1SGEBRD(1) LAPACK routine (version 3.2) SGEBRD(1)
2
6 SGEBRD - reduces a general real M-by-N matrix A to upper or lower bidi‐
7 agonal 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. If m >=
20 n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
21
23 M (input) INTEGER
24 The number of rows in the matrix A. M >= 0.
25 N (input) INTEGER
27 The number of columns in the matrix A. N >= 0.
28 A (input/output) REAL array, dimension (LDA,N)
30 On entry, the M-by-N general matrix to be reduced. On exit, if
31 m >= n, the diagonal and the first superdiagonal are overwrit‐
32 ten with the upper bidiagonal matrix B; the elements below the
33 diagonal, with the array TAUQ, represent the orthogonal matrix
34 Q as a product of elementary reflectors, and the elements above
35 the first superdiagonal, with the array TAUP, represent the
36 orthogonal matrix P as a product of elementary reflectors; if m
37 < n, the diagonal and the first subdiagonal are overwritten
38 with the lower bidiagonal matrix B; the elements below the
39 first subdiagonal, with the array TAUQ, represent the orthogo‐
40 nal matrix Q as a product of elementary reflectors, and the
41 elements above the diagonal, with the array TAUP, represent the
42 orthogonal matrix P as a product of elementary reflectors. See
43 Further Details. LDA (input) INTEGER The leading dimension
44 of the array A. LDA >= max(1,M).
45 D (output) REAL array, dimension (min(M,N))
47 The diagonal elements of the bidiagonal matrix B: D(i) =
48 A(i,i).
49 E (output) REAL array, dimension (min(M,N)-1)
51 The off-diagonal elements of the bidiagonal matrix B: if m >=
52 n, E(i) = A(i,i+1) for i = 1,2,...,n-1; if m < n, E(i) =
53 A(i+1,i) for i = 1,2,...,m-1.
54 TAUQ (output) REAL array dimension (min(M,N))
56 The scalar factors of the elementary reflectors which represent
57 the orthogonal matrix Q. See Further Details. TAUP (output)
58 REAL array, dimension (min(M,N)) The scalar factors of the ele‐
59 mentary reflectors which represent the orthogonal matrix P. See
60 Further Details. WORK (workspace/output) REAL array, dimen‐
61 sion (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the
62 optimal LWORK.
63 LWORK (input) INTEGER
65 The length of the array WORK. LWORK >= max(1,M,N). For opti‐
66 mum performance LWORK >= (M+N)*NB, where NB is the optimal
67 blocksize. If LWORK = -1, then a workspace query is assumed;
68 the routine only calculates the optimal size of the WORK array,
69 returns this value as the first entry of the WORK array, and no
70 error message related to LWORK is issued by XERBLA.
71 INFO (output) INTEGER
73 = 0: successful exit
74 < 0: if INFO = -i, the i-th argument had an illegal value.
75
77 The matrices Q and P are represented as products of elementary reflec‐
78 tors:
79 If m >= n,
80 Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1) Each H(i)
81 and G(i) has the form:
82 H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' where tauq
83 and taup are real scalars, and v and u are real vectors; v(1:i-1) = 0,
84 v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i); u(1:i) = 0,
85 u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n); tauq is
86 stored in TAUQ(i) and taup in TAUP(i).
87 If m < n,
88 Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m) Each H(i)
89 and G(i) has the form:
90 H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' where tauq
91 and taup are real scalars, and v and u are real vectors; v(1:i) = 0,
92 v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i); u(1:i-1) = 0,
93 u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n); tauq is stored
94 in TAUQ(i) and taup in TAUP(i).
95 The contents of A on exit are illustrated by the following examples: m
96 = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
97 ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
98 ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
99 ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
100 ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
101 ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
102 ( v1 v2 v3 v4 v5 )
103 where d and e denote diagonal and off-diagonal elements of B, vi
104 denotes an element of the vector defining H(i), and ui an element of
105 the vector defining G(i).
106 LAPACK routine (version 3.2) November 2008 SGEBRD(1)