LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
dlabrd.f
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1 *> \brief \b DLABRD 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 DLABRD( 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 A( LDA, * ), D( * ), E( * ), TAUP( * ),
29 * \$ TAUQ( * ), X( LDX, * ), Y( LDY, * )
30 * ..
31 *
32 *
33 *> \par Purpose:
34 * =============
35 *>
36 *> \verbatim
37 *>
38 *> DLABRD reduces the first NB rows and columns of a real general
39 *> m by n matrix A to upper or lower bidiagonal form by an orthogonal
40 *> transformation Q**T * A * P, and returns the matrices X and Y which
41 *> are needed to apply the transformation to the unreduced part of A.
42 *>
43 *> If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
44 *> bidiagonal form.
45 *>
46 *> This is an auxiliary routine called by DGEBRD
47 *> \endverbatim
48 *
49 * Arguments:
50 * ==========
51 *
52 *> \param[in] M
53 *> \verbatim
54 *> M is INTEGER
55 *> The number of rows in the matrix A.
56 *> \endverbatim
57 *>
58 *> \param[in] N
59 *> \verbatim
60 *> N is INTEGER
61 *> The number of columns in the matrix A.
62 *> \endverbatim
63 *>
64 *> \param[in] NB
65 *> \verbatim
66 *> NB is INTEGER
67 *> The number of leading rows and columns of A to be reduced.
68 *> \endverbatim
69 *>
70 *> \param[in,out] A
71 *> \verbatim
72 *> A is DOUBLE PRECISION array, dimension (LDA,N)
73 *> On entry, the m by n general matrix to be reduced.
74 *> On exit, the first NB rows and columns of the matrix are
75 *> overwritten; the rest of the array is unchanged.
76 *> If m >= n, elements on and below the diagonal in the first NB
77 *> columns, with the array TAUQ, represent the orthogonal
78 *> matrix Q as a product of elementary reflectors; and
79 *> elements above the diagonal in the first NB rows, with the
80 *> array TAUP, represent the orthogonal matrix P as a product
81 *> of elementary reflectors.
82 *> If m < n, elements below the diagonal in the first NB
83 *> columns, with the array TAUQ, represent the orthogonal
84 *> matrix Q as a product of elementary reflectors, and
85 *> elements on and above the diagonal in the first NB rows,
86 *> with the array TAUP, represent the orthogonal matrix P as
87 *> a product of elementary reflectors.
88 *> See Further Details.
89 *> \endverbatim
90 *>
91 *> \param[in] LDA
92 *> \verbatim
93 *> LDA is INTEGER
94 *> The leading dimension of the array A. LDA >= max(1,M).
95 *> \endverbatim
96 *>
97 *> \param[out] D
98 *> \verbatim
99 *> D is DOUBLE PRECISION array, dimension (NB)
100 *> The diagonal elements of the first NB rows and columns of
101 *> the reduced matrix. D(i) = A(i,i).
102 *> \endverbatim
103 *>
104 *> \param[out] E
105 *> \verbatim
106 *> E is DOUBLE PRECISION array, dimension (NB)
107 *> The off-diagonal elements of the first NB rows and columns of
108 *> the reduced matrix.
109 *> \endverbatim
110 *>
111 *> \param[out] TAUQ
112 *> \verbatim
113 *> TAUQ is DOUBLE PRECISION array dimension (NB)
114 *> The scalar factors of the elementary reflectors which
115 *> represent the orthogonal matrix Q. See Further Details.
116 *> \endverbatim
117 *>
118 *> \param[out] TAUP
119 *> \verbatim
120 *> TAUP is DOUBLE PRECISION array, dimension (NB)
121 *> The scalar factors of the elementary reflectors which
122 *> represent the orthogonal matrix P. See Further Details.
123 *> \endverbatim
124 *>
125 *> \param[out] X
126 *> \verbatim
127 *> X is DOUBLE PRECISION array, dimension (LDX,NB)
128 *> The m-by-nb matrix X required to update the unreduced part
129 *> of A.
130 *> \endverbatim
131 *>
132 *> \param[in] LDX
133 *> \verbatim
134 *> LDX is INTEGER
135 *> The leading dimension of the array X. LDX >= max(1,M).
136 *> \endverbatim
137 *>
138 *> \param[out] Y
139 *> \verbatim
140 *> Y is DOUBLE PRECISION array, dimension (LDY,NB)
141 *> The n-by-nb matrix Y required to update the unreduced part
142 *> of A.
143 *> \endverbatim
144 *>
145 *> \param[in] LDY
146 *> \verbatim
147 *> LDY is INTEGER
148 *> The leading dimension of the array Y. LDY >= max(1,N).
149 *> \endverbatim
150 *
151 * Authors:
152 * ========
153 *
154 *> \author Univ. of Tennessee
155 *> \author Univ. of California Berkeley
156 *> \author Univ. of Colorado Denver
157 *> \author NAG Ltd.
158 *
159 *> \date September 2012
160 *
161 *> \ingroup doubleOTHERauxiliary
162 *
163 *> \par Further Details:
164 * =====================
165 *>
166 *> \verbatim
167 *>
168 *> The matrices Q and P are represented as products of elementary
169 *> reflectors:
170 *>
171 *> Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb)
172 *>
173 *> Each H(i) and G(i) has the form:
174 *>
175 *> H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
176 *>
177 *> where tauq and taup are real scalars, and v and u are real 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**T 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**T - X*U**T.
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 dlabrd( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y,
211  \$ ldy )
212 *
213 * -- LAPACK auxiliary routine (version 3.4.2) --
214 * -- LAPACK is a software package provided by Univ. of Tennessee, --
215 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
216 * September 2012
217 *
218 * .. Scalar Arguments ..
219  INTEGER LDA, LDX, LDY, M, N, NB
220 * ..
221 * .. Array Arguments ..
222  DOUBLE PRECISION A( lda, * ), D( * ), E( * ), TAUP( * ),
223  \$ tauq( * ), x( ldx, * ), y( ldy, * )
224 * ..
225 *
226 * =====================================================================
227 *
228 * .. Parameters ..
229  DOUBLE PRECISION ZERO, ONE
230  parameter ( zero = 0.0d0, one = 1.0d0 )
231 * ..
232 * .. Local Scalars ..
233  INTEGER I
234 * ..
235 * .. External Subroutines ..
236  EXTERNAL dgemv, dlarfg, dscal
237 * ..
238 * .. Intrinsic Functions ..
239  INTRINSIC min
240 * ..
241 * .. Executable Statements ..
242 *
243 * Quick return if possible
244 *
245  IF( m.LE.0 .OR. n.LE.0 )
246  \$ RETURN
247 *
248  IF( m.GE.n ) THEN
249 *
250 * Reduce to upper bidiagonal form
251 *
252  DO 10 i = 1, nb
253 *
254 * Update A(i:m,i)
255 *
256  CALL dgemv( 'No transpose', m-i+1, i-1, -one, a( i, 1 ),
257  \$ lda, y( i, 1 ), ldy, one, a( i, i ), 1 )
258  CALL dgemv( 'No transpose', m-i+1, i-1, -one, x( i, 1 ),
259  \$ ldx, a( 1, i ), 1, one, a( i, i ), 1 )
260 *
261 * Generate reflection Q(i) to annihilate A(i+1:m,i)
262 *
263  CALL dlarfg( m-i+1, a( i, i ), a( min( i+1, m ), i ), 1,
264  \$ tauq( i ) )
265  d( i ) = a( i, i )
266  IF( i.LT.n ) THEN
267  a( i, i ) = one
268 *
269 * Compute Y(i+1:n,i)
270 *
271  CALL dgemv( 'Transpose', m-i+1, n-i, one, a( i, i+1 ),
272  \$ lda, a( i, i ), 1, zero, y( i+1, i ), 1 )
273  CALL dgemv( 'Transpose', m-i+1, i-1, one, a( i, 1 ), lda,
274  \$ a( i, i ), 1, zero, y( 1, i ), 1 )
275  CALL dgemv( 'No transpose', n-i, i-1, -one, y( i+1, 1 ),
276  \$ ldy, y( 1, i ), 1, one, y( i+1, i ), 1 )
277  CALL dgemv( 'Transpose', m-i+1, i-1, one, x( i, 1 ), ldx,
278  \$ a( i, i ), 1, zero, y( 1, i ), 1 )
279  CALL dgemv( 'Transpose', i-1, n-i, -one, a( 1, i+1 ),
280  \$ lda, y( 1, i ), 1, one, y( i+1, i ), 1 )
281  CALL dscal( n-i, tauq( i ), y( i+1, i ), 1 )
282 *
283 * Update A(i,i+1:n)
284 *
285  CALL dgemv( 'No transpose', n-i, i, -one, y( i+1, 1 ),
286  \$ ldy, a( i, 1 ), lda, one, a( i, i+1 ), lda )
287  CALL dgemv( 'Transpose', i-1, n-i, -one, a( 1, i+1 ),
288  \$ lda, x( i, 1 ), ldx, one, a( i, i+1 ), lda )
289 *
290 * Generate reflection P(i) to annihilate A(i,i+2:n)
291 *
292  CALL dlarfg( n-i, a( i, i+1 ), a( i, min( i+2, n ) ),
293  \$ lda, taup( i ) )
294  e( i ) = a( i, i+1 )
295  a( i, i+1 ) = one
296 *
297 * Compute X(i+1:m,i)
298 *
299  CALL dgemv( 'No transpose', m-i, n-i, one, a( i+1, i+1 ),
300  \$ lda, a( i, i+1 ), lda, zero, x( i+1, i ), 1 )
301  CALL dgemv( 'Transpose', n-i, i, one, y( i+1, 1 ), ldy,
302  \$ a( i, i+1 ), lda, zero, x( 1, i ), 1 )
303  CALL dgemv( 'No transpose', m-i, i, -one, a( i+1, 1 ),
304  \$ lda, x( 1, i ), 1, one, x( i+1, i ), 1 )
305  CALL dgemv( 'No transpose', i-1, n-i, one, a( 1, i+1 ),
306  \$ lda, a( i, i+1 ), lda, zero, x( 1, i ), 1 )
307  CALL dgemv( 'No transpose', m-i, i-1, -one, x( i+1, 1 ),
308  \$ ldx, x( 1, i ), 1, one, x( i+1, i ), 1 )
309  CALL dscal( m-i, taup( i ), x( i+1, i ), 1 )
310  END IF
311  10 CONTINUE
312  ELSE
313 *
314 * Reduce to lower bidiagonal form
315 *
316  DO 20 i = 1, nb
317 *
318 * Update A(i,i:n)
319 *
320  CALL dgemv( 'No transpose', n-i+1, i-1, -one, y( i, 1 ),
321  \$ ldy, a( i, 1 ), lda, one, a( i, i ), lda )
322  CALL dgemv( 'Transpose', i-1, n-i+1, -one, a( 1, i ), lda,
323  \$ x( i, 1 ), ldx, one, a( i, i ), lda )
324 *
325 * Generate reflection P(i) to annihilate A(i,i+1:n)
326 *
327  CALL dlarfg( n-i+1, a( i, i ), a( i, min( i+1, n ) ), lda,
328  \$ taup( i ) )
329  d( i ) = a( i, i )
330  IF( i.LT.m ) THEN
331  a( i, i ) = one
332 *
333 * Compute X(i+1:m,i)
334 *
335  CALL dgemv( 'No transpose', m-i, n-i+1, one, a( i+1, i ),
336  \$ lda, a( i, i ), lda, zero, x( i+1, i ), 1 )
337  CALL dgemv( 'Transpose', n-i+1, i-1, one, y( i, 1 ), ldy,
338  \$ a( i, i ), lda, zero, x( 1, i ), 1 )
339  CALL dgemv( 'No transpose', m-i, i-1, -one, a( i+1, 1 ),
340  \$ lda, x( 1, i ), 1, one, x( i+1, i ), 1 )
341  CALL dgemv( 'No transpose', i-1, n-i+1, one, a( 1, i ),
342  \$ lda, a( i, i ), lda, zero, x( 1, i ), 1 )
343  CALL dgemv( 'No transpose', m-i, i-1, -one, x( i+1, 1 ),
344  \$ ldx, x( 1, i ), 1, one, x( i+1, i ), 1 )
345  CALL dscal( m-i, taup( i ), x( i+1, i ), 1 )
346 *
347 * Update A(i+1:m,i)
348 *
349  CALL dgemv( 'No transpose', m-i, i-1, -one, a( i+1, 1 ),
350  \$ lda, y( i, 1 ), ldy, one, a( i+1, i ), 1 )
351  CALL dgemv( 'No transpose', m-i, i, -one, x( i+1, 1 ),
352  \$ ldx, a( 1, i ), 1, one, a( i+1, i ), 1 )
353 *
354 * Generate reflection Q(i) to annihilate A(i+2:m,i)
355 *
356  CALL dlarfg( m-i, a( i+1, i ), a( min( i+2, m ), i ), 1,
357  \$ tauq( i ) )
358  e( i ) = a( i+1, i )
359  a( i+1, i ) = one
360 *
361 * Compute Y(i+1:n,i)
362 *
363  CALL dgemv( 'Transpose', m-i, n-i, one, a( i+1, i+1 ),
364  \$ lda, a( i+1, i ), 1, zero, y( i+1, i ), 1 )
365  CALL dgemv( 'Transpose', m-i, i-1, one, a( i+1, 1 ), lda,
366  \$ a( i+1, i ), 1, zero, y( 1, i ), 1 )
367  CALL dgemv( 'No transpose', n-i, i-1, -one, y( i+1, 1 ),
368  \$ ldy, y( 1, i ), 1, one, y( i+1, i ), 1 )
369  CALL dgemv( 'Transpose', m-i, i, one, x( i+1, 1 ), ldx,
370  \$ a( i+1, i ), 1, zero, y( 1, i ), 1 )
371  CALL dgemv( 'Transpose', i, n-i, -one, a( 1, i+1 ), lda,
372  \$ y( 1, i ), 1, one, y( i+1, i ), 1 )
373  CALL dscal( n-i, tauq( i ), y( i+1, i ), 1 )
374  END IF
375  20 CONTINUE
376  END IF
377  RETURN
378 *
379 * End of DLABRD
380 *
381  END
subroutine dgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
DGEMV
Definition: dgemv.f:158
subroutine dlabrd(M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY)
DLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.
Definition: dlabrd.f:212
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:55
subroutine dlarfg(N, ALPHA, X, INCX, TAU)
DLARFG generates an elementary reflector (Householder matrix).
Definition: dlarfg.f:108