Eigen 矩陣定義#include <Eigen/Dense>Matrix<double, 3, 3> A; // Fixed rows and cols. Same as Matrix3d.Matrix<double, 3, Dynamic> B; // Fixed rows, dynamic cols.Matrix<double, Dynamic, Dynamic> C; // Full dynamic. Same as MatrixXd.Matrix<double, 3, 3, RowMajor> E; // Row major; default is column-major.Matrix3f P, Q, R; // 3x3 float matrix.Vector3f x, y, z; // 3x1 float matrix.RowVector3f a, b, c; // 1x3 float matrix.VectorXd v; // Dynamic column vector of doubles// Eigen // Matlab // commentsx.size() // length(x) // vector sizeC.rows() // size(C,1) // number of rowsC.cols() // size(C,2) // number of columnsx(i) // x(i+1) // Matlab is 1-basedC(i,j) // C(i+1,j+1) //Eigen 基礎使用// Basic usage// Eigen // Matlab // commentsx.size() // length(x) // vector sizeC.rows() // size(C,1) // number of rowsC.cols() // size(C,2) // number of columnsx(i) // x(i+1) // Matlab is 1-basedC(i, j) // C(i+1,j+1) //A.resize(4, 4); // Runtime error if assertions are on.B.resize(4, 9); // Runtime error if assertions are on.A.resize(3, 3); // Ok; size didn't change.B.resize(3, 9); // Ok; only dynamic cols changed.A << 1, 2, 3, // Initialize A. The elements can also be4, 5, 6, // matrices, which are stacked along cols7, 8, 9; // and then the rows are stacked.B << A, A, A; // B is three horizontally stacked A's.A.fill(10); // Fill A with all 10's.Eigen 特殊矩陣生成// Eigen // MatlabMatrixXd::Identity(rows,cols) // eye(rows,cols)C.setIdentity(rows,cols) // C = eye(rows,cols)MatrixXd::Zero(rows,cols) // zeros(rows,cols)C.setZero(rows,cols) // C = ones(rows,cols)MatrixXd::Ones(rows,cols) // ones(rows,cols)C.setOnes(rows,cols) // C = ones(rows,cols)MatrixXd::Random(rows,cols) // rand(rows,cols)*2-1 // MatrixXd::Random returns uniform random numbers in (-1, 1).C.setRandom(rows,cols) // C = rand(rows,cols)*2-1VectorXd::LinSpaced(size,low,high) // linspace(low,high,size)'v.setLinSpaced(size,low,high) // v = linspace(low,high,size)'Eigen 矩陣分塊// Matrix slicing and blocks. All expressions listed here are read/write.// Templated size versions are faster. Note that Matlab is 1-based (a size N// vector is x(1)...x(N)).// Eigen // Matlabx.head(n) // x(1:n)x.head<n>() // x(1:n)x.tail(n) // x(end - n + 1: end)x.tail<n>() // x(end - n + 1: end)x.segment(i, n) // x(i+1 : i+n)x.segment<n>(i) // x(i+1 : i+n)P.block(i, j, rows, cols) // P(i+1 : i+rows, j+1 : j+cols)P.block<rows, cols>(i, j) // P(i+1 : i+rows, j+1 : j+cols)P.row(i) // P(i+1, :)P.col(j) // P(:, j+1)P.leftCols<cols>() // P(:, 1:cols)P.leftCols(cols) // P(:, 1:cols)P.middleCols<cols>(j) // P(:, j+1:j+cols)P.middleCols(j, cols) // P(:, j+1:j+cols)P.rightCols<cols>() // P(:, end-cols+1:end)P.rightCols(cols) // P(:, end-cols+1:end)P.topRows<rows>() // P(1:rows, :)P.topRows(rows) // P(1:rows, :)P.middleRows<rows>(i) // P(i+1:i+rows, :)P.middleRows(i, rows) // P(i+1:i+rows, :)P.bottomRows<rows>() // P(end-rows+1:end, :)P.bottomRows(rows) // P(end-rows+1:end, :)P.topLeftCorner(rows, cols) // P(1:rows, 1:cols)P.topRightCorner(rows, cols) // P(1:rows, end-cols+1:end)P.bottomLeftCorner(rows, cols) // P(end-rows+1:end, 1:cols)P.bottomRightCorner(rows, cols) // P(end-rows+1:end, end-cols+1:end)P.topLeftCorner<rows,cols>() // P(1:rows, 1:cols)P.topRightCorner<rows,cols>() // P(1:rows, end-cols+1:end)P.bottomLeftCorner<rows,cols>() // P(end-rows+1:end, 1:cols)P.bottomRightCorner<rows,cols>() // P(end-rows+1:end, end-cols+1:end)Eigen 矩陣元素交換// Of particular note is Eigen's swap function which is highly optimized.// Eigen // MatlabR.row(i) = P.col(j); // R(i, :) = P(:, i)R.col(j1).swap(mat1.col(j2)); // R(:, [j1 j2]) = R(:, [j2, j1])Eigen 矩陣轉置// Views, transpose, etc; all read-write except for .adjoint().// Eigen // MatlabR.adjoint() // R'R.transpose() // R.' or conj(R')R.diagonal() // diag(R)x.asDiagonal() // diag(x)R.transpose().colwise().reverse(); // rot90(R)R.conjugate() // conj(R)Eigen 矩陣乘積// All the same as Matlab, but matlab doesn't have *= style operators.// Matrix-vector. Matrix-matrix. Matrix-scalar.y = M*x; R = P*Q; R = P*s;a = b*M; R = P - Q; R = s*P;a *= M; R = P + Q; R = P/s;R *= Q; R = s*P;R += Q; R *= s;R -= Q; R /= s;Eigen 矩陣單個元素操作// Vectorized operations on each element independently// Eigen // MatlabR = P.cwiseProduct(Q); // R = P .* QR = P.array() * s.array();// R = P .* sR = P.cwiseQuotient(Q); // R = P ./ QR = P.array() / Q.array();// R = P ./ QR = P.array() + s.array();// R = P + sR = P.array() - s.array();// R = P - sR.array() += s; // R = R + sR.array() -= s; // R = R - sR.array() < Q.array(); // R < QR.array() <= Q.array(); // R <= QR.cwiseInverse(); // 1 ./ PR.array().inverse(); // 1 ./ PR.array().sin() // sin(P)R.array().cos() // cos(P)R.array().pow(s) // P .^ sR.array().square() // P .^ 2R.array().cube() // P .^ 3R.cwiseSqrt() // sqrt(P)R.array().sqrt() // sqrt(P)R.array().exp() // exp(P)R.array().log() // log(P)R.cwiseMax(P) // max(R, P)R.array().max(P.array()) // max(R, P)R.cwiseMin(P) // min(R, P)R.array().min(P.array()) // min(R, P)R.cwiseAbs() // abs(P)R.array().abs() // abs(P)R.cwiseAbs2() // abs(P.^2)R.array().abs2() // abs(P.^2)(R.array() < s).select(P,Q); // (R < s ? P : Q)Eigen 矩陣化簡// Reductions.int r, c;// Eigen // MatlabR.minCoeff() // min(R(:))R.maxCoeff() // max(R(:))s = R.minCoeff(&r, &c) // [s, i] = min(R(:)); [r, c] = ind2sub(size(R), i);s = R.maxCoeff(&r, &c) // [s, i] = max(R(:)); [r, c] = ind2sub(size(R), i);R.sum() // sum(R(:))R.colwise().sum() // sum(R)R.rowwise().sum() // sum(R, 2) or sum(R')'R.prod() // prod(R(:))R.colwise().prod() // prod(R)R.rowwise().prod() // prod(R, 2) or prod(R')'R.trace() // trace(R)R.all() // all(R(:))R.colwise().all() // all(R)R.rowwise().all() // all(R, 2)R.any() // any(R(:))R.colwise().any() // any(R)R.rowwise().any() // any(R, 2)Eigen 矩陣點乘// Dot products, norms, etc.// Eigen // Matlabx.norm() // norm(x). Note that norm(R) doesn't work in Eigen.x.squaredNorm() // dot(x, x) Note the equivalence is not true for complexx.dot(y) // dot(x, y)x.cross(y) // cross(x, y) Requires #include <Eigen/Geometry>Eigen 矩陣類型轉換 Type conversion// Eigen // MatlabA.cast<double>(); // double(A)A.cast<float>(); // single(A)A.cast<int>(); // int32(A)A.real(); // real(A)A.imag(); // imag(A)// if the original type equals destination type, no work is doneEigen 求解線性方程組 Ax = b// Solve Ax = b. Result stored in x. Matlab: x = A \ b.x = A.ldlt().solve(b)); // A sym. p.s.d. #include <Eigen/Cholesky>x = A.llt() .solve(b)); // A sym. p.d. #include <Eigen/Cholesky>x = A.lu() .solve(b)); // Stable and fast. #include <Eigen/LU>x = A.qr() .solve(b)); // No pivoting. #include <Eigen/QR>x = A.svd() .solve(b)); // Stable, slowest. #include <Eigen/SVD>// .ldlt() -> .matrixL() and .matrixD()// .llt() -> .matrixL()// .lu() -> .matrixL() and .matrixU()// .qr() -> .matrixQ() and .matrixR()// .svd() -> .matrixU(), .singularValues(), and .matrixV()Eigen 矩陣特征值// Eigenvalue problems// Eigen // MatlabA.eigenvalues(); // eig(A);EigenSolver<Matrix3d> eig(A); // [vec val] = eig(A)eig.eigenvalues(); // diag(val)eig.eigenvectors(); // vec// For self-adjoint matrices use SelfAdjointEigenSolver<>

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