linbox

These are problem oriented drivers providing simple interfaces to the linear algebra algorithms. More...
Modules  
Characteristic polynomial  
NO DOC YET.  
Determinant  
NO DOC YET.  
Minimal polynomial  
NO DOC YET.  
Nullspace  
NO DOC YET.  
Rank  
NO DOC YET.  
Recuded forms  
NO DOC YET.  
System solving  
NO DOC YET.  
Files  
file  det.h 
NO DOC.  
file  methods.h 
NO DOC.  
file  solutiontags.h 
This allows files defining objects that have traits concerning several solutions to get the tags and traits with one inclusion.  
file  solve.h 
System Solving.  
Functions  
template<class Blackbox , class MyMethod >  
Blackbox::Field::Element &  lif_cra_det (typename Blackbox::Field::Element &d, const Blackbox &A, const RingCategories::IntegerTag &tag, const MyMethod &M) 
Compute the determinant of A over the integers. More...  
template<class Blackbox , class DetMethod , class DomainCategory >  
Blackbox::Field::Element &  det (typename Blackbox::Field::Element &d, const Blackbox &A, const DomainCategory &tag, const DetMethod &Meth) 
Compute the determinant of A. More...  
template<class Field >  
Field::Element &  detin (typename Field::Element &d, BlasMatrix< Field > &A) 
Rank of Blackbox A . More...  
template<class VecPoly , class Blackbox >  
vector< Poly< Field > > &  invariants (VecPol &S, const Blackbox< Field > &A, Method &M=Method::Hybrid) 
Compute the Frobenius form invariant factors of A. More...  
template<class Blackbox , class Method , class DomainCategory >  
unsigned long &  rank (unsigned long &r, const Blackbox &A, const DomainCategory &tag, const Method &M) 
Compute the rank of a linear transform A over a field by selected method. More...  
template<class Blackbox >  
unsigned long &  rank (unsigned long &r, const Blackbox &A) 
Compute the rank of a linear transform A over a field. More...  
template<class Blackbox , class Method >  
unsigned long &  rank (unsigned long &r, const Blackbox &A, const Method &M) 
Compute the rank of a linear transform A over a field. More...  
template<class Output , class Blackbox , class MyMethod >  
Output &  smithForm (Output &S, const Blackbox &A, const MyMethod &M) 
Compute the Smith form of A. More...  
template<class Blackbox , class MyMethod >  
Blackbox::Field::Element &  valence (typename Blackbox::Field::Element &v, const Blackbox &A, const MyMethod &M) 
Compute the valence of A. More...  
These are problem oriented drivers providing simple interfaces to the linear algebra algorithms.
The arguments to each are discussed in detail in the documentation of each function. The optional method parameter is discussed in general terms just below the list.
rank(r, A[, method])
r gets rank of A determinant(d, A[, method])
d gets determinant of A solve(x, A, b[, method[, status]])
get x such that Ax = b, random solution if sigular and consistent, x = 0 if inconsistent (also inconsistency is indicated in the status). The method parameter may include specifications that can make the computation faster. These include symmetry, nonsingularity, and that an arbitrary solution is acceptable rather than a random sample. minpoly(m, A[, method, status])
m gets minimal polynomial of A charpoly(c, A[, method, status])
c gets characteristic polynomial of A smithForm(S, A[, method])
S gets Smith Normal Form of A. Smith form is unitarily equivalent to A and is diag(s_{1} .. s_{n}) such that s_{i}  s_{i+1}.
frobeniusForm(F, A, method, status)
F gets Frobenius (Rational) normal form of A. Frobenius form is similar to A and is a direct sum of companion matrices C_{fi} such that, f_{i} divides f_{i+1}.
signature(s, A, method)
bool isPositiveDefinite(A, method_symm)
bool isPositiveSemidefinite(A, method_symm)
The algorithms are actively under development, and there is an algorithm choice to be made for each function. The choice is determined by an optional argument to the function which is a method object. The class of the method object determines the approach used and data in the object may help determine algorithm variants at run time. This makes it easy to try out and compare several approaches for a given matrix.
The first method classes to consider are the Blackbox, Elimination, and Hybrid classes. Blackbox
and Elimination
methods may be used for virtually every solution function.
Blackbox
, Elimination
, or a mix of them based on matrix properties checked at runtime. It is the default method. Thus for each problem (e.g. rank) the function call without a method, rank(r, A)
, is equivalent to a call using the default Hybrid object, rank(r, A, Method::Hybrid())
.Hybrid algorithms are under development and not in final form. The method used under the Hybrid designation may be a sophisticated hybrid algorithm, a very basic one, or simply our guess of the most widely useful of the available algorithms. The basic hybrid method is to use the elimination approach if the matrix is small or in a dense representation, and a blackbox method otherwise. A dense representation is one which inherits from DenseMatrix
or BlasMatrix
. Small means both row and column dimensions are less than 10^{3}. The threshold values can be changed in the hybrid method object. For instance
Additional method classes exist to focus in greater detail on the method. For instance, in the Blackbox
situation there are the Wiedemann, BlockWiedemann, Lanczos, and BlockLanczos approaches, to name a few. Actually, for each solution function, the Blackbox
class simply chooses from among these the best one as currently implemented. They may be used specifically to override that default. Thus rank(r, A,Blackbox())
uses the Wiedemann method. But rank(r, A, BlockLanczos())
could be called as well.
Also choice of preconditioner may be possible either at runtime by putting an indicator in the method object or at compile time by using a more special class of method such as BlockWiedemannButterfly.
Other method classes focus on algorithms attuned to specific matrix properties. For instance, there may be HybridSymmetric, EliminationSymmetric, and BlackboxSymmetric. Using these method choices, several functions are about twice as fast as those where the symmetry is unspecified. To get the same effect as a runtime choice, objects of the basic method classes may contain an indicator of symmetry,
Not every implemented algorithm is available through solution functions and their methods, but this is a good starting point. See algorithms for the other possibilities and LinBox::Method
for available methods.
If the status parameter is given, it is used for output of side information. For instance the solve function will set it's isConsistent()
property indicating whether or not the system was found to be consistent. Also an upper bound on the probability of error is given by errorBound()
. In the future, it may even provide consistency certificates and cofactors for normal forms.
Blackbox::Field::Element& LinBox::lif_cra_det  (  typename Blackbox::Field::Element &  d, 
const Blackbox &  A,  
const RingCategories::IntegerTag &  tag,  
const MyMethod &  M  
) 
Compute the determinant of A over the integers.
The determinant of a linear operator A, represented as a black box, is computed over the integers.
This variant is a hybrid between Last Invariant Factor and Chinese Remaindering It performs several determinants mod primes Then switches to the LIF method, producing a factor of det. It then comes back to the CRA if necessary to compute the remaining (usually small) factor of the determinant.
d  Field element into which to store the result 
A  Black box of which to compute the determinant 
tag  explicit over the integers 
M  may be a Method::BlasElimination (default) or a Method::Wiedemann. 
Blackbox::Field::Element& LinBox::det  (  typename Blackbox::Field::Element &  d, 
const Blackbox &  A,  
const DomainCategory &  tag,  
const DetMethod &  Meth  
) 
Compute the determinant of A.
The determinant of a linear operator A, represented as a black box, is computed over the ring or field of A.
d  Field element into which to store the result 
A  Black box of which to compute the determinant 
tag  optional tag. Specifies Integer, Rational or modular ring/field 
M  optional method. The default is Method::Hybrid(), Other options include Blackbox, Elimination, Wiedemann, BlasElimination and SparseElimination. Sometimes it helps to indicate properties of the matrix in the method object (for instance symmetry). See class Method for details. 
Field::Element& LinBox::detin  (  typename Field::Element &  d, 
BlasMatrix< Field > &  A  
) 
Rank of Blackbox A
.
A will be modified.
[out]  d  determinant of A . 
A  this BlasMatrix matrix will be modified in place in the process. 
d
vector<Poly<Field> >& LinBox::invariants  (  VecPol &  S, 
const Blackbox< Field > &  A,  
Method &  M = Method::Hybrid 

) 
Compute the Frobenius form invariant factors of A.
The first of them is the minimal polynomial, each divides the previous, and the product is the characteristic polynomial of A.
[out]  S  a vector of the invariant factors. 
A  Square matrix over a field.  
M  The default method is Hybrid. For sparse and structured matrices, a block blackbox method is used For dense matrices the approach is Krylov style. 

inline 
Compute the rank of a linear transform A over a field by selected method.
For very large and/or very sparse matrices the Wiedemann method will be faster (and it is memory efficient). For some sparse matrices SparseElimination may outperform Wiedemann. For small or dense matrices BlasElimination will be faster.
[out]  r  output rank of A. 
[in]  A  linear transform, member of any blackbox class. 
[in]  M  may be a Method::Hybrid (the default), a Method::Wiedemann , a Method::BlasElimination , or a Method::SparseElimination .. 
tag  UNDOC 

inline 
Compute the rank of a linear transform A over a field.
The default method is Wiedemann(), using diagonal preconditioning and the minpoly. For small or dense matrices BlasElimination will be faster.
A  linear transform, member of any blackbox class.  
[out]  r  rank of A 
r
rank of A
.

inline 
Compute the rank of a linear transform A over a field.
The default method is Wiedemann()
, using diagonal preconditioning and the minpoly. For small or dense matrices BlasElimination
will be faster.
r
rank of A
. A  linear transform, member of any blackbox class.  
[out]  r  rank of A 
M  method (see ???) 
Output& LinBox::smithForm  (  Output &  S, 
const Blackbox &  A,  
const MyMethod &  M  
) 
Compute the Smith form of A.
The Smith form of a linear operator A, represented as a black box, is computed over a representation of or .
[out]  S  a list of invariant/repcount pairs. 
A  Matrix of which to compute the Smith form  
M  may be a Method::Hybrid (default), which uses the algorithms/smithformadaptive. 
Blackbox::Field::Element& LinBox::valence  (  typename Blackbox::Field::Element &  v, 
const Blackbox &  A,  
const MyMethod &  M  
) 
Compute the valence of A.
The valence of a linear operator A, represented as a black box, is computed over the ring or field of A.
v  Field element into which to store the result 
A  Black box of which to compute the determinant 
M  may is a Method. 