{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "df1bd85d",
   "metadata": {},
   "source": [
    "# decomposeQR\n",
    "This tutorial shows how to decompose a matrix A in an orthogonal matrix Q and an upper\n",
    "triangular matrix R using QR Householder decomposition with the TDecompQRH class.\n",
    "The matrix same matrix as in this example is used: https://en.wikipedia.org/wiki/QR_decomposition#Example_2\n",
    "\n",
    "\n",
    "**Author:**   \n",
    "<i><small>This notebook tutorial was automatically generated with <a href= \"https://github.com/root-project/root/blob/master/documentation/doxygen/converttonotebook.py\">ROOTBOOK-izer</a> from the macro found in the ROOT repository  on Tuesday, May 19, 2026 at 08:26 PM.</small></i>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "794e1aac",
   "metadata": {
    "collapsed": false,
    "execution": {
     "iopub.execute_input": "2026-05-19T20:26:05.605693Z",
     "iopub.status.busy": "2026-05-19T20:26:05.605559Z",
     "iopub.status.idle": "2026-05-19T20:26:06.096315Z",
     "shell.execute_reply": "2026-05-19T20:26:06.095936Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "initial matrox A \n",
      "\n",
      "3x3 matrix is as follows\n",
      "\n",
      "     |      0    |      1    |      2    |\n",
      "--------------------------------------------\n",
      "   0 |         12         -51           4 \n",
      "   1 |          6         167         -68 \n",
      "   2 |         -4          24         -41 \n",
      "\n",
      "Orthogonal Q matrix \n"
     ]
    }
   ],
   "source": [
    "const int n = 3;\n",
    "\n",
    "\n",
    "double a[] = {12, -51, 4, 6, 167, -68, -4, 24, -41};\n",
    "\n",
    "TMatrixT<double> A(3, 3, a);\n",
    "\n",
    "std::cout << \"initial matrox A \" << std::endl;\n",
    "\n",
    "A.Print();\n",
    "\n",
    "TDecompQRH decomp(A);\n",
    "\n",
    "bool ret = decomp.Decompose();\n",
    "\n",
    "std::cout << \"Orthogonal Q matrix \" << std::endl;"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fbec9db4",
   "metadata": {},
   "source": [
    "note that decomp.GetQ()  returns an intenrnal matrix which is not Q defined as A = QR"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "882d1799",
   "metadata": {
    "collapsed": false,
    "execution": {
     "iopub.execute_input": "2026-05-19T20:26:06.103794Z",
     "iopub.status.busy": "2026-05-19T20:26:06.103664Z",
     "iopub.status.idle": "2026-05-19T20:26:06.311037Z",
     "shell.execute_reply": "2026-05-19T20:26:06.310390Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "3x3 matrix is as follows\n",
      "\n",
      "     |      0    |      1    |      2    |\n",
      "--------------------------------------------\n",
      "   0 |    -0.8571      0.3943      0.3314 \n",
      "   1 |    -0.4286     -0.9029    -0.03429 \n",
      "   2 |     0.2857     -0.1714      0.9429 \n",
      "\n",
      "Upper Triangular R matrix \n",
      "\n",
      "3x3 matrix is as follows\n",
      "\n",
      "     |      0    |      1    |      2    |\n",
      "--------------------------------------------\n",
      "   0 |        -14         -21          14 \n",
      "   1 |          0        -175          70 \n",
      "   2 |          0           0         -35 \n",
      "\n"
     ]
    }
   ],
   "source": [
    "auto Q = decomp.GetOrthogonalMatrix();\n",
    "Q.Print();\n",
    "\n",
    "std::cout << \"Upper Triangular R matrix \" << std::endl;\n",
    "auto R = decomp.GetR();\n",
    "\n",
    "R.Print();"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f98b8e66",
   "metadata": {},
   "source": [
    "check that we have a correct Q-R decomposition"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "b198dfe3",
   "metadata": {
    "collapsed": false,
    "execution": {
     "iopub.execute_input": "2026-05-19T20:26:06.312583Z",
     "iopub.status.busy": "2026-05-19T20:26:06.312468Z",
     "iopub.status.idle": "2026-05-19T20:26:06.514976Z",
     "shell.execute_reply": "2026-05-19T20:26:06.514446Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Computed A matrix from Q * R \n",
      "\n",
      "3x3 matrix is as follows\n",
      "\n",
      "     |      0    |      1    |      2    |\n",
      "--------------------------------------------\n",
      "   0 |         12         -51           4 \n",
      "   1 |          6         167         -68 \n",
      "   2 |         -4          24         -41 \n",
      "\n"
     ]
    }
   ],
   "source": [
    "TMatrixT<double> compA = Q * R;\n",
    "\n",
    "std::cout << \"Computed A matrix from Q * R \" << std::endl;\n",
    "compA.Print();\n",
    "\n",
    "for (int i = 0; i < A.GetNrows(); ++i) {\n",
    "   for (int j = 0; j < A.GetNcols(); ++j) {\n",
    "      if (!TMath::AreEqualAbs( compA(i,j), A(i,j), 1.E-6) )\n",
    "         Error(\"decomposeQR\",\"Reconstrate matrix is not equal to the original : %f different than %f\",compA(i,j),A(i,j));\n",
    "   }\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e56cef12",
   "metadata": {},
   "source": [
    "chech also that Q is orthogonal (Q^T * Q = I)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "b616d7b1",
   "metadata": {
    "collapsed": false,
    "execution": {
     "iopub.execute_input": "2026-05-19T20:26:06.516805Z",
     "iopub.status.busy": "2026-05-19T20:26:06.516690Z",
     "iopub.status.idle": "2026-05-19T20:26:06.718953Z",
     "shell.execute_reply": "2026-05-19T20:26:06.718246Z"
    }
   },
   "outputs": [],
   "source": [
    "auto QT = Q;\n",
    "QT.Transpose(Q);\n",
    "\n",
    "auto qtq = QT * Q;\n",
    "for (int i = 0; i < Q.GetNrows(); ++i) {\n",
    "   for (int j = 0; j < Q.GetNcols(); ++j) {\n",
    "      if ((i == j && !TMath::AreEqualAbs(qtq(i, i), 1., 1.E-6)) ||\n",
    "          (i != j && !TMath::AreEqualAbs(qtq(i, j), 0., 1.E-6))) {\n",
    "         Error(\"decomposeQR\", \"Q matrix is not orthogonal \");\n",
    "         qtq.Print();\n",
    "         break;\n",
    "      }\n",
    "   }\n",
    "}"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "ROOT C++",
   "language": "c++",
   "name": "root"
  },
  "language_info": {
   "codemirror_mode": "text/x-c++src",
   "file_extension": ".C",
   "mimetype": " text/x-c++src",
   "name": "c++"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}