer square root.

Useful for fast fixed-point computations.

#include <vigra/mathutil.hxx>
Namespace: vigra

Unsigned integer square root.

Useful for fast fixed-point computations.

#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, but with the same type as t.

#include <vigra/mathutil.hxx>
Namespace: vigra

int vigra::signi | ( | T | t | ) |

The integer sign function.

Returns 1, 0, or -1 depending on the sign of t, converted to int.

#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

bool vigra::even | ( | int | t | ) |

Check if an integer is even.

Defined for all integral types.

bool vigra::odd | ( | int | t | ) |

Check if an integer is odd.

Defined for all integral types.

NormTraits<T>::SquaredNormType vigra::squaredNorm | ( | T const & | t | ) |

The squared norm of a numerical object.

For scalar types: equals `vigra::sq(t)`
. For vectorial types: equals `vigra::dot(t, t)`
. For complex types: equals `vigra::sq(t.real()) + vigra::sq(t.imag())`
. For matrix types: results in the squared Frobenius norm (sum of squares of the matrix elements).

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

void vigra::symmetric2x2Eigenvalues | ( | T | a00,
|  | T | a01,
|  | T | a11,
|  | T * | r0,
|  | T * | r1
| ) |  |

Compute the eigenvalues of a 2x2 real symmetric matrix.

This uses the analytical eigenvalue formula

#include <vigra/mathutil.hxx>
Namespace: vigra

void vigra::symmetric3x3Eigenvalues | ( | T | a00,
|  | T | a01,
|  | T | a02,
|  | T | a11,
|  | T | a12,
|  | T | a22,
|  | T * | r0,
|  | T * | r1,
|  | T * | r2
| ) |  |

Compute the eigenvalues of a 3x3 real symmetric matrix.

This uses a numerically stable version of the analytical eigenvalue formula according to

David Eberly: "Eigensystems for 3 × 3 Symmetric Matrices (Revisited)", Geometric Tools Documentation, 2006

#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::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

REAL vigra::legendre | ( | unsigned int | l,
|  | int | m,
|  | REAL | x
| ) |  |

Associated Legendre polynomial.

Computes the value of the associated Legendre polynomial of order `l, m` for argument `x`. `x` must be in the range `[-1.0, 1.0]`, otherwise an exception is thrown. The standard Legendre polynomials are the special case `m == 0`.

#include <vigra/mathutil.hxx>
Namespace: vigra

REAL vigra::legendre | ( | unsigned int | l,
|  | REAL | x
| ) |  |

Legendre polynomial.

Computes the value of the Legendre polynomial of order `l` for argument `x`. `x` must be in the range `[-1.0, 1.0]`, otherwise an exception is thrown.

#include <vigra/mathutil.hxx>
Namespace: vigra

REAL vigra::sin_pi | ( | REAL | x | ) |

sin(pi*x).

Essentially calls `std::sin(M_PI*x)` but uses a more accurate implementation to make sure that `sin_pi(1.0) == 0.0` (which does not hold for `std::sin(M_PI)` due to round-off error), and `sin_pi(0.5) == 1.0`.

#include <vigra/mathutil.hxx>
Namespace: vigra

REAL vigra::cos_pi | ( | REAL | x | ) |

cos(pi*x).

Essentially calls `std::cos(M_PI*x)` but uses a more accurate implementation to make sure that `cos_pi(1.0) == -1.0` and `cos_pi(0.5) == 0.0`.

#include <vigra/mathutil.hxx>
Namespace: vigra

double vigra::gamma | ( | double | x | ) |

The gamma function.

This function implements the algorithm from
Zhang and Jin: "Computation of Special Functions", John Wiley and Sons, 1996.

The argument must be <= 171.0 and cannot be zero or a negative integer. An exception is thrown when these conditions are violated.

#include <vigra/mathutil.hxx>
Namespace: vigra

Examples:
  total_variation.cxx.

double vigra::loggamma | ( | double | x | ) |

The natural logarithm of the gamma function.

This function is based on a free implementation by Sun Microsystems, Inc., see sourceware.org archive. It can be removed once all compilers support the new C99 math functions.

The argument must be positive and < 1e30. An exception is thrown when these conditions are violated.

#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

void vigra::convexHull | ( | const PointArray1 & | points,
|  | PointArray2 & | convex_hull
| ) |  |

Compute convex hull of a 2D polygon.

The input array points contains a (not necessarily ordered) set of 2D points whose convex hull is to be computed. The array's `value_type` (i.e. the point type) must be compatible with std::vector (in particular, it must support indexing, copying, and have `size() == 2`). The points of the convex hull will be appended to the output array convex_hull (which must support `std::back_inserter(convex_hull)`). Since the convex hull is a closed polygon, the first and last point of the output will be the same (i.e. the first point will simply be inserted at the end again). The points of the convex hull will be ordered counter-clockwise, starting with the leftmost point of the input. The function implements Andrew's Monotone Chain algorithm.

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

function (functor) call

© Ullrich Köthe (ullrich.koethe@iwr.uni-heidelberg.de) Heidelberg Collaboratory for Image Processing, University of Heidelberg, Germany | html generated using doxygen and Python vigra 1.9.0 (Tue Nov 6 2012)