1SLABRD(1) LAPACK auxiliary routine (version 3.2) SLABRD(1)
2
6 SLABRD - 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
10
12 SUBROUTINE SLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY )
13 INTEGER LDA, LDX, LDY, M, N, NB
15 REAL A( LDA, * ), D( * ), E( * ), TAUP( * ), TAUQ( * ),
17 X( LDX, * ), Y( LDY, * )
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20 SLABRD 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 SGEBRD
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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) REAL 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) REAL 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) REAL array, dimension (NB)
59 The off-diagonal elements of the first NB rows and columns of
60 the reduced matrix.
61 TAUQ (output) REAL array dimension (NB)
63 The scalar factors of the elementary reflectors which represent
64 the orthogonal matrix Q. See Further Details. TAUP (output)
65 REAL array, dimension (NB) The scalar factors of the elementary
66 reflectors which represent the orthogonal matrix P. See Further
67 Details. X (output) REAL array, dimension (LDX,NB) The
68 m-by-nb matrix X required to update the unreduced part of A.
69 LDX (input) INTEGER
71 The leading dimension of the array X. LDX >= M.
72 Y (output) REAL array, dimension (LDY,NB)
74 The n-by-nb matrix Y required to update the unreduced part of
75 A.
76 LDY (input) INTEGER
78 The leading dimension of the array Y. LDY >= N.
79
81 The matrices Q and P are represented as products of elementary reflec‐
82 tors:
83 Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb) Each H(i)
84 and G(i) has the form:
85 H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' where tauq
86 and taup are real scalars, and v and u are real vectors. If m >= n,
87 v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in A(i:m,i);
88 u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
89 tauq is stored in TAUQ(i) and taup in TAUP(i). If m < n, v(1:i) = 0,
90 v(i+1) = 1, and v(i+1:m) is stored on exit in A(i+2:m,i); u(1:i-1) = 0,
91 u(i) = 1, and u(i:n) is stored on exit in A(i,i+1:n); tauq is stored in
92 TAUQ(i) and taup in TAUP(i). The elements of the vectors v and u
93 together form the m-by-nb matrix V and the nb-by-n matrix U' which are
94 needed, with X and Y, to apply the transformation to the unreduced part
95 of the matrix, using a block update of the form: A := A - V*Y' - X*U'.
96 The contents of A on exit are illustrated by the following examples
97 with nb = 2:
98 m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
99 ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 )
100 ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 )
101 ( v1 v2 a a a ) ( v1 1 a a a a )
102 ( v1 v2 a a a ) ( v1 v2 a a a a )
103 ( v1 v2 a a a ) ( v1 v2 a a a a )
104 ( v1 v2 a a a )
105 where a denotes an element of the original matrix which is unchanged,
106 vi denotes an element of the vector defining H(i), and ui an element of
107 the vector defining G(i).
108 LAPACK auxiliary routine (versionNo3v.e2m)ber 2008 SLABRD(1)