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Reference Guide
invertMatrix.C
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1/// \file
2/// \ingroup tutorial_matrix
3/// \notebook -nodraw
4/// This macro shows several ways to invert a matrix . Each method
5/// is a trade-off between accuracy of the inversion and speed.
6/// Which method to chose depends on "how well-behaved" the matrix is.
7/// This is best checked through a call to Condition(), available in each
8/// decomposition class. A second possibility (less preferred) would be to
9/// check the determinant
10///
11/// #### USAGE
12///
13/// This macro can be executed with Cling or ACLIC
14/// - via the interpretor, do
15/// ~~~{.cpp}
16/// root > .x invertMatrix.C
17/// ~~~
18/// - via ACLIC
19/// ~~~{.cpp}
20/// root > gSystem->Load("libMatrix");
21/// root > .x invertMatrix.C+
22/// ~~~
23///
24/// \macro_output
25/// \macro_code
26///
27/// \author Eddy Offermann
28
29#include <iostream>
30#include "TMath.h"
31#include "TMatrixD.h"
32#include "TMatrixDLazy.h"
33#include "TVectorD.h"
34#include "TDecompLU.h"
35#include "TDecompSVD.h"
36
37void invertMatrix(Int_t msize=6)
38{
39 if (msize < 2 || msize > 10) {
40 std::cout << "2 <= msize <= 10" <<std::endl;
41 return;
42 }
43 std::cout << "--------------------------------------------------------" <<std::endl;
44 std::cout << "Inversion results for a ("<<msize<<","<<msize<<") matrix" <<std::endl;
45 std::cout << "For each inversion procedure we check the maximum size " <<std::endl;
46 std::cout << "of the off-diagonal elements of Inv(A) * A " <<std::endl;
47 std::cout << "--------------------------------------------------------" <<std::endl;
48
49 TMatrixD H_square = THilbertMatrixD(msize,msize);
50
51 // ### 1. InvertFast(Double_t *det=0)
52 // It is identical to Invert() for sizes > 6 x 6 but for smaller sizes, the
53 // inversion is performed according to Cramer's rule by explicitly calculating
54 // all Jacobi's sub-determinants . For instance for a 6 x 6 matrix this means:
55 // \# of 5 x 5 determinant : 36
56 // \# of 4 x 4 determinant : 75
57 // \# of 3 x 3 determinant : 80
58 // \# of 2 x 2 determinant : 45 (see TMatrixD/FCramerInv.cxx)
59 //
60 // The only "quality" control in this process is to check whether the 6 x 6
61 // determinant is unequal 0 . But speed gains are significant compared to Invert() ,
62 // up to an order of magnitude for sizes <= 4 x 4
63 //
64 // The inversion is done "in place", so the original matrix will be overwritten
65 // If a pointer to a Double_t is supplied the determinant is calculated
66 //
67
68 std::cout << "1. Use .InvertFast(&det)" <<std::endl;
69 if (msize > 6)
70 std::cout << " for ("<<msize<<","<<msize<<") this is identical to .Invert(&det)" <<std::endl;
71
72 Double_t det1;
73 TMatrixD H1 = H_square;
74 H1.InvertFast(&det1);
75
76 // Get the maximum off-diagonal matrix value . One way to do this is to set the
77 // diagonal to zero .
78
79 TMatrixD U1(H1,TMatrixD::kMult,H_square);
80 TMatrixDDiag diag1(U1); diag1 = 0.0;
81 const Double_t U1_max_offdiag = (U1.Abs()).Max();
82 std::cout << " Maximum off-diagonal = " << U1_max_offdiag << std::endl;
83 std::cout << " Determinant = " << det1 << std::endl;
84
85 // ### 2. Invert(Double_t *det=0)
86 // Again the inversion is performed in place .
87 // It consists out of a sequence of calls to the decomposition classes . For instance
88 // for the general dense matrix TMatrixD the LU decomposition is invoked:
89 // - The matrix is decomposed using a scheme according to Crout which involves
90 // "implicit partial pivoting", see for instance Num. Recip. (we have also available
91 // a decomposition scheme that does not the scaling and is therefore even slightly
92 // faster but less stable)
93 // With each decomposition, a tolerance has to be specified . If this tolerance
94 // requirement is not met, the matrix is regarded as being singular. The value
95 // passed to this decomposition, is the data member fTol of the matrix . Its
96 // default value is DBL_EPSILON, which is defined as the smallest number so that
97 // 1+DBL_EPSILON > 1
98 // - The last step is a standard forward/backward substitution .
99 //
100 // It is important to realize that both InvertFast() and Invert() are "one-shot" deals , speed
101 // comes at a price . If something goes wrong because the matrix is (near) singular, you have
102 // overwritten your original matrix and no factorization is available anymore to get more
103 // information like condition number or change the tolerance number .
104 //
105 // All other calls in the matrix classes involving inversion like the ones with the "smart"
106 // constructors (kInverted,kInvMult...) use this inversion method .
107 //
108
109 std::cout << "2. Use .Invert(&det)" << std::endl;
110
111 Double_t det2;
112 TMatrixD H2 = H_square;
113 H2.Invert(&det2);
114
115 TMatrixD U2(H2,TMatrixD::kMult,H_square);
116 TMatrixDDiag diag2(U2); diag2 = 0.0;
117 const Double_t U2_max_offdiag = (U2.Abs()).Max();
118 std::cout << " Maximum off-diagonal = " << U2_max_offdiag << std::endl;
119 std::cout << " Determinant = " << det2 << std::endl;
120
121 // ### 3. Inversion through LU decomposition
122 // The (default) algorithms used are similar to 2. (Not identical because in 2, the whole
123 // calculation is done "in-place". Here the original matrix is copied (so more memory
124 // management => slower) and several operations can be performed without having to repeat
125 // the decomposition step .
126 // Inverting a matrix is nothing else than solving a set of equations where the rhs is given
127 // by the unit matrix, so the steps to take are identical to those solving a linear equation :
128 //
129
130 std::cout << "3. Use TDecompLU" << std::endl;
131
132 TMatrixD H3 = H_square;
133 TDecompLU lu(H_square);
134
135 // Any operation that requires a decomposition will trigger it . The class keeps
136 // an internal state so that following operations will not perform the decomposition again
137 // unless the matrix is changed through SetMatrix(..)
138 // One might want to proceed more cautiously by invoking first Decompose() and check its
139 // return value before proceeding....
140
141 lu.Invert(H3);
142 Double_t d1_lu; Double_t d2_lu;
143 lu.Det(d1_lu,d2_lu);
144 Double_t det3 = d1_lu*TMath::Power(2.,d2_lu);
145
146 TMatrixD U3(H3,TMatrixD::kMult,H_square);
147 TMatrixDDiag diag3(U3); diag3 = 0.0;
148 const Double_t U3_max_offdiag = (U3.Abs()).Max();
149 std::cout << " Maximum off-diagonal = " << U3_max_offdiag << std::endl;
150 std::cout << " Determinant = " << det3 << std::endl;
151
152 // ### 4. Inversion through SVD decomposition
153 // For SVD and QRH, the (n x m) matrix does only have to fulfill n >=m . In case n > m
154 // a pseudo-inverse is calculated
155 std::cout << "4. Use TDecompSVD on non-square matrix" << std::endl;
156
157 TMatrixD H_nsquare = THilbertMatrixD(msize,msize-1);
158
159 TDecompSVD svd(H_nsquare);
160
161 TMatrixD H4 = svd.Invert();
162 Double_t d1_svd; Double_t d2_svd;
163 svd.Det(d1_svd,d2_svd);
164 Double_t det4 = d1_svd*TMath::Power(2.,d2_svd);
165
166 TMatrixD U4(H4,TMatrixD::kMult,H_nsquare);
167 TMatrixDDiag diag4(U4); diag4 = 0.0;
168 const Double_t U4_max_offdiag = (U4.Abs()).Max();
169 std::cout << " Maximum off-diagonal = " << U4_max_offdiag << std::endl;
170 std::cout << " Determinant = " << det4 << std::endl;
171}
int Int_t
Definition: RtypesCore.h:41
double Double_t
Definition: RtypesCore.h:55
THilbertMatrixT< Double_t > THilbertMatrixD
Definition: TMatrixDLazy.h:35
LU Decomposition class.
Definition: TDecompLU.h:24
Single Value Decomposition class.
Definition: TDecompSVD.h:24
TMatrixT< Element > & Invert(Double_t *det=0)
Invert the matrix and calculate its determinant.
Definition: TMatrixT.cxx:1396
TMatrixT< Element > & InvertFast(Double_t *det=0)
Invert the matrix and calculate its determinant, however upto (6x6) a fast Cramer inversion is used .
Definition: TMatrixT.cxx:1410
Short_t Max(Short_t a, Short_t b)
Definition: TMathBase.h:212
LongDouble_t Power(LongDouble_t x, LongDouble_t y)
Definition: TMath.h:723