LAPACK  3.9.1
LAPACK: Linear Algebra PACKage
zlabrd.f
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1 *> \brief \b ZLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
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16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE ZLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y,
22 * LDY )
23 *
24 * .. Scalar Arguments ..
25 * INTEGER LDA, LDX, LDY, M, N, NB
26 * ..
27 * .. Array Arguments ..
28 * DOUBLE PRECISION D( * ), E( * )
29 * COMPLEX*16 A( LDA, * ), TAUP( * ), TAUQ( * ), X( LDX, * ),
30 * $ Y( LDY, * )
31 * ..
32 *
33 *
34 *> \par Purpose:
35 * =============
36 *>
37 *> \verbatim
38 *>
39 *> ZLABRD reduces the first NB rows and columns of a complex general
40 *> m by n matrix A to upper or lower real bidiagonal form by a unitary
41 *> transformation Q**H * A * P, and returns the matrices X and Y which
42 *> are needed to apply the transformation to the unreduced part of A.
43 *>
44 *> If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
45 *> bidiagonal form.
46 *>
47 *> This is an auxiliary routine called by ZGEBRD
48 *> \endverbatim
49 *
50 * Arguments:
51 * ==========
52 *
53 *> \param[in] M
54 *> \verbatim
55 *> M is INTEGER
56 *> The number of rows in the matrix A.
57 *> \endverbatim
58 *>
59 *> \param[in] N
60 *> \verbatim
61 *> N is INTEGER
62 *> The number of columns in the matrix A.
63 *> \endverbatim
64 *>
65 *> \param[in] NB
66 *> \verbatim
67 *> NB is INTEGER
68 *> The number of leading rows and columns of A to be reduced.
69 *> \endverbatim
70 *>
71 *> \param[in,out] A
72 *> \verbatim
73 *> A is COMPLEX*16 array, dimension (LDA,N)
74 *> On entry, the m by n general matrix to be reduced.
75 *> On exit, the first NB rows and columns of the matrix are
76 *> overwritten; the rest of the array is unchanged.
77 *> If m >= n, elements on and below the diagonal in the first NB
78 *> columns, with the array TAUQ, represent the unitary
79 *> matrix Q as a product of elementary reflectors; and
80 *> elements above the diagonal in the first NB rows, with the
81 *> array TAUP, represent the unitary matrix P as a product
82 *> of elementary reflectors.
83 *> If m < n, elements below the diagonal in the first NB
84 *> columns, with the array TAUQ, represent the unitary
85 *> matrix Q as a product of elementary reflectors, and
86 *> elements on and above the diagonal in the first NB rows,
87 *> with the array TAUP, represent the unitary matrix P as
88 *> a product of elementary reflectors.
89 *> See Further Details.
90 *> \endverbatim
91 *>
92 *> \param[in] LDA
93 *> \verbatim
94 *> LDA is INTEGER
95 *> The leading dimension of the array A. LDA >= max(1,M).
96 *> \endverbatim
97 *>
98 *> \param[out] D
99 *> \verbatim
100 *> D is DOUBLE PRECISION array, dimension (NB)
101 *> The diagonal elements of the first NB rows and columns of
102 *> the reduced matrix. D(i) = A(i,i).
103 *> \endverbatim
104 *>
105 *> \param[out] E
106 *> \verbatim
107 *> E is DOUBLE PRECISION array, dimension (NB)
108 *> The off-diagonal elements of the first NB rows and columns of
109 *> the reduced matrix.
110 *> \endverbatim
111 *>
112 *> \param[out] TAUQ
113 *> \verbatim
114 *> TAUQ is COMPLEX*16 array, dimension (NB)
115 *> The scalar factors of the elementary reflectors which
116 *> represent the unitary matrix Q. See Further Details.
117 *> \endverbatim
118 *>
119 *> \param[out] TAUP
120 *> \verbatim
121 *> TAUP is COMPLEX*16 array, dimension (NB)
122 *> The scalar factors of the elementary reflectors which
123 *> represent the unitary matrix P. See Further Details.
124 *> \endverbatim
125 *>
126 *> \param[out] X
127 *> \verbatim
128 *> X is COMPLEX*16 array, dimension (LDX,NB)
129 *> The m-by-nb matrix X required to update the unreduced part
130 *> of A.
131 *> \endverbatim
132 *>
133 *> \param[in] LDX
134 *> \verbatim
135 *> LDX is INTEGER
136 *> The leading dimension of the array X. LDX >= max(1,M).
137 *> \endverbatim
138 *>
139 *> \param[out] Y
140 *> \verbatim
141 *> Y is COMPLEX*16 array, dimension (LDY,NB)
142 *> The n-by-nb matrix Y required to update the unreduced part
143 *> of A.
144 *> \endverbatim
145 *>
146 *> \param[in] LDY
147 *> \verbatim
148 *> LDY is INTEGER
149 *> The leading dimension of the array Y. LDY >= max(1,N).
150 *> \endverbatim
151 *
152 * Authors:
153 * ========
154 *
155 *> \author Univ. of Tennessee
156 *> \author Univ. of California Berkeley
157 *> \author Univ. of Colorado Denver
158 *> \author NAG Ltd.
159 *
160 *> \ingroup complex16OTHERauxiliary
161 *
162 *> \par Further Details:
163 * =====================
164 *>
165 *> \verbatim
166 *>
167 *> The matrices Q and P are represented as products of elementary
168 *> reflectors:
169 *>
170 *> Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb)
171 *>
172 *> Each H(i) and G(i) has the form:
173 *>
174 *> H(i) = I - tauq * v * v**H and G(i) = I - taup * u * u**H
175 *>
176 *> where tauq and taup are complex scalars, and v and u are complex
177 *> vectors.
178 *>
179 *> If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
180 *> A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in
181 *> A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
182 *>
183 *> If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
184 *> A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
185 *> A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
186 *>
187 *> The elements of the vectors v and u together form the m-by-nb matrix
188 *> V and the nb-by-n matrix U**H which are needed, with X and Y, to apply
189 *> the transformation to the unreduced part of the matrix, using a block
190 *> update of the form: A := A - V*Y**H - X*U**H.
191 *>
192 *> The contents of A on exit are illustrated by the following examples
193 *> with nb = 2:
194 *>
195 *> m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
196 *>
197 *> ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 )
198 *> ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 )
199 *> ( v1 v2 a a a ) ( v1 1 a a a a )
200 *> ( v1 v2 a a a ) ( v1 v2 a a a a )
201 *> ( v1 v2 a a a ) ( v1 v2 a a a a )
202 *> ( v1 v2 a a a )
203 *>
204 *> where a denotes an element of the original matrix which is unchanged,
205 *> vi denotes an element of the vector defining H(i), and ui an element
206 *> of the vector defining G(i).
207 *> \endverbatim
208 *>
209 * =====================================================================
210  SUBROUTINE zlabrd( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y,
211  $ LDY )
212 *
213 * -- LAPACK auxiliary routine --
214 * -- LAPACK is a software package provided by Univ. of Tennessee, --
215 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
216 *
217 * .. Scalar Arguments ..
218  INTEGER LDA, LDX, LDY, M, N, NB
219 * ..
220 * .. Array Arguments ..
221  DOUBLE PRECISION D( * ), E( * )
222  COMPLEX*16 A( LDA, * ), TAUP( * ), TAUQ( * ), X( LDX, * ),
223  $ y( ldy, * )
224 * ..
225 *
226 * =====================================================================
227 *
228 * .. Parameters ..
229  COMPLEX*16 ZERO, ONE
230  parameter( zero = ( 0.0d+0, 0.0d+0 ),
231  $ one = ( 1.0d+0, 0.0d+0 ) )
232 * ..
233 * .. Local Scalars ..
234  INTEGER I
235  COMPLEX*16 ALPHA
236 * ..
237 * .. External Subroutines ..
238  EXTERNAL zgemv, zlacgv, zlarfg, zscal
239 * ..
240 * .. Intrinsic Functions ..
241  INTRINSIC min
242 * ..
243 * .. Executable Statements ..
244 *
245 * Quick return if possible
246 *
247  IF( m.LE.0 .OR. n.LE.0 )
248  $ RETURN
249 *
250  IF( m.GE.n ) THEN
251 *
252 * Reduce to upper bidiagonal form
253 *
254  DO 10 i = 1, nb
255 *
256 * Update A(i:m,i)
257 *
258  CALL zlacgv( i-1, y( i, 1 ), ldy )
259  CALL zgemv( 'No transpose', m-i+1, i-1, -one, a( i, 1 ),
260  $ lda, y( i, 1 ), ldy, one, a( i, i ), 1 )
261  CALL zlacgv( i-1, y( i, 1 ), ldy )
262  CALL zgemv( 'No transpose', m-i+1, i-1, -one, x( i, 1 ),
263  $ ldx, a( 1, i ), 1, one, a( i, i ), 1 )
264 *
265 * Generate reflection Q(i) to annihilate A(i+1:m,i)
266 *
267  alpha = a( i, i )
268  CALL zlarfg( m-i+1, alpha, a( min( i+1, m ), i ), 1,
269  $ tauq( i ) )
270  d( i ) = alpha
271  IF( i.LT.n ) THEN
272  a( i, i ) = one
273 *
274 * Compute Y(i+1:n,i)
275 *
276  CALL zgemv( 'Conjugate transpose', m-i+1, n-i, one,
277  $ a( i, i+1 ), lda, a( i, i ), 1, zero,
278  $ y( i+1, i ), 1 )
279  CALL zgemv( 'Conjugate transpose', m-i+1, i-1, one,
280  $ a( i, 1 ), lda, a( i, i ), 1, zero,
281  $ y( 1, i ), 1 )
282  CALL zgemv( 'No transpose', n-i, i-1, -one, y( i+1, 1 ),
283  $ ldy, y( 1, i ), 1, one, y( i+1, i ), 1 )
284  CALL zgemv( 'Conjugate transpose', m-i+1, i-1, one,
285  $ x( i, 1 ), ldx, a( i, i ), 1, zero,
286  $ y( 1, i ), 1 )
287  CALL zgemv( 'Conjugate transpose', i-1, n-i, -one,
288  $ a( 1, i+1 ), lda, y( 1, i ), 1, one,
289  $ y( i+1, i ), 1 )
290  CALL zscal( n-i, tauq( i ), y( i+1, i ), 1 )
291 *
292 * Update A(i,i+1:n)
293 *
294  CALL zlacgv( n-i, a( i, i+1 ), lda )
295  CALL zlacgv( i, a( i, 1 ), lda )
296  CALL zgemv( 'No transpose', n-i, i, -one, y( i+1, 1 ),
297  $ ldy, a( i, 1 ), lda, one, a( i, i+1 ), lda )
298  CALL zlacgv( i, a( i, 1 ), lda )
299  CALL zlacgv( i-1, x( i, 1 ), ldx )
300  CALL zgemv( 'Conjugate transpose', i-1, n-i, -one,
301  $ a( 1, i+1 ), lda, x( i, 1 ), ldx, one,
302  $ a( i, i+1 ), lda )
303  CALL zlacgv( i-1, x( i, 1 ), ldx )
304 *
305 * Generate reflection P(i) to annihilate A(i,i+2:n)
306 *
307  alpha = a( i, i+1 )
308  CALL zlarfg( n-i, alpha, a( i, min( i+2, n ) ), lda,
309  $ taup( i ) )
310  e( i ) = alpha
311  a( i, i+1 ) = one
312 *
313 * Compute X(i+1:m,i)
314 *
315  CALL zgemv( 'No transpose', m-i, n-i, one, a( i+1, i+1 ),
316  $ lda, a( i, i+1 ), lda, zero, x( i+1, i ), 1 )
317  CALL zgemv( 'Conjugate transpose', n-i, i, one,
318  $ y( i+1, 1 ), ldy, a( i, i+1 ), lda, zero,
319  $ x( 1, i ), 1 )
320  CALL zgemv( 'No transpose', m-i, i, -one, a( i+1, 1 ),
321  $ lda, x( 1, i ), 1, one, x( i+1, i ), 1 )
322  CALL zgemv( 'No transpose', i-1, n-i, one, a( 1, i+1 ),
323  $ lda, a( i, i+1 ), lda, zero, x( 1, i ), 1 )
324  CALL zgemv( 'No transpose', m-i, i-1, -one, x( i+1, 1 ),
325  $ ldx, x( 1, i ), 1, one, x( i+1, i ), 1 )
326  CALL zscal( m-i, taup( i ), x( i+1, i ), 1 )
327  CALL zlacgv( n-i, a( i, i+1 ), lda )
328  END IF
329  10 CONTINUE
330  ELSE
331 *
332 * Reduce to lower bidiagonal form
333 *
334  DO 20 i = 1, nb
335 *
336 * Update A(i,i:n)
337 *
338  CALL zlacgv( n-i+1, a( i, i ), lda )
339  CALL zlacgv( i-1, a( i, 1 ), lda )
340  CALL zgemv( 'No transpose', n-i+1, i-1, -one, y( i, 1 ),
341  $ ldy, a( i, 1 ), lda, one, a( i, i ), lda )
342  CALL zlacgv( i-1, a( i, 1 ), lda )
343  CALL zlacgv( i-1, x( i, 1 ), ldx )
344  CALL zgemv( 'Conjugate transpose', i-1, n-i+1, -one,
345  $ a( 1, i ), lda, x( i, 1 ), ldx, one, a( i, i ),
346  $ lda )
347  CALL zlacgv( i-1, x( i, 1 ), ldx )
348 *
349 * Generate reflection P(i) to annihilate A(i,i+1:n)
350 *
351  alpha = a( i, i )
352  CALL zlarfg( n-i+1, alpha, a( i, min( i+1, n ) ), lda,
353  $ taup( i ) )
354  d( i ) = alpha
355  IF( i.LT.m ) THEN
356  a( i, i ) = one
357 *
358 * Compute X(i+1:m,i)
359 *
360  CALL zgemv( 'No transpose', m-i, n-i+1, one, a( i+1, i ),
361  $ lda, a( i, i ), lda, zero, x( i+1, i ), 1 )
362  CALL zgemv( 'Conjugate transpose', n-i+1, i-1, one,
363  $ y( i, 1 ), ldy, a( i, i ), lda, zero,
364  $ x( 1, i ), 1 )
365  CALL zgemv( 'No transpose', m-i, i-1, -one, a( i+1, 1 ),
366  $ lda, x( 1, i ), 1, one, x( i+1, i ), 1 )
367  CALL zgemv( 'No transpose', i-1, n-i+1, one, a( 1, i ),
368  $ lda, a( i, i ), lda, zero, x( 1, i ), 1 )
369  CALL zgemv( 'No transpose', m-i, i-1, -one, x( i+1, 1 ),
370  $ ldx, x( 1, i ), 1, one, x( i+1, i ), 1 )
371  CALL zscal( m-i, taup( i ), x( i+1, i ), 1 )
372  CALL zlacgv( n-i+1, a( i, i ), lda )
373 *
374 * Update A(i+1:m,i)
375 *
376  CALL zlacgv( i-1, y( i, 1 ), ldy )
377  CALL zgemv( 'No transpose', m-i, i-1, -one, a( i+1, 1 ),
378  $ lda, y( i, 1 ), ldy, one, a( i+1, i ), 1 )
379  CALL zlacgv( i-1, y( i, 1 ), ldy )
380  CALL zgemv( 'No transpose', m-i, i, -one, x( i+1, 1 ),
381  $ ldx, a( 1, i ), 1, one, a( i+1, i ), 1 )
382 *
383 * Generate reflection Q(i) to annihilate A(i+2:m,i)
384 *
385  alpha = a( i+1, i )
386  CALL zlarfg( m-i, alpha, a( min( i+2, m ), i ), 1,
387  $ tauq( i ) )
388  e( i ) = alpha
389  a( i+1, i ) = one
390 *
391 * Compute Y(i+1:n,i)
392 *
393  CALL zgemv( 'Conjugate transpose', m-i, n-i, one,
394  $ a( i+1, i+1 ), lda, a( i+1, i ), 1, zero,
395  $ y( i+1, i ), 1 )
396  CALL zgemv( 'Conjugate transpose', m-i, i-1, one,
397  $ a( i+1, 1 ), lda, a( i+1, i ), 1, zero,
398  $ y( 1, i ), 1 )
399  CALL zgemv( 'No transpose', n-i, i-1, -one, y( i+1, 1 ),
400  $ ldy, y( 1, i ), 1, one, y( i+1, i ), 1 )
401  CALL zgemv( 'Conjugate transpose', m-i, i, one,
402  $ x( i+1, 1 ), ldx, a( i+1, i ), 1, zero,
403  $ y( 1, i ), 1 )
404  CALL zgemv( 'Conjugate transpose', i, n-i, -one,
405  $ a( 1, i+1 ), lda, y( 1, i ), 1, one,
406  $ y( i+1, i ), 1 )
407  CALL zscal( n-i, tauq( i ), y( i+1, i ), 1 )
408  ELSE
409  CALL zlacgv( n-i+1, a( i, i ), lda )
410  END IF
411  20 CONTINUE
412  END IF
413  RETURN
414 *
415 * End of ZLABRD
416 *
417  END
subroutine zscal(N, ZA, ZX, INCX)
ZSCAL
Definition: zscal.f:78
subroutine zgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZGEMV
Definition: zgemv.f:158
subroutine zlacgv(N, X, INCX)
ZLACGV conjugates a complex vector.
Definition: zlacgv.f:74
subroutine zlabrd(M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY)
ZLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.
Definition: zlabrd.f:212
subroutine zlarfg(N, ALPHA, X, INCX, TAU)
ZLARFG generates an elementary reflector (Householder matrix).
Definition: zlarfg.f:106