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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2003 Ferdinando Ametrano
Copyright (C) 2007 Klaus Spanderen
Copyright (C) 2007 Neil Firth
This file is part of QuantLib, a free-software/open-source library
for financial quantitative analysts and developers - http://quantlib.org/
QuantLib is free software: you can redistribute it and/or modify it
under the terms of the QuantLib license. You should have received a
copy of the license along with this program; if not, please email
<quantlib-dev@lists.sf.net>. The license is also available online at
<http://quantlib.org/license.shtml>.
This program is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the license for more details.
*/
#include "matrices.hpp"
#include "utilities.hpp"
#include <ql/math/matrix.hpp>
#include <ql/math/matrixutilities/pseudosqrt.hpp>
#include <ql/math/matrixutilities/svd.hpp>
#include <ql/math/matrixutilities/symmetricschurdecomposition.hpp>
using namespace QuantLib;
using namespace boost::unit_test_framework;
QL_BEGIN_TEST_LOCALS(MatricesTest)
Size N;
Matrix M1, M2, M3, M4, M5, M6, I;
Real norm(const Array& v) {
return std::sqrt(DotProduct(v,v));
}
Real norm(const Matrix& m) {
Real sum = 0.0;
for (Size i=0; i<m.rows(); i++)
for (Size j=0; j<m.columns(); j++)
sum += m[i][j]*m[i][j];
return std::sqrt(sum);
}
void setup() {
N = 3;
M1 = M2 = I = Matrix(N,N);
M3 = Matrix(3,4);
M4 = Matrix(4,3);
M5 = Matrix(4, 4, 0.0);
M6 = Matrix(4, 4, 0.0);
M1[0][0] = 1.0; M1[0][1] = 0.9; M1[0][2] = 0.7;
M1[1][0] = 0.9; M1[1][1] = 1.0; M1[1][2] = 0.4;
M1[2][0] = 0.7; M1[2][1] = 0.4; M1[2][2] = 1.0;
M2[0][0] = 1.0; M2[0][1] = 0.9; M2[0][2] = 0.7;
M2[1][0] = 0.9; M2[1][1] = 1.0; M2[1][2] = 0.3;
M2[2][0] = 0.7; M2[2][1] = 0.3; M2[2][2] = 1.0;
I[0][0] = 1.0; I[0][1] = 0.0; I[0][2] = 0.0;
I[1][0] = 0.0; I[1][1] = 1.0; I[1][2] = 0.0;
I[2][0] = 0.0; I[2][1] = 0.0; I[2][2] = 1.0;
M3[0][0] = 1; M3[0][1] = 2; M3[0][2] = 3; M3[0][3] = 4;
M3[1][0] = 2; M3[1][1] = 0; M3[1][2] = 2; M3[1][3] = 1;
M3[2][0] = 0; M3[2][1] = 1; M3[2][2] = 0; M3[2][3] = 0;
M4[0][0] = 1; M4[0][1] = 2; M4[0][2] = 400;
M4[1][0] = 2; M4[1][1] = 0; M4[1][2] = 1;
M4[2][0] = 30; M4[2][1] = 2; M4[2][2] = 0;
M4[3][0] = 2; M4[3][1] = 0; M4[3][2] = 1.05;
// from Higham - nearest correlation matrix
M5[0][0] = 2; M5[0][1] = -1; M5[0][2] = 0.0; M5[0][3] = 0.0;
M5[1][0] = M5[0][1]; M5[1][1] = 2; M5[1][2] = -1; M5[1][3] = 0.0;
M5[2][0] = M5[0][2]; M5[2][1] = M5[1][2]; M5[2][2] = 2; M5[2][3] = -1;
M5[3][0] = M5[0][3]; M5[3][1] = M5[1][3]; M5[3][2] = M5[2][3]; M5[3][3] = 2;
// from Higham - nearest correlation matrix to M5
M6[0][0] = 1; M6[0][1] = -0.8084124981; M6[0][2] = 0.1915875019; M6[0][3] = 0.106775049;
M6[1][0] = M6[0][1]; M6[1][1] = 1; M6[1][2] = -0.6562326948; M6[1][3] = M6[0][2];
M6[2][0] = M6[0][2]; M6[2][1] = M6[1][2]; M6[2][2] = 1; M6[2][3] = M6[0][1];
M6[3][0] = M6[0][3]; M6[3][1] = M6[1][3]; M6[3][2] = M6[2][3]; M6[3][3] = 1;
}
QL_END_TEST_LOCALS(MatricesTest)
void MatricesTest::testEigenvectors() {
BOOST_MESSAGE("Testing eigenvalues and eigenvectors calculation...");
setup();
Matrix testMatrices[] = { M1, M2 };
for (Size k=0; k<LENGTH(testMatrices); k++) {
Matrix& M = testMatrices[k];
SymmetricSchurDecomposition dec(M);
Array eigenValues = dec.eigenvalues();
Matrix eigenVectors = dec.eigenvectors();
Real minHolder = QL_MAX_REAL;
for (Size i=0; i<N; i++) {
Array v(N);
for (Size j=0; j<N; j++)
v[j] = eigenVectors[j][i];
// check definition
Array a = M*v;
Array b = eigenValues[i]*v;
if (norm(a-b) > 1.0e-15)
BOOST_FAIL("Eigenvector definition not satisfied");
// check decreasing ordering
if (eigenValues[i] >= minHolder) {
BOOST_FAIL("Eigenvalues not ordered: " << eigenValues);
} else
minHolder = eigenValues[i];
}
// check normalization
Matrix m = eigenVectors * transpose(eigenVectors);
if (norm(m-I) > 1.0e-15)
BOOST_FAIL("Eigenvector not normalized");
}
}
void MatricesTest::testSqrt() {
BOOST_MESSAGE("Testing matricial square root...");
setup();
Matrix m = pseudoSqrt(M1, SalvagingAlgorithm::None);
Matrix temp = m*transpose(m);
Real error = norm(temp - M1);
Real tolerance = 1.0e-12;
if (error>tolerance) {
BOOST_FAIL("Matrix square root calculation failed\n"
<< "original matrix:\n" << M1
<< "pseudoSqrt:\n" << m
<< "pseudoSqrt*pseudoSqrt:\n" << temp
<< "\nerror: " << error
<< "\ntolerance: " << tolerance);
}
}
void MatricesTest::testHighamSqrt() {
BOOST_MESSAGE("Testing Higham matricial square root...");
setup();
Matrix tempSqrt = pseudoSqrt(M5, SalvagingAlgorithm::Higham);
Matrix ansSqrt = pseudoSqrt(M6, SalvagingAlgorithm::None);
Real error = norm(ansSqrt - tempSqrt);
Real tolerance = 1.0e-4;
if (error>tolerance) {
BOOST_FAIL("Higham matrix correction failed\n"
<< "original matrix:\n" << M5
<< "pseudoSqrt:\n" << tempSqrt
<< "should be:\n" << ansSqrt
<< "\nerror: " << error
<< "\ntolerance: " << tolerance);
}
}
void MatricesTest::testSVD() {
BOOST_MESSAGE("Testing singular value decomposition...");
setup();
Real tol = 1.0e-12;
Matrix testMatrices[] = { M1, M2, M3, M4 };
for (Size j = 0; j < LENGTH(testMatrices); j++) {
// m >= n required (rows >= columns)
Matrix& A = testMatrices[j];
SVD svd(A);
// U is m x n
Matrix U = svd.U();
// s is n long
Array s = svd.singularValues();
// S is n x n
Matrix S = svd.S();
// V is n x n
Matrix V = svd.V();
for (Size i=0; i < S.rows(); i++) {
if (S[i][i] != s[i])
BOOST_FAIL("S not consistent with s");
}
// tests
Matrix U_Utranspose = transpose(U)*U;
if (norm(U_Utranspose-I) > tol)
BOOST_FAIL("U not orthogonal (norm of U^T*U-I = "
<< norm(U_Utranspose-I) << ")");
Matrix V_Vtranspose = transpose(V)*V;
if (norm(V_Vtranspose-I) > tol)
BOOST_FAIL("V not orthogonal (norm of V^T*V-I = "
<< norm(V_Vtranspose-I) << ")");
Matrix A_reconstructed = U * S * transpose(V);
if (norm(A_reconstructed-A) > tol)
BOOST_FAIL("Product does not recover A: (norm of U*S*V^T-A = "
<< norm(A_reconstructed-A) << ")");
}
}
void MatricesTest::testInverse() {
BOOST_MESSAGE("Testing inverse calculation...");
setup();
Real tol = 1.0e-12;
Matrix testMatrices[] = { M1, M2, I };
for (Size j = 0; j < LENGTH(testMatrices); j++) {
Matrix& A = testMatrices[j];
Matrix invA = inverse(A);
Matrix I1 = invA*A;
Matrix I2 = A*invA;
if (norm(I1 - I) > tol)
BOOST_FAIL("inverse(A)*A does not recover unit matrix (norm = "
<< norm(I1-I) << ")");
if (norm(I2 - I) > tol)
BOOST_FAIL("A*inverse(A) does not recover unit matrix (norm = "
<< norm(I1-I) << ")");
}
}
test_suite* MatricesTest::suite() {
test_suite* suite = BOOST_TEST_SUITE("Matrix tests");
suite->add(BOOST_TEST_CASE(&MatricesTest::testEigenvectors));
suite->add(BOOST_TEST_CASE(&MatricesTest::testSqrt));
suite->add(BOOST_TEST_CASE(&MatricesTest::testSVD));
suite->add(BOOST_TEST_CASE(&MatricesTest::testInverse));
suite->add(BOOST_TEST_CASE(&MatricesTest::testHighamSqrt));
return suite;
}
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