LAPACK  3.9.0
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
6 * http://www.netlib.org/lapack/explore-html/
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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 *> \date June 2017
161 *
162 *> \ingroup complex16OTHERauxiliary
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 zlabrd( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y,
213  $ LDY )
214 *
215 * -- LAPACK auxiliary routine (version 3.7.1) --
216 * -- LAPACK is a software package provided by Univ. of Tennessee, --
217 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
218 * June 2017
219 *
220 * .. Scalar Arguments ..
221  INTEGER LDA, LDX, LDY, M, N, NB
222 * ..
223 * .. Array Arguments ..
224  DOUBLE PRECISION D( * ), E( * )
225  COMPLEX*16 A( LDA, * ), TAUP( * ), TAUQ( * ), X( LDX, * ),
226  $ y( ldy, * )
227 * ..
228 *
229 * =====================================================================
230 *
231 * .. Parameters ..
232  COMPLEX*16 ZERO, ONE
233  parameter( zero = ( 0.0d+0, 0.0d+0 ),
234  $ one = ( 1.0d+0, 0.0d+0 ) )
235 * ..
236 * .. Local Scalars ..
237  INTEGER I
238  COMPLEX*16 ALPHA
239 * ..
240 * .. External Subroutines ..
241  EXTERNAL zgemv, zlacgv, zlarfg, zscal
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 zlacgv( i-1, y( i, 1 ), ldy )
262  CALL zgemv( 'No transpose', m-i+1, i-1, -one, a( i, 1 ),
263  $ lda, y( i, 1 ), ldy, one, a( i, i ), 1 )
264  CALL zlacgv( i-1, y( i, 1 ), ldy )
265  CALL zgemv( '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 zlarfg( 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 zgemv( '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 zgemv( '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 zgemv( '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 zgemv( '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 zgemv( '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 zscal( n-i, tauq( i ), y( i+1, i ), 1 )
294 *
295 * Update A(i,i+1:n)
296 *
297  CALL zlacgv( n-i, a( i, i+1 ), lda )
298  CALL zlacgv( i, a( i, 1 ), lda )
299  CALL zgemv( 'No transpose', n-i, i, -one, y( i+1, 1 ),
300  $ ldy, a( i, 1 ), lda, one, a( i, i+1 ), lda )
301  CALL zlacgv( i, a( i, 1 ), lda )
302  CALL zlacgv( i-1, x( i, 1 ), ldx )
303  CALL zgemv( '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 zlacgv( 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 zlarfg( n-i, alpha, a( i, min( i+2, n ) ), lda,
312  $ taup( i ) )
313  e( i ) = alpha
314  a( i, i+1 ) = one
315 *
316 * Compute X(i+1:m,i)
317 *
318  CALL zgemv( '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 zgemv( '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 zgemv( 'No transpose', m-i, i, -one, a( i+1, 1 ),
324  $ lda, x( 1, i ), 1, one, x( i+1, i ), 1 )
325  CALL zgemv( '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 zgemv( '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 zscal( m-i, taup( i ), x( i+1, i ), 1 )
330  CALL zlacgv( 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 zlacgv( n-i+1, a( i, i ), lda )
342  CALL zlacgv( i-1, a( i, 1 ), lda )
343  CALL zgemv( 'No transpose', n-i+1, i-1, -one, y( i, 1 ),
344  $ ldy, a( i, 1 ), lda, one, a( i, i ), lda )
345  CALL zlacgv( i-1, a( i, 1 ), lda )
346  CALL zlacgv( i-1, x( i, 1 ), ldx )
347  CALL zgemv( '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 zlacgv( 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 zlarfg( 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 zgemv( '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 zgemv( '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 zgemv( '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 zgemv( 'No transpose', i-1, n-i+1, one, a( 1, i ),
371  $ lda, a( i, i ), lda, zero, x( 1, i ), 1 )
372  CALL zgemv( '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 zscal( m-i, taup( i ), x( i+1, i ), 1 )
375  CALL zlacgv( n-i+1, a( i, i ), lda )
376 *
377 * Update A(i+1:m,i)
378 *
379  CALL zlacgv( i-1, y( i, 1 ), ldy )
380  CALL zgemv( '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 zlacgv( i-1, y( i, 1 ), ldy )
383  CALL zgemv( '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 zlarfg( 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 zgemv( '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 zgemv( '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 zgemv( '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 zgemv( '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 zgemv( '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 zscal( n-i, tauq( i ), y( i+1, i ), 1 )
411  ELSE
412  CALL zlacgv( n-i+1, a( i, i ), lda )
413  END IF
414  20 CONTINUE
415  END IF
416  RETURN
417 *
418 * End of ZLABRD
419 *
420  END
zlacgv
subroutine zlacgv(N, X, INCX)
ZLACGV conjugates a complex vector.
Definition: zlacgv.f:76
zgemv
subroutine zgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZGEMV
Definition: zgemv.f:160
zlabrd
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:214
zlarfg
subroutine zlarfg(N, ALPHA, X, INCX, TAU)
ZLARFG generates an elementary reflector (Householder matrix).
Definition: zlarfg.f:108
zscal
subroutine zscal(N, ZA, ZX, INCX)
ZSCAL
Definition: zscal.f:80