1CGEBD2(1) LAPACK routine (version 3.1) CGEBD2(1)
2
6 CGEBD2 - a complex general m by n matrix A to upper or lower real bidi‐
7 agonal form B by a unitary transformation
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10 SUBROUTINE CGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )
11 INTEGER INFO, LDA, M, N
13 REAL D( * ), E( * )
15 COMPLEX A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * )
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19 CGEBD2 reduces a complex general m by n matrix A to upper or lower real
20 bidiagonal form B by a unitary transformation: Q' * 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 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) REAL array, dimension (min(M,N))
50 The diagonal elements of the bidiagonal matrix B: D(i) =
51 A(i,i).
52 E (output) REAL 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 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 array, dimension (min(M,N)) The scalar factors of the
62 elementary reflectors which represent the unitary matrix P. See
63 Further Details. WORK (workspace) COMPLEX array, dimension
64 (max(M,N))
65 INFO (output) INTEGER
67 = 0: successful exit
68 < 0: if INFO = -i, the i-th argument had an illegal value.
69
71 The matrices Q and P are represented as products of elementary reflec‐
72 tors:
73 If m >= n,
75 Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
77 Each H(i) and G(i) has the form:
79 H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'
81 where tauq and taup are complex scalars, and v and u are complex vec‐
83 tors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in
84 A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in
85 A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
86 If m < n,
88 Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
90 Each H(i) and G(i) has the form:
92 H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'
94 where tauq and taup are complex scalars, v and u are complex vectors;
96 v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
97 u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
98 tauq is stored in TAUQ(i) and taup in TAUP(i).
99 The contents of A on exit are illustrated by the following examples:
101 m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
103 ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
105 ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
106 ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
107 ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
108 ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
109 ( v1 v2 v3 v4 v5 )
110 where d and e denote diagonal and off-diagonal elements of B, vi
112 denotes an element of the vector defining H(i), and ui an element of
113 the vector defining G(i).
114 LAPACK routine (version 3.1) November 2006 CGEBD2(1)