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SMatrix.h
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1 // @(#)root/smatrix:$Id$
2 // Author: T. Glebe, L. Moneta, J. Palacios 2005
3 
4 #ifndef ROOT_Math_SMatrix
5 #define ROOT_Math_SMatrix
6 
7 /*********************************************************************************
8 //
9 // source:
10 //
11 // type: source code
12 //
13 // created: 20. Mar 2001
14 //
15 // author: Thorsten Glebe
16 // HERA-B Collaboration
17 // Max-Planck-Institut fuer Kernphysik
18 // Saupfercheckweg 1
19 // 69117 Heidelberg
20 // Germany
21 // E-mail: T.Glebe@mpi-hd.mpg.de
22 //
23 // Description: A fixed size two dimensional Matrix class
24 //
25 // changes:
26 // 20 Mar 2001 (TG) creation
27 // 21 Mar 2001 (TG) added operators +=, -=, *=, /=
28 // 26 Mar 2001 (TG) place_in_row(), place_in_col() added
29 // 02 Apr 2001 (TG) non-const Array() added
30 // 03 Apr 2001 (TG) invert() added
31 // 07 Apr 2001 (TG) CTOR from SVertex (dyadic product) added
32 // 09 Apr 2001 (TG) CTOR from array added
33 // 11 Apr 2001 (TG) rows(), cols(), size() replaced by rows, cols, size
34 // 25 Mai 2001 (TG) row(), col() added
35 // 04 Sep 2001 (TG) moved inlined functions to .icc file
36 // 11 Jan 2002 (TG) added operator==(), operator!=()
37 // 14 Jan 2002 (TG) added more operator==(), operator!=(), operator>(), operator<()
38 //
39 ***************************************************************************/
40 // for platform specific configurations
41 
42 #include "Math/MConfig.h"
43 
44 #include <iosfwd>
45 
46 
47 
48 
49 //doxygen tag
50 
51 /**
52  @defgroup SMatrixGroup SMatrix
53  @ingroup Math
54 
55  \ref SMatrixPage for high performance vector and matrix computations.
56  Classes representing Matrices and Vectors of arbitrary type and dimension and related functions.
57  For a detailed description and usage examples see:
58  <ul>
59  <li>\ref SMatrixPage home page
60  <li>\ref SVectorDoc
61  <li>\ref SMatrixDoc
62  <li>\ref MatVecFunctions
63  </ul>
64 
65 
66 */
67 
68 /**
69  @defgroup SMatrixSVector Matrix and Vector classes
70 
71  @ingroup SMatrixGroup
72 
73  Classes representing Matrices and Vectors of arbitrary type and dimension.
74  For a detailed description and usage examples see:
75  <ul>
76  <li>\ref SVectorDoc
77  <li>\ref SMatrixDoc
78  <li>\ref MatVecFunctions
79  </ul>
80 
81 */
82 
83 
84 #include "Math/Expression.h"
86 
87 
88 namespace ROOT {
89 
90 namespace Math {
91 
92 
93 template <class T, unsigned int D> class SVector;
94 
95 struct SMatrixIdentity { };
96 struct SMatrixNoInit { };
97 
98 //__________________________________________________________________________
99 /**
100  SMatrix: a generic fixed size D1 x D2 Matrix class.
101  The class is template on the scalar type, on the matrix sizes:
102  D1 = number of rows and D2 = number of columns
103  amd on the representation storage type.
104  By default the representation is MatRepStd<T,D1,D2> (standard D1xD2 of type T),
105  but it can be of type MatRepSym<T,D> for symmetric matrices DxD, where the storage is only
106  D*(D+1)/2.
107 
108  See \ref SMatrixDoc.
109 
110  Original author is Thorsten Glebe
111  HERA-B Collaboration, MPI Heidelberg (Germany)
112 
113  @ingroup SMatrixSVector
114 
115  @authors T. Glebe, L. Moneta and J. Palacios
116 */
117 //==============================================================================
118 // SMatrix: column-wise storage
119 //==============================================================================
120 template <class T,
121  unsigned int D1,
122  unsigned int D2 = D1,
123  class R=MatRepStd<T, D1, D2> >
124 class SMatrix {
125 public:
126  /** @name --- Typedefs --- */
127 
128  /** contained scalar type */
129  typedef T value_type;
130 
131  /** storage representation type */
132  typedef R rep_type;
133 
134  /** STL iterator interface. */
135  typedef T* iterator;
136 
137  /** STL const_iterator interface. */
138  typedef const T* const_iterator;
139 
140 
141 
142  /** @name --- Constructors and Assignment --- */
143 
144  /**
145  Default constructor:
146  */
147  SMatrix();
148  ///
149  /**
150  construct from without initialization
151  */
152  inline SMatrix( SMatrixNoInit ){}
153 
154  /**
155  construct from an identity matrix
156  */
158  /**
159  copy constructor (from a matrix of the same representation
160  */
161  SMatrix(const SMatrix<T,D1,D2,R>& rhs);
162  /**
163  construct from a matrix with different representation.
164  Works only from symmetric to general and not viceversa.
165  */
166  template <class R2>
167  SMatrix(const SMatrix<T,D1,D2,R2>& rhs);
168 
169  /**
170  Construct from an expression.
171  In case of symmetric matrices does not work if expression is of type general
172  matrices. In case one needs to force the assignment from general to symmetric, one can use the
173  ROOT::Math::AssignSym::Evaluate function.
174  */
175  template <class A, class R2>
176  SMatrix(const Expr<A,T,D1,D2,R2>& rhs);
177 
178 
179  /**
180  Constructor with STL iterator interface. The data will be copied into the matrix
181  \param begin start iterator position
182  \param end end iterator position
183  \param triang if true only the triangular lower/upper part of the matrix is filled from the iterators
184  \param lower if true the lower triangular part is filled
185 
186  Size of the matrix must match size of the iterators, if triang is false, otherwise the size of the
187  triangular block. In the case of symmetric matrices triang is considered always to be true
188  (what-ever the user specifies) and the size of the iterators must be equal to the size of the
189  triangular block, which is the number of independent elements of a symmetric matrix: N*(N+1)/2
190 
191  */
192  template<class InputIterator>
193  SMatrix(InputIterator begin, InputIterator end, bool triang = false, bool lower = true);
194 
195  /**
196  Constructor with STL iterator interface. The data will be copied into the matrix
197  \param begin start iterator position
198  \param size iterator size
199  \param triang if true only the triangular lower/upper part of the matrix is filled from the iterators
200  \param lower if true the lower triangular part is filled
201 
202  Size of the iterators must not be larger than the size of the matrix representation.
203  In the case of symmetric matrices the size is N*(N+1)/2.
204 
205  */
206  template<class InputIterator>
207  SMatrix(InputIterator begin, unsigned int size, bool triang = false, bool lower = true);
208 
209  /**
210  constructor of a symmetrix a matrix from a SVector containing the lower (upper)
211  triangular part.
212  */
213 #ifndef UNSUPPORTED_TEMPLATE_EXPRESSION
214  SMatrix(const SVector<T, D1*(D2+1)/2> & v, bool lower = true );
215 #else
216  template<unsigned int N>
217  SMatrix(const SVector<T,N> & v, bool lower = true );
218 #endif
219 
220 
221  /**
222  Construct from a scalar value (only for size 1 matrices)
223  */
224  explicit SMatrix(const T& rhs);
225 
226  /**
227  Assign from another compatible matrix.
228  Possible Symmetirc to general but NOT vice-versa
229  */
230  template <class M>
231  SMatrix<T,D1,D2,R>& operator=(const M& rhs);
232 
233  /**
234  Assign from a matrix expression
235  */
236  template <class A, class R2>
238 
239  /**
240  Assign from an identity matrix
241  */
243 
244  /**
245  Assign from a scalar value (only for size 1 matrices)
246  */
247  SMatrix<T,D1,D2,R>& operator=(const T& rhs);
248 
249  /** @name --- Matrix dimension --- */
250 
251  /**
252  Enumeration defining the matrix dimension,
253  number of rows, columns and size = rows*columns)
254  */
255  enum {
256  /// return no. of matrix rows
257  kRows = D1,
258  /// return no. of matrix columns
259  kCols = D2,
260  /// return no of elements: rows*columns
261  kSize = D1*D2
262  };
263 
264  /** @name --- Access functions --- */
265 
266  /** access the parse tree with the index starting from zero and
267  following the C convention for the order in accessing
268  the matrix elements.
269  Same convention for general and symmetric matrices.
270  */
271  T apply(unsigned int i) const;
272 
273  /// return read-only pointer to internal array
274  const T* Array() const;
275  /// return pointer to internal array
276  T* Array();
277 
278  /** @name --- STL-like interface ---
279  The iterators access the matrix element in the order how they are
280  stored in memory. The C (row-major) convention is used, and in the
281  case of symmetric matrices the iterator spans only the lower diagonal
282  block. For example for a symmetric 3x3 matrices the order of the 6
283  elements \f${a_0,...a_5}\f$ is:
284  \f[
285  M = \left( \begin{array}{ccc}
286  a_0 & a_1 & a_3 \\
287  a_1 & a_2 & a_4 \\
288  a_3 & a_4 & a_5 \end{array} \right)
289  \f]
290  */
291 
292  /** STL iterator interface. */
293  iterator begin();
294 
295  /** STL iterator interface. */
296  iterator end();
297 
298  /** STL const_iterator interface. */
299  const_iterator begin() const;
300 
301  /** STL const_iterator interface. */
302  const_iterator end() const;
303 
304  /**
305  Set matrix elements with STL iterator interface. The data will be copied into the matrix
306  \param begin start iterator position
307  \param end end iterator position
308  \param triang if true only the triangular lower/upper part of the matrix is filled from the iterators
309  \param lower if true the lower triangular part is filled
310 
311  Size of the matrix must match size of the iterators, if triang is false, otherwise the size of the
312  triangular block. In the case of symmetric matrices triang is considered always to be true
313  (what-ever the user specifies) and the size of the iterators must be equal to the size of the
314  triangular block, which is the number of independent elements of a symmetric matrix: N*(N+1)/2
315 
316  */
317  template<class InputIterator>
318  void SetElements(InputIterator begin, InputIterator end, bool triang = false, bool lower = true);
319 
320  /**
321  Constructor with STL iterator interface. The data will be copied into the matrix
322  \param begin start iterator position
323  \param size iterator size
324  \param triang if true only the triangular lower/upper part of the matrix is filled from the iterators
325  \param lower if true the lower triangular part is filled
326 
327  Size of the iterators must not be larger than the size of the matrix representation.
328  In the case of symmetric matrices the size is N*(N+1)/2.
329 
330  */
331  template<class InputIterator>
332  void SetElements(InputIterator begin, unsigned int size, bool triang = false, bool lower = true);
333 
334 
335  /** @name --- Operators --- */
336  /// element wise comparison
337  bool operator==(const T& rhs) const;
338  /// element wise comparison
339  bool operator!=(const T& rhs) const;
340  /// element wise comparison
341  template <class R2>
342  bool operator==(const SMatrix<T,D1,D2,R2>& rhs) const;
343  /// element wise comparison
344  bool operator!=(const SMatrix<T,D1,D2,R>& rhs) const;
345  /// element wise comparison
346  template <class A, class R2>
347  bool operator==(const Expr<A,T,D1,D2,R2>& rhs) const;
348  /// element wise comparison
349  template <class A, class R2>
350  bool operator!=(const Expr<A,T,D1,D2,R2>& rhs) const;
351 
352  /// element wise comparison
353  bool operator>(const T& rhs) const;
354  /// element wise comparison
355  bool operator<(const T& rhs) const;
356  /// element wise comparison
357  template <class R2>
358  bool operator>(const SMatrix<T,D1,D2,R2>& rhs) const;
359  /// element wise comparison
360  template <class R2>
361  bool operator<(const SMatrix<T,D1,D2,R2>& rhs) const;
362  /// element wise comparison
363  template <class A, class R2>
364  bool operator>(const Expr<A,T,D1,D2,R2>& rhs) const;
365  /// element wise comparison
366  template <class A, class R2>
367  bool operator<(const Expr<A,T,D1,D2,R2>& rhs) const;
368 
369  /**
370  read only access to matrix element, with indices starting from 0
371  */
372  const T& operator()(unsigned int i, unsigned int j) const;
373  /**
374  read/write access to matrix element with indices starting from 0
375  */
376  T& operator()(unsigned int i, unsigned int j);
377 
378  /**
379  read only access to matrix element, with indices starting from 0.
380  Function will check index values and it will assert if they are wrong
381  */
382  const T& At(unsigned int i, unsigned int j) const;
383  /**
384  read/write access to matrix element with indices starting from 0.
385  Function will check index values and it will assert if they are wrong
386  */
387  T& At(unsigned int i, unsigned int j);
388 
389 
390  // helper class for implementing the m[i][j] operator
391 
392  class SMatrixRow {
393  public:
394  SMatrixRow ( SMatrix<T,D1,D2,R> & rhs, unsigned int i ) :
395  fMat(&rhs), fRow(i)
396  {}
397  T & operator[](int j) { return (*fMat)(fRow,j); }
398  private:
400  unsigned int fRow;
401  };
402 
404  public:
405  SMatrixRow_const ( const SMatrix<T,D1,D2,R> & rhs, unsigned int i ) :
406  fMat(&rhs), fRow(i)
407  {}
408 
409  const T & operator[](int j) const { return (*fMat)(fRow, j); }
410 
411  private:
413  unsigned int fRow;
414  };
415 
416  /**
417  read only access to matrix element, with indices starting from 0 : m[i][j]
418  */
419  SMatrixRow_const operator[](unsigned int i) const { return SMatrixRow_const(*this, i); }
420  /**
421  read/write access to matrix element with indices starting from 0 : m[i][j]
422  */
423  SMatrixRow operator[](unsigned int i) { return SMatrixRow(*this, i); }
424 
425 
426  /**
427  addition with a scalar
428  */
429  SMatrix<T,D1,D2,R>&operator+=(const T& rhs);
430 
431  /**
432  addition with another matrix of any compatible representation
433  */
434  template <class R2>
436 
437  /**
438  addition with a compatible matrix expression
439  */
440  template <class A, class R2>
442 
443  /**
444  subtraction with a scalar
445  */
446  SMatrix<T,D1,D2,R>& operator-=(const T& rhs);
447 
448  /**
449  subtraction with another matrix of any compatible representation
450  */
451  template <class R2>
452  SMatrix<T,D1,D2,R>&operator-=(const SMatrix<T,D1,D2,R2>& rhs);
453 
454  /**
455  subtraction with a compatible matrix expression
456  */
457  template <class A, class R2>
458  SMatrix<T,D1,D2,R>& operator-=(const Expr<A,T,D1,D2,R2>& rhs);
459 
460  /**
461  multiplication with a scalar
462  */
463  SMatrix<T,D1,D2,R>& operator*=(const T& rhs);
464 
465 #ifndef __CINT__
466 
467 
468  /**
469  multiplication with another compatible matrix (it is a real matrix multiplication)
470  Note that this operation does not avid to create a temporary to store intermidiate result
471  */
472  template <class R2>
473  SMatrix<T,D1,D2,R>& operator*=(const SMatrix<T,D1,D2,R2>& rhs);
474 
475  /**
476  multiplication with a compatible matrix expression (it is a real matrix multiplication)
477  */
478  template <class A, class R2>
479  SMatrix<T,D1,D2,R>& operator*=(const Expr<A,T,D1,D2,R2>& rhs);
480 
481 #endif
482 
483  /**
484  division with a scalar
485  */
486  SMatrix<T,D1,D2,R>& operator/=(const T& rhs);
487 
488 
489 
490  /** @name --- Linear Algebra Functions --- */
491 
492  /**
493  Invert a square Matrix ( this method changes the current matrix).
494  Return true if inversion is successfull.
495  The method used for general square matrices is the LU factorization taken from Dinv routine
496  from the CERNLIB (written in C++ from CLHEP authors)
497  In case of symmetric matrices Bunch-Kaufman diagonal pivoting method is used
498  (The implementation is the one written by the CLHEP authors)
499  */
500  bool Invert();
501 
502  /**
503  Invert a square Matrix and returns a new matrix. In case the inversion fails
504  the current matrix is returned.
505  \param ifail . ifail will be set to 0 when inversion is successfull.
506  See ROOT::Math::SMatrix::Invert for the inversion algorithm
507  */
508  SMatrix<T,D1,D2,R> Inverse(int & ifail ) const;
509 
510  /**
511  Fast Invertion of a square Matrix ( this method changes the current matrix).
512  Return true if inversion is successfull.
513  The method used is based on direct inversion using the Cramer rule for
514  matrices upto 5x5. Afterwards the same default algorithm of Invert() is used.
515  Note that this method is faster but can suffer from much larger numerical accuracy
516  when the condition of the matrix is large
517  */
518  bool InvertFast();
519 
520  /**
521  Invert a square Matrix and returns a new matrix. In case the inversion fails
522  the current matrix is returned.
523  \param ifail . ifail will be set to 0 when inversion is successfull.
524  See ROOT::Math::SMatrix::InvertFast for the inversion algorithm
525  */
526  SMatrix<T,D1,D2,R> InverseFast(int & ifail ) const;
527 
528  /**
529  Invertion of a symmetric positive defined Matrix using Choleski decomposition.
530  ( this method changes the current matrix).
531  Return true if inversion is successfull.
532  The method used is based on Choleski decomposition
533  A compile error is given if the matrix is not of type symmetric and a run-time failure if the
534  matrix is not positive defined.
535  For solving a linear system, it is possible to use also the function
536  ROOT::Math::SolveChol(matrix, vector) which will be faster than performing the inversion
537  */
538  bool InvertChol();
539 
540  /**
541  Invert of a symmetric positive defined Matrix using Choleski decomposition.
542  A compile error is given if the matrix is not of type symmetric and a run-time failure if the
543  matrix is not positive defined.
544  In case the inversion fails the current matrix is returned.
545  \param ifail . ifail will be set to 0 when inversion is successfull.
546  See ROOT::Math::SMatrix::InvertChol for the inversion algorithm
547  */
548  SMatrix<T,D1,D2,R> InverseChol(int & ifail ) const;
549 
550  /**
551  determinant of square Matrix via Dfact.
552  Return true when the calculation is successfull.
553  \param det will contain the calculated determinant value
554  \b Note: this will destroy the contents of the Matrix!
555  */
556  bool Det(T& det);
557 
558  /**
559  determinant of square Matrix via Dfact.
560  Return true when the calculation is successfull.
561  \param det will contain the calculated determinant value
562  \b Note: this will preserve the content of the Matrix!
563  */
564  bool Det2(T& det) const;
565 
566 
567  /** @name --- Matrix Slice Functions --- */
568 
569  /// place a vector in a Matrix row
570  template <unsigned int D>
571  SMatrix<T,D1,D2,R>& Place_in_row(const SVector<T,D>& rhs,
572  unsigned int row,
573  unsigned int col);
574  /// place a vector expression in a Matrix row
575  template <class A, unsigned int D>
576  SMatrix<T,D1,D2,R>& Place_in_row(const VecExpr<A,T,D>& rhs,
577  unsigned int row,
578  unsigned int col);
579  /// place a vector in a Matrix column
580  template <unsigned int D>
581  SMatrix<T,D1,D2,R>& Place_in_col(const SVector<T,D>& rhs,
582  unsigned int row,
583  unsigned int col);
584  /// place a vector expression in a Matrix column
585  template <class A, unsigned int D>
586  SMatrix<T,D1,D2,R>& Place_in_col(const VecExpr<A,T,D>& rhs,
587  unsigned int row,
588  unsigned int col);
589  /// place a matrix in this matrix
590  template <unsigned int D3, unsigned int D4, class R2>
591  SMatrix<T,D1,D2,R>& Place_at(const SMatrix<T,D3,D4,R2>& rhs,
592  unsigned int row,
593  unsigned int col);
594  /// place a matrix expression in this matrix
595  template <class A, unsigned int D3, unsigned int D4, class R2>
596  SMatrix<T,D1,D2,R>& Place_at(const Expr<A,T,D3,D4,R2>& rhs,
597  unsigned int row,
598  unsigned int col);
599 
600  /**
601  return a full Matrix row as a vector (copy the content in a new vector)
602  */
603  SVector<T,D2> Row(unsigned int therow) const;
604 
605  /**
606  return a full Matrix column as a vector (copy the content in a new vector)
607  */
608  SVector<T,D1> Col(unsigned int thecol) const;
609 
610  /**
611  return a slice of therow as a vector starting at the colum value col0 until col0+N,
612  where N is the size of the vector (SubVector::kSize )
613  Condition col0+N <= D2
614  */
615  template <class SubVector>
616  SubVector SubRow(unsigned int therow, unsigned int col0 = 0 ) const;
617 
618  /**
619  return a slice of the column as a vector starting at the row value row0 until row0+Dsub.
620  where N is the size of the vector (SubVector::kSize )
621  Condition row0+N <= D1
622  */
623  template <class SubVector>
624  SubVector SubCol(unsigned int thecol, unsigned int row0 = 0) const;
625 
626  /**
627  return a submatrix with the upper left corner at the values (row0, col0) and with sizes N1, N2
628  where N1 and N2 are the dimension of the sub-matrix (SubMatrix::kRows and SubMatrix::kCols )
629  Condition row0+N1 <= D1 && col0+N2 <=D2
630  */
631  template <class SubMatrix >
632  SubMatrix Sub(unsigned int row0, unsigned int col0) const;
633 
634  /**
635  return diagonal elements of a matrix as a Vector.
636  It works only for squared matrices D1 == D2, otherwise it will produce a compile error
637  */
638  SVector<T,D1> Diagonal() const;
639 
640  /**
641  Set the diagonal elements from a Vector
642  Require that vector implements ::kSize since a check (statically) is done on
643  diagonal size == vector size
644  */
645  template <class Vector>
646  void SetDiagonal(const Vector & v);
647 
648  /**
649  return the trace of a matrix
650  Sum of the diagonal elements
651  */
652  T Trace() const;
653 
654 
655  /**
656  return the upper Triangular block of the matrices (including the diagonal) as
657  a vector of sizes N = D1 * (D1 + 1)/2.
658  It works only for square matrices with D1==D2, otherwise it will produce a compile error
659  */
660 #ifndef UNSUPPORTED_TEMPLATE_EXPRESSION
661  SVector<T, D1 * (D2 +1)/2> UpperBlock() const;
662 #else
663  template<class SubVector>
664  SubVector UpperBlock() const;
665 #endif
666 
667  /**
668  return the lower Triangular block of the matrices (including the diagonal) as
669  a vector of sizes N = D1 * (D1 + 1)/2.
670  It works only for square matrices with D1==D2, otherwise it will produce a compile error
671  */
672 #ifndef UNSUPPORTED_TEMPLATE_EXPRESSION
673  SVector<T, D1 * (D2 +1)/2> LowerBlock() const;
674 #else
675  template<class SubVector>
676  SubVector LowerBlock() const;
677 #endif
678 
679 
680  /** @name --- Other Functions --- */
681 
682  /**
683  Function to check if a matrix is sharing same memory location of the passed pointer
684  This function is used by the expression templates to avoid the alias problem during
685  expression evaluation. When the matrix is in use, for example in operations
686  like A = B * A, a temporary object storing the intermediate result is automatically
687  created when evaluating the expression.
688 
689  */
690  bool IsInUse(const T* p) const;
691 
692  // submatrices
693 
694  /// Print: used by operator<<()
695  std::ostream& Print(std::ostream& os) const;
696 
697 
698 
699 
700 public:
701 
702  /** @name --- Data Member --- */
703 
704  /**
705  Matrix Storage Object containing matrix data
706  */
707  R fRep;
708 
709 }; // end of class SMatrix
710 
711 
712 
713 
714 //==============================================================================
715 // operator<<
716 //==============================================================================
717 template <class T, unsigned int D1, unsigned int D2, class R>
718 inline std::ostream& operator<<(std::ostream& os, const ROOT::Math::SMatrix<T,D1,D2,R>& rhs) {
719  return rhs.Print(os);
720 }
721 
722 
723  } // namespace Math
724 
725 } // namespace ROOT
726 
727 
728 
729 
730 
731 
732 #ifndef __CINT__
733 
734 #include "Math/SMatrix.icc"
735 
736 #include "Math/MatrixFunctions.h"
737 
738 #endif //__CINT__
739 
740 #endif /* ROOT_Math_SMatrix */
Bool_t operator<(const TDatime &d1, const TDatime &d2)
Definition: TDatime.h:106
int Invert(LASymMatrix &)
Definition: LaInverse.cxx:22
Namespace for new ROOT classes and functions.
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double T(double x)
Definition: ChebyshevPol.h:34
SMatrixRow_const operator[](unsigned int i) const
read only access to matrix element, with indices starting from 0 : m[i][j]
Definition: SMatrix.h:419
T * iterator
STL iterator interface.
Definition: SMatrix.h:135
SMatrix< T, D1, D2, R > * fMat
Definition: SMatrix.h:399
TRObject operator()(const T1 &t1) const
ABObj< sym, MatrixInverse< sym, ABObj< sym, LASymMatrix, double >, double >, double > Inverse(const ABObj< sym, LASymMatrix, double > &obj)
LAPACK Algebra functions specialize the Invert function for LASymMatrix.
Definition: LaInverse.h:27
const T * const_iterator
STL const_iterator interface.
Definition: SMatrix.h:138
SMatrix: a generic fixed size D1 x D2 Matrix class.
Bool_t operator!=(const TDatime &d1, const TDatime &d2)
Definition: TDatime.h:104
R rep_type
storage representation type
Definition: SMatrix.h:132
Expression wrapper class for Matrix objects.
Definition: Expression.h:134
R fRep
Matrix Storage Object containing matrix data.
Definition: SMatrix.h:707
Bool_t operator>(const TDatime &d1, const TDatime &d2)
Definition: TDatime.h:110
SMatrixRow operator[](unsigned int i)
read/write access to matrix element with indices starting from 0 : m[i][j]
Definition: SMatrix.h:423
const char * Array
SVector< double, 2 > v
Definition: Dict.h:5
SMatrix(SMatrixNoInit)
construct from without initialization
Definition: SMatrix.h:152
SMatrixRow_const(const SMatrix< T, D1, D2, R > &rhs, unsigned int i)
Definition: SMatrix.h:405
T value_type
contained scalar type
Definition: SMatrix.h:129
SMatrixRow(SMatrix< T, D1, D2, R > &rhs, unsigned int i)
Definition: SMatrix.h:394
void Print(std::ostream &os, const OptionType &opt)
const SMatrix< T, D1, D2, R > * fMat
Definition: SMatrix.h:412
const T & operator[](int j) const
Definition: SMatrix.h:409
Namespace for new Math classes and functions.
Binding & operator=(OUT(*fun)(void))
Expression wrapper class for Vector objects.
Definition: Expression.h:64
Bool_t operator==(const TDatime &d1, const TDatime &d2)
Definition: TDatime.h:102
decltype(auto) constexpr apply(F &&f, Tuple &&t)
TRandom3 R
a TMatrixD.
Definition: testIO.cxx:28
std::string & operator+=(std::string &left, const TString &right)
Definition: TString.h:456
SVector: a generic fixed size Vector class.