LAPACK  3.10.1
LAPACK: Linear Algebra PACKage
slabrd.f
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1 *> \brief \b SLABRD 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 SLABRD( 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 A( LDA, * ), D( * ), E( * ), TAUP( * ),
29 * $ TAUQ( * ), X( LDX, * ), Y( LDY, * )
30 * ..
31 *
32 *
33 *> \par Purpose:
34 * =============
35 *>
36 *> \verbatim
37 *>
38 *> SLABRD 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 SGEBRD
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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 *> \ingroup realOTHERauxiliary
160 *
161 *> \par Further Details:
162 * =====================
163 *>
164 *> \verbatim
165 *>
166 *> The matrices Q and P are represented as products of elementary
167 *> reflectors:
168 *>
169 *> Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb)
170 *>
171 *> Each H(i) and G(i) has the form:
172 *>
173 *> H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
174 *>
175 *> where tauq and taup are real scalars, and v and u are real vectors.
176 *>
177 *> If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
178 *> A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in
179 *> A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
180 *>
181 *> If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
182 *> A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i: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 *> The elements of the vectors v and u together form the m-by-nb matrix
186 *> V and the nb-by-n matrix U**T which are needed, with X and Y, to apply
187 *> the transformation to the unreduced part of the matrix, using a block
188 *> update of the form: A := A - V*Y**T - X*U**T.
189 *>
190 *> The contents of A on exit are illustrated by the following examples
191 *> with nb = 2:
192 *>
193 *> m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
194 *>
195 *> ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 )
196 *> ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 )
197 *> ( v1 v2 a a a ) ( v1 1 a a a a )
198 *> ( v1 v2 a a a ) ( v1 v2 a a a a )
199 *> ( v1 v2 a a a ) ( v1 v2 a a a a )
200 *> ( v1 v2 a a a )
201 *>
202 *> where a denotes an element of the original matrix which is unchanged,
203 *> vi denotes an element of the vector defining H(i), and ui an element
204 *> of the vector defining G(i).
205 *> \endverbatim
206 *>
207 * =====================================================================
208  SUBROUTINE slabrd( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y,
209  $ LDY )
210 *
211 * -- LAPACK auxiliary routine --
212 * -- LAPACK is a software package provided by Univ. of Tennessee, --
213 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
214 *
215 * .. Scalar Arguments ..
216  INTEGER LDA, LDX, LDY, M, N, NB
217 * ..
218 * .. Array Arguments ..
219  REAL A( LDA, * ), D( * ), E( * ), TAUP( * ),
220  $ tauq( * ), x( ldx, * ), y( ldy, * )
221 * ..
222 *
223 * =====================================================================
224 *
225 * .. Parameters ..
226  REAL ZERO, ONE
227  parameter( zero = 0.0e0, one = 1.0e0 )
228 * ..
229 * .. Local Scalars ..
230  INTEGER I
231 * ..
232 * .. External Subroutines ..
233  EXTERNAL sgemv, slarfg, sscal
234 * ..
235 * .. Intrinsic Functions ..
236  INTRINSIC min
237 * ..
238 * .. Executable Statements ..
239 *
240 * Quick return if possible
241 *
242  IF( m.LE.0 .OR. n.LE.0 )
243  $ RETURN
244 *
245  IF( m.GE.n ) THEN
246 *
247 * Reduce to upper bidiagonal form
248 *
249  DO 10 i = 1, nb
250 *
251 * Update A(i:m,i)
252 *
253  CALL sgemv( 'No transpose', m-i+1, i-1, -one, a( i, 1 ),
254  $ lda, y( i, 1 ), ldy, one, a( i, i ), 1 )
255  CALL sgemv( 'No transpose', m-i+1, i-1, -one, x( i, 1 ),
256  $ ldx, a( 1, i ), 1, one, a( i, i ), 1 )
257 *
258 * Generate reflection Q(i) to annihilate A(i+1:m,i)
259 *
260  CALL slarfg( m-i+1, a( i, i ), a( min( i+1, m ), i ), 1,
261  $ tauq( i ) )
262  d( i ) = a( i, i )
263  IF( i.LT.n ) THEN
264  a( i, i ) = one
265 *
266 * Compute Y(i+1:n,i)
267 *
268  CALL sgemv( 'Transpose', m-i+1, n-i, one, a( i, i+1 ),
269  $ lda, a( i, i ), 1, zero, y( i+1, i ), 1 )
270  CALL sgemv( 'Transpose', m-i+1, i-1, one, a( i, 1 ), lda,
271  $ a( i, i ), 1, zero, y( 1, i ), 1 )
272  CALL sgemv( 'No transpose', n-i, i-1, -one, y( i+1, 1 ),
273  $ ldy, y( 1, i ), 1, one, y( i+1, i ), 1 )
274  CALL sgemv( 'Transpose', m-i+1, i-1, one, x( i, 1 ), ldx,
275  $ a( i, i ), 1, zero, y( 1, i ), 1 )
276  CALL sgemv( 'Transpose', i-1, n-i, -one, a( 1, i+1 ),
277  $ lda, y( 1, i ), 1, one, y( i+1, i ), 1 )
278  CALL sscal( n-i, tauq( i ), y( i+1, i ), 1 )
279 *
280 * Update A(i,i+1:n)
281 *
282  CALL sgemv( 'No transpose', n-i, i, -one, y( i+1, 1 ),
283  $ ldy, a( i, 1 ), lda, one, a( i, i+1 ), lda )
284  CALL sgemv( 'Transpose', i-1, n-i, -one, a( 1, i+1 ),
285  $ lda, x( i, 1 ), ldx, one, a( i, i+1 ), lda )
286 *
287 * Generate reflection P(i) to annihilate A(i,i+2:n)
288 *
289  CALL slarfg( n-i, a( i, i+1 ), a( i, min( i+2, n ) ),
290  $ lda, taup( i ) )
291  e( i ) = a( i, i+1 )
292  a( i, i+1 ) = one
293 *
294 * Compute X(i+1:m,i)
295 *
296  CALL sgemv( 'No transpose', m-i, n-i, one, a( i+1, i+1 ),
297  $ lda, a( i, i+1 ), lda, zero, x( i+1, i ), 1 )
298  CALL sgemv( 'Transpose', n-i, i, one, y( i+1, 1 ), ldy,
299  $ a( i, i+1 ), lda, zero, x( 1, i ), 1 )
300  CALL sgemv( 'No transpose', m-i, i, -one, a( i+1, 1 ),
301  $ lda, x( 1, i ), 1, one, x( i+1, i ), 1 )
302  CALL sgemv( 'No transpose', i-1, n-i, one, a( 1, i+1 ),
303  $ lda, a( i, i+1 ), lda, zero, x( 1, i ), 1 )
304  CALL sgemv( 'No transpose', m-i, i-1, -one, x( i+1, 1 ),
305  $ ldx, x( 1, i ), 1, one, x( i+1, i ), 1 )
306  CALL sscal( m-i, taup( i ), x( i+1, i ), 1 )
307  END IF
308  10 CONTINUE
309  ELSE
310 *
311 * Reduce to lower bidiagonal form
312 *
313  DO 20 i = 1, nb
314 *
315 * Update A(i,i:n)
316 *
317  CALL sgemv( 'No transpose', n-i+1, i-1, -one, y( i, 1 ),
318  $ ldy, a( i, 1 ), lda, one, a( i, i ), lda )
319  CALL sgemv( 'Transpose', i-1, n-i+1, -one, a( 1, i ), lda,
320  $ x( i, 1 ), ldx, one, a( i, i ), lda )
321 *
322 * Generate reflection P(i) to annihilate A(i,i+1:n)
323 *
324  CALL slarfg( n-i+1, a( i, i ), a( i, min( i+1, n ) ), lda,
325  $ taup( i ) )
326  d( i ) = a( i, i )
327  IF( i.LT.m ) THEN
328  a( i, i ) = one
329 *
330 * Compute X(i+1:m,i)
331 *
332  CALL sgemv( 'No transpose', m-i, n-i+1, one, a( i+1, i ),
333  $ lda, a( i, i ), lda, zero, x( i+1, i ), 1 )
334  CALL sgemv( 'Transpose', n-i+1, i-1, one, y( i, 1 ), ldy,
335  $ a( i, i ), lda, zero, x( 1, i ), 1 )
336  CALL sgemv( 'No transpose', m-i, i-1, -one, a( i+1, 1 ),
337  $ lda, x( 1, i ), 1, one, x( i+1, i ), 1 )
338  CALL sgemv( 'No transpose', i-1, n-i+1, one, a( 1, i ),
339  $ lda, a( i, i ), lda, zero, x( 1, i ), 1 )
340  CALL sgemv( 'No transpose', m-i, i-1, -one, x( i+1, 1 ),
341  $ ldx, x( 1, i ), 1, one, x( i+1, i ), 1 )
342  CALL sscal( m-i, taup( i ), x( i+1, i ), 1 )
343 *
344 * Update A(i+1:m,i)
345 *
346  CALL sgemv( 'No transpose', m-i, i-1, -one, a( i+1, 1 ),
347  $ lda, y( i, 1 ), ldy, one, a( i+1, i ), 1 )
348  CALL sgemv( 'No transpose', m-i, i, -one, x( i+1, 1 ),
349  $ ldx, a( 1, i ), 1, one, a( i+1, i ), 1 )
350 *
351 * Generate reflection Q(i) to annihilate A(i+2:m,i)
352 *
353  CALL slarfg( m-i, a( i+1, i ), a( min( i+2, m ), i ), 1,
354  $ tauq( i ) )
355  e( i ) = a( i+1, i )
356  a( i+1, i ) = one
357 *
358 * Compute Y(i+1:n,i)
359 *
360  CALL sgemv( 'Transpose', m-i, n-i, one, a( i+1, i+1 ),
361  $ lda, a( i+1, i ), 1, zero, y( i+1, i ), 1 )
362  CALL sgemv( 'Transpose', m-i, i-1, one, a( i+1, 1 ), lda,
363  $ a( i+1, i ), 1, zero, y( 1, i ), 1 )
364  CALL sgemv( 'No transpose', n-i, i-1, -one, y( i+1, 1 ),
365  $ ldy, y( 1, i ), 1, one, y( i+1, i ), 1 )
366  CALL sgemv( 'Transpose', m-i, i, one, x( i+1, 1 ), ldx,
367  $ a( i+1, i ), 1, zero, y( 1, i ), 1 )
368  CALL sgemv( 'Transpose', i, n-i, -one, a( 1, i+1 ), lda,
369  $ y( 1, i ), 1, one, y( i+1, i ), 1 )
370  CALL sscal( n-i, tauq( i ), y( i+1, i ), 1 )
371  END IF
372  20 CONTINUE
373  END IF
374  RETURN
375 *
376 * End of SLABRD
377 *
378  END
subroutine slarfg(N, ALPHA, X, INCX, TAU)
SLARFG generates an elementary reflector (Householder matrix).
Definition: slarfg.f:106
subroutine slabrd(M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY)
SLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.
Definition: slabrd.f:210
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:79
subroutine sgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SGEMV
Definition: sgemv.f:156