1SGEBD2(1) LAPACK routine (version 3.1) SGEBD2(1)
2
6 SGEBD2 - a real general m by n matrix A to upper or lower bidiagonal
7 form B by an orthogonal transformation
8
10 SUBROUTINE SGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )
11 INTEGER INFO, LDA, M, N
13 REAL A( LDA, * ), D( * ), E( * ), TAUP( * ), TAUQ( * ),
15 WORK( * )
16
18 SGEBD2 reduces a real general m by n matrix A to upper or lower bidiag‐
19 onal form B by an orthogonal transformation: Q' * A * P = B.
20 If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
22
25 M (input) INTEGER
26 The number of rows in the matrix A. M >= 0.
27 N (input) INTEGER
29 The number of columns in the matrix A. N >= 0.
30 A (input/output) REAL array, dimension (LDA,N)
32 On entry, the m by n general matrix to be reduced. On exit, if
33 m >= n, the diagonal and the first superdiagonal are overwrit‐
34 ten with the upper bidiagonal matrix B; the elements below the
35 diagonal, with the array TAUQ, represent the orthogonal matrix
36 Q as a product of elementary reflectors, and the elements above
37 the first superdiagonal, with the array TAUP, represent the
38 orthogonal matrix P as a product of elementary reflectors; if m
39 < n, the diagonal and the first subdiagonal are overwritten
40 with the lower bidiagonal matrix B; the elements below the
41 first subdiagonal, with the array TAUQ, represent the orthogo‐
42 nal matrix Q as a product of elementary reflectors, and the
43 elements above the diagonal, with the array TAUP, represent the
44 orthogonal matrix P as a product of elementary reflectors. See
45 Further Details. LDA (input) INTEGER The leading dimension
46 of the array A. LDA >= max(1,M).
47 D (output) REAL array, dimension (min(M,N))
49 The diagonal elements of the bidiagonal matrix B: D(i) =
50 A(i,i).
51 E (output) REAL array, dimension (min(M,N)-1)
53 The off-diagonal elements of the bidiagonal matrix B: if m >=
54 n, E(i) = A(i,i+1) for i = 1,2,...,n-1; if m < n, E(i) =
55 A(i+1,i) for i = 1,2,...,m-1.
56 TAUQ (output) REAL array dimension (min(M,N))
58 The scalar factors of the elementary reflectors which represent
59 the orthogonal matrix Q. See Further Details. TAUP (output)
60 REAL array, dimension (min(M,N)) The scalar factors of the ele‐
61 mentary reflectors which represent the orthogonal matrix P. See
62 Further Details. WORK (workspace) REAL array, dimension
63 (max(M,N))
64 INFO (output) INTEGER
66 = 0: successful exit.
67 < 0: if INFO = -i, the i-th argument had an illegal value.
68
70 The matrices Q and P are represented as products of elementary reflec‐
71 tors:
72 If m >= n,
74 Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
76 Each H(i) and G(i) has the form:
78 H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'
80 where tauq and taup are real scalars, and v and u are real vectors;
82 v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
83 u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
84 tauq is stored in TAUQ(i) and taup in TAUP(i).
85 If m < n,
87 Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
89 Each H(i) and G(i) has the form:
91 H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'
93 where tauq and taup are real scalars, and v and u are real vectors;
95 v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
96 u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
97 tauq is stored in TAUQ(i) and taup in TAUP(i).
98 The contents of A on exit are illustrated by the following examples:
100 m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
102 ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
104 ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
105 ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
106 ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
107 ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
108 ( v1 v2 v3 v4 v5 )
109 where d and e denote diagonal and off-diagonal elements of B, vi
111 denotes an element of the vector defining H(i), and ui an element of
112 the vector defining G(i).
113 LAPACK routine (version 3.1) November 2006 SGEBD2(1)