1SGEBRD(1) LAPACK routine (version 3.1) SGEBRD(1)
2
6 SGEBRD - a general real M-by-N matrix A to upper or lower bidiagonal
7 form B by an orthogonal transformation
8
10 SUBROUTINE SGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO )
11 INTEGER INFO, LDA, LWORK, M, N
13 REAL A( LDA, * ), D( * ), E( * ), TAUP( * ), TAUQ( * ),
15 WORK( * )
16
18 SGEBRD reduces a general real M-by-N matrix A to upper or lower bidiag‐
19 onal form B by an orthogonal transformation: Q**T * 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/output) REAL array, dimen‐
63 sion (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the
64 optimal LWORK.
65 LWORK (input) INTEGER
67 The length of the array WORK. LWORK >= max(1,M,N). For opti‐
68 mum performance LWORK >= (M+N)*NB, where NB is the optimal
69 blocksize.
70 If LWORK = -1, then a workspace query is assumed; the routine
72 only calculates the optimal size of the WORK array, returns
73 this value as the first entry of the WORK array, and no error
74 message related to LWORK is issued by XERBLA.
75 INFO (output) INTEGER
77 = 0: successful exit
78 < 0: if INFO = -i, the i-th argument had an illegal value.
79
81 The matrices Q and P are represented as products of elementary reflec‐
82 tors:
83 If m >= n,
85 Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
87 Each H(i) and G(i) has the form:
89 H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'
91 where tauq and taup are real scalars, and v and u are real vectors;
93 v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
94 u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
95 tauq is stored in TAUQ(i) and taup in TAUP(i).
96 If m < n,
98 Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
100 Each H(i) and G(i) has the form:
102 H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'
104 where tauq and taup are real scalars, and v and u are real vectors;
106 v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
107 u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
108 tauq is stored in TAUQ(i) and taup in TAUP(i).
109 The contents of A on exit are illustrated by the following examples:
111 m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
113 ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
115 ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
116 ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
117 ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
118 ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
119 ( v1 v2 v3 v4 v5 )
120 where d and e denote diagonal and off-diagonal elements of B, vi
122 denotes an element of the vector defining H(i), and ui an element of
123 the vector defining G(i).
124 LAPACK routine (version 3.1) November 2006 SGEBRD(1)