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tql2.cpp

This tutorial explains how one can use the tql-algorithm to compute the eigenvalues of tridiagonal matrices.

We start with including the necessary headers:

// include necessary system headers
#include <iostream>
#include <fstream>
#include <limits>
#include <string>
#include <iomanip>
#define VIENNACL_WITH_UBLAS
//include basic scalar and vector types of ViennaCL
#include "viennacl/scalar.hpp"
#include "viennacl/vector.hpp"
#include "viennacl/compressed_matrix.hpp"
#include "viennacl/io/matrix_market.hpp"
#include "viennacl/linalg/qr-method.hpp"
#include "viennacl/linalg/qr-method-common.hpp"
#include "viennacl/linalg/host_based/matrix_operations.hpp"
// Use the shortcut 'ublas::' instead of 'boost::numeric::ublas::'
namespace ublas = boost::numeric::ublas;
// Run in single precision. Change to double precision if provided by your GPU.
typedef float ScalarType;

We generate a symmetric tridiagonal matrix with known eigenvalues, call the tql-algorithm, and then print the results.

int main()
{
std::size_t sz = 10;
std::cout << "Compute eigenvalues and eigenvectors of matrix of size " << sz << "-by-" << sz << std::endl << std::endl;
std::vector<ScalarType> d(sz), e(sz);
// Initialize diagonal and superdiagonal elements of the tridiagonal matrix
d[0] = 1; e[0] = 0;
d[1] = 2; e[1] = 4;
d[2] =-4; e[2] = 5;
d[3] = 6; e[3] = 1;
d[4] = 3; e[4] = 2;
d[5] = 4; e[5] =-3;
d[6] = 7; e[6] = 5;
d[7] = 9; e[7] = 1;
d[8] = 3; e[8] = 5;
d[9] = 8; e[9] = 2;

Initialize the matrix Q as the identity matrix. It will hold the eigenvectors.

viennacl::matrix<ScalarType> Q = viennacl::identity_matrix<ScalarType>(sz);

Compute the eigenvalues and eigenvectors

viennacl::linalg::tql2(Q, d, e);

Print the results:

std::cout << "Eigenvalues: " << std::endl;
for (unsigned int i = 0; i < d.size(); i++)
std::cout << std::setprecision(6) << std::fixed << d[i] << " ";
std::cout << std::endl;
std::cout << std::endl;
std::cout << "Eigenvectors corresponding to the eigenvalues above are the columns: " << std::endl << std::endl;
std::cout << Q << std::endl;

That's it. Print success message and exit.

std::cout << std::endl <<"--------TUTORIAL COMPLETED----------" << std::endl;
return EXIT_SUCCESS;
}

## Full Example Code

/* =========================================================================
Copyright (c) 2010-2016, Institute for Microelectronics,
Institute for Analysis and Scientific Computing,
TU Wien.
Portions of this software are copyright by UChicago Argonne, LLC.
-----------------
ViennaCL - The Vienna Computing Library
-----------------
Project Head: Karl Rupp rupp@iue.tuwien.ac.at
(A list of authors and contributors can be found in the PDF manual)
License: MIT (X11), see file LICENSE in the base directory
============================================================================= */
// include necessary system headers
#include <iostream>
#include <fstream>
#include <limits>
#include <string>
#include <iomanip>
#define VIENNACL_WITH_UBLAS
//include basic scalar and vector types of ViennaCL
#include "viennacl/scalar.hpp"
#include "viennacl/vector.hpp"
#include "viennacl/compressed_matrix.hpp"
#include "viennacl/io/matrix_market.hpp"
#include "viennacl/linalg/qr-method.hpp"
#include "viennacl/linalg/qr-method-common.hpp"
#include "viennacl/linalg/host_based/matrix_operations.hpp"
// Use the shortcut 'ublas::' instead of 'boost::numeric::ublas::'
namespace ublas = boost::numeric::ublas;
// Run in single precision. Change to double precision if provided by your GPU.
typedef float ScalarType;
int main()
{
std::size_t sz = 10;
std::cout << "Compute eigenvalues and eigenvectors of matrix of size " << sz << "-by-" << sz << std::endl << std::endl;
std::vector<ScalarType> d(sz), e(sz);
// Initialize diagonal and superdiagonal elements of the tridiagonal matrix
d[0] = 1; e[0] = 0;
d[1] = 2; e[1] = 4;
d[2] =-4; e[2] = 5;
d[3] = 6; e[3] = 1;
d[4] = 3; e[4] = 2;
d[5] = 4; e[5] =-3;
d[6] = 7; e[6] = 5;
d[7] = 9; e[7] = 1;
d[8] = 3; e[8] = 5;
d[9] = 8; e[9] = 2;
viennacl::matrix<ScalarType> Q = viennacl::identity_matrix<ScalarType>(sz);
viennacl::linalg::tql2(Q, d, e);
std::cout << "Eigenvalues: " << std::endl;
for (unsigned int i = 0; i < d.size(); i++)
std::cout << std::setprecision(6) << std::fixed << d[i] << " ";
std::cout << std::endl;
std::cout << std::endl;
std::cout << "Eigenvectors corresponding to the eigenvalues above are the columns: " << std::endl << std::endl;
std::cout << Q << std::endl;
std::cout << std::endl <<"--------TUTORIAL COMPLETED----------" << std::endl;
return EXIT_SUCCESS;
}