Constructor.

Parameters

[in]	v	Matrix containing the essential parts of the Householder vectors
[in]	h	Vector containing the Householder coefficients

Constructs the Householder sequence with coefficients given by h and vectors given by v. The i-th Householder coefficient \( h_i \) is given by h(i) and the essential part of the i-th Householder vector \( v_i \) is given by v(k,i) with k > i (the subdiagonal part of the i-th column). If v has fewer columns than rows, then the Householder sequence contains as many Householder reflections as there are columns.

Note

The HouseholderSequence object stores v and h by reference.

Example:

Matrix3d v = Matrix3d::Random();
cout << "The matrix v is:" << endl;
cout << v << endl;
 
Vector3d v0(1, v(1,0), v(2,0));
cout << "The first Householder vector is: v_0 = " << v0.transpose() << endl;
Vector3d v1(0, 1, v(2,1));
cout << "The second Householder vector is: v_1 = " << v1.transpose()  << endl;
Vector3d v2(0, 0, 1);
cout << "The third Householder vector is: v_2 = " << v2.transpose() << endl;
 
Vector3d h = Vector3d::Random();
cout << "The Householder coefficients are: h = " << h.transpose() << endl;
 
Matrix3d H0 = Matrix3d::Identity() - h(0) * v0 * v0.adjoint();
cout << "The first Householder reflection is represented by H_0 = " << endl;
cout << H0 << endl;
Matrix3d H1 = Matrix3d::Identity() - h(1) * v1 * v1.adjoint();
cout << "The second Householder reflection is represented by H_1 = " << endl;
cout << H1 << endl;
Matrix3d H2 = Matrix3d::Identity() - h(2) * v2 * v2.adjoint();
cout << "The third Householder reflection is represented by H_2 = " << endl;
cout << H2 << endl;
cout << "Their product is H_0 H_1 H_2 = " << endl;
cout << H0 * H1 * H2 << endl;
 
HouseholderSequence<Matrix3d, Vector3d> hhSeq(v, h);
Matrix3d hhSeqAsMatrix(hhSeq);
cout << "If we construct a HouseholderSequence from v and h" << endl;
cout << "and convert it to a matrix, we get:" << endl;
cout << hhSeqAsMatrix << endl;

Output:

The matrix v is:
  0.68  0.597  -0.33
-0.211  0.823  0.536
 0.566 -0.605 -0.444
The first Householder vector is: v_0 =      1 -0.211  0.566
The second Householder vector is: v_1 =      0      1 -0.605
The third Householder vector is: v_2 = 0 0 1
The Householder coefficients are: h =   0.108 -0.0452   0.258
The first Householder reflection is represented by H_0 = 
  0.892  0.0228 -0.0611
 0.0228   0.995  0.0129
-0.0611  0.0129   0.965
The second Householder reflection is represented by H_1 = 
      1       0       0
      0    1.05 -0.0273
      0 -0.0273    1.02
The third Householder reflection is represented by H_2 = 
    1     0     0
    0     1     0
    0     0 0.742
Their product is H_0 H_1 H_2 = 
  0.892  0.0255 -0.0466
 0.0228    1.04 -0.0105
-0.0611 -0.0129   0.728
If we construct a HouseholderSequence from v and h
and convert it to a matrix, we get:
  0.892  0.0255 -0.0466
 0.0228    1.04 -0.0105
-0.0611 -0.0129   0.728

See also

setLength(), setShift()

Number of columns of transformation viewed as a matrix.

Returns

Number of columns

This equals the dimension of the space that the transformation acts on.

Returns

an expression of the complex conjugate of *this if Cond==true, returns *this otherwise.

Essential part of a Householder vector.

Parameters

[in]

k

Index of Householder reflection

Returns

Vector containing non-trivial entries of k-th Householder vector

This function returns the essential part of the Householder vector \( v_i \). This is a vector of length \( n-i \) containing the last \( n-i \) entries of the vector

\[ v_i = [\underbrace{0, \ldots, 0}_{i-1\mbox{ zeros}}, 1, \underbrace{*, \ldots,*}_{n-i\mbox{ arbitrary entries}} ]. \]

The index \( i \) equals k + shift(), corresponding to the k-th column of the matrix v passed to the constructor.

See also

setShift(), shift()

Computes the product of a Householder sequence with a matrix.

Parameters

[in]

other

Matrix being multiplied.

Returns

Expression object representing the product.

This function computes \( HM \) where \( H \) is the Householder sequence represented by *this and \( M \) is the matrix other.

Number of rows of transformation viewed as a matrix.

Returns

Number of rows

This equals the dimension of the space that the transformation acts on.

Sets the length of the Householder sequence.

Parameters

[in]

length

New value for the length.

By default, the length \( n \) of the Householder sequence \( H = H_0 H_1 \ldots H_{n-1} \) is set to the number of columns of the matrix v passed to the constructor, or the number of rows if that is smaller. After this function is called, the length equals length.

See also

length()

Sets the shift of the Householder sequence.

Parameters

[in]

shift

New value for the shift.

By default, a HouseholderSequence object represents \( H = H_0 H_1 \ldots H_{n-1} \) and the i-th column of the matrix v passed to the constructor corresponds to the i-th Householder reflection. After this function is called, the object represents \( H = H_{\mathrm{shift}} H_{\mathrm{shift}+1} \ldots H_{n-1} \) and the i-th column of v corresponds to the (shift+i)-th Householder reflection.

See also

shift()