Deprecated:

since Eigen 3.3

Default constructor.

Parameters

[in]

size

Positive integer, size of the matrix whose tridiagonal decomposition will be computed.

The default constructor is useful in cases in which the user intends to perform decompositions via compute(). The size parameter is only used as a hint. It is not an error to give a wrong size, but it may impair performance.

See also

compute() for an example.

Constructor; computes tridiagonal decomposition of given matrix.

Parameters

[in]

matrix

Selfadjoint matrix whose tridiagonal decomposition is to be computed.

This constructor calls compute() to compute the tridiagonal decomposition.

Example:

MatrixXd X = MatrixXd::Random(5,5);
MatrixXd A = X + X.transpose();
cout << "Here is a random symmetric 5x5 matrix:" << endl << A << endl << endl;
Tridiagonalization<MatrixXd> triOfA(A);
MatrixXd Q = triOfA.matrixQ();
cout << "The orthogonal matrix Q is:" << endl << Q << endl;
MatrixXd T = triOfA.matrixT();
cout << "The tridiagonal matrix T is:" << endl << T << endl << endl;
cout << "Q * T * Q^T = " << endl << Q * T * Q.transpose() << endl;

Output:

Here is a random symmetric 5x5 matrix:
  1.36 -0.816  0.521   1.43 -0.144
-0.816 -0.659  0.794 -0.173 -0.406
 0.521  0.794 -0.541  0.461  0.179
  1.43 -0.173  0.461  -1.43  0.822
-0.144 -0.406  0.179  0.822  -1.37

The orthogonal matrix Q is:
       1        0        0        0        0
       0   -0.471    0.127   -0.671   -0.558
       0    0.301   -0.195    0.437   -0.825
       0    0.825   0.0459   -0.563 -0.00872
       0  -0.0832   -0.971   -0.202   0.0922
The tridiagonal matrix T is:
  1.36   1.73      0      0      0
  1.73   -1.2 -0.966      0      0
     0 -0.966  -1.28  0.214      0
     0      0  0.214  -1.69  0.345
     0      0      0  0.345  0.164

Q * T * Q^T = 
  1.36 -0.816  0.521   1.43 -0.144
-0.816 -0.659  0.794 -0.173 -0.406
 0.521  0.794 -0.541  0.461  0.179
  1.43 -0.173  0.461  -1.43  0.822
-0.144 -0.406  0.179  0.822  -1.37

Computes tridiagonal decomposition of given matrix.

Parameters

[in]

matrix

Selfadjoint matrix whose tridiagonal decomposition is to be computed.

Returns

Reference to *this

The tridiagonal decomposition is computed by bringing the columns of the matrix successively in the required form using Householder reflections. The cost is \( 4n^3/3 \) flops, where \( n \) denotes the size of the given matrix.

This method reuses of the allocated data in the Tridiagonalization object, if the size of the matrix does not change.

Example:

Tridiagonalization<MatrixXf> tri;
MatrixXf X = MatrixXf::Random(4,4);
MatrixXf A = X + X.transpose();
tri.compute(A);
cout << "The matrix T in the tridiagonal decomposition of A is: " << endl;
cout << tri.matrixT() << endl;
tri.compute(2*A); // re-use tri to compute eigenvalues of 2A
cout << "The matrix T in the tridiagonal decomposition of 2A is: " << endl;
cout << tri.matrixT() << endl;

Output:

The matrix T in the tridiagonal decomposition of A is: 
  1.36 -0.704      0      0
-0.704 0.0147   1.71      0
     0   1.71  0.856  0.641
     0      0  0.641 -0.506
The matrix T in the tridiagonal decomposition of 2A is: 
  2.72  -1.41      0      0
 -1.41 0.0294   3.43      0
     0   3.43   1.71   1.28
     0      0   1.28  -1.01

Returns the diagonal of the tridiagonal matrix T in the decomposition.

Returns

expression representing the diagonal of T

Precondition

Either the constructor Tridiagonalization(const MatrixType&) or the member function compute(const MatrixType&) has been called before to compute the tridiagonal decomposition of a matrix.

Example:

MatrixXcd X = MatrixXcd::Random(4,4);
MatrixXcd A = X + X.adjoint();
cout << "Here is a random self-adjoint 4x4 matrix:" << endl << A << endl << endl;
 
Tridiagonalization<MatrixXcd> triOfA(A);
MatrixXd T = triOfA.matrixT();
cout << "The tridiagonal matrix T is:" << endl << T << endl << endl;
 
cout << "We can also extract the diagonals of T directly ..." << endl;
VectorXd diag = triOfA.diagonal();
cout << "The diagonal is:" << endl << diag << endl; 
VectorXd subdiag = triOfA.subDiagonal();
cout << "The subdiagonal is:" << endl << subdiag << endl;

Output:

Here is a random self-adjoint 4x4 matrix:
    (-0.422,0)  (0.705,-1.01) (-0.17,-0.552) (0.338,-0.357)
  (0.705,1.01)      (0.515,0) (0.241,-0.446)   (0.05,-1.64)
 (-0.17,0.552)  (0.241,0.446)      (-1.03,0)  (0.0449,1.72)
 (0.338,0.357)    (0.05,1.64) (0.0449,-1.72)       (1.36,0)

The tridiagonal matrix T is:
-0.422  -1.45      0      0
 -1.45   1.01  -1.42      0
     0  -1.42    1.8   -1.2
     0      0   -1.2  -1.96

We can also extract the diagonals of T directly ...
The diagonal is:
-0.422
  1.01
   1.8
 -1.96
The subdiagonal is:
-1.45
-1.42
 -1.2

See also

matrixT(), subDiagonal()

Returns the Householder coefficients.

Returns

a const reference to the vector of Householder coefficients

Precondition

Either the constructor Tridiagonalization(const MatrixType&) or the member function compute(const MatrixType&) has been called before to compute the tridiagonal decomposition of a matrix.

The Householder coefficients allow the reconstruction of the matrix \( Q \) in the tridiagonal decomposition from the packed data.

Example:

Matrix4d X = Matrix4d::Random(4,4);
Matrix4d A = X + X.transpose();
cout << "Here is a random symmetric 4x4 matrix:" << endl << A << endl;
Tridiagonalization<Matrix4d> triOfA(A);
Vector3d hc = triOfA.householderCoefficients();
cout << "The vector of Householder coefficients is:" << endl << hc << endl;

Output:

Here is a random symmetric 4x4 matrix:
   1.36   0.612   0.122   0.326
  0.612   -1.21  -0.222   0.563
  0.122  -0.222 -0.0904    1.16
  0.326   0.563    1.16    1.66
The vector of Householder coefficients is:
1.87
1.24
   0

See also

packedMatrix(), Householder module

Returns the unitary matrix Q in the decomposition.

Returns

object representing the matrix Q

Precondition

Either the constructor Tridiagonalization(const MatrixType&) or the member function compute(const MatrixType&) has been called before to compute the tridiagonal decomposition of a matrix.

This function returns a light-weight object of template class HouseholderSequence. You can either apply it directly to a matrix or you can convert it to a matrix of type MatrixType.

See also

Tridiagonalization(const MatrixType&) for an example, matrixT(), class HouseholderSequence

Returns an expression of the tridiagonal matrix T in the decomposition.

Returns

expression object representing the matrix T

Precondition

Either the constructor Tridiagonalization(const MatrixType&) or the member function compute(const MatrixType&) has been called before to compute the tridiagonal decomposition of a matrix.

Currently, this function can be used to extract the matrix T from internal data and copy it to a dense matrix object. In most cases, it may be sufficient to directly use the packed matrix or the vector expressions returned by diagonal() and subDiagonal() instead of creating a new dense copy matrix with this function.

See also

Tridiagonalization(const MatrixType&) for an example, matrixQ(), packedMatrix(), diagonal(), subDiagonal()

Returns the internal representation of the decomposition.

Returns

a const reference to a matrix with the internal representation of the decomposition.

Precondition

Either the constructor Tridiagonalization(const MatrixType&) or the member function compute(const MatrixType&) has been called before to compute the tridiagonal decomposition of a matrix.

The returned matrix contains the following information:

• the strict upper triangular part is equal to the input matrix A.
• the diagonal and lower sub-diagonal represent the real tridiagonal symmetric matrix T.
• the rest of the lower part contains the Householder vectors that, combined with Householder coefficients returned by householderCoefficients(), allows to reconstruct the matrix Q as \( Q = H_{N-1} \ldots H_1 H_0 \). Here, the matrices \( H_i \) are the Householder transformations \( H_i = (I - h_i v_i v_i^T) \) where \( h_i \) is the \( i \)th Householder coefficient and \( v_i \) is the Householder vector defined by \( v_i = [ 0, \ldots, 0, 1, M(i+2,i), \ldots, M(N-1,i) ]^T \) with M the matrix returned by this function.

See LAPACK for further details on this packed storage.

Example:

Matrix4d X = Matrix4d::Random(4,4);
Matrix4d A = X + X.transpose();
cout << "Here is a random symmetric 4x4 matrix:" << endl << A << endl;
Tridiagonalization<Matrix4d> triOfA(A);
Matrix4d pm = triOfA.packedMatrix();
cout << "The packed matrix M is:" << endl << pm << endl;
cout << "The diagonal and subdiagonal corresponds to the matrix T, which is:" 
     << endl << triOfA.matrixT() << endl;

Output:

Here is a random symmetric 4x4 matrix:
   1.36   0.612   0.122   0.326
  0.612   -1.21  -0.222   0.563
  0.122  -0.222 -0.0904    1.16
  0.326   0.563    1.16    1.66
The packed matrix M is:
  1.36  0.612  0.122  0.326
-0.704 0.0147 -0.222  0.563
0.0925   1.71  0.856   1.16
 0.248  0.785  0.641 -0.506
The diagonal and subdiagonal corresponds to the matrix T, which is:
  1.36 -0.704      0      0
-0.704 0.0147   1.71      0
     0   1.71  0.856  0.641
     0      0  0.641 -0.506

See also

householderCoefficients()

Returns the subdiagonal of the tridiagonal matrix T in the decomposition.

Returns

expression representing the subdiagonal of T

Precondition

Either the constructor Tridiagonalization(const MatrixType&) or the member function compute(const MatrixType&) has been called before to compute the tridiagonal decomposition of a matrix.

See also

diagonal() for an example, matrixT()