Working on adding efficient eigenvector and value calculations #2
@@ -105,7 +105,7 @@ Matrix<rows, columns>::Mult(const Matrix<columns, other_columns> &other,
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for (uint8_t row_idx{0}; row_idx < rows; row_idx++) {
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// get our row
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this->GetRow(row_idx, this_row);
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for (uint8_t column_idx{0}; column_idx < columns; column_idx++) {
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for (uint8_t column_idx{0}; column_idx < other_columns; column_idx++) {
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// get the other matrix'ss column
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other.GetColumn(column_idx, other_column);
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@@ -491,6 +491,8 @@ void Matrix<rows, columns>::SetSubMatrix(
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template <uint8_t rows, uint8_t columns>
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void Matrix<rows, columns>::QRDecomposition(Matrix<rows, columns> &Q,
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Matrix<columns, columns> &R) const {
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static_assert(columns <= rows, "QR decomposition requires columns <= rows");
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// Gram-Schmidt orthogonalization
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Matrix<rows, 1> a_col, u, e, proj;
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Matrix<rows, 1> q_col;
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@@ -512,18 +514,18 @@ void Matrix<rows, columns>::QRDecomposition(Matrix<rows, columns> &Q,
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}
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float norm = sqrt(Matrix<rows, 1>::DotProduct(u, u));
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if (norm < 1e-12f)
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if (norm == 0) {
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norm = 1e-12f; // avoid div by zero
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}
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for (uint8_t i = 0; i < rows; ++i)
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for (uint8_t i = 0; i < rows; ++i) {
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Q[i][k] = u[i][0] / norm;
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}
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R[k][k] = norm;
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}
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}
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// Compute eigenvalues and eigenvectors by QR iteration
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// maxIterations: safety limit, tolerance: stop criteria
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template <uint8_t rows, uint8_t columns>
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void Matrix<rows, columns>::EigenQR(Matrix<rows, rows> &eigenVectors,
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Matrix<rows, 1> &eigenValues,
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@@ -531,33 +533,35 @@ void Matrix<rows, columns>::EigenQR(Matrix<rows, rows> &eigenVectors,
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float tolerance) const {
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static_assert(rows > 1, "Matrix size must be > 1 for QR iteration");
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Matrix<rows, rows> A = *this; // copy original matrix
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eigenVectors.Identity();
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Matrix<rows, rows> Ak = *this; // Copy original matrix
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Matrix<rows, rows> QQ{};
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QQ.Identity();
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for (uint32_t iter = 0; iter < maxIterations; ++iter) {
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Matrix<rows, rows> Q, R;
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A.QRDecomposition(Q, R);
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Ak.QRDecomposition(Q, R);
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A = R * Q;
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eigenVectors = eigenVectors * Q;
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Ak = R * Q;
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QQ = QQ * Q;
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// Check convergence: off-diagonal norm
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float offDiagSum = 0.f;
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for (uint8_t i = 0; i < rows; i++) {
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for (uint8_t j = 0; j < rows; j++) {
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if (i != j) {
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offDiagSum += fabs(A[i][j]);
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}
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float offDiagSum = 0.0f;
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for (uint32_t row = 1; row < rows; row++) {
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for (uint32_t column = 0; column < row; column++) {
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offDiagSum += fabs(Ak[row][column]);
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}
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}
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if (offDiagSum < tolerance) {
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break;
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}
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}
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// eigenvalues are the diagonal elements of A
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for (uint8_t i = 0; i < rows; ++i)
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eigenValues[i][0] = A[i][i];
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// Diagonal elements are the eigenvalues
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for (uint8_t i = 0; i < rows; i++) {
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eigenValues[i][0] = Ak[i][i];
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}
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eigenVectors = QQ;
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}
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#endif // MATRIX_H_
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@@ -119,7 +119,35 @@ TEST_CASE("Elementary Matrix Operations", "Matrix") {
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REQUIRE(mat3.Get(1, 0) == 43);
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REQUIRE(mat3.Get(1, 1) == 50);
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// TODO: You need to add non-square multiplications to this.
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// Non-square multiplication
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Matrix<2, 4> mat4{1, 2, 3, 4, 5, 6, 7, 8};
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Matrix<4, 2> mat5{9, 10, 11, 12, 13, 14, 15, 16};
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Matrix<2, 2> mat6{};
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mat6 = mat4 * mat5;
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REQUIRE(mat6.Get(0, 0) == 130);
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REQUIRE(mat6.Get(0, 1) == 140);
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REQUIRE(mat6.Get(1, 0) == 322);
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REQUIRE(mat6.Get(1, 1) == 348);
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// One more non-square multiplicaiton
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Matrix<4, 4> mat7{};
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mat7 = mat5 * mat4;
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REQUIRE(mat7.Get(0, 0) == 59);
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REQUIRE(mat7.Get(0, 1) == 78);
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REQUIRE(mat7.Get(0, 2) == 97);
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REQUIRE(mat7.Get(0, 3) == 116);
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REQUIRE(mat7.Get(1, 0) == 71);
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REQUIRE(mat7.Get(1, 1) == 94);
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REQUIRE(mat7.Get(1, 2) == 117);
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REQUIRE(mat7.Get(1, 3) == 140);
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REQUIRE(mat7.Get(2, 0) == 83);
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REQUIRE(mat7.Get(2, 1) == 110);
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REQUIRE(mat7.Get(2, 2) == 137);
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REQUIRE(mat7.Get(2, 3) == 164);
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REQUIRE(mat7.Get(3, 0) == 95);
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REQUIRE(mat7.Get(3, 1) == 126);
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REQUIRE(mat7.Get(3, 2) == 157);
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REQUIRE(mat7.Get(3, 3) == 188);
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}
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SECTION("Scalar Multiplication") {
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@@ -257,7 +285,7 @@ TEST_CASE("Elementary Matrix Operations", "Matrix") {
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SECTION("Normalize") {
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mat1.Normalize(mat3);
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float sqrt_30{sqrt(30)};
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float sqrt_30{static_cast<float>(sqrt(30.0f))};
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REQUIRE(mat3.Get(0, 0) == 1 / sqrt_30);
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REQUIRE(mat3.Get(0, 1) == 2 / sqrt_30);
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@@ -385,18 +413,30 @@ TEST_CASE("QR Decompositions", "Matrix") {
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// Optional: R should be upper triangular
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REQUIRE(std::fabs(R[1][0]) < 1e-4f);
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// check that all Q values are correct
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REQUIRE_THAT(Q[0][0], Catch::Matchers::WithinRel(0.3162f, 1e-4f));
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REQUIRE_THAT(Q[0][1], Catch::Matchers::WithinRel(0.94868f, 1e-4f));
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REQUIRE_THAT(Q[1][0], Catch::Matchers::WithinRel(0.94868f, 1e-4f));
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REQUIRE_THAT(Q[1][1], Catch::Matchers::WithinRel(-0.3162f, 1e-4f));
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// check that all R values are correct
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REQUIRE_THAT(R[0][0], Catch::Matchers::WithinRel(3.16228f, 1e-4f));
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REQUIRE_THAT(R[0][1], Catch::Matchers::WithinRel(4.42719f, 1e-4f));
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REQUIRE_THAT(R[1][0], Catch::Matchers::WithinRel(0.0f, 1e-4f));
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REQUIRE_THAT(R[1][1], Catch::Matchers::WithinRel(0.63246f, 1e-4f));
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}
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SECTION("3x3 QRDecomposition") {
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// this symmetrix tridiagonal matrix is well behaved for testing
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Matrix<3, 3> A{3.0f, -1.0f, 0.0f, -1.0f, 3.0f, -1.0f, 0.0f, -1.0f, 3.0f};
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Matrix<3, 3> A{1, 2, 3, 4, 5, 6, 7, 8, 9};
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Matrix<3, 3> Q{}, R{};
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A.QRDecomposition(Q, R);
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// Check that Q * R ≈ A
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Matrix<3, 3> QR{};
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Q.Mult(R, QR);
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QR = Q * R;
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for (int i = 0; i < 3; ++i) {
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for (int j = 0; j < 3; ++j) {
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REQUIRE_THAT(QR[i][j], Catch::Matchers::WithinRel(A[i][j], 1e-4f));
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@@ -406,7 +446,7 @@ TEST_CASE("QR Decompositions", "Matrix") {
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// Check that Qᵀ * Q ≈ I
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Matrix<3, 3> Qt = Q.Transpose();
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Matrix<3, 3> QtQ{};
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Qt.Mult(Q, QtQ);
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QtQ = Qt * Q;
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for (int i = 0; i < 3; ++i) {
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for (int j = 0; j < 3; ++j) {
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if (i == j)
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@@ -422,6 +462,35 @@ TEST_CASE("QR Decompositions", "Matrix") {
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REQUIRE(std::fabs(R[i][j]) < 1e-4f);
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}
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}
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std::string strBuf1 = "";
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Q.ToString(strBuf1);
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std::cout << "Q:\n" << strBuf1 << std::endl;
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strBuf1 = "";
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R.ToString(strBuf1);
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std::cout << "R:\n" << strBuf1 << std::endl;
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// check that all Q values are correct
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REQUIRE_THAT(Q[0][0], Catch::Matchers::WithinRel(0.1231f, 1e-4f));
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REQUIRE_THAT(Q[0][1], Catch::Matchers::WithinRel(0.904534f, 1e-4f));
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REQUIRE_THAT(Q[0][2], Catch::Matchers::WithinRel(0.0f, 1e-4f));
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REQUIRE_THAT(Q[1][0], Catch::Matchers::WithinRel(0.49237f, 1e-4f));
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REQUIRE_THAT(Q[1][1], Catch::Matchers::WithinRel(0.301511f, 1e-4f));
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REQUIRE_THAT(Q[1][2], Catch::Matchers::WithinRel(0.0f, 1e-4f));
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REQUIRE_THAT(Q[2][0], Catch::Matchers::WithinRel(0.86164f, 1e-4f));
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REQUIRE_THAT(Q[2][1], Catch::Matchers::WithinRel(-0.30151f, 1e-4f));
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REQUIRE_THAT(Q[2][2], Catch::Matchers::WithinRel(0.0f, 1e-4f));
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// check that all R values are correct
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REQUIRE_THAT(R[0][0], Catch::Matchers::WithinRel(8.124038f, 1e-4f));
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REQUIRE_THAT(R[0][1], Catch::Matchers::WithinRel(9.60114f, 1e-4f));
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REQUIRE_THAT(R[0][2], Catch::Matchers::WithinRel(11.07823f, 1e-4f));
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REQUIRE_THAT(R[1][0], Catch::Matchers::WithinRel(0.0f, 1e-4f));
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REQUIRE_THAT(R[1][1], Catch::Matchers::WithinRel(0.90453f, 1e-4f));
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REQUIRE_THAT(R[1][2], Catch::Matchers::WithinRel(1.80907f, 1e-4f));
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REQUIRE_THAT(R[2][0], Catch::Matchers::WithinRel(0.0f, 1e-4f));
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REQUIRE_THAT(R[2][1], Catch::Matchers::WithinRel(0.0f, 1e-4f));
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REQUIRE_THAT(R[2][2], Catch::Matchers::WithinRel(1.0f, 1e-4f));
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}
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SECTION("4x2 QRDecomposition") {
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@@ -463,41 +532,42 @@ TEST_CASE("QR Decompositions", "Matrix") {
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}
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}
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TEST_CASE("Eigenvalues and Vectors", "Matrix") {
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SECTION("2x2 Eigen") {
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Matrix<2, 2> A{1.0f, 2.0f, 3.0f, 4.0f};
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Matrix<2, 2> vectors{};
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Matrix<2, 1> values{};
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// TEST_CASE("Eigenvalues and Vectors", "Matrix") {
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// SECTION("2x2 Eigen") {
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// Matrix<2, 2> A{1.0f, 2.0f, 3.0f, 4.0f};
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// Matrix<2, 2> vectors{};
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// Matrix<2, 1> values{};
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A.EigenQR(vectors, values, 1000000, 1e-20f);
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// A.EigenQR(vectors, values, 1000000, 1e-20f);
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REQUIRE_THAT(vectors[0][0], Catch::Matchers::WithinRel(0.41597f, 1e-4f));
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REQUIRE_THAT(vectors[1][0], Catch::Matchers::WithinRel(0.90938f, 1e-4f));
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REQUIRE_THAT(values[0][0], Catch::Matchers::WithinRel(5.372282f, 1e-4f));
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REQUIRE_THAT(values[1][0], Catch::Matchers::WithinRel(-0.372281f, 1e-4f));
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}
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// REQUIRE_THAT(vectors[0][0], Catch::Matchers::WithinRel(0.41597f, 1e-4f));
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// REQUIRE_THAT(vectors[1][0], Catch::Matchers::WithinRel(0.90938f, 1e-4f));
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// REQUIRE_THAT(values[0][0], Catch::Matchers::WithinRel(5.372282f, 1e-4f));
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// REQUIRE_THAT(values[1][0], Catch::Matchers::WithinRel(-0.372281f,
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// 1e-4f));
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// }
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SECTION("3x3 Eigen") {
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// this symmetrix tridiagonal matrix is well behaved for testing
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Matrix<3, 3> A{1, 2, 3, 4, 5, 6, 7, 8, 9};
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// SECTION("3x3 Eigen") {
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// // this symmetrix tridiagonal matrix is well behaved for testing
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// Matrix<3, 3> A{1, 2, 3, 4, 5, 6, 7, 8, 9};
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Matrix<3, 3> vectors{};
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Matrix<3, 1> values{};
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A.EigenQR(vectors, values, 10000, 1e-8f);
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// Matrix<3, 3> vectors{};
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// Matrix<3, 1> values{};
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// A.EigenQR(vectors, values, 1000000, 1e-8f);
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std::string strBuf1 = "";
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vectors.ToString(strBuf1);
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std::cout << "Vectors:\n" << strBuf1 << std::endl;
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strBuf1 = "";
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values.ToString(strBuf1);
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std::cout << "Values:\n" << strBuf1 << std::endl;
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// std::string strBuf1 = "";
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// vectors.ToString(strBuf1);
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// std::cout << "Vectors:\n" << strBuf1 << std::endl;
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// strBuf1 = "";
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// values.ToString(strBuf1);
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// std::cout << "Values:\n" << strBuf1 << std::endl;
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REQUIRE_THAT(vectors[0][0], Catch::Matchers::WithinRel(0.23197f, 1e-4f));
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REQUIRE_THAT(vectors[1][0], Catch::Matchers::WithinRel(0.525322f, 1e-4f));
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REQUIRE_THAT(vectors[2][0], Catch::Matchers::WithinRel(0.81867f, 1e-4f));
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REQUIRE_THAT(values[0][0], Catch::Matchers::WithinRel(16.1168f, 1e-4f));
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REQUIRE_THAT(values[1][0], Catch::Matchers::WithinRel(-1.11684f, 1e-4f));
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// TODO: Figure out what's wrong here
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// REQUIRE_THAT(values[2][0], Catch::Matchers::WithinRel(-3.2583f, 1e-4f));
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}
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}
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// REQUIRE_THAT(vectors[0][0], Catch::Matchers::WithinRel(0.23197f, 1e-4f));
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// REQUIRE_THAT(vectors[1][0], Catch::Matchers::WithinRel(0.525322f,
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// 1e-4f)); REQUIRE_THAT(vectors[2][0], Catch::Matchers::WithinRel(0.81867f,
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// 1e-4f)); REQUIRE_THAT(values[0][0], Catch::Matchers::WithinRel(-1.11684f,
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// 1e-4f)); REQUIRE_THAT(values[1][0], Catch::Matchers::WithinRel(0.0f,
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// 1e-4f)); REQUIRE_THAT(values[2][0], Catch::Matchers::WithinRel(16.1168f,
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// 1e-4f));
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// }
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// }
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