Last active
November 9, 2024 13:28
-
-
Save gocarlos/c91237b02c120c6319612e42fa196d77 to your computer and use it in GitHub Desktop.
Cheat sheet for the linear algebra library Eigen: http://eigen.tuxfamily.org/
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
| // A simple quickref for Eigen. Add anything that's missing. | |
| // Main author: Keir Mierle | |
| #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 | |
| double s; | |
| // Basic usage | |
| // Eigen // Matlab // comments | |
| x.size() // length(x) // vector size | |
| C.rows() // size(C,1) // number of rows | |
| C.cols() // size(C,2) // number of columns | |
| x(i) // x(i+1) // Matlab is 1-based | |
| C(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 be | |
| 4, 5, 6, // matrices, which are stacked along cols | |
| 7, 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 // Matlab | |
| MatrixXd::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 = zeros(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)*2-1 | |
| VectorXd::LinSpaced(size,low,high) // linspace(low,high,size)' | |
| v.setLinSpaced(size,low,high) // v = linspace(low,high,size)' | |
| VectorXi::LinSpaced(((hi-low)/step)+1, // low:step:hi | |
| low,low+step*(size-1)) // | |
| // 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 // Matlab | |
| x.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) | |
| // Of particular note is Eigen's swap function which is highly optimized. | |
| // Eigen // Matlab | |
| R.row(i) = P.col(j); // R(i, :) = P(:, j) | |
| R.col(j1).swap(mat1.col(j2)); // R(:, [j1 j2]) = R(:, [j2, j1]) | |
| // Views, transpose, etc; | |
| // Eigen // Matlab | |
| R.adjoint() // R' | |
| R.transpose() // R.' or conj(R') // Read-write | |
| R.diagonal() // diag(R) // Read-write | |
| x.asDiagonal() // diag(x) | |
| R.transpose().colwise().reverse() // rot90(R) // Read-write | |
| R.rowwise().reverse() // fliplr(R) | |
| R.colwise().reverse() // flipud(R) | |
| R.replicate(i,j) // repmat(P,i,j) | |
| // 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; | |
| // Vectorized operations on each element independently | |
| // Eigen // Matlab | |
| R = P.cwiseProduct(Q); // R = P .* Q | |
| R = P.array() * s.array(); // R = P .* s | |
| R = P.cwiseQuotient(Q); // R = P ./ Q | |
| R = P.array() / Q.array(); // R = P ./ Q | |
| R = P.array() + s.array(); // R = P + s | |
| R = P.array() - s.array(); // R = P - s | |
| R.array() += s; // R = R + s | |
| R.array() -= s; // R = R - s | |
| R.array() < Q.array(); // R < Q | |
| R.array() <= Q.array(); // R <= Q | |
| R.cwiseInverse(); // 1 ./ P | |
| R.array().inverse(); // 1 ./ P | |
| R.array().sin() // sin(P) | |
| R.array().cos() // cos(P) | |
| R.array().pow(s) // P .^ s | |
| R.array().square() // P .^ 2 | |
| R.array().cube() // P .^ 3 | |
| R.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) | |
| R = (Q.array()==0).select(P,A) // R(Q==0) = P(Q==0) | |
| R = P.unaryExpr(ptr_fun(func)) // R = arrayfun(func, P) // with: scalar func(const scalar &x); | |
| // Reductions. | |
| int r, c; | |
| // Eigen // Matlab | |
| R.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) | |
| // Dot products, norms, etc. | |
| // Eigen // Matlab | |
| x.norm() // norm(x). Note that norm(R) doesn't work in Eigen. | |
| x.squaredNorm() // dot(x, x) Note the equivalence is not true for complex | |
| x.dot(y) // dot(x, y) | |
| x.cross(y) // cross(x, y) Requires #include <Eigen/Geometry> | |
| //// Type conversion | |
| // Eigen // Matlab | |
| A.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 done | |
| // Note that for most operations Eigen requires all operands to have the same type: | |
| MatrixXf F = MatrixXf::Zero(3,3); | |
| A += F; // illegal in Eigen. In Matlab A = A+F is allowed | |
| A += F.cast<double>(); // F converted to double and then added (generally, conversion happens on-the-fly) | |
| // Eigen can map existing memory into Eigen matrices. | |
| float array[3]; | |
| Vector3f::Map(array).fill(10); // create a temporary Map over array and sets entries to 10 | |
| int data[4] = {1, 2, 3, 4}; | |
| Matrix2i mat2x2(data); // copies data into mat2x2 | |
| Matrix2i::Map(data) = 2*mat2x2; // overwrite elements of data with 2*mat2x2 | |
| MatrixXi::Map(data, 2, 2) += mat2x2; // adds mat2x2 to elements of data (alternative syntax if size is not know at compile time) | |
| // 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() | |
| // Eigenvalue problems | |
| // Eigen // Matlab | |
| A.eigenvalues(); // eig(A); | |
| EigenSolver<Matrix3d> eig(A); // [vec val] = eig(A) | |
| eig.eigenvalues(); // diag(val) | |
| eig.eigenvectors(); // vec | |
| // For self-adjoint matrices use SelfAdjointEigenSolver<> |
Excellent gist thank you so much for this 👍
Other ways to initialize matrices:
static const Eigen::Matrix<double,7,6> a {
{0, 0, 0, 0, 0, 0},
{1.0/5.0, 0, 0, 0, 0, 0},
{3.0/40.0, 9.0/40.0, 0, 0, 0, 0},
{44.0/45.0, -56.0/15.0, 32.0/9.0, 0, 0, 0},
{19372.0/6561.0, -25360.0/2187.0, 64448.0/6561.0, -212.0/729.0, 0, 0},
{9017.0/3168.0, -355.0/33.0, 46732.0/5247.0, 49.0/176.0, -5103.0/18656.0, 0},
{35.0/384.0, 0.0, 500.0/1113.0, 125.0/192.0, -2187.0/6784.0, 11.0/84.0},
};or
static const Eigen::Matrix<double,7,6> a = {
{0, 0, 0, 0, 0, 0},
{1.0/5.0, 0, 0, 0, 0, 0},
{3.0/40.0, 9.0/40.0, 0, 0, 0, 0},
{44.0/45.0, -56.0/15.0, 32.0/9.0, 0, 0, 0},
{19372.0/6561.0, -25360.0/2187.0, 64448.0/6561.0, -212.0/729.0, 0, 0},
{9017.0/3168.0, -355.0/33.0, 46732.0/5247.0, 49.0/176.0, -5103.0/18656.0, 0},
{35.0/384.0, 0.0, 500.0/1113.0, 125.0/192.0, -2187.0/6784.0, 11.0/84.0},
};or
MatrixXd a;
a = Eigen::Matrix<double,7,6> {
{0, 0, 0, 0, 0, 0},
{1.0/5.0, 0, 0, 0, 0, 0},
{3.0/40.0, 9.0/40.0, 0, 0, 0, 0},
{44.0/45.0, -56.0/15.0, 32.0/9.0, 0, 0, 0},
{19372.0/6561.0, -25360.0/2187.0, 64448.0/6561.0, -212.0/729.0, 0, 0},
{9017.0/3168.0, -355.0/33.0, 46732.0/5247.0, 49.0/176.0, -5103.0/18656.0, 0},
{35.0/384.0, 0.0, 500.0/1113.0, 125.0/192.0, -2187.0/6784.0, 11.0/84.0},
};
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment
Really good gist. And actually more helpful than the tuxfamily documentation. It makes sense to add numpy column