Notebook

decomposeQR¶

This tutorial shows how to decompose a matrix A in an orthogonal matrix Q and an upper triangular matrix R using QR Householder decomposition with the TDecompQRH class. The matrix same matrix as in this example is used: https://en.wikipedia.org/wiki/QR_decomposition#Example_2

Author:
This notebook tutorial was automatically generated with ROOTBOOK-izer from the macro found in the ROOT repository on Friday, March 24, 2023 at 10:56 AM.

In [1]:

const int n = 3;

double a[] = {12, -51, 4, 6, 167, -68, -4, 24, -41};

TMatrixT<double> A(3, 3, a);

std::cout << "initial matrox A " << std::endl;

A.Print();

TDecompQRH decomp(A);

bool ret = decomp.Decompose();

std::cout << "Orthogonal Q matrix " << std::endl;

initial matrox A 

3x3 matrix is as follows

     |      0    |      1    |      2    |
--------------------------------------------
   0 |         12         -51           4 
   1 |          6         167         -68 
   2 |         -4          24         -41 

Orthogonal Q matrix

note that decomp.GetQ() returns an intenrnal matrix which is not Q defined as A = QR

In [2]:

auto Q = decomp.GetOrthogonalMatrix();
Q.Print();

std::cout << "Upper Triangular R matrix " << std::endl;
auto R = decomp.GetR();

R.Print();

3x3 matrix is as follows

     |      0    |      1    |      2    |
--------------------------------------------
   0 |    -0.8571      0.3943      0.3314 
   1 |    -0.4286     -0.9029    -0.03429 
   2 |     0.2857     -0.1714      0.9429 

Upper Triangular R matrix 

3x3 matrix is as follows

     |      0    |      1    |      2    |
--------------------------------------------
   0 |        -14         -21          14 
   1 |          0        -175          70 
   2 |          0           0         -35

check that we have a correct Q-R decomposition

In [3]:

TMatrixT<double> compA = Q * R;

std::cout << "Computed A matrix from Q * R " << std::endl;
compA.Print();

for (int i = 0; i < A.GetNrows(); ++i) {
   for (int j = 0; j < A.GetNcols(); ++j) {
      if (!TMath::AreEqualAbs( compA(i,j), A(i,j), 1.E-6) )
         Error("decomposeQR","Reconstrate matrix is not equal to the original : %f different than %f",compA(i,j),A(i,j));
   }
}

Computed A matrix from Q * R 

3x3 matrix is as follows

     |      0    |      1    |      2    |
--------------------------------------------
   0 |         12         -51           4 
   1 |          6         167         -68 
   2 |         -4          24         -41

chech also that Q is orthogonal (Q^T * Q = I)

In [4]:

auto QT = Q;
QT.Transpose(Q);

auto qtq = QT * Q;
for (int i = 0; i < Q.GetNrows(); ++i) {
   for (int j = 0; j < Q.GetNcols(); ++j) {
      if ((i == j && !TMath::AreEqualAbs(qtq(i, i), 1., 1.E-6)) ||
          (i != j && !TMath::AreEqualAbs(qtq(i, j), 0., 1.E-6))) {
         Error("decomposeQR", "Q matrix is not orthogonal ");
         qtq.Print();
         break;
      }
   }
}