ReduceToBidiagonal

Reduces a general real or complex m-by-n matrix A to upper or lower bidiagonal form B by an orthogonal transformation: Q**T * A * P = B. If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal. Lapack function GEBRD.

Computing for type matrix<double>

Computing for type matrix<float>

Computing for type matrix<complex>

Computing for type matrix<complexf>

Parameters

B

[out]  Upper or lower bidiagonal matrix.

reflect_qp

[out]  Transformed matrix A. If m >= n, the diagonal and the first superdiagonal are overwritten with the upper bidiagonal matrix B; the elements below the diagonal, with the vector tau_q, represent the orthogonal matrix Q as a product of elementary reflectors, and the elements above the first superdiagonal, with the vector tau_p, represent the orthogonal matrix P as a product of elementary reflectors; if m < n, the diagonal and the first subdiagonal are overwritten with the lower bidiagonal matrix B; the elements below the first subdiagonal, with the vector tau_q, represent the orthogonal matrix Q as a product of elementary reflectors, and the elements above the diagonal, with the vector tau_p, represent the orthogonal matrix P as a product of elementary reflectors.

tau_q

[out] Vector of the scalar factors of the elementary reflectors which represent the orthogonal matrix Q.

tau_p

[out] Vector of the scalar factors of the elementary reflectors which represent the orthogonal matrix P.

Return Value

Return true if successful, otherwise false in case of an error.

Note

The matrices Q and P are represented as products of elementary reflectors:

If m >= n,

    Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)

Each H(i) and G(i) has the form:

    H(i) = I - tauq * v * v**T  and G(i) = I - taup * u * u**T

where tauq and taup are scalars, and v and u are vectors;

v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);

u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);

tauq is stored in tau_q(i) and taup in tau_p(i).

If m < n,

    Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)

Each H(i) and G(i) has the form:

    H(i) = I - tauq * v * v**T  and G(i) = I - taup * u * u**T

where tauq and taup are scalars, and v and u are vectors;

v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);

u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);

tauq is stored in tau_q(i) and taup in tau_p(i).

The contents of A on exit (output matrix reflect_qp) are illustrated by the following examples:

where d and e denote diagonal and off-diagonal elements of B, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i).

Matrices Q and P can be produced with ReflectBidiagonalToQP method.