1DGEBD2(1) LAPACK routine (version 3.2) DGEBD2(1)
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6 DGEBD2 - reduces a real general m by n matrix A to upper or lower bidi‐
7 agonal form B by an orthogonal transformation
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10 SUBROUTINE DGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )
11 INTEGER INFO, LDA, M, N
13 DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
15 TAUQ( * ), WORK( * )
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18 DGEBD2 reduces a real general m by n matrix A to upper or lower bidiag‐
19 onal form B by an orthogonal transformation: Q' * A * P = B. If m >=
20 n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
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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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (min(M,N))
47 The diagonal elements of the bidiagonal matrix B: D(i) =
48 A(i,i).
49 E (output) DOUBLE PRECISION 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) DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (min(M,N)) The scalar factors
59 of the elementary reflectors which represent the orthogonal
60 matrix P. See Further Details. WORK (workspace) DOUBLE PRE‐
61 CISION array, dimension (max(M,N))
62 INFO (output) INTEGER
64 = 0: successful exit.
65 < 0: if INFO = -i, the i-th argument had an illegal value.
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68 The matrices Q and P are represented as products of elementary reflec‐
69 tors:
70 If m >= n,
71 Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1) Each H(i)
72 and G(i) has the form:
73 H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' where tauq
74 and taup are real scalars, and v and u are real vectors; v(1:i-1) = 0,
75 v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i); u(1:i) = 0,
76 u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n); tauq is
77 stored in TAUQ(i) and taup in TAUP(i).
78 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 real scalars, and v and u are real 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 DGEBD2(1)