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template<typename T>
void gebd2(integer *m, integer *n, T *a, integer *lda, T *d, T *e, T *tauq, T *taup, T *work, integer *info)# Reduction to bidiagonal form (unblocked algorithm)
Purpose:
Reduction of a general real 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.
Further Details
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 real scalars, and V and u are real 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 tauq(i) and taup in taup(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 real scalars, and V and u are real 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 tauq(i) and taup in taup(i). The contents of A on exit are illustrated by the following examples: m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): ( d e u1 u1 u1) ( d u1 u1 u1 u1 u1) ( v1 d e u2 u2) ( e d u2 u2 u2 u2) ( v1 v2 d e u3) ( v1 e d u3 u3 u3) ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4) ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5) ( v1 v2 v3 v4 v5) 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).
- Parameters:
m – [in]
m is integer*
The number of rows in the matrix a. m >= 0.
n – [in]
n is integer*
The number of columns in the matrix a. n >= 0.
a – [inout]
a is float/double array, dimension (lda,n)
On entry, the m-by-n general matrix to be reduced.
On exit,
if m >= n, the diagonal and the first superdiagonal are overwritten with the upper bidiagonal matrix b; the elements below the diagonal, with the array tauq, represent the orthogonal matrix Q as a product of elementary reflectors, and the elements above the first superdiagonal, with the array taup, 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 array tauq, represent the orthogonal matrix Q as a product of elementary reflectors, and the elements above the diagonal, with the array taup, represent the orthogonal matrix P as a product of elementary reflectors.
See Further Details.
lda – [in]
lda is integer*
The leading dimension of the array a. lda >= fla_max(1,m).
d – [out]
d is float/double array, dimension (min(m,n))
The diagonal elements of the bidiagonal matrix b:
D(i) = A(i,i).e – [out]
e is float/double array, dimension (min(m,n)-1)
The off-diagonal elements of the bidiagonal matrix b:
if m >= n, E(i) = A(i,i+1) for i = 1,2,…,n-1;
if m < n, E(i) = A(i+1,i) for i = 1,2,…,m-1.
tauq – [out]
tauq is float/double array, dimension (min(m,n))
The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. See Further Details.
taup – [out]
taup is float/double array, dimension (min(m,n))
The scalar factors of the elementary reflectors which represent the orthogonal matrix P.
See Further Details.WORK – [out] WORK is COMPLEX array, dimension (fla_max(M,N))
INFO – [out]
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.