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 real 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.

Example:

MatrixXd A = MatrixXd::Random(6,6);
cout << "Here is a random 6x6 matrix, A:" << endl << A << endl << endl;
 
RealSchur<MatrixXd> schur(A);
cout << "The orthogonal matrix U is:" << endl << schur.matrixU() << endl;
cout << "The quasi-triangular matrix T is:" << endl << schur.matrixT() << endl << endl;
 
MatrixXd U = schur.matrixU();
MatrixXd T = schur.matrixT();
cout << "U * T * U^T = " << endl << U * T * U.transpose() << endl;

Output:

Here is a random 6x6 matrix, A:
   0.68   -0.33   -0.27  -0.717  -0.687  0.0259
 -0.211   0.536  0.0268   0.214  -0.198   0.678
  0.566  -0.444   0.904  -0.967   -0.74   0.225
  0.597   0.108   0.832  -0.514  -0.782  -0.408
  0.823 -0.0452   0.271  -0.726   0.998   0.275
 -0.605   0.258   0.435   0.608  -0.563  0.0486

The orthogonal matrix U is:
  0.348  -0.754 0.00435  -0.351  0.0146   0.432
  -0.16  -0.266  -0.747   0.457  -0.366  0.0571
  0.505  -0.157  0.0746   0.644   0.518  -0.177
  0.703   0.324  -0.409  -0.349  -0.187  -0.275
  0.296   0.372    0.24   0.324  -0.379   0.684
 -0.126   0.305   -0.46  -0.161   0.647   0.485
The quasi-triangular matrix T is:
   -0.2   -1.83   0.864   0.271    1.09   0.139
  0.647   0.298 -0.0536   0.676  -0.288  0.0231
      0       0   0.967  -0.201  -0.429   0.847
      0       0       0   0.353   0.603   0.694
      0       0       0       0   0.572   -1.03
      0       0       0       0  0.0184   0.664

U * T * U^T = 
   0.68   -0.33   -0.27  -0.717  -0.687  0.0259
 -0.211   0.536  0.0268   0.214  -0.198   0.678
  0.566  -0.444   0.904  -0.967   -0.74   0.225
  0.597   0.108   0.832  -0.514  -0.782  -0.408
  0.823 -0.0452   0.271  -0.726   0.998   0.275
 -0.605   0.258   0.435   0.608  -0.563  0.0486

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 Francis QR iterations with implicit double shift. The cost of computing the Schur decomposition depends on the number of iterations; as a rough guide, it may be taken to be \(25n^3\) flops if computeU is true and \(10n^3\) flops if computeU is false.

Example:

MatrixXf A = MatrixXf::Random(4,4);
RealSchur<MatrixXf> schur(4);
schur.compute(A, /* computeU = */ false);
cout << "The matrix T in the decomposition of A is:" << endl << schur.matrixT() << endl;
schur.compute(A.inverse(), /* computeU = */ false);
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:
 0.523 -0.698  0.148  0.742
 0.475  0.986 -0.793  0.721
     0      0  -0.28  -0.77
     0      0 0.0145 -0.367
The matrix T in the decomposition of A^(-1) is:
-3.06 -4.57 -5.97  5.48
0.168 -2.62 -3.27   3.9
    0     0 0.427 0.573
    0     0 -1.05  1.35

See also

compute(const MatrixType&, bool, Index)

Computes Schur decomposition of a Hessenberg matrix H = Z T Z^T.

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 quasi-triangular matrix in the Schur decomposition.

Returns

A const reference to the matrix T.

Precondition

Either the constructor RealSchur(const MatrixType&, bool) or the member function compute(const MatrixType&, bool) has been called before to compute the Schur decomposition of a matrix.

See also

RealSchur(const MatrixType&, bool) for an example

Returns the orthogonal matrix in the Schur decomposition.

Returns

A const reference to the matrix U.

Precondition

Either the constructor RealSchur(const MatrixType&, bool) or the member function compute(const MatrixType&, bool) has been called before to compute the Schur decomposition of a matrix, and computeU was set to true (the default value).

See also

RealSchur(const MatrixType&, bool) for an example

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 40.