LAPACK  3.6.0
LAPACK: Linear Algebra PACKage
slagge.f
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1 *> \brief \b SLAGGE
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 * Definition:
9 * ===========
10 *
11 * SUBROUTINE SLAGGE( M, N, KL, KU, D, A, LDA, ISEED, WORK, INFO )
12 *
13 * .. Scalar Arguments ..
14 * INTEGER INFO, KL, KU, LDA, M, N
15 * ..
16 * .. Array Arguments ..
17 * INTEGER ISEED( 4 )
18 * REAL A( LDA, * ), D( * ), WORK( * )
19 * ..
20 *
21 *
22 *> \par Purpose:
23 * =============
24 *>
25 *> \verbatim
26 *>
27 *> SLAGGE generates a real general m by n matrix A, by pre- and post-
28 *> multiplying a real diagonal matrix D with random orthogonal matrices:
29 *> A = U*D*V. The lower and upper bandwidths may then be reduced to
30 *> kl and ku by additional orthogonal transformations.
31 *> \endverbatim
32 *
33 * Arguments:
34 * ==========
35 *
36 *> \param[in] M
37 *> \verbatim
38 *> M is INTEGER
39 *> The number of rows of the matrix A. M >= 0.
40 *> \endverbatim
41 *>
42 *> \param[in] N
43 *> \verbatim
44 *> N is INTEGER
45 *> The number of columns of the matrix A. N >= 0.
46 *> \endverbatim
47 *>
48 *> \param[in] KL
49 *> \verbatim
50 *> KL is INTEGER
51 *> The number of nonzero subdiagonals within the band of A.
52 *> 0 <= KL <= M-1.
53 *> \endverbatim
54 *>
55 *> \param[in] KU
56 *> \verbatim
57 *> KU is INTEGER
58 *> The number of nonzero superdiagonals within the band of A.
59 *> 0 <= KU <= N-1.
60 *> \endverbatim
61 *>
62 *> \param[in] D
63 *> \verbatim
64 *> D is REAL array, dimension (min(M,N))
65 *> The diagonal elements of the diagonal matrix D.
66 *> \endverbatim
67 *>
68 *> \param[out] A
69 *> \verbatim
70 *> A is REAL array, dimension (LDA,N)
71 *> The generated m by n matrix A.
72 *> \endverbatim
73 *>
74 *> \param[in] LDA
75 *> \verbatim
76 *> LDA is INTEGER
77 *> The leading dimension of the array A. LDA >= M.
78 *> \endverbatim
79 *>
80 *> \param[in,out] ISEED
81 *> \verbatim
82 *> ISEED is INTEGER array, dimension (4)
83 *> On entry, the seed of the random number generator; the array
84 *> elements must be between 0 and 4095, and ISEED(4) must be
85 *> odd.
86 *> On exit, the seed is updated.
87 *> \endverbatim
88 *>
89 *> \param[out] WORK
90 *> \verbatim
91 *> WORK is REAL array, dimension (M+N)
92 *> \endverbatim
93 *>
94 *> \param[out] INFO
95 *> \verbatim
96 *> INFO is INTEGER
97 *> = 0: successful exit
98 *> < 0: if INFO = -i, the i-th argument had an illegal value
99 *> \endverbatim
100 *
101 * Authors:
102 * ========
103 *
104 *> \author Univ. of Tennessee
105 *> \author Univ. of California Berkeley
106 *> \author Univ. of Colorado Denver
107 *> \author NAG Ltd.
108 *
109 *> \date November 2015
110 *
111 *> \ingroup real_matgen
112 *
113 * =====================================================================
114  SUBROUTINE slagge( M, N, KL, KU, D, A, LDA, ISEED, WORK, INFO )
115 *
116 * -- LAPACK auxiliary routine (version 3.6.0) --
117 * -- LAPACK is a software package provided by Univ. of Tennessee, --
118 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
119 * November 2015
120 *
121 * .. Scalar Arguments ..
122  INTEGER INFO, KL, KU, LDA, M, N
123 * ..
124 * .. Array Arguments ..
125  INTEGER ISEED( 4 )
126  REAL A( lda, * ), D( * ), WORK( * )
127 * ..
128 *
129 * =====================================================================
130 *
131 * .. Parameters ..
132  REAL ZERO, ONE
133  parameter( zero = 0.0e+0, one = 1.0e+0 )
134 * ..
135 * .. Local Scalars ..
136  INTEGER I, J
137  REAL TAU, WA, WB, WN
138 * ..
139 * .. External Subroutines ..
140  EXTERNAL sgemv, sger, slarnv, sscal, xerbla
141 * ..
142 * .. Intrinsic Functions ..
143  INTRINSIC max, min, sign
144 * ..
145 * .. External Functions ..
146  REAL SNRM2
147  EXTERNAL snrm2
148 * ..
149 * .. Executable Statements ..
150 *
151 * Test the input arguments
152 *
153  info = 0
154  IF( m.LT.0 ) THEN
155  info = -1
156  ELSE IF( n.LT.0 ) THEN
157  info = -2
158  ELSE IF( kl.LT.0 .OR. kl.GT.m-1 ) THEN
159  info = -3
160  ELSE IF( ku.LT.0 .OR. ku.GT.n-1 ) THEN
161  info = -4
162  ELSE IF( lda.LT.max( 1, m ) ) THEN
163  info = -7
164  END IF
165  IF( info.LT.0 ) THEN
166  CALL xerbla( 'SLAGGE', -info )
167  RETURN
168  END IF
169 *
170 * initialize A to diagonal matrix
171 *
172  DO 20 j = 1, n
173  DO 10 i = 1, m
174  a( i, j ) = zero
175  10 CONTINUE
176  20 CONTINUE
177  DO 30 i = 1, min( m, n )
178  a( i, i ) = d( i )
179  30 CONTINUE
180 *
181 * Quick exit if the user wants a diagonal matrix
182 *
183  IF(( kl .EQ. 0 ).AND.( ku .EQ. 0)) RETURN
184 *
185 * pre- and post-multiply A by random orthogonal matrices
186 *
187  DO 40 i = min( m, n ), 1, -1
188  IF( i.LT.m ) THEN
189 *
190 * generate random reflection
191 *
192  CALL slarnv( 3, iseed, m-i+1, work )
193  wn = snrm2( m-i+1, work, 1 )
194  wa = sign( wn, work( 1 ) )
195  IF( wn.EQ.zero ) THEN
196  tau = zero
197  ELSE
198  wb = work( 1 ) + wa
199  CALL sscal( m-i, one / wb, work( 2 ), 1 )
200  work( 1 ) = one
201  tau = wb / wa
202  END IF
203 *
204 * multiply A(i:m,i:n) by random reflection from the left
205 *
206  CALL sgemv( 'Transpose', m-i+1, n-i+1, one, a( i, i ), lda,
207  $ work, 1, zero, work( m+1 ), 1 )
208  CALL sger( m-i+1, n-i+1, -tau, work, 1, work( m+1 ), 1,
209  $ a( i, i ), lda )
210  END IF
211  IF( i.LT.n ) THEN
212 *
213 * generate random reflection
214 *
215  CALL slarnv( 3, iseed, n-i+1, work )
216  wn = snrm2( n-i+1, work, 1 )
217  wa = sign( wn, work( 1 ) )
218  IF( wn.EQ.zero ) THEN
219  tau = zero
220  ELSE
221  wb = work( 1 ) + wa
222  CALL sscal( n-i, one / wb, work( 2 ), 1 )
223  work( 1 ) = one
224  tau = wb / wa
225  END IF
226 *
227 * multiply A(i:m,i:n) by random reflection from the right
228 *
229  CALL sgemv( 'No transpose', m-i+1, n-i+1, one, a( i, i ),
230  $ lda, work, 1, zero, work( n+1 ), 1 )
231  CALL sger( m-i+1, n-i+1, -tau, work( n+1 ), 1, work, 1,
232  $ a( i, i ), lda )
233  END IF
234  40 CONTINUE
235 *
236 * Reduce number of subdiagonals to KL and number of superdiagonals
237 * to KU
238 *
239  DO 70 i = 1, max( m-1-kl, n-1-ku )
240  IF( kl.LE.ku ) THEN
241 *
242 * annihilate subdiagonal elements first (necessary if KL = 0)
243 *
244  IF( i.LE.min( m-1-kl, n ) ) THEN
245 *
246 * generate reflection to annihilate A(kl+i+1:m,i)
247 *
248  wn = snrm2( m-kl-i+1, a( kl+i, i ), 1 )
249  wa = sign( wn, a( kl+i, i ) )
250  IF( wn.EQ.zero ) THEN
251  tau = zero
252  ELSE
253  wb = a( kl+i, i ) + wa
254  CALL sscal( m-kl-i, one / wb, a( kl+i+1, i ), 1 )
255  a( kl+i, i ) = one
256  tau = wb / wa
257  END IF
258 *
259 * apply reflection to A(kl+i:m,i+1:n) from the left
260 *
261  CALL sgemv( 'Transpose', m-kl-i+1, n-i, one,
262  $ a( kl+i, i+1 ), lda, a( kl+i, i ), 1, zero,
263  $ work, 1 )
264  CALL sger( m-kl-i+1, n-i, -tau, a( kl+i, i ), 1, work, 1,
265  $ a( kl+i, i+1 ), lda )
266  a( kl+i, i ) = -wa
267  END IF
268 *
269  IF( i.LE.min( n-1-ku, m ) ) THEN
270 *
271 * generate reflection to annihilate A(i,ku+i+1:n)
272 *
273  wn = snrm2( n-ku-i+1, a( i, ku+i ), lda )
274  wa = sign( wn, a( i, ku+i ) )
275  IF( wn.EQ.zero ) THEN
276  tau = zero
277  ELSE
278  wb = a( i, ku+i ) + wa
279  CALL sscal( n-ku-i, one / wb, a( i, ku+i+1 ), lda )
280  a( i, ku+i ) = one
281  tau = wb / wa
282  END IF
283 *
284 * apply reflection to A(i+1:m,ku+i:n) from the right
285 *
286  CALL sgemv( 'No transpose', m-i, n-ku-i+1, one,
287  $ a( i+1, ku+i ), lda, a( i, ku+i ), lda, zero,
288  $ work, 1 )
289  CALL sger( m-i, n-ku-i+1, -tau, work, 1, a( i, ku+i ),
290  $ lda, a( i+1, ku+i ), lda )
291  a( i, ku+i ) = -wa
292  END IF
293  ELSE
294 *
295 * annihilate superdiagonal elements first (necessary if
296 * KU = 0)
297 *
298  IF( i.LE.min( n-1-ku, m ) ) THEN
299 *
300 * generate reflection to annihilate A(i,ku+i+1:n)
301 *
302  wn = snrm2( n-ku-i+1, a( i, ku+i ), lda )
303  wa = sign( wn, a( i, ku+i ) )
304  IF( wn.EQ.zero ) THEN
305  tau = zero
306  ELSE
307  wb = a( i, ku+i ) + wa
308  CALL sscal( n-ku-i, one / wb, a( i, ku+i+1 ), lda )
309  a( i, ku+i ) = one
310  tau = wb / wa
311  END IF
312 *
313 * apply reflection to A(i+1:m,ku+i:n) from the right
314 *
315  CALL sgemv( 'No transpose', m-i, n-ku-i+1, one,
316  $ a( i+1, ku+i ), lda, a( i, ku+i ), lda, zero,
317  $ work, 1 )
318  CALL sger( m-i, n-ku-i+1, -tau, work, 1, a( i, ku+i ),
319  $ lda, a( i+1, ku+i ), lda )
320  a( i, ku+i ) = -wa
321  END IF
322 *
323  IF( i.LE.min( m-1-kl, n ) ) THEN
324 *
325 * generate reflection to annihilate A(kl+i+1:m,i)
326 *
327  wn = snrm2( m-kl-i+1, a( kl+i, i ), 1 )
328  wa = sign( wn, a( kl+i, i ) )
329  IF( wn.EQ.zero ) THEN
330  tau = zero
331  ELSE
332  wb = a( kl+i, i ) + wa
333  CALL sscal( m-kl-i, one / wb, a( kl+i+1, i ), 1 )
334  a( kl+i, i ) = one
335  tau = wb / wa
336  END IF
337 *
338 * apply reflection to A(kl+i:m,i+1:n) from the left
339 *
340  CALL sgemv( 'Transpose', m-kl-i+1, n-i, one,
341  $ a( kl+i, i+1 ), lda, a( kl+i, i ), 1, zero,
342  $ work, 1 )
343  CALL sger( m-kl-i+1, n-i, -tau, a( kl+i, i ), 1, work, 1,
344  $ a( kl+i, i+1 ), lda )
345  a( kl+i, i ) = -wa
346  END IF
347  END IF
348 *
349  IF (i .LE. n) THEN
350  DO 50 j = kl + i + 1, m
351  a( j, i ) = zero
352  50 CONTINUE
353  END IF
354 *
355  IF (i .LE. m) THEN
356  DO 60 j = ku + i + 1, n
357  a( i, j ) = zero
358  60 CONTINUE
359  END IF
360  70 CONTINUE
361  RETURN
362 *
363 * End of SLAGGE
364 *
365  END
subroutine sgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SGEMV
Definition: sgemv.f:158
subroutine sger(M, N, ALPHA, X, INCX, Y, INCY, A, LDA)
SGER
Definition: sger.f:132
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine slagge(M, N, KL, KU, D, A, LDA, ISEED, WORK, INFO)
SLAGGE
Definition: slagge.f:115
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:55
subroutine slarnv(IDIST, ISEED, N, X)
SLARNV returns a vector of random numbers from a uniform or normal distribution.
Definition: slarnv.f:99