Regina Calculation Engine

Underlying mathematical gruntwork. More...
Classes  
class  regina::NCyclotomic 
Represents an element of a cyclotomic field. More...  
class  regina::NNativeInteger< bytes > 
A wrapper class for a native, fixedprecision integer type of the given size. More...  
struct  regina::InfinityBase< supportInfinity > 
Internal base classes for use with NIntegerBase, templated on whether we should support infinity as an allowed value. More...  
class  regina::NIntegerBase< supportInfinity > 
Represents an arbitrary precision integer. More...  
class  regina::NMatrix< T > 
Represents a matrix of elements of the given type T. More...  
class  regina::NMatrixRing< T > 
Represents a matrix of elements from a given ring T. More...  
class  regina::NMatrix2 
Represents a 2by2 integer matrix. More...  
class  regina::NMatrixInt 
Represents a matrix of arbitrary precision integers. More...  
class  regina::NPerm< int > 
Represents a permutation of {0,1,...,n1}. More...  
class  regina::NPerm< 2 > 
Represents a permutation of {0,1}. More...  
class  regina::NPerm< 3 > 
Represents a permutation of {0,1,2}. More...  
class  regina::NPerm< 4 > 
Represents a permutation of {0,1,2,3}. More...  
class  regina::NPerm< 5 > 
Represents a permutation of {0,1,2,3,4}. More...  
class  regina::NPolynomial< T > 
Represents a singlevariable polynomial with coefficients of type T. More...  
class  regina::NPrimes 
A helper class for finding primes and factorising integers. More...  
class  regina::NRational 
Represents an arbitrary precision rational number. More...  
class  regina::NRay 
A fast class for storing a ray rooted at the origin whose coordinates are rational. More...  
class  regina::NVector< T > 
An optimised vector class of elements from a given ring T. More...  
Typedefs  
typedef NIntegerBase< true >  regina::NLargeInteger 
NLargeInteger is a typedef for NIntegerBase<true>, which offers arbitrary precision integers with support for infinity. More...  
typedef NIntegerBase< false >  regina::NInteger 
NInteger is a typedef for NIntegerBase<false>, which offers arbitrary precision integers without support for infinity. More...  
typedef NNativeInteger< sizeof(long)>  regina::NNativeLong 
NNativeLong is a typedef for the NNativeInteger template class whose underlying integer type is a native long. More...  
typedef NPerm< 2 >  regina::NPerm2 
A convenience typedef for NPerm<2>. More...  
typedef NPerm< 3 >  regina::NPerm3 
A convenience typedef for NPerm<3>. More...  
typedef NPerm< 4 >  regina::NPerm4 
A convenience typedef for NPerm<4>. More...  
typedef NPerm< 5 >  regina::NPerm5 
A convenience typedef for NPerm<5>. More...  
Functions  
template<class R >  
bool  regina::isZero (R x) 
Determines whether the given real number is zero. More...  
template<class R >  
bool  regina::isNonZero (R x) 
Determines whether the given real number is nonzero. More...  
template<class R >  
bool  regina::isPositive (R x) 
Determines whether the given real number is strictly positive. More...  
template<class R >  
bool  regina::isNegative (R x) 
Determines whether the given real number is strictly negative. More...  
template<class R >  
bool  regina::isNonNegative (R x) 
Determines whether the given real number is nonnegative. More...  
template<class R >  
bool  regina::isNonPositive (R x) 
Determines whether the given real number is nonpositive. More...  
REGINA_API void  regina::smithNormalForm (NMatrixInt &matrix) 
Transforms the given integer matrix into Smith normal form. More...  
REGINA_API void  regina::smithNormalForm (NMatrixInt &matrix, NMatrixInt &rowSpaceBasis, NMatrixInt &rowSpaceBasisInv, NMatrixInt &colSpaceBasis, NMatrixInt &colSpaceBasisInv) 
A Smith normal form algorithm that also returns change of basis matrices. More...  
REGINA_API void  regina::metricalSmithNormalForm (NMatrixInt &matrix, NMatrixInt *rowSpaceBasis=0, NMatrixInt *rowSpaceBasisInv=0, NMatrixInt *colSpaceBasis=0, NMatrixInt *colSpaceBasisInv=0) 
An alternative Smith normal form algorithm that also returns change of basis matrices. More...  
REGINA_API unsigned  regina::rowBasis (NMatrixInt &matrix) 
Find a basis for the row space of the given matrix. More...  
REGINA_API unsigned  regina::rowBasisAndOrthComp (NMatrixInt &input, NMatrixInt &complement) 
Finds a basis for the row space of the given matrix, as well as an "incremental" basis for its orthogonal complement. More...  
REGINA_API void  regina::columnEchelonForm (NMatrixInt &M, NMatrixInt &R, NMatrixInt &Ri, const std::vector< unsigned > &rowList) 
Transforms a given matrix into column echelon form with respect to a collection of rows. More...  
REGINA_API std::unique_ptr < NMatrixInt >  regina::preImageOfLattice (const NMatrixInt &hom, const std::vector< NLargeInteger > &sublattice) 
Given a homomorphism from Z^n to Z^k and a sublattice of Z^k, compute the preimage of this sublattice under this homomorphism. More...  
REGINA_API std::unique_ptr < NMatrixInt >  regina::torsionAutInverse (const NMatrixInt &input, const std::vector< NLargeInteger > &invF) 
Given an automorphism of an abelian group, this procedure computes the inverse automorphism. More...  
template<bool supportInfinity>  
std::ostream &  regina::operator<< (std::ostream &out, const NIntegerBase< supportInfinity > &i) 
Writes the given integer to the given output stream. More...  
template<bool supportInfinity>  
NIntegerBase< supportInfinity >  regina::operator+ (long lhs, const NIntegerBase< supportInfinity > &rhs) 
Adds the given native integer to the given large integer. More...  
template<bool supportInfinity>  
NIntegerBase< supportInfinity >  regina::operator* (long lhs, const NIntegerBase< supportInfinity > &rhs) 
Multiplies the given native integer with the given large integer. More...  
template<int bytes>  
std::ostream &  regina::operator<< (std::ostream &out, const NNativeInteger< bytes > &i) 
Writes the given integer to the given output stream. More...  
REGINA_API std::ostream &  regina::operator<< (std::ostream &out, const NMatrix2 &mat) 
Writes the given matrix to the given output stream. More...  
REGINA_API bool  regina::simpler (const NMatrix2 &m1, const NMatrix2 &m2) 
Determines whether the first given matrix is more aesthetically pleasing than the second. More...  
REGINA_API bool  regina::simpler (const NMatrix2 &pair1first, const NMatrix2 &pair1second, const NMatrix2 &pair2first, const NMatrix2 &pair2second) 
Determines whether the first given pair of matrices is more aesthetically pleasing than the second pair. More...  
constexpr char  regina::digit (int i) 
Returns the character used to express the integer i in a permutation. More...  
constexpr int64_t  regina::factorial (int n) 
Returns the factorial of n. More...  
template<int n>  
std::ostream &  regina::operator<< (std::ostream &out, const NPerm< n > &p) 
Writes a string representation of the given permutation to the given output stream. More...  
REGINA_API std::ostream &  regina::operator<< (std::ostream &out, const NRational &rat) 
Writes the given rational to the given output stream. More...  
REGINA_API long  regina::reducedMod (long k, long modBase) 
Reduces k modulo modBase to give the smallest possible absolute value. More...  
REGINA_API long  regina::gcd (long a, long b) 
Calculates the greatest common divisor of two signed integers. More...  
REGINA_API long  regina::gcdWithCoeffs (long a, long b, long &u, long &v) 
Calculates the greatest common divisor of two given integers and finds the smallest coefficients with which these integers combine to give their gcd. More...  
REGINA_API long  regina::lcm (long a, long b) 
Calculates the lowest common multiple of two signed integers. More...  
REGINA_API unsigned long  regina::modularInverse (unsigned long n, unsigned long k) 
Calculates the multiplicative inverse of one integer modulo another. More...  
REGINA_DEPRECATED REGINA_API void  regina::factorise (unsigned long n, std::list< unsigned long > &factors) 
Calculates the prime factorisation of the given integer. More...  
REGINA_DEPRECATED REGINA_API void  regina::primesUpTo (const NLargeInteger &roof, std::list< NLargeInteger > &primes) 
Determines all primes up to and including the given upper bound. More...  
template<class T >  
std::ostream &  regina::operator<< (std::ostream &out, const NVector< T > &vector) 
Writes the given vector to the given output stream. More...  
REGINA_DEPRECATED REGINA_API NPerm5  regina::perm4to5 (const NPerm4 &p) 
Deprecated routine that converts a 4element permutation to a 5element permutation. More...  
REGINA_DEPRECATED REGINA_API NPerm4  regina::perm5to4 (const NPerm5 &p) 
Deprecated routine that expresses the given 5element permutation as a 4element permutation. More...  
REGINA_DEPRECATED REGINA_API NPerm4  regina::perm3to4 (const NPerm3 &p) 
Deprecated routine that converts a 3element permutation to a 4element permutation. More...  
REGINA_DEPRECATED REGINA_API NPerm3  regina::perm4to3 (const NPerm4 &p) 
Deprecated routine that expresses the given 4element permutation as a 3element permutation. More...  
REGINA_DEPRECATED REGINA_API NPerm5  regina::perm3to5 (const NPerm3 &p) 
Deprecated routine that converts a 3element permutation to a 5element permutation. More...  
REGINA_DEPRECATED REGINA_API NPerm3  regina::perm5to3 (const NPerm5 &p) 
Deprecated routine that expresses the given 5element permutation as a 3element permutation. More...  
regina::NMatrixInt::NMatrixInt (unsigned long rows, unsigned long cols)  
Creates a new matrix of the given size. More...  
regina::NMatrixInt::NMatrixInt (const NMatrixInt &cloneMe)  
Creates a new matrix that is a clone of the given matrix. More...  
Variables  
REGINA_API const double  regina::epsilon 
A very small positive real designed to accommodate for rounding error. More...  
static T  regina::NMatrixRing< T >::zero 
Zero in the underlying ring. More...  
static T  regina::NMatrixRing< T >::one 
One (the multiplicative identity) in the underlying ring. More...  
static T  regina::NVector< T >::zero 
Zero in the underlying number system. More...  
static T  regina::NVector< T >::one 
One in the underlying number system. More...  
static T  regina::NVector< T >::minusOne 
Negative one in the underlying number system. More...  
Underlying mathematical gruntwork.
typedef NIntegerBase<false> regina::NInteger 
NInteger is a typedef for NIntegerBase<false>, which offers arbitrary precision integers without support for infinity.
typedef NIntegerBase<true> regina::NLargeInteger 
NLargeInteger is a typedef for NIntegerBase<true>, which offers arbitrary precision integers with support for infinity.
typedef NNativeInteger<sizeof(long)> regina::NNativeLong 
NNativeLong is a typedef for the NNativeInteger template class whose underlying integer type is a native long.
typedef NPerm<2> regina::NPerm2 
A convenience typedef for NPerm<2>.
typedef NPerm<3> regina::NPerm3 
A convenience typedef for NPerm<3>.
typedef NPerm<4> regina::NPerm4 
A convenience typedef for NPerm<4>.
typedef NPerm<5> regina::NPerm5 
A convenience typedef for NPerm<5>.
REGINA_API void regina::columnEchelonForm  (  NMatrixInt &  M, 
NMatrixInt &  R,  
NMatrixInt &  Ri,  
const std::vector< unsigned > &  rowList  
) 
Transforms a given matrix into column echelon form with respect to a collection of rows.
Given the matrix M and the list rowList of rows from M, this algorithm puts M in column echelon form with respect to the rows in rowList. The only purpose of rowList is to clarify and/or weaken precisely what is meant by "column echelon form"; all rows of M are affected by the resulting column operations that take place.
This routine also returns the corresponding change of coordinate matrices R and Ri:
original_M * R = final_M
and final_M * Ri = original_M
(and of course final_M
is in column echelon form with respect to the given row list).Our convention is that a matrix is in column echelon form if:
By a "zero column" here we simply mean "zero for every row in \a rowList". Likewise, by "first nonzero entry" we mean "first row in \a rowList with a nonzero entry".
In a pinch, you can also use this routine to compute the inverse of an invertible square matrix.
M  the matrix to reduce. 
R  used to return the rowreduction matrix, as described above. 
Ri  used to return the inverse of R. 
rowList  the rows to pay attention to. This list must contain distinct integers, all between 0 and M.rows()1 inclusive. The integers may appear in any order (though changing the order will change the resulting column echelon form). 

inline 
Returns the character used to express the integer i in a permutation.
the  integer to represent; this must be between 0 and 35 inclusive. 

inline 
Returns the factorial of n.
n  any nonnegative integer; this must be at most 20 (since otherwise the factorial will overflow). 
REGINA_DEPRECATED REGINA_API void regina::factorise  (  unsigned long  n, 
std::list< unsigned long > &  factors  
) 
Calculates the prime factorisation of the given integer.
All the prime factors will be inserted into the given list. The algorithm used is very neanderthal and should only be used with reasonably sized integers. Don't use it to do RSA!
If a prime factor is repeated, it will be inserted multiple times into the list. The primes in the list are not guaranteed to appear in any specific order, nor are multiple occurrences of the same prime guaranteed to appear together.
Note that once finished the list will contain the prime factors as well as whatever happened to be in the list before this function was called.
n  the integer to factorise. 
factors  the list into which prime factors will be inserted. 
REGINA_API long regina::gcd  (  long  a, 
long  b  
) 
Calculates the greatest common divisor of two signed integers.
This routine is not recursive.
Although the arguments may be negative, the result is guaranteed to be nonnegative. As a special case, gcd(0,0) is considered to be zero.
a  one of the two integers to work with. 
b  the other integer with which to work. 
REGINA_API long regina::gcdWithCoeffs  (  long  a, 
long  b,  
long &  u,  
long &  v  
) 
Calculates the greatest common divisor of two given integers and finds the smallest coefficients with which these integers combine to give their gcd.
This routine is not recursive.
Note that the given integers need not be nonnegative. However, the gcd returned is guaranteed to be nonnegative. As a special case, gcd(0,0) is considered to be zero.
If d is the gcd of a and b, the values placed in u and v will be those for which u*a + v*b = d
, abs(a)/d < v*sign(b) <= 0
and 1 <= u*sign(a) <= abs(b)/d
.
In the special case where one of the given integers is zero, the corresponding coefficient will also be zero and the other coefficient will be 1 or 1 so that u*a + v*b = d
still holds. If both given integers are zero, both of the coefficients will be set to zero.
a  one of the integers to work with. 
b  the other integer with which to work. 
u  a variable into which the final coefficient of a will be placed. 
v  a variable into which the final coefficient of b will be placed. 

inline 
Determines whether the given real number is strictly negative.
Any number within regina::epsilon of zero is considered to be zero.
x  the number to examine. 
true
if and only if the given number is strictly negative.

inline 
Determines whether the given real number is nonnegative.
Any number within regina::epsilon of zero is considered to be zero.
x  the number to examine. 
true
if and only if the given number is nonnegative.

inline 
Determines whether the given real number is nonpositive.
Any number within regina::epsilon of zero is considered to be zero.
x  the number to examine. 
true
if and only if the given number is nonpositive.

inline 
Determines whether the given real number is nonzero.
Any number within regina::epsilon of zero is considered to be zero.
x  the number to examine. 
true
if and only if the given number is approximately nonzero.

inline 
Determines whether the given real number is strictly positive.
Any number within regina::epsilon of zero is considered to be zero.
x  the number to examine. 
true
if and only if the given number is strictly positive.

inline 
Determines whether the given real number is zero.
Any number within regina::epsilon of zero is considered to be zero.
x  the number to examine. 
true
if and only if the given number is approximately zero. REGINA_API long regina::lcm  (  long  a, 
long  b  
) 
Calculates the lowest common multiple of two signed integers.
Although the arguments may be negative, the result is guaranteed to be nonnegative.
If either of the arguments is zero, the return value will also be zero.
Regarding possible overflow: This routine does not create any temporary integers that are larger than the final LCM.
a  one of the two integers to work with. 
b  the other integer with which to work. 
REGINA_API void regina::metricalSmithNormalForm  (  NMatrixInt &  matrix, 
NMatrixInt *  rowSpaceBasis = 0 , 

NMatrixInt *  rowSpaceBasisInv = 0 , 

NMatrixInt *  colSpaceBasis = 0 , 

NMatrixInt *  colSpaceBasisInv = 0 

) 
An alternative Smith normal form algorithm that also returns change of basis matrices.
This routine may be preferable for extremely large matrices. This is a variant of HafnerMcCurley and HavasHoltRees's description of pivoting methods.
The only input argument is matrix. The four remaining arguments (the change of basis matrices), if passed, will be refilled, though they must be constructed with the correct dimensions as seen in the preconditions below. All five arguments are used to return information as follows.
Let M be the initial value of matrix, and let S be the Smith normal form of M. After this routine exits:
colSpaceBasis * M * rowSpaceBasis = S
;colSpaceBasisInv * S * rowSpaceBasisInv = M
;colSpaceBasis * colSpaceBasisInv
and rowSpaceBasis * rowSpaceBasisInv
are both identity matrices.Thus, one obtains the Smith normal form the original matrix by multiplying on the left by ColSpaceBasis and on the right by RowSpaceBasis.
matrix  the original matrix to put into Smith Normal Form (this need not be square). When the algorithm terminates, this matrix is in its Smith Normal Form. 
rowSpaceBasis  used to return a change of basis matrix (see above for details). This is optional; you may pass a null pointer instead. 
rowSpaceBasisInv  used to return the inverse of rowSpaceBasis. This is optional; you may pass a null pointer instead. 
colSpaceBasis  used to return a change of basis matrix (see above for details). This is optional; you may pass a null pointer instead. 
colSpaceBasisInv  used to return the inverse of colSpaceBasis. This is optional; you may pass a null pointer instead. 
REGINA_API unsigned long regina::modularInverse  (  unsigned long  n, 
unsigned long  k  
) 
Calculates the multiplicative inverse of one integer modulo another.
The inverse returned will be between 0 and n1 inclusive.
n  the modular base in which to work. 
k  the number whose multiplicative inverse should be found. 
k * v == 1 (mod n)
.

inline 
Creates a new matrix of the given size.
All entries will be initialised to zero.
rows  the number of rows in the new matrix. 
cols  the number of columns in the new matrix. 

inline 
Creates a new matrix that is a clone of the given matrix.
cloneMe  the matrix to clone. 

inline 
Multiplies the given native integer with the given large integer.
If the large integer is infinite, the result will also be infinity.
lhs  the native integer to multiply. 
rhs  the large integer to multiply. 

inline 
Adds the given native integer to the given large integer.
If the large integer is infinite, the result will also be infinity.
lhs  the native integer to add. 
rhs  the large integer to add. 

inline 
Writes the given matrix to the given output stream.
The matrix will be written entirely on a single line, with the first row followed by the second row.
out  the output stream to which to write. 
mat  the matrix to write. 
std::ostream& regina::operator<<  (  std::ostream &  out, 
const NVector< T > &  vector  
) 
Writes the given vector to the given output stream.
The vector will be written on a single line with elements separated by a single space. No newline will be written.
out  the output stream to which to write. 
vector  the vector to write. 
REGINA_API std::ostream& regina::operator<<  (  std::ostream &  out, 
const NRational &  rat  
) 
Writes the given rational to the given output stream.
Infinity will be written as Inf
. Undefined will be written as Undef
. A rational with denominator one will be written as a single integer. All other rationals will be written in the form r/s
.
out  the output stream to which to write. 
rat  the rational to write. 

inline 
Writes a string representation of the given permutation to the given output stream.
The format will be the same as is used by NPerm::str().
out  the output stream to which to write. 
p  the permutation to write. 
n  the number of objects being permuted. This must be between 3 and 16 inclusive. 
std::ostream& regina::operator<<  (  std::ostream &  out, 
const NIntegerBase< supportInfinity > &  i  
) 
Writes the given integer to the given output stream.
out  the output stream to which to write. 
i  the integer to write. 

inline 
Writes the given integer to the given output stream.
out  the output stream to which to write. 
i  the integer to write. 

inline 
Deprecated routine that converts a 3element permutation to a 4element permutation.
The resulting 4element permutation will map 3 to 3, and will map 0, 1 and 2 to their respective images under p.
p  the given 3element permutation. 

inline 
Deprecated routine that converts a 3element permutation to a 5element permutation.
The resulting 5element permutation will map 3 to 3 and 4 to 4, and will map 0, 1 and 2 to their respective images under p.
p  the given 3element permutation. 

inline 
Deprecated routine that expresses the given 4element permutation as a 3element permutation.
The resulting 3element permutation will map 0, 1 and 2 to their respective images under p. It is assumed that the image of 3 is 3 under p; otherwise this conversion cannot be performed.
p  the given 4element permutation. 

inline 
Deprecated routine that converts a 4element permutation to a 5element permutation.
The resulting 5element permutation will map 4 to 4, and will map 0, 1, 2 and 3 to their respective images under p.
p  the given 4element permutation. 

inline 
Deprecated routine that expresses the given 5element permutation as a 3element permutation.
The resulting 3element permutation will map 0, 1 and 2 to their respective images under p. It is assumed that maps
3,4 to 3,4 (but it may do so in either order); otherwise this conversion cannot be performed.
p  the given 5element permutation. 

inline 
Deprecated routine that expresses the given 5element permutation as a 4element permutation.
The resulting 4element permutation will map 0, 1, 2 and 3 to their respective images under p. It is assumed that the image of 4 is 4 under p; otherwise this conversion cannot be performed.
p  the given 5element permutation. 
REGINA_API std::unique_ptr<NMatrixInt> regina::preImageOfLattice  (  const NMatrixInt &  hom, 
const std::vector< NLargeInteger > &  sublattice  
) 
Given a homomorphism from Z^n to Z^k and a sublattice of Z^k, compute the preimage of this sublattice under this homomorphism.
The homomorphism from Z^n to Z^k is described by the given k by n matrix hom. The sublattice is of the form (p1 Z) * (p2 Z) * ... * (pk Z)
, where the nonnegative integers p1, ..., pk are passed in the given list sublattice.
An equivalent problem is to consider hom to be a homomorphism from Z^n to Z_p1 + ... + Z_pk; this routine then finds the kernel of this homomorphism.
The preimage of the sublattice (equivalently, the kernel described above) is some rank n lattice in Z^n. This algorithm finds and returns a basis for the lattice.
hom  the matrix representing the homomorphism from Z^n to Z^k; this must be a k by n matrix. 
sublattice  a list of length k describing the sublattice of Z^k; the elements of this list must be the nonnegative integers p1, ..., pk as described above. 
REGINA_DEPRECATED REGINA_API void regina::primesUpTo  (  const NLargeInteger &  roof, 
std::list< NLargeInteger > &  primes  
) 
Determines all primes up to and including the given upper bound.
All the primes found will be inserted into the given list in increasing order.
The algorithm currently used is fairly neanderthal.
roof  the upper bound up to which primes will be found. 
primes  the list into which the primes will be inserted. 
REGINA_API long regina::reducedMod  (  long  k, 
long  modBase  
) 
Reduces k modulo modBase to give the smallest possible absolute value.
For instance, reducedMod(4,10) = 4
but reducedMod(6,10) = 4
. In the case of a tie, the positive solution is taken.
k  the number to reduce modulo modBase. 
modBase  the modular base in which to work. 
REGINA_API unsigned regina::rowBasis  (  NMatrixInt &  matrix  ) 
Find a basis for the row space of the given matrix.
This routine will rearrange the rows of the given matrix so that the first rank rows form a basis for the row space (where rank is the rank of the matrix). The rank itself will be returned. No other changes will be made to the matrix aside from swapping rows.
Although this routine takes an integer matrix (and only uses integer operations), we consider the row space to be over the rationals. That is, although we never divide, we act as though we could if we wanted to.
matrix  the matrix to examine and rearrange. 
REGINA_API unsigned regina::rowBasisAndOrthComp  (  NMatrixInt &  input, 
NMatrixInt &  complement  
) 
Finds a basis for the row space of the given matrix, as well as an "incremental" basis for its orthogonal complement.
This routine takes an (r by c) matrix input, as well as a square (c by c) matrix complement, and does the following:
This routine can help with larger procedures that need to build up a row space and simultaneously cut down the complement one dimension at a time.
Although this routine takes integer matrices (and only uses integer operations), we consider all bases to be over the rationals. That is, although we never divide, we act as though we could if we wanted to.
input  the input matrix whose row space we will describe; this matrix will be changed (though only by swapping rows). 
complement  the square matrix that will be refilled with the "incremental" basis for the orthogonal complement of input. 
REGINA_API bool regina::simpler  (  const NMatrix2 &  m1, 
const NMatrix2 &  m2  
) 
Determines whether the first given matrix is more aesthetically pleasing than the second.
The way in which this judgement is made is purely aesthetic on the part of the author, and is subject to change in future versions of Regina.
m1  the first matrix to examine. 
m2  the second matrix to examine. 
true
if m1 is deemed to be more pleasing than m2, or false
if either the matrices are equal or m2 is more pleasing than m1. REGINA_API bool regina::simpler  (  const NMatrix2 &  pair1first, 
const NMatrix2 &  pair1second,  
const NMatrix2 &  pair2first,  
const NMatrix2 &  pair2second  
) 
Determines whether the first given pair of matrices is more aesthetically pleasing than the second pair.
The way in which this judgement is made is purely aesthetic on the part of the author, and is subject to change in future versions of Regina.
Note that pairs are ordered, so the pair (M, N) may be more (or perhaps less) pleasing than the pair (N, M).
pair1first  the first matrix of the first pair to examine. 
pair1second  the second matrix of the first pair to examine. 
pair2first  the first matrix of the second pair to examine. 
pair2second  the second matrix of the second pair to examine. 
true
if the first pair is deemed to be more pleasing than the second pair, or false
if either the ordered pairs are equal or the second pair is more pleasing than the first. REGINA_API void regina::smithNormalForm  (  NMatrixInt &  matrix  ) 
Transforms the given integer matrix into Smith normal form.
Note that the given matrix need not be square and need not be of full rank.
Reading down the diagonal, the final Smith normal form will have a series of nonnegative, nondecreasing invariant factors followed by zeroes. "Invariant factor" refers to the convention that the ith term divides the (i+1)th term, and so they are unique.
The algorithm used is due to Hafner and McCurley (1991). It does not use modular arithmetic to control the intermediate coefficient explosion.
matrix  the matrix to transform. 
REGINA_API void regina::smithNormalForm  (  NMatrixInt &  matrix, 
NMatrixInt &  rowSpaceBasis,  
NMatrixInt &  rowSpaceBasisInv,  
NMatrixInt &  colSpaceBasis,  
NMatrixInt &  colSpaceBasisInv  
) 
A Smith normal form algorithm that also returns change of basis matrices.
This is a modification of the oneargument smithNormalForm(NMatrixInt&). As well as converting the given matrix matrix into Smith normal form, it also returns the appropriate changeofbasis matrices corresponding to all the row and column operations that were performed.
The only input argument is matrix. The four remaining arguments (the change of basis matrices) will be refilled, though they must be constructed with the correct dimensions as seen in the preconditions below. All five arguments are used to return information as follows.
Let M be the initial value of matrix, and let S be the Smith normal form of M. After this routine exits:
colSpaceBasis * M * rowSpaceBasis = S
;colSpaceBasisInv * S * rowSpaceBasisInv = M
;colSpaceBasis * colSpaceBasisInv
and rowSpaceBasis * rowSpaceBasisInv
are both identity matrices.Thus, one obtains the Smith normal form the original matrix by multiplying on the left by ColSpaceBasis and on the right by RowSpaceBasis.
matrix  the original matrix to put into Smith Normal Form (this need not be square). When the algorithm terminates, this matrix is in its Smith Normal Form. 
rowSpaceBasis  used to return a change of basis matrix (see above for details). 
rowSpaceBasisInv  used to return the inverse of rowSpaceBasis. 
colSpaceBasis  used to return a change of basis matrix (see above for details). 
colSpaceBasisInv  used to return the inverse of colSpaceBasis. 
REGINA_API std::unique_ptr<NMatrixInt> regina::torsionAutInverse  (  const NMatrixInt &  input, 
const std::vector< NLargeInteger > &  invF  
) 
Given an automorphism of an abelian group, this procedure computes the inverse automorphism.
The abelian group is of the form Z_p1 + Z_p2 + ... + Z_pn
. The input is an nbyn matrix A which represents a lift of the automorphism to just some nbyn matrix. Specifically, you have a little commutative diagram with Z^n –A–> Z^n
covering the automorphism of Z_p1 + Z_p2 + ... + Z_pn
, where the maps down are the direct sum of the standard quotients Z –> Z_pi
. So if you want this procedure to give you meaningful output, A must be a lift of a genuine automorphism of Z_p1 + ... + Z_pn
.
input  the nbyn matrix A, which must be a lift of a genuine automorphism as described above. 
invF  the list p1, p2, ..., pn. 
REGINA_API const double regina::epsilon 
A very small positive real designed to accommodate for rounding error.
Any two numbers within epsilon of each other are considered to be equal by the generic zerotesting and signtesting routines defined in this file (isZero(), isPositive(), isNonNegative() and so on).

static 
Negative one in the underlying number system.
This would be const
if it weren't for the fact that some compilers don't like this. It should never be modified!

static 
One in the underlying number system.
This would be const
if it weren't for the fact that some compilers don't like this. It should never be modified!

static 
One (the multiplicative identity) in the underlying ring.
This would be const
if it weren't for the fact that some compilers don't like this. It should never be modified!

static 
Zero in the underlying number system.
This would be const
if it weren't for the fact that some compilers don't like this. It should never be modified!

static 
Zero in the underlying ring.
This would be const
if it weren't for the fact that some compilers don't like this. It should never be modified!