LAPACK  3.4.2 LAPACK: Linear Algebra PACKage
clabrd.f
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1 *> \brief \b CLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE CLABRD( 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 * REAL D( * ), E( * )
29 * COMPLEX A( LDA, * ), TAUP( * ), TAUQ( * ), X( LDX, * ),
30 * \$ Y( LDY, * )
31 * ..
32 *
33 *
34 *> \par Purpose:
35 * =============
36 *>
37 *> \verbatim
38 *>
39 *> CLABRD 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 CGEBRD
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 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 REAL 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 REAL 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 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 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 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 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 *> \date September 2012
161 *
162 *> \ingroup complexOTHERauxiliary
163 *
164 *> \par Further Details:
165 * =====================
166 *>
167 *> \verbatim
168 *>
169 *> The matrices Q and P are represented as products of elementary
170 *> reflectors:
171 *>
172 *> Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb)
173 *>
174 *> Each H(i) and G(i) has the form:
175 *>
176 *> H(i) = I - tauq * v * v**H and G(i) = I - taup * u * u**H
177 *>
178 *> where tauq and taup are complex scalars, and v and u are complex
179 *> vectors.
180 *>
181 *> If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
182 *> A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in
183 *> A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
184 *>
185 *> If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
186 *> A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
187 *> A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
188 *>
189 *> The elements of the vectors v and u together form the m-by-nb matrix
190 *> V and the nb-by-n matrix U**H which are needed, with X and Y, to apply
191 *> the transformation to the unreduced part of the matrix, using a block
192 *> update of the form: A := A - V*Y**H - X*U**H.
193 *>
194 *> The contents of A on exit are illustrated by the following examples
195 *> with nb = 2:
196 *>
197 *> m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
198 *>
199 *> ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 )
200 *> ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 )
201 *> ( v1 v2 a a a ) ( v1 1 a a a a )
202 *> ( v1 v2 a a a ) ( v1 v2 a a a a )
203 *> ( v1 v2 a a a ) ( v1 v2 a a a a )
204 *> ( v1 v2 a a a )
205 *>
206 *> where a denotes an element of the original matrix which is unchanged,
207 *> vi denotes an element of the vector defining H(i), and ui an element
208 *> of the vector defining G(i).
209 *> \endverbatim
210 *>
211 * =====================================================================
212  SUBROUTINE clabrd( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y,
213  \$ ldy )
214 *
215 * -- LAPACK auxiliary routine (version 3.4.2) --
216 * -- LAPACK is a software package provided by Univ. of Tennessee, --
217 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
218 * September 2012
219 *
220 * .. Scalar Arguments ..
221  INTEGER lda, ldx, ldy, m, n, nb
222 * ..
223 * .. Array Arguments ..
224  REAL d( * ), e( * )
225  COMPLEX a( lda, * ), taup( * ), tauq( * ), x( ldx, * ),
226  \$ y( ldy, * )
227 * ..
228 *
229 * =====================================================================
230 *
231 * .. Parameters ..
232  COMPLEX zero, one
233  parameter( zero = ( 0.0e+0, 0.0e+0 ),
234  \$ one = ( 1.0e+0, 0.0e+0 ) )
235 * ..
236 * .. Local Scalars ..
237  INTEGER i
238  COMPLEX alpha
239 * ..
240 * .. External Subroutines ..
241  EXTERNAL cgemv, clacgv, clarfg, cscal
242 * ..
243 * .. Intrinsic Functions ..
244  INTRINSIC min
245 * ..
246 * .. Executable Statements ..
247 *
248 * Quick return if possible
249 *
250  IF( m.LE.0 .OR. n.LE.0 )
251  \$ return
252 *
253  IF( m.GE.n ) THEN
254 *
255 * Reduce to upper bidiagonal form
256 *
257  DO 10 i = 1, nb
258 *
259 * Update A(i:m,i)
260 *
261  CALL clacgv( i-1, y( i, 1 ), ldy )
262  CALL cgemv( 'No transpose', m-i+1, i-1, -one, a( i, 1 ),
263  \$ lda, y( i, 1 ), ldy, one, a( i, i ), 1 )
264  CALL clacgv( i-1, y( i, 1 ), ldy )
265  CALL cgemv( 'No transpose', m-i+1, i-1, -one, x( i, 1 ),
266  \$ ldx, a( 1, i ), 1, one, a( i, i ), 1 )
267 *
268 * Generate reflection Q(i) to annihilate A(i+1:m,i)
269 *
270  alpha = a( i, i )
271  CALL clarfg( m-i+1, alpha, a( min( i+1, m ), i ), 1,
272  \$ tauq( i ) )
273  d( i ) = alpha
274  IF( i.LT.n ) THEN
275  a( i, i ) = one
276 *
277 * Compute Y(i+1:n,i)
278 *
279  CALL cgemv( 'Conjugate transpose', m-i+1, n-i, one,
280  \$ a( i, i+1 ), lda, a( i, i ), 1, zero,
281  \$ y( i+1, i ), 1 )
282  CALL cgemv( 'Conjugate transpose', m-i+1, i-1, one,
283  \$ a( i, 1 ), lda, a( i, i ), 1, zero,
284  \$ y( 1, i ), 1 )
285  CALL cgemv( 'No transpose', n-i, i-1, -one, y( i+1, 1 ),
286  \$ ldy, y( 1, i ), 1, one, y( i+1, i ), 1 )
287  CALL cgemv( 'Conjugate transpose', m-i+1, i-1, one,
288  \$ x( i, 1 ), ldx, a( i, i ), 1, zero,
289  \$ y( 1, i ), 1 )
290  CALL cgemv( 'Conjugate transpose', i-1, n-i, -one,
291  \$ a( 1, i+1 ), lda, y( 1, i ), 1, one,
292  \$ y( i+1, i ), 1 )
293  CALL cscal( n-i, tauq( i ), y( i+1, i ), 1 )
294 *
295 * Update A(i,i+1:n)
296 *
297  CALL clacgv( n-i, a( i, i+1 ), lda )
298  CALL clacgv( i, a( i, 1 ), lda )
299  CALL cgemv( 'No transpose', n-i, i, -one, y( i+1, 1 ),
300  \$ ldy, a( i, 1 ), lda, one, a( i, i+1 ), lda )
301  CALL clacgv( i, a( i, 1 ), lda )
302  CALL clacgv( i-1, x( i, 1 ), ldx )
303  CALL cgemv( 'Conjugate transpose', i-1, n-i, -one,
304  \$ a( 1, i+1 ), lda, x( i, 1 ), ldx, one,
305  \$ a( i, i+1 ), lda )
306  CALL clacgv( i-1, x( i, 1 ), ldx )
307 *
308 * Generate reflection P(i) to annihilate A(i,i+2:n)
309 *
310  alpha = a( i, i+1 )
311  CALL clarfg( n-i, alpha, a( i, min( i+2, n ) ),
312  \$ lda, taup( i ) )
313  e( i ) = alpha
314  a( i, i+1 ) = one
315 *
316 * Compute X(i+1:m,i)
317 *
318  CALL cgemv( 'No transpose', m-i, n-i, one, a( i+1, i+1 ),
319  \$ lda, a( i, i+1 ), lda, zero, x( i+1, i ), 1 )
320  CALL cgemv( 'Conjugate transpose', n-i, i, one,
321  \$ y( i+1, 1 ), ldy, a( i, i+1 ), lda, zero,
322  \$ x( 1, i ), 1 )
323  CALL cgemv( 'No transpose', m-i, i, -one, a( i+1, 1 ),
324  \$ lda, x( 1, i ), 1, one, x( i+1, i ), 1 )
325  CALL cgemv( 'No transpose', i-1, n-i, one, a( 1, i+1 ),
326  \$ lda, a( i, i+1 ), lda, zero, x( 1, i ), 1 )
327  CALL cgemv( 'No transpose', m-i, i-1, -one, x( i+1, 1 ),
328  \$ ldx, x( 1, i ), 1, one, x( i+1, i ), 1 )
329  CALL cscal( m-i, taup( i ), x( i+1, i ), 1 )
330  CALL clacgv( n-i, a( i, i+1 ), lda )
331  END IF
332  10 continue
333  ELSE
334 *
335 * Reduce to lower bidiagonal form
336 *
337  DO 20 i = 1, nb
338 *
339 * Update A(i,i:n)
340 *
341  CALL clacgv( n-i+1, a( i, i ), lda )
342  CALL clacgv( i-1, a( i, 1 ), lda )
343  CALL cgemv( 'No transpose', n-i+1, i-1, -one, y( i, 1 ),
344  \$ ldy, a( i, 1 ), lda, one, a( i, i ), lda )
345  CALL clacgv( i-1, a( i, 1 ), lda )
346  CALL clacgv( i-1, x( i, 1 ), ldx )
347  CALL cgemv( 'Conjugate transpose', i-1, n-i+1, -one,
348  \$ a( 1, i ), lda, x( i, 1 ), ldx, one, a( i, i ),
349  \$ lda )
350  CALL clacgv( i-1, x( i, 1 ), ldx )
351 *
352 * Generate reflection P(i) to annihilate A(i,i+1:n)
353 *
354  alpha = a( i, i )
355  CALL clarfg( n-i+1, alpha, a( i, min( i+1, n ) ), lda,
356  \$ taup( i ) )
357  d( i ) = alpha
358  IF( i.LT.m ) THEN
359  a( i, i ) = one
360 *
361 * Compute X(i+1:m,i)
362 *
363  CALL cgemv( 'No transpose', m-i, n-i+1, one, a( i+1, i ),
364  \$ lda, a( i, i ), lda, zero, x( i+1, i ), 1 )
365  CALL cgemv( 'Conjugate transpose', n-i+1, i-1, one,
366  \$ y( i, 1 ), ldy, a( i, i ), lda, zero,
367  \$ x( 1, i ), 1 )
368  CALL cgemv( 'No transpose', m-i, i-1, -one, a( i+1, 1 ),
369  \$ lda, x( 1, i ), 1, one, x( i+1, i ), 1 )
370  CALL cgemv( 'No transpose', i-1, n-i+1, one, a( 1, i ),
371  \$ lda, a( i, i ), lda, zero, x( 1, i ), 1 )
372  CALL cgemv( 'No transpose', m-i, i-1, -one, x( i+1, 1 ),
373  \$ ldx, x( 1, i ), 1, one, x( i+1, i ), 1 )
374  CALL cscal( m-i, taup( i ), x( i+1, i ), 1 )
375  CALL clacgv( n-i+1, a( i, i ), lda )
376 *
377 * Update A(i+1:m,i)
378 *
379  CALL clacgv( i-1, y( i, 1 ), ldy )
380  CALL cgemv( 'No transpose', m-i, i-1, -one, a( i+1, 1 ),
381  \$ lda, y( i, 1 ), ldy, one, a( i+1, i ), 1 )
382  CALL clacgv( i-1, y( i, 1 ), ldy )
383  CALL cgemv( 'No transpose', m-i, i, -one, x( i+1, 1 ),
384  \$ ldx, a( 1, i ), 1, one, a( i+1, i ), 1 )
385 *
386 * Generate reflection Q(i) to annihilate A(i+2:m,i)
387 *
388  alpha = a( i+1, i )
389  CALL clarfg( m-i, alpha, a( min( i+2, m ), i ), 1,
390  \$ tauq( i ) )
391  e( i ) = alpha
392  a( i+1, i ) = one
393 *
394 * Compute Y(i+1:n,i)
395 *
396  CALL cgemv( 'Conjugate transpose', m-i, n-i, one,
397  \$ a( i+1, i+1 ), lda, a( i+1, i ), 1, zero,
398  \$ y( i+1, i ), 1 )
399  CALL cgemv( 'Conjugate transpose', m-i, i-1, one,
400  \$ a( i+1, 1 ), lda, a( i+1, i ), 1, zero,
401  \$ y( 1, i ), 1 )
402  CALL cgemv( 'No transpose', n-i, i-1, -one, y( i+1, 1 ),
403  \$ ldy, y( 1, i ), 1, one, y( i+1, i ), 1 )
404  CALL cgemv( 'Conjugate transpose', m-i, i, one,
405  \$ x( i+1, 1 ), ldx, a( i+1, i ), 1, zero,
406  \$ y( 1, i ), 1 )
407  CALL cgemv( 'Conjugate transpose', i, n-i, -one,
408  \$ a( 1, i+1 ), lda, y( 1, i ), 1, one,
409  \$ y( i+1, i ), 1 )
410  CALL cscal( n-i, tauq( i ), y( i+1, i ), 1 )
411  ELSE
412  CALL clacgv( n-i+1, a( i, i ), lda )
413  END IF
414  20 continue
415  END IF
416  return
417 *
418 * End of CLABRD
419 *
420  END