1ZLABRD(1) LAPACK auxiliary routine (version 3.1) ZLABRD(1)
2
6 ZLABRD - the first NB rows and columns of a complex general m by n
7 matrix A to upper or lower real bidiagonal form by a unitary transfor‐
8 mation 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 ZLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY )
13 INTEGER LDA, LDX, LDY, M, N, NB
15 DOUBLE PRECISION D( * ), E( * )
17 COMPLEX*16 A( LDA, * ), TAUP( * ), TAUQ( * ), X( LDX, * ), Y(
19 LDY, * )
20
22 ZLABRD reduces the first NB rows and columns of a complex general m by
23 n matrix A to upper or lower real bidiagonal form by a unitary trans‐
24 formation Q' * A * P, and returns the matrices X and Y which are needed
25 to apply the transformation to the unreduced part of A.
26 If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
28 bidiagonal form.
29 This is an auxiliary routine called by ZGEBRD
31
34 M (input) INTEGER
35 The number of rows in the matrix A.
36 N (input) INTEGER
38 The number of columns in the matrix A.
39 NB (input) INTEGER
41 The number of leading rows and columns of A to be reduced.
42 A (input/output) COMPLEX*16 array, dimension (LDA,N)
44 On entry, the m by n general matrix to be reduced. On exit,
45 the first NB rows and columns of the matrix are overwritten;
46 the rest of the array is unchanged. If m >= n, elements on and
47 below the diagonal in the first NB columns, with the array
48 TAUQ, represent the unitary matrix Q as a product of elementary
49 reflectors; and elements above the diagonal in the first NB
50 rows, with the array TAUP, represent the unitary matrix P as a
51 product of elementary reflectors. If m < n, elements below the
52 diagonal in the first NB columns, with the array TAUQ, repre‐
53 sent the unitary matrix Q as a product of elementary reflec‐
54 tors, and elements on and above the diagonal in the first NB
55 rows, with the array TAUP, represent the unitary matrix P as a
56 product of elementary reflectors. See Further Details. LDA
57 (input) INTEGER The leading dimension of the array A. LDA >=
58 max(1,M).
59 D (output) DOUBLE PRECISION array, dimension (NB)
61 The diagonal elements of the first NB rows and columns of the
62 reduced matrix. D(i) = A(i,i).
63 E (output) DOUBLE PRECISION array, dimension (NB)
65 The off-diagonal elements of the first NB rows and columns of
66 the reduced matrix.
67 TAUQ (output) COMPLEX*16 array dimension (NB)
69 The scalar factors of the elementary reflectors which represent
70 the unitary matrix Q. See Further Details. TAUP (output)
71 COMPLEX*16 array, dimension (NB) The scalar factors of the ele‐
72 mentary reflectors which represent the unitary matrix P. See
73 Further Details. X (output) COMPLEX*16 array, dimension
74 (LDX,NB) The m-by-nb matrix X required to update the unreduced
75 part of A.
76 LDX (input) INTEGER
78 The leading dimension of the array X. LDX >= max(1,M).
79 Y (output) COMPLEX*16 array, dimension (LDY,NB)
81 The n-by-nb matrix Y required to update the unreduced part of
82 A.
83 LDY (input) INTEGER
85 The leading dimension of the array Y. LDY >= max(1,N).
86
88 The matrices Q and P are represented as products of elementary reflec‐
89 tors:
90 Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb)
92 Each H(i) and G(i) has the form:
94 H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'
96 where tauq and taup are complex scalars, and v and u are complex vec‐
98 tors.
99 If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
101 A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in
102 A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
103 If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
105 A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
106 A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
107 The elements of the vectors v and u together form the m-by-nb matrix V
109 and the nb-by-n matrix U' which are needed, with X and Y, to apply the
110 transformation to the unreduced part of the matrix, using a block
111 update of the form: A := A - V*Y' - X*U'.
112 The contents of A on exit are illustrated by the following examples
114 with nb = 2:
115 m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
117 ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 )
119 ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 )
120 ( v1 v2 a a a ) ( v1 1 a a a a )
121 ( v1 v2 a a a ) ( v1 v2 a a a a )
122 ( v1 v2 a a a ) ( v1 v2 a a a a )
123 ( v1 v2 a a a )
124 where a denotes an element of the original matrix which is unchanged,
126 vi denotes an element of the vector defining H(i), and ui an element of
127 the vector defining G(i).
128 LAPACK auxiliary routine (versionNo3v.e1m)ber 2006 ZLABRD(1)