1ZLABRD(1) LAPACK auxiliary routine (version 3.2) ZLABRD(1)
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6 ZLABRD - reduces the first NB rows and columns of a complex general m
7 by n matrix A to upper or lower real bidiagonal form by a unitary
8 transformation Q' * A * P, and returns the matrices X and Y which are
9 needed to 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. If m >= n, A
26 is reduced to upper bidiagonal form; if m < n, to lower bidiagonal
27 form.
28 This is an auxiliary routine called by ZGEBRD
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31 M (input) INTEGER
32 The number of rows in the matrix A.
33 N (input) INTEGER
35 The number of columns in the matrix A.
36 NB (input) INTEGER
38 The number of leading rows and columns of A to be reduced.
39 A (input/output) COMPLEX*16 array, dimension (LDA,N)
41 On entry, the m by n general matrix to be reduced. On exit,
42 the first NB rows and columns of the matrix are overwritten;
43 the rest of the array is unchanged. If m >= n, elements on and
44 below the diagonal in the first NB columns, with the array
45 TAUQ, represent the unitary matrix Q as a product of elementary
46 reflectors; and elements above the diagonal in the first NB
47 rows, with the array TAUP, represent the unitary matrix P as a
48 product of elementary reflectors. If m < n, elements below the
49 diagonal in the first NB columns, with the array TAUQ, repre‐
50 sent the unitary matrix Q as a product of elementary reflec‐
51 tors, and elements on and above the diagonal in the first NB
52 rows, with the array TAUP, represent the unitary matrix P as a
53 product of elementary reflectors. See Further Details. LDA
54 (input) INTEGER The leading dimension of the array A. LDA >=
55 max(1,M).
56 D (output) DOUBLE PRECISION array, dimension (NB)
58 The diagonal elements of the first NB rows and columns of the
59 reduced matrix. D(i) = A(i,i).
60 E (output) DOUBLE PRECISION array, dimension (NB)
62 The off-diagonal elements of the first NB rows and columns of
63 the reduced matrix.
64 TAUQ (output) COMPLEX*16 array dimension (NB)
66 The scalar factors of the elementary reflectors which represent
67 the unitary matrix Q. See Further Details. TAUP (output)
68 COMPLEX*16 array, dimension (NB) The scalar factors of the ele‐
69 mentary reflectors which represent the unitary matrix P. See
70 Further Details. X (output) COMPLEX*16 array, dimension
71 (LDX,NB) The m-by-nb matrix X required to update the unreduced
72 part of A.
73 LDX (input) INTEGER
75 The leading dimension of the array X. LDX >= max(1,M).
76 Y (output) COMPLEX*16 array, dimension (LDY,NB)
78 The n-by-nb matrix Y required to update the unreduced part of
79 A.
80 LDY (input) INTEGER
82 The leading dimension of the array Y. LDY >= max(1,N).
83
85 The matrices Q and P are represented as products of elementary reflec‐
86 tors:
87 Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb) Each H(i)
88 and G(i) has the form:
89 H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' where tauq
90 and taup are complex scalars, and v and u are complex vectors.
91 If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
92 A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in
93 A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i). If m < n,
94 v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in A(i+2:m,i);
95 u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in A(i,i+1:n);
96 tauq is stored in TAUQ(i) and taup in TAUP(i). The elements of the
97 vectors v and u together form the m-by-nb matrix V and the nb-by-n
98 matrix U' which are needed, with X and Y, to apply the transformation
99 to the unreduced part of the matrix, using a block update of the form:
100 A := A - V*Y' - X*U'.
101 The contents of A on exit are illustrated by the following examples
102 with nb = 2:
103 m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
104 ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 )
105 ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 )
106 ( v1 v2 a a a ) ( v1 1 a a a a )
107 ( v1 v2 a a a ) ( v1 v2 a a a a )
108 ( v1 v2 a a a ) ( v1 v2 a a a a )
109 ( v1 v2 a a a )
110 where a denotes an element of the original matrix which is unchanged,
111 vi denotes an element of the vector defining H(i), and ui an element of
112 the vector defining G(i).
113 LAPACK auxiliary routine (versionNo3v.e2m)ber 2008 ZLABRD(1)