1DLABRD(1) LAPACK auxiliary routine (version 3.2) DLABRD(1)
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6 DLABRD - reduces the first NB rows and columns of a real general m by n
7 matrix A to upper or lower bidiagonal form by an orthogonal transforma‐
8 tion Q' * A * P, and returns the matrices X and Y which are needed to
9 apply the transformation to the unreduced part of A
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12 SUBROUTINE DLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY )
13 INTEGER LDA, LDX, LDY, M, N, NB
15 DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
17 TAUQ( * ), X( LDX, * ), Y( LDY, * )
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20 DLABRD reduces the first NB rows and columns of a real general m by n
21 matrix A to upper or lower bidiagonal form by an orthogonal transforma‐
22 tion Q' * A * P, and returns the matrices X and Y which are needed to
23 apply the transformation to the unreduced part of A. If m >= n, A is
24 reduced to upper bidiagonal form; if m < n, to lower bidiagonal form.
25 This is an auxiliary routine called by DGEBRD
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28 M (input) INTEGER
29 The number of rows in the matrix A.
30 N (input) INTEGER
32 The number of columns in the matrix A.
33 NB (input) INTEGER
35 The number of leading rows and columns of A to be reduced.
36 A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
38 On entry, the m by n general matrix to be reduced. On exit,
39 the first NB rows and columns of the matrix are overwritten;
40 the rest of the array is unchanged. If m >= n, elements on and
41 below the diagonal in the first NB columns, with the array
42 TAUQ, represent the orthogonal matrix Q as a product of elemen‐
43 tary reflectors; and elements above the diagonal in the first
44 NB rows, with the array TAUP, represent the orthogonal matrix P
45 as a product of elementary reflectors. If m < n, elements
46 below the diagonal in the first NB columns, with the array
47 TAUQ, represent the orthogonal matrix Q as a product of elemen‐
48 tary reflectors, and elements on and above the diagonal in the
49 first NB rows, with the array TAUP, represent the orthogonal
50 matrix P as a product of elementary reflectors. See Further
51 Details. LDA (input) INTEGER The leading dimension of the
52 array A. LDA >= max(1,M).
53 D (output) DOUBLE PRECISION array, dimension (NB)
55 The diagonal elements of the first NB rows and columns of the
56 reduced matrix. D(i) = A(i,i).
57 E (output) DOUBLE PRECISION array, dimension (NB)
59 The off-diagonal elements of the first NB rows and columns of
60 the reduced matrix.
61 TAUQ (output) DOUBLE PRECISION array dimension (NB)
63 The scalar factors of the elementary reflectors which represent
64 the orthogonal matrix Q. See Further Details. TAUP (output)
65 DOUBLE PRECISION array, dimension (NB) The scalar factors of
66 the elementary reflectors which represent the orthogonal matrix
67 P. See Further Details. X (output) DOUBLE PRECISION
68 array, dimension (LDX,NB) The m-by-nb matrix X required to
69 update the unreduced part of A.
70 LDX (input) INTEGER
72 The leading dimension of the array X. LDX >= M.
73 Y (output) DOUBLE PRECISION array, dimension (LDY,NB)
75 The n-by-nb matrix Y required to update the unreduced part of
76 A.
77 LDY (input) INTEGER
79 The leading dimension of the array Y. LDY >= N.
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82 The matrices Q and P are represented as products of elementary reflec‐
83 tors:
84 Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb) Each H(i)
85 and G(i) has the form:
86 H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' where tauq
87 and taup are real scalars, and v and u are real vectors. If m >= n,
88 v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in A(i:m,i);
89 u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
90 tauq is stored in TAUQ(i) and taup in TAUP(i). If m < n, v(1:i) = 0,
91 v(i+1) = 1, and v(i+1:m) is stored on exit in A(i+2:m,i); u(1:i-1) = 0,
92 u(i) = 1, and u(i:n) is stored on exit in A(i,i+1:n); tauq is stored in
93 TAUQ(i) and taup in TAUP(i). The elements of the vectors v and u
94 together form the m-by-nb matrix V and the nb-by-n matrix U' which are
95 needed, with X and Y, to apply the transformation to the unreduced part
96 of the matrix, using a block update of the form: A := A - V*Y' - X*U'.
97 The contents of A on exit are illustrated by the following examples
98 with nb = 2:
99 m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
100 ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 )
101 ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 )
102 ( v1 v2 a a a ) ( v1 1 a a a a )
103 ( v1 v2 a a a ) ( v1 v2 a a a a )
104 ( v1 v2 a a a ) ( v1 v2 a a a a )
105 ( v1 v2 a a a )
106 where a denotes an element of the original matrix which is unchanged,
107 vi denotes an element of the vector defining H(i), and ui an element of
108 the vector defining G(i).
109 LAPACK auxiliary routine (versionNo3v.e2m)ber 2008 DLABRD(1)