Quellcode-Bibliothek PaStiXSupport.h Sprache: C

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2012 Désiré Nuentsa-Wakam <desire.nuentsa_wakam@inria.fr>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_PASTIXSUPPORT_H
#define EIGEN_PASTIXSUPPORT_H

namespace Eigen {

#if defined(DCOMPLEX)
  #define PASTIX_COMPLEX  COMPLEX
  #define PASTIX_DCOMPLEX DCOMPLEX
#else
  #define PASTIX_COMPLEX  std::complex<float>
  #define PASTIX_DCOMPLEX std::complex<double>
#endif

/** \ingroup PaStiXSupport_Module
  * \brief Interface to the PaStix solver
  *
  * This class is used to solve the linear systems A.X = B via the PaStix library.
  * The matrix can be either real or complex, symmetric or not.
  *
  * \sa TutorialSparseDirectSolvers
  */
template<typename _MatrixType, bool IsStrSym = false> class PastixLU;
template<typename _MatrixType, int Options> class PastixLLT;
template<typename _MatrixType, int Options> class PastixLDLT;

namespace internal
{

  template<class Pastix> struct pastix_traits;

  template<typename _MatrixType>
  struct pastix_traits< PastixLU<_MatrixType> >
  {
    typedef _MatrixType MatrixType;
    typedef typename _MatrixType::Scalar Scalar;
    typedef typename _MatrixType::RealScalar RealScalar;
    typedef typename _MatrixType::StorageIndex StorageIndex;
  };

  template<typename _MatrixType, int Options>
  struct pastix_traits< PastixLLT<_MatrixType,Options> >
  {
    typedef _MatrixType MatrixType;
    typedef typename _MatrixType::Scalar Scalar;
    typedef typename _MatrixType::RealScalar RealScalar;
    typedef typename _MatrixType::StorageIndex StorageIndex;
  };

  template<typename _MatrixType, int Options>
  struct pastix_traits< PastixLDLT<_MatrixType,Options> >
  {
    typedef _MatrixType MatrixType;
    typedef typename _MatrixType::Scalar Scalar;
    typedef typename _MatrixType::RealScalar RealScalar;
    typedef typename _MatrixType::StorageIndex StorageIndex;
  };

  inline void eigen_pastix(pastix_data_t **pastix_data, int pastix_comm, int n, int *ptr, int *idx, float *vals, int *perm, int * invp, float *x, int nbrhs, int *iparm, double *dparm)
  {
    if (n == 0) { ptr = NULL; idx = NULL; vals = NULL; }
    if (nbrhs == 0) {x = NULL; nbrhs=1;}
    s_pastix(pastix_data, pastix_comm, n, ptr, idx, vals, perm, invp, x, nbrhs, iparm, dparm);
  }

  inline void eigen_pastix(pastix_data_t **pastix_data, int pastix_comm, int n, int *ptr, int *idx, double *vals, int *perm, int * invp, double *x, int nbrhs, int *iparm, double *dparm)
  {
    if (n == 0) { ptr = NULL; idx = NULL; vals = NULL; }
    if (nbrhs == 0) {x = NULL; nbrhs=1;}
    d_pastix(pastix_data, pastix_comm, n, ptr, idx, vals, perm, invp, x, nbrhs, iparm, dparm);
  }

  inline void eigen_pastix(pastix_data_t **pastix_data, int pastix_comm, int n, int *ptr, int *idx, std::complex<float> *vals, int *perm, int * invp, std::complex<float> *x, int nbrhs, int *iparm, double *dparm)
  {
    if (n == 0) { ptr = NULL; idx = NULL; vals = NULL; }
    if (nbrhs == 0) {x = NULL; nbrhs=1;}
    c_pastix(pastix_data, pastix_comm, n, ptr, idx, reinterpret_cast<PASTIX_COMPLEX*>(vals), perm, invp, reinterpret_cast<PASTIX_COMPLEX*>(x), nbrhs, iparm, dparm);
  }

  inline void eigen_pastix(pastix_data_t **pastix_data, int pastix_comm, int n, int *ptr, int *idx, std::complex<double> *vals, int *perm, int * invp, std::complex<double> *x, int nbrhs, int *iparm, double *dparm)
  {
    if (n == 0) { ptr = NULL; idx = NULL; vals = NULL; }
    if (nbrhs == 0) {x = NULL; nbrhs=1;}
    z_pastix(pastix_data, pastix_comm, n, ptr, idx, reinterpret_cast<PASTIX_DCOMPLEX*>(vals), perm, invp, reinterpret_cast<PASTIX_DCOMPLEX*>(x), nbrhs, iparm, dparm);
  }

  // Convert the matrix  to Fortran-style Numbering
  template <typename MatrixType>
  void c_to_fortran_numbering (MatrixType& mat)
  {
    if ( !(mat.outerIndexPtr()[0]) )
    {
      int i;
      for(i = 0; i <= mat.rows(); ++i)
        ++mat.outerIndexPtr()[i];
      for(i = 0; i < mat.nonZeros(); ++i)
        ++mat.innerIndexPtr()[i];
    }
  }

  // Convert to C-style Numbering
  template <typename MatrixType>
  void fortran_to_c_numbering (MatrixType& mat)
  {
    // Check the Numbering
    if ( mat.outerIndexPtr()[0] == 1 )
    { // Convert to C-style numbering
      int i;
      for(i = 0; i <= mat.rows(); ++i)
        --mat.outerIndexPtr()[i];
      for(i = 0; i < mat.nonZeros(); ++i)
        --mat.innerIndexPtr()[i];
    }
  }
}

// This is the base class to interface with PaStiX functions.
// Users should not used this class directly.
template <class Derived>
class PastixBase : public SparseSolverBase<Derived>
{
  protected:
    typedef SparseSolverBase<Derived> Base;
    using Base::derived;
    using Base::m_isInitialized;
  public:
    using Base::_solve_impl;

    typedef typename internal::pastix_traits<Derived>::MatrixType _MatrixType;
    typedef _MatrixType MatrixType;
    typedef typename MatrixType::Scalar Scalar;
    typedef typename MatrixType::RealScalar RealScalar;
    typedef typename MatrixType::StorageIndex StorageIndex;
    typedef Matrix<Scalar,Dynamic,1> Vector;
    typedef SparseMatrix<Scalar, ColMajor> ColSpMatrix;
    enum {
      ColsAtCompileTime = MatrixType::ColsAtCompileTime,
      MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
    };

  public:

    PastixBase() : m_initisOk(false), m_analysisIsOk(false), m_factorizationIsOk(false), m_pastixdata(0), m_size(0)
    {
      init();
    }

    ~PastixBase()
    {
      clean();
    }

    template<typename Rhs,typename Dest>
    bool _solve_impl(const MatrixBase<Rhs> &b, MatrixBase<Dest> &x) const;

    /** Returns a reference to the integer vector IPARM of PaStiX parameters
      * to modify the default parameters.
      * The statistics related to the different phases of factorization and solve are saved here as well
      * \sa analyzePattern() factorize()
      */
    Array<StorageIndex,IPARM_SIZE,1>& iparm()
    {
      return m_iparm;
    }

    /** Return a reference to a particular index parameter of the IPARM vector
     * \sa iparm()
     */

    int& iparm(int idxparam)
    {
      return m_iparm(idxparam);
    }

     /** Returns a reference to the double vector DPARM of PaStiX parameters
      * The statistics related to the different phases of factorization and solve are saved here as well
      * \sa analyzePattern() factorize()
      */
    Array<double,DPARM_SIZE,1>& dparm()
    {
      return m_dparm;
    }


    /** Return a reference to a particular index parameter of the DPARM vector
     * \sa dparm()
     */
    double& dparm(int idxparam)
    {
      return m_dparm(idxparam);
    }

    inline Index cols() const { return m_size; }
    inline Index rows() const { return m_size; }

     /** \brief Reports whether previous computation was successful.
      *
      * \returns \c Success if computation was successful,
      *          \c NumericalIssue if the PaStiX reports a problem
      *          \c InvalidInput if the input matrix is invalid
      *
      * \sa iparm()
      */
    ComputationInfo info() const
    {
      eigen_assert(m_isInitialized && "Decomposition is not initialized.");
      return m_info;
    }

  protected:

    // Initialize the Pastix data structure, check the matrix
    void init();

    // Compute the ordering and the symbolic factorization
    void analyzePattern(ColSpMatrix& mat);

    // Compute the numerical factorization
    void factorize(ColSpMatrix& mat);

    // Free all the data allocated by Pastix
    void clean()
    {
      eigen_assert(m_initisOk && "The Pastix structure should be allocated first");
      m_iparm(IPARM_START_TASK) = API_TASK_CLEAN;
      m_iparm(IPARM_END_TASK) = API_TASK_CLEAN;
      internal::eigen_pastix(&m_pastixdata, MPI_COMM_WORLD, 0, 0, 0, (Scalar*)0,
                             m_perm.data(), m_invp.data(), 0, 0, m_iparm.data(), m_dparm.data());
    }

    void compute(ColSpMatrix& mat);

    int m_initisOk;
    int m_analysisIsOk;
    int m_factorizationIsOk;
    mutable ComputationInfo m_info;
    mutable pastix_data_t *m_pastixdata; // Data structure for pastix
    mutable int m_comm; // The MPI communicator identifier
    mutable Array<int,IPARM_SIZE,1> m_iparm; // integer vector for the input parameters
    mutable Array<double,DPARM_SIZE,1> m_dparm; // Scalar vector for the input parameters
    mutable Matrix<StorageIndex,Dynamic,1> m_perm;  // Permutation vector
    mutable Matrix<StorageIndex,Dynamic,1> m_invp;  // Inverse permutation vector
    mutable int m_size; // Size of the matrix
};

/** Initialize the PaStiX data structure.
   *A first call to this function fills iparm and dparm with the default PaStiX parameters
   * \sa iparm() dparm()
   */
template <class Derived>
void PastixBase<Derived>::init()
{
  m_size = 0;
  m_iparm.setZero(IPARM_SIZE);
  m_dparm.setZero(DPARM_SIZE);

  m_iparm(IPARM_MODIFY_PARAMETER) = API_NO;
  pastix(&m_pastixdata, MPI_COMM_WORLD,
         0, 0, 0, 0,
         0, 0, 0, 1, m_iparm.data(), m_dparm.data());

  m_iparm[IPARM_MATRIX_VERIFICATION] = API_NO;
  m_iparm[IPARM_VERBOSE]             = API_VERBOSE_NOT;
  m_iparm[IPARM_ORDERING]            = API_ORDER_SCOTCH;
  m_iparm[IPARM_INCOMPLETE]          = API_NO;
  m_iparm[IPARM_OOC_LIMIT]           = 2000;
  m_iparm[IPARM_RHS_MAKING]          = API_RHS_B;
  m_iparm(IPARM_MATRIX_VERIFICATION) = API_NO;

  m_iparm(IPARM_START_TASK) = API_TASK_INIT;
  m_iparm(IPARM_END_TASK) = API_TASK_INIT;
  internal::eigen_pastix(&m_pastixdata, MPI_COMM_WORLD, 0, 0, 0, (Scalar*)0,
                         0, 0, 0, 0, m_iparm.data(), m_dparm.data());

  // Check the returned error
  if(m_iparm(IPARM_ERROR_NUMBER)) {
    m_info = InvalidInput;
    m_initisOk = false;
  }
  else {
    m_info = Success;
    m_initisOk = true;
  }
}

template <class Derived>
void PastixBase<Derived>::compute(ColSpMatrix& mat)
{
  eigen_assert(mat.rows() == mat.cols() && "The input matrix should be squared");

  analyzePattern(mat);
  factorize(mat);

  m_iparm(IPARM_MATRIX_VERIFICATION) = API_NO;
}

template <class Derived>
void PastixBase<Derived>::analyzePattern(ColSpMatrix& mat)
{
  eigen_assert(m_initisOk && "The initialization of PaSTiX failed");

  // clean previous calls
  if(m_size>0)
    clean();

  m_size = internal::convert_index<int>(mat.rows());
  m_perm.resize(m_size);
  m_invp.resize(m_size);

  m_iparm(IPARM_START_TASK) = API_TASK_ORDERING;
  m_iparm(IPARM_END_TASK) = API_TASK_ANALYSE;
  internal::eigen_pastix(&m_pastixdata, MPI_COMM_WORLD, m_size, mat.outerIndexPtr(), mat.innerIndexPtr(),
               mat.valuePtr(), m_perm.data(), m_invp.data(), 0, 0, m_iparm.data(), m_dparm.data());

  // Check the returned error
  if(m_iparm(IPARM_ERROR_NUMBER))
  {
    m_info = NumericalIssue;
    m_analysisIsOk = false;
  }
  else
  {
    m_info = Success;
    m_analysisIsOk = true;
  }
}

template <class Derived>
void PastixBase<Derived>::factorize(ColSpMatrix& mat)
{
//   if(&m_cpyMat != &mat) m_cpyMat = mat;
  eigen_assert(m_analysisIsOk && "The analysis phase should be called before the factorization phase");
  m_iparm(IPARM_START_TASK) = API_TASK_NUMFACT;
  m_iparm(IPARM_END_TASK) = API_TASK_NUMFACT;
  m_size = internal::convert_index<int>(mat.rows());

  internal::eigen_pastix(&m_pastixdata, MPI_COMM_WORLD, m_size, mat.outerIndexPtr(), mat.innerIndexPtr(),
               mat.valuePtr(), m_perm.data(), m_invp.data(), 0, 0, m_iparm.data(), m_dparm.data());

  // Check the returned error
  if(m_iparm(IPARM_ERROR_NUMBER))
  {
    m_info = NumericalIssue;
    m_factorizationIsOk = false;
    m_isInitialized = false;
  }
  else
  {
    m_info = Success;
    m_factorizationIsOk = true;
    m_isInitialized = true;
  }
}

/* Solve the system */
template<typename Base>
template<typename Rhs,typename Dest>
bool PastixBase<Base>::_solve_impl(const MatrixBase<Rhs> &b, MatrixBase<Dest> &x) const
{
  eigen_assert(m_isInitialized && "The matrix should be factorized first");
  EIGEN_STATIC_ASSERT((Dest::Flags&RowMajorBit)==0,
                     THIS_METHOD_IS_ONLY_FOR_COLUMN_MAJOR_MATRICES);
  int rhs = 1;

  x = b; /* on return, x is overwritten by the computed solution */

  for (int i = 0; i < b.cols(); i++){
    m_iparm[IPARM_START_TASK]          = API_TASK_SOLVE;
    m_iparm[IPARM_END_TASK]            = API_TASK_REFINE;

    internal::eigen_pastix(&m_pastixdata, MPI_COMM_WORLD, internal::convert_index<int>(x.rows()), 0, 0, 0,
                           m_perm.data(), m_invp.data(), &x(0, i), rhs, m_iparm.data(), m_dparm.data());
  }

  // Check the returned error
  m_info = m_iparm(IPARM_ERROR_NUMBER)==0 ? Success : NumericalIssue;

  return m_iparm(IPARM_ERROR_NUMBER)==0;
}

/** \ingroup PaStiXSupport_Module
  * \class PastixLU
  * \brief Sparse direct LU solver based on PaStiX library
  *
  * This class is used to solve the linear systems A.X = B with a supernodal LU
  * factorization in the PaStiX library. The matrix A should be squared and nonsingular
  * PaStiX requires that the matrix A has a symmetric structural pattern.
  * This interface can symmetrize the input matrix otherwise.
  * The vectors or matrices X and B can be either dense or sparse.
  *
  * \tparam _MatrixType the type of the sparse matrix A, it must be a SparseMatrix<>
  * \tparam IsStrSym Indicates if the input matrix has a symmetric pattern, default is false
  * NOTE : Note that if the analysis and factorization phase are called separately,
  * the input matrix will be symmetrized at each call, hence it is advised to
  * symmetrize the matrix in a end-user program and set \p IsStrSym to true
  *
  * \implsparsesolverconcept
  *
  * \sa \ref TutorialSparseSolverConcept, class SparseLU
  *
  */
template<typename _MatrixType, bool IsStrSym>
class PastixLU : public PastixBase< PastixLU<_MatrixType> >
{
  public:
    typedef _MatrixType MatrixType;
    typedef PastixBase<PastixLU<MatrixType> > Base;
    typedef typename Base::ColSpMatrix ColSpMatrix;
    typedef typename MatrixType::StorageIndex StorageIndex;

  public:
    PastixLU() : Base()
    {
      init();
    }

    explicit PastixLU(const MatrixType& matrix):Base()
    {
      init();
      compute(matrix);
    }
    /** Compute the LU supernodal factorization of \p matrix.
      * iparm and dparm can be used to tune the PaStiX parameters.
      * see the PaStiX user's manual
      * \sa analyzePattern() factorize()
      */
    void compute (const MatrixType& matrix)
    {
      m_structureIsUptodate = false;
      ColSpMatrix temp;
      grabMatrix(matrix, temp);
      Base::compute(temp);
    }
    /** Compute the LU symbolic factorization of \p matrix using its sparsity pattern.
      * Several ordering methods can be used at this step. See the PaStiX user's manual.
      * The result of this operation can be used with successive matrices having the same pattern as \p matrix
      * \sa factorize()
      */
    void analyzePattern(const MatrixType& matrix)
    {
      m_structureIsUptodate = false;
      ColSpMatrix temp;
      grabMatrix(matrix, temp);
      Base::analyzePattern(temp);
    }

    /** Compute the LU supernodal factorization of \p matrix
      * WARNING The matrix \p matrix should have the same structural pattern
      * as the same used in the analysis phase.
      * \sa analyzePattern()
      */
    void factorize(const MatrixType& matrix)
    {
      ColSpMatrix temp;
      grabMatrix(matrix, temp);
      Base::factorize(temp);
    }
  protected:

    void init()
    {
      m_structureIsUptodate = false;
      m_iparm(IPARM_SYM) = API_SYM_NO;
      m_iparm(IPARM_FACTORIZATION) = API_FACT_LU;
    }

    void grabMatrix(const MatrixType& matrix, ColSpMatrix& out)
    {
      if(IsStrSym)
        out = matrix;
      else
      {
        if(!m_structureIsUptodate)
        {
          // update the transposed structure
          m_transposedStructure = matrix.transpose();

          // Set the elements of the matrix to zero
          for (Index j=0; j<m_transposedStructure.outerSize(); ++j)
            for(typename ColSpMatrix::InnerIterator it(m_transposedStructure, j); it; ++it)
              it.valueRef() = 0.0;

          m_structureIsUptodate = true;
        }

        out = m_transposedStructure + matrix;
      }
      internal::c_to_fortran_numbering(out);
    }

    using Base::m_iparm;
    using Base::m_dparm;

    ColSpMatrix m_transposedStructure;
    bool m_structureIsUptodate;
};

/** \ingroup PaStiXSupport_Module
  * \class PastixLLT
  * \brief A sparse direct supernodal Cholesky (LLT) factorization and solver based on the PaStiX library
  *
  * This class is used to solve the linear systems A.X = B via a LL^T supernodal Cholesky factorization
  * available in the PaStiX library. The matrix A should be symmetric and positive definite
  * WARNING Selfadjoint complex matrices are not supported in the current version of PaStiX
  * The vectors or matrices X and B can be either dense or sparse
  *
  * \tparam MatrixType the type of the sparse matrix A, it must be a SparseMatrix<>
  * \tparam UpLo The part of the matrix to use : Lower or Upper. The default is Lower as required by PaStiX
  *
  * \implsparsesolverconcept
  *
  * \sa \ref TutorialSparseSolverConcept, class SimplicialLLT
  */
template<typename _MatrixType, int _UpLo>
class PastixLLT : public PastixBase< PastixLLT<_MatrixType, _UpLo> >
{
  public:
    typedef _MatrixType MatrixType;
    typedef PastixBase<PastixLLT<MatrixType, _UpLo> > Base;
    typedef typename Base::ColSpMatrix ColSpMatrix;

  public:
    enum { UpLo = _UpLo };
    PastixLLT() : Base()
    {
      init();
    }

    explicit PastixLLT(const MatrixType& matrix):Base()
    {
      init();
      compute(matrix);
    }

    /** Compute the L factor of the LL^T supernodal factorization of \p matrix
      * \sa analyzePattern() factorize()
      */
    void compute (const MatrixType& matrix)
    {
      ColSpMatrix temp;
      grabMatrix(matrix, temp);
      Base::compute(temp);
    }

     /** Compute the LL^T symbolic factorization of \p matrix using its sparsity pattern
      * The result of this operation can be used with successive matrices having the same pattern as \p matrix
      * \sa factorize()
      */
    void analyzePattern(const MatrixType& matrix)
    {
      ColSpMatrix temp;
      grabMatrix(matrix, temp);
      Base::analyzePattern(temp);
    }
      /** Compute the LL^T supernodal numerical factorization of \p matrix
        * \sa analyzePattern()
        */
    void factorize(const MatrixType& matrix)
    {
      ColSpMatrix temp;
      grabMatrix(matrix, temp);
      Base::factorize(temp);
    }
  protected:
    using Base::m_iparm;

    void init()
    {
      m_iparm(IPARM_SYM) = API_SYM_YES;
      m_iparm(IPARM_FACTORIZATION) = API_FACT_LLT;
    }

    void grabMatrix(const MatrixType& matrix, ColSpMatrix& out)
    {
      out.resize(matrix.rows(), matrix.cols());
      // Pastix supports only lower, column-major matrices
      out.template selfadjointView<Lower>() = matrix.template selfadjointView<UpLo>();
      internal::c_to_fortran_numbering(out);
    }
};

/** \ingroup PaStiXSupport_Module
  * \class PastixLDLT
  * \brief A sparse direct supernodal Cholesky (LLT) factorization and solver based on the PaStiX library
  *
  * This class is used to solve the linear systems A.X = B via a LDL^T supernodal Cholesky factorization
  * available in the PaStiX library. The matrix A should be symmetric and positive definite
  * WARNING Selfadjoint complex matrices are not supported in the current version of PaStiX
  * The vectors or matrices X and B can be either dense or sparse
  *
  * \tparam MatrixType the type of the sparse matrix A, it must be a SparseMatrix<>
  * \tparam UpLo The part of the matrix to use : Lower or Upper. The default is Lower as required by PaStiX
  *
  * \implsparsesolverconcept
  *
  * \sa \ref TutorialSparseSolverConcept, class SimplicialLDLT
  */
template<typename _MatrixType, int _UpLo>
class PastixLDLT : public PastixBase< PastixLDLT<_MatrixType, _UpLo> >
{
  public:
    typedef _MatrixType MatrixType;
    typedef PastixBase<PastixLDLT<MatrixType, _UpLo> > Base;
    typedef typename Base::ColSpMatrix ColSpMatrix;

  public:
    enum { UpLo = _UpLo };
    PastixLDLT():Base()
    {
      init();
    }

    explicit PastixLDLT(const MatrixType& matrix):Base()
    {
      init();
      compute(matrix);
    }

    /** Compute the L and D factors of the LDL^T factorization of \p matrix
      * \sa analyzePattern() factorize()
      */
    void compute (const MatrixType& matrix)
    {
      ColSpMatrix temp;
      grabMatrix(matrix, temp);
      Base::compute(temp);
    }

    /** Compute the LDL^T symbolic factorization of \p matrix using its sparsity pattern
      * The result of this operation can be used with successive matrices having the same pattern as \p matrix
      * \sa factorize()
      */
    void analyzePattern(const MatrixType& matrix)
    {
      ColSpMatrix temp;
      grabMatrix(matrix, temp);
      Base::analyzePattern(temp);
    }
    /** Compute the LDL^T supernodal numerical factorization of \p matrix
      *
      */
    void factorize(const MatrixType& matrix)
    {
      ColSpMatrix temp;
      grabMatrix(matrix, temp);
      Base::factorize(temp);
    }

  protected:
    using Base::m_iparm;

    void init()
    {
      m_iparm(IPARM_SYM) = API_SYM_YES;
      m_iparm(IPARM_FACTORIZATION) = API_FACT_LDLT;
    }

    void grabMatrix(const MatrixType& matrix, ColSpMatrix& out)
    {
      // Pastix supports only lower, column-major matrices
      out.resize(matrix.rows(), matrix.cols());
      out.template selfadjointView<Lower>() = matrix.template selfadjointView<UpLo>();
      internal::c_to_fortran_numbering(out);
    }
};

} // end namespace Eigen

#endif

quality92%

¤ Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.0.3Bemerkung: (vorverarbeitet) ¤

*Bot Zugriff

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.