1CGEBD2(1) LAPACK routine (version 3.2) CGEBD2(1)
2
6 CGEBD2 - reduces a complex general m by n matrix A to upper or lower
7 real bidiagonal 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. If m >=
21 n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
22
24 M (input) INTEGER
25 The number of rows in the matrix A. M >= 0.
26 N (input) INTEGER
28 The number of columns in the matrix A. N >= 0.
29 A (input/output) COMPLEX array, dimension (LDA,N)
31 On entry, the m by n general matrix to be reduced. On exit, if
32 m >= n, the diagonal and the first superdiagonal are overwrit‐
33 ten with the upper bidiagonal matrix B; the elements below the
34 diagonal, with the array TAUQ, represent the unitary matrix Q
35 as a product of elementary reflectors, and the elements above
36 the first superdiagonal, with the array TAUP, represent the
37 unitary matrix P as a product of elementary reflectors; if m <
38 n, the diagonal and the first subdiagonal are overwritten with
39 the lower bidiagonal matrix B; the elements below the first
40 subdiagonal, with the array TAUQ, represent the unitary matrix
41 Q as a product of elementary reflectors, and the elements above
42 the diagonal, with the array TAUP, represent the unitary matrix
43 P as a product of elementary reflectors. See Further Details.
44 LDA (input) INTEGER The leading dimension of the array A.
45 LDA >= max(1,M).
46 D (output) REAL array, dimension (min(M,N))
48 The diagonal elements of the bidiagonal matrix B: D(i) =
49 A(i,i).
50 E (output) REAL array, dimension (min(M,N)-1)
52 The off-diagonal elements of the bidiagonal matrix B: if m >=
53 n, E(i) = A(i,i+1) for i = 1,2,...,n-1; if m < n, E(i) =
54 A(i+1,i) for i = 1,2,...,m-1.
55 TAUQ (output) COMPLEX array dimension (min(M,N))
57 The scalar factors of the elementary reflectors which represent
58 the unitary matrix Q. See Further Details. TAUP (output)
59 COMPLEX array, dimension (min(M,N)) The scalar factors of the
60 elementary reflectors which represent the unitary matrix P. See
61 Further Details. WORK (workspace) COMPLEX array, dimension
62 (max(M,N))
63 INFO (output) INTEGER
65 = 0: successful exit
66 < 0: if INFO = -i, the i-th argument had an illegal value.
67
69 The matrices Q and P are represented as products of elementary reflec‐
70 tors:
71 If m >= n,
72 Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1) Each H(i)
73 and G(i) has the form:
74 H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' where tauq
75 and taup are complex scalars, and v and u are complex vectors; v(1:i-1)
76 = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i); u(1:i) =
77 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n); tauq is
78 stored in TAUQ(i) and taup in TAUP(i). If m < n,
79 Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m) Each H(i)
80 and G(i) has the form:
81 H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' where tauq
82 and taup are complex scalars, v and u are complex vectors; v(1:i) = 0,
83 v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i); u(1:i-1) = 0,
84 u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n); tauq is stored
85 in TAUQ(i) and taup in TAUP(i).
86 The contents of A on exit are illustrated by the following examples: m
87 = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
88 ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
89 ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
90 ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
91 ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
92 ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
93 ( v1 v2 v3 v4 v5 )
94 where d and e denote diagonal and off-diagonal elements of B, vi
95 denotes an element of the vector defining H(i), and ui an element of
96 the vector defining G(i).
97 LAPACK routine (version 3.2) November 2008 CGEBD2(1)