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least-squares.cpp

This tutorial shows how least Squares problems for matrices from ViennaCL or Boost.uBLAS can be solved solved.

We start with including the respective header files:

// activate ublas support in ViennaCL
#define VIENNACL_WITH_UBLAS
//
// include necessary system headers
//
#include <iostream>
// Boost headers
#include <boost/numeric/ublas/triangular.hpp>
#include <boost/numeric/ublas/vector.hpp>
#include <boost/numeric/ublas/vector_proxy.hpp>
#include <boost/numeric/ublas/matrix.hpp>
#include <boost/numeric/ublas/matrix_proxy.hpp>
#include <boost/numeric/ublas/lu.hpp>
#include <boost/numeric/ublas/io.hpp>
// ViennaCL headers
#include "viennacl/vector.hpp"
#include "viennacl/matrix.hpp"
#include "viennacl/matrix_proxy.hpp"
#include "viennacl/linalg/qr.hpp"
#include "viennacl/linalg/lu.hpp"
#include "viennacl/linalg/direct_solve.hpp"

The minimization problem of finding x such that is solved as follows:

- Compute the QR-factorization of A = QR.
- Compute for the equivalent minimization problem .
- Solve the triangular system , where is the upper square matrix of R.

int main (int, const char **)
{
typedef float ScalarType; //feel free to change this to 'double' if supported by your hardware
typedef boost::numeric::ublas::matrix<ScalarType> MatrixType;
typedef boost::numeric::ublas::vector<ScalarType> VectorType;
typedef viennacl::matrix<ScalarType, viennacl::column_major> VCLMatrixType;
typedef viennacl::vector<ScalarType> VCLVectorType;

Create vectors and matrices with data:

VectorType ublas_b(4);
ublas_b(0) = -4;
ublas_b(1) = 2;
ublas_b(2) = 5;
ublas_b(3) = -1;
MatrixType ublas_A(4, 3);
ublas_A(0, 0) = 2; ublas_A(0, 1) = -1; ublas_A(0, 2) = 1;
ublas_A(1, 0) = 1; ublas_A(1, 1) = -5; ublas_A(1, 2) = 2;
ublas_A(2, 0) = -3; ublas_A(2, 1) = 1; ublas_A(2, 2) = -4;
ublas_A(3, 0) = 1; ublas_A(3, 1) = -1; ublas_A(3, 2) = 1;

Setup the matrix and vector with ViennaCL objects and copy the data from the uBLAS objects:

VCLVectorType vcl_b(ublas_b.size());
VCLMatrixType vcl_A(ublas_A.size1(), ublas_A.size2());
viennacl::copy(ublas_b, vcl_b);
viennacl::copy(ublas_A, vcl_A);

## Option 1: Using Boost.uBLAS

The implementation in ViennaCL accepts both uBLAS and ViennaCL types. We start with a single-threaded implementation using Boost.uBLAS.

std::cout << "--- Boost.uBLAS ---" << std::endl;

The first (and computationally most expensive) step is to compute the QR factorization of A. Since we do not need A later, we directly overwrite A with the householder reflectors and the upper triangular matrix R. The returned vector holds the scalar coefficients (betas) for the Householder reflections

std::vector<ScalarType> ublas_betas = viennacl::linalg::inplace_qr(ublas_A);

Compute the modified RHS of the minimization problem from the QR factorization, but do not form explicitly: b' := Q^T b

viennacl::linalg::inplace_qr_apply_trans_Q(ublas_A, ublas_betas, ublas_b);

Final step: triangular solve: Rx = b'', where b'' are the first three entries in b' We only need the upper left square part of A, which defines the upper triangular matrix R

boost::numeric::ublas::range ublas_range(0, 3);
boost::numeric::ublas::matrix_range<MatrixType> ublas_R(ublas_A, ublas_range, ublas_range);
boost::numeric::ublas::vector_range<VectorType> ublas_b2(ublas_b, ublas_range);
boost::numeric::ublas::inplace_solve(ublas_R, ublas_b2, boost::numeric::ublas::upper_tag());
std::cout << "Result: " << ublas_b2 << std::endl;

## Option 2: Use ViennaCL types

ViennaCL is used for the computationally intensive BLAS 3 computations. Boost.uBLAS is used for the panel factorization on the host (CPU).

/* =========================================================================
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
============================================================================= */
// activate ublas support in ViennaCL
#define VIENNACL_WITH_UBLAS
//
// include necessary system headers
//
#include <iostream>
// Boost headers
#include <boost/numeric/ublas/triangular.hpp>
#include <boost/numeric/ublas/vector.hpp>
#include <boost/numeric/ublas/vector_proxy.hpp>
#include <boost/numeric/ublas/matrix.hpp>
#include <boost/numeric/ublas/matrix_proxy.hpp>
#include <boost/numeric/ublas/lu.hpp>
#include <boost/numeric/ublas/io.hpp>
// ViennaCL headers
#include "viennacl/vector.hpp"
#include "viennacl/matrix.hpp"
#include "viennacl/matrix_proxy.hpp"
#include "viennacl/linalg/qr.hpp"
#include "viennacl/linalg/lu.hpp"
#include "viennacl/linalg/direct_solve.hpp"
int main (int, const char **)
{
typedef float ScalarType; //feel free to change this to 'double' if supported by your hardware
typedef boost::numeric::ublas::matrix<ScalarType> MatrixType;
typedef boost::numeric::ublas::vector<ScalarType> VectorType;
typedef viennacl::matrix<ScalarType, viennacl::column_major> VCLMatrixType;
typedef viennacl::vector<ScalarType> VCLVectorType;
VectorType ublas_b(4);
ublas_b(0) = -4;
ublas_b(1) = 2;
ublas_b(2) = 5;
ublas_b(3) = -1;
MatrixType ublas_A(4, 3);
ublas_A(0, 0) = 2; ublas_A(0, 1) = -1; ublas_A(0, 2) = 1;
ublas_A(1, 0) = 1; ublas_A(1, 1) = -5; ublas_A(1, 2) = 2;
ublas_A(2, 0) = -3; ublas_A(2, 1) = 1; ublas_A(2, 2) = -4;
ublas_A(3, 0) = 1; ublas_A(3, 1) = -1; ublas_A(3, 2) = 1;
VCLVectorType vcl_b(ublas_b.size());
VCLMatrixType vcl_A(ublas_A.size1(), ublas_A.size2());
viennacl::copy(ublas_b, vcl_b);
viennacl::copy(ublas_A, vcl_A);
std::cout << "--- Boost.uBLAS ---" << std::endl;
std::vector<ScalarType> ublas_betas = viennacl::linalg::inplace_qr(ublas_A);
viennacl::linalg::inplace_qr_apply_trans_Q(ublas_A, ublas_betas, ublas_b);
boost::numeric::ublas::range ublas_range(0, 3);
boost::numeric::ublas::matrix_range<MatrixType> ublas_R(ublas_A, ublas_range, ublas_range);
boost::numeric::ublas::vector_range<VectorType> ublas_b2(ublas_b, ublas_range);
boost::numeric::ublas::inplace_solve(ublas_R, ublas_b2, boost::numeric::ublas::upper_tag());
std::cout << "Result: " << ublas_b2 << std::endl;
std::cout << "--- ViennaCL (hybrid implementation) ---" << std::endl;
std::vector<ScalarType> hybrid_betas = viennacl::linalg::inplace_qr(vcl_A);
viennacl::linalg::inplace_qr_apply_trans_Q(vcl_A, hybrid_betas, vcl_b);
viennacl::range vcl_range(0, 3);
viennacl::matrix_range<VCLMatrixType> vcl_R(vcl_A, vcl_range, vcl_range);
viennacl::vector_range<VCLVectorType> vcl_b2(vcl_b, vcl_range);
viennacl::linalg::inplace_solve(vcl_R, vcl_b2, viennacl::linalg::upper_tag());
std::cout << "Result: " << vcl_b2 << std::endl;
std::cout << "!!!! TUTORIAL COMPLETED SUCCESSFULLY !!!!" << std::endl;
return EXIT_SUCCESS;
}