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  •    4年前 (2014-11-29)  Eigen |   抢沙发  19 
    文章评分 0 次,平均分 0.0

    1.求方程的特征根

    // Solve Ax = b. Result stored in x. Matlab: x = A \ b.
    bool solved;
    solved = A.ldlt().solve(b, &x));  // A sym. p.s.d.    #include <Eigen/Cholesky>
    solved = A.llt() .solve(b, &x));  // A sym. p.d.      #include <Eigen/Cholesky>
    solved = A.lu()  .solve(b, &x));  // Stable and fast. #include <Eigen/LU>
    solved = A.qr()  .solve(b, &x));  // No pivoting.     #include <Eigen/QR>
    solved = A.svd() .solve(b, &x));  // Stable, slowest. #include <Eigen/SVD>
    // .ldlt() -> .matrixL() and .matrixD()
    // .llt()  -> .matrixL()
    // .lu()   -> .matrixL() and .matrixU()
    // .qr()   -> .matrixQ() and .matrixR()
    // .svd()  -> .matrixU(), .singularValues(), and .matrixV()

    2.求特征值与特征向量

    // Eigenvalue problems
    // Eigen                          // Matlab
    A.eigenvalues();                  // eig(A);
    EigenSolver<Matrix3d> eig(A);     // [vec val] = eig(A)
    eig.eigenvalues();                // diag(val)
    eig.eigenvectors();               // ve

    3.其他

    // A simple quickref for Eigen. Add anything that's missing.
    // Main author: Keir Mierle
    
    #include <Eigen/Core>
    #include <Eigen/Array>
    
    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.
    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.
    A.setRandom();    // Fill A with uniform random numbers in (-1, 1).
                      // Requires #include <Eigen/Array>.
    A.setIdentity();  // Fill A with the identity.
    
    // 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)                          // N = rows(x); x(N - n: N)
    x.tail<n>()                        // N = rows(x); x(N - n: N)
    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.topLeftCorner(rows, cols)        // P(1:rows, 1:cols)
    P.topRightCorner(rows, cols)       // [m n]=size(P); P(1:rows, n-cols+1:n)
    P.bottomLeftCorner(rows, cols)     // [m n]=size(P); P(m-rows+1:m, 1:cols)
    P.bottomRightCorner(rows, cols)    // [m n]=size(P); P(m-rows+1:m, n-cols+1:n)
    P.topLeftCorner<rows,cols>()       // P(1:rows, 1:cols)
    P.topRightCorner<rows,cols>()      // [m n]=size(P); P(1:rows, n-cols+1:n)
    P.bottomLeftCorner<rows,cols>()    // [m n]=size(P); P(m-rows+1:m, 1:cols)
    P.bottomRightCorner<rows,cols>()   // [m n]=size(P); P(m-rows+1:m, n-cols+1:n)
    
    // Of particular note is Eigen's swap function which is highly optimized.
    // Eigen                           // Matlab
    R.row(i) = P.col(j);               // R(i, :) = P(:, i)
    R.col(j1).swap(mat1.col(j2));      // R(:, [j1 j2]) = R(:, [j2, j1])
    
    // Views, transpose, etc; all read-write except for .adjoint().
    // Eigen                           // Matlab
    R.adjoint()                        // R'
    R.transpose()                      // R.' or conj(R')
    R.diagonal()                       // diag(R)
    x.asDiagonal()                     // diag(x)
    
    // 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
     // (most require #include <Eigen/Array>)
    // 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)
    
    // Reductions.
    int r, c;
    // Eigen                  // Matlab
    R.minCoeff()              // min(R(:))
    R.maxCoeff()              // max(R(:))
    s = R.minCoeff(&r, &c)    // [aa, bb] = min(R); [cc, dd] = min(aa);
                              // r = bb(dd); c = dd; s = cc
    s = R.maxCoeff(&r, &c)    // [aa, bb] = max(R); [cc, dd] = max(aa);
                              // row = bb(dd); col = dd; s = cc
    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>
    
    // Eigen can map existing memory into Eigen matrices.
    float array[3];
    Map<Vector3f>(array, 3).fill(10);
    int data[4] = 1, 2, 3, 4;
    Matrix2i mat2x2(data);
    MatrixXi mat2x2 = Map<Matrix2i>(data);
    MatrixXi mat2x2 = Map<MatrixXi>(data, 2, 2);
    
    // Solve Ax = b. Result stored in x. Matlab: x = A \ b.
    bool solved;
    solved = A.ldlt().solve(b, &x));  // A sym. p.s.d.    #include <Eigen/Cholesky>
    solved = A.llt() .solve(b, &x));  // A sym. p.d.      #include <Eigen/Cholesky>
    solved = A.lu()  .solve(b, &x));  // Stable and fast. #include <Eigen/LU>
    solved = A.qr()  .solve(b, &x));  // No pivoting.     #include <Eigen/QR>
    solved = A.svd() .solve(b, &x));  // 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
    
     

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