1ZGEBRD(1) LAPACK routine (version 3.1) ZGEBRD(1)
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6 ZGEBRD - a general complex M-by-N matrix A to upper or lower bidiagonal
7 form B by a unitary transformation
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10 SUBROUTINE ZGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO )
11 INTEGER INFO, LDA, LWORK, M, N
13 DOUBLE PRECISION D( * ), E( * )
15 COMPLEX*16 A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * )
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19 ZGEBRD reduces a general complex M-by-N matrix A to upper or lower
20 bidiagonal form B by a unitary transformation: Q**H * A * P = B.
21 If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
23
26 M (input) INTEGER
27 The number of rows in the matrix A. M >= 0.
28 N (input) INTEGER
30 The number of columns in the matrix A. N >= 0.
31 A (input/output) COMPLEX*16 array, dimension (LDA,N)
33 On entry, the M-by-N general matrix to be reduced. On exit, if
34 m >= n, the diagonal and the first superdiagonal are overwrit‐
35 ten with the upper bidiagonal matrix B; the elements below the
36 diagonal, with the array TAUQ, represent the unitary matrix Q
37 as a product of elementary reflectors, and the elements above
38 the first superdiagonal, with the array TAUP, represent the
39 unitary matrix P as a product of elementary reflectors; if m <
40 n, the diagonal and the first subdiagonal are overwritten with
41 the lower bidiagonal matrix B; the elements below the first
42 subdiagonal, with the array TAUQ, represent the unitary matrix
43 Q as a product of elementary reflectors, and the elements above
44 the diagonal, with the array TAUP, represent the unitary matrix
45 P as a product of elementary reflectors. See Further Details.
46 LDA (input) INTEGER The leading dimension of the array A.
47 LDA >= max(1,M).
48 D (output) DOUBLE PRECISION array, dimension (min(M,N))
50 The diagonal elements of the bidiagonal matrix B: D(i) =
51 A(i,i).
52 E (output) DOUBLE PRECISION array, dimension (min(M,N)-1)
54 The off-diagonal elements of the bidiagonal matrix B: if m >=
55 n, E(i) = A(i,i+1) for i = 1,2,...,n-1; if m < n, E(i) =
56 A(i+1,i) for i = 1,2,...,m-1.
57 TAUQ (output) COMPLEX*16 array dimension (min(M,N))
59 The scalar factors of the elementary reflectors which represent
60 the unitary matrix Q. See Further Details. TAUP (output)
61 COMPLEX*16 array, dimension (min(M,N)) The scalar factors of
62 the elementary reflectors which represent the unitary matrix P.
63 See Further Details. WORK (workspace/output) COMPLEX*16
64 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1)
65 returns the optimal LWORK.
66 LWORK (input) INTEGER
68 The length of the array WORK. LWORK >= max(1,M,N). For opti‐
69 mum performance LWORK >= (M+N)*NB, where NB is the optimal
70 blocksize.
71 If LWORK = -1, then a workspace query is assumed; the routine
73 only calculates the optimal size of the WORK array, returns
74 this value as the first entry of the WORK array, and no error
75 message related to LWORK is issued by XERBLA.
76 INFO (output) INTEGER
78 = 0: successful exit.
79 < 0: if INFO = -i, the i-th argument had an illegal value.
80
82 The matrices Q and P are represented as products of elementary reflec‐
83 tors:
84 If m >= n,
86 Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
88 Each H(i) and G(i) has the form:
90 H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'
92 where tauq and taup are complex scalars, and v and u are complex vec‐
94 tors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in
95 A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in
96 A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
97 If m < n,
99 Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
101 Each H(i) and G(i) has the form:
103 H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'
105 where tauq and taup are complex scalars, and v and u are complex vec‐
107 tors; v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in
108 A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in
109 A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
110 The contents of A on exit are illustrated by the following examples:
112 m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
114 ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
116 ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
117 ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
118 ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
119 ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
120 ( v1 v2 v3 v4 v5 )
121 where d and e denote diagonal and off-diagonal elements of B, vi
123 denotes an element of the vector defining H(i), and ui an element of
124 the vector defining G(i).
125 LAPACK routine (version 3.1) November 2006 ZGEBRD(1)