Type for the matrices in the Schur decomposition.

This is a square matrix with entries of type ComplexScalar. The size is the same as the size of _MatrixType.

Complex scalar type for _MatrixType.

This is std::complex<Scalar> if Scalar is real (e.g., float or double) and just Scalar if Scalar is complex.

Deprecated:

since Eigen 3.3

Default constructor.

Parameters

[in]

size

Positive integer, size of the matrix whose Schur 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 Schur decomposition of given matrix.

Parameters

[in]	matrix	Square matrix whose Schur decomposition is to be computed.
[in]	computeU	If true, both T and U are computed; if false, only T is computed.

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

See also

matrixT() and matrixU() for examples.

Computes Schur decomposition of given matrix.

Parameters

[in]	matrix	Square matrix whose Schur decomposition is to be computed.
[in]	computeU	If true, both T and U are computed; if false, only T is computed.

Returns

Reference to *this

The Schur decomposition is computed by first reducing the matrix to Hessenberg form using the class HessenbergDecomposition. The Hessenberg matrix is then reduced to triangular form by performing QR iterations with a single shift. The cost of computing the Schur decomposition depends on the number of iterations; as a rough guide, it may be taken on the number of iterations; as a rough guide, it may be taken to be \(25n^3\) complex flops, or \(10n^3\) complex flops if computeU is false.

Example:

MatrixXcf A = MatrixXcf::Random(4,4);
ComplexSchur<MatrixXcf> schur(4);
schur.compute(A);
cout << "The matrix T in the decomposition of A is:" << endl << schur.matrixT() << endl;
schur.compute(A.inverse());
cout << "The matrix T in the decomposition of A^(-1) is:" << endl << schur.matrixT() << endl;

Output:

The matrix T in the decomposition of A is:
 (-0.691,-1.63)  (0.763,-0.144) (-0.104,-0.836) (-0.462,-0.378)
          (0,0)   (-0.758,1.22)  (-0.65,-0.772)  (-0.244,0.113)
          (0,0)           (0,0)   (0.137,0.505) (0.0687,-0.404)
          (0,0)           (0,0)           (0,0)   (1.52,-0.402)
The matrix T in the decomposition of A^(-1) is:
    (0.501,-1.84)    (-1.01,-0.984)       (0.636,1.3)    (-0.676,0.352)
            (0,0)   (-0.369,-0.593)     (0.0733,0.18) (-0.0658,-0.0263)
            (0,0)             (0,0)    (-0.222,0.521)    (-0.191,0.121)
            (0,0)             (0,0)             (0,0)     (0.614,0.162)

See also

compute(const MatrixType&, bool, Index)

Compute Schur decomposition from a given Hessenberg matrix.

Parameters

[in]	matrixH	Matrix in Hessenberg form H
[in]	matrixQ	orthogonal matrix Q that transform a matrix A to H : A = Q H Q^T
	computeU	Computes the matriX U of the Schur vectors

Returns

Reference to *this

This routine assumes that the matrix is already reduced in Hessenberg form matrixH using either the class HessenbergDecomposition or another mean. It computes the upper quasi-triangular matrix T of the Schur decomposition of H When computeU is true, this routine computes the matrix U such that A = U T U^T = (QZ) T (QZ)^T = Q H Q^T where A is the initial matrix

NOTE Q is referenced if computeU is true; so, if the initial orthogonal matrix is not available, the user should give an identity matrix (Q.setIdentity())

See also

compute(const MatrixType&, bool)

Reports whether previous computation was successful.

Returns

Success if computation was successful, NoConvergence otherwise.

Returns the triangular matrix in the Schur decomposition.

Returns

A const reference to the matrix T.

It is assumed that either the constructor ComplexSchur(const MatrixType& matrix, bool computeU) or the member function compute(const MatrixType& matrix, bool computeU) has been called before to compute the Schur decomposition of a matrix.

Note that this function returns a plain square matrix. If you want to reference only the upper triangular part, use:

schur.matrixT().triangularView<Upper>() 

Example:

MatrixXcf A = MatrixXcf::Random(4,4);
cout << "Here is a random 4x4 matrix, A:" << endl << A << endl << endl;
ComplexSchur<MatrixXcf> schurOfA(A, false); // false means do not compute U
cout << "The triangular matrix T is:" << endl << schurOfA.matrixT() << endl;

Output:

Here is a random 4x4 matrix, A:
  (-0.211,0.68)  (0.108,-0.444)   (0.435,0.271) (-0.198,-0.687)
  (0.597,0.566) (0.258,-0.0452)  (0.214,-0.717)  (-0.782,-0.74)
 (-0.605,0.823)  (0.0268,-0.27) (-0.514,-0.967)  (-0.563,0.998)
  (0.536,-0.33)   (0.832,0.904)  (0.608,-0.726)  (0.678,0.0259)

The triangular matrix T is:
 (-0.691,-1.63)  (0.763,-0.144) (-0.104,-0.836) (-0.462,-0.378)
          (0,0)   (-0.758,1.22)  (-0.65,-0.772)  (-0.244,0.113)
          (0,0)           (0,0)   (0.137,0.505) (0.0687,-0.404)
          (0,0)           (0,0)           (0,0)   (1.52,-0.402)

Returns the unitary matrix in the Schur decomposition.

Returns

A const reference to the matrix U.

It is assumed that either the constructor ComplexSchur(const MatrixType& matrix, bool computeU) or the member function compute(const MatrixType& matrix, bool computeU) has been called before to compute the Schur decomposition of a matrix, and that computeU was set to true (the default value).

Example:

MatrixXcf A = MatrixXcf::Random(4,4);
cout << "Here is a random 4x4 matrix, A:" << endl << A << endl << endl;
ComplexSchur<MatrixXcf> schurOfA(A);
cout << "The unitary matrix U is:" << endl << schurOfA.matrixU() << endl;

Output:

Here is a random 4x4 matrix, A:
  (-0.211,0.68)  (0.108,-0.444)   (0.435,0.271) (-0.198,-0.687)
  (0.597,0.566) (0.258,-0.0452)  (0.214,-0.717)  (-0.782,-0.74)
 (-0.605,0.823)  (0.0268,-0.27) (-0.514,-0.967)  (-0.563,0.998)
  (0.536,-0.33)   (0.832,0.904)  (0.608,-0.726)  (0.678,0.0259)

The unitary matrix U is:
 (-0.122,0.271)   (0.354,0.255)    (-0.7,0.321) (0.0909,-0.346)
   (0.247,0.23)  (0.435,-0.395)   (0.184,-0.38)  (0.492,-0.347)
(0.859,-0.0877)  (0.00469,0.21) (-0.256,0.0163)   (0.133,0.355)
 (-0.116,0.195) (-0.484,-0.432)  (-0.183,0.359)   (0.559,0.231)

Sets the maximum number of iterations allowed.

If not specified by the user, the maximum number of iterations is m_maxIterationsPerRow times the size of the matrix.

Maximum number of iterations per row.

If not otherwise specified, the maximum number of iterations is this number times the size of the matrix. It is currently set to 30.