LAPACK  3.10.0 LAPACK: Linear Algebra PACKage
sorhr_col.f
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1 *> \brief \b SORHR_COL
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
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15 *> [TXT]</a>
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SORHR_COL( M, N, NB, A, LDA, T, LDT, D, INFO )
22 *
23 * .. Scalar Arguments ..
24 * INTEGER INFO, LDA, LDT, M, N, NB
25 * ..
26 * .. Array Arguments ..
27 * REAL A( LDA, * ), D( * ), T( LDT, * )
28 * ..
29 *
30 *> \par Purpose:
31 * =============
32 *>
33 *> \verbatim
34 *>
35 *> SORHR_COL takes an M-by-N real matrix Q_in with orthonormal columns
36 *> as input, stored in A, and performs Householder Reconstruction (HR),
37 *> i.e. reconstructs Householder vectors V(i) implicitly representing
38 *> another M-by-N matrix Q_out, with the property that Q_in = Q_out*S,
39 *> where S is an N-by-N diagonal matrix with diagonal entries
40 *> equal to +1 or -1. The Householder vectors (columns V(i) of V) are
41 *> stored in A on output, and the diagonal entries of S are stored in D.
42 *> Block reflectors are also returned in T
43 *> (same output format as SGEQRT).
44 *> \endverbatim
45 *
46 * Arguments:
47 * ==========
48 *
49 *> \param[in] M
50 *> \verbatim
51 *> M is INTEGER
52 *> The number of rows of the matrix A. M >= 0.
53 *> \endverbatim
54 *>
55 *> \param[in] N
56 *> \verbatim
57 *> N is INTEGER
58 *> The number of columns of the matrix A. M >= N >= 0.
59 *> \endverbatim
60 *>
61 *> \param[in] NB
62 *> \verbatim
63 *> NB is INTEGER
64 *> The column block size to be used in the reconstruction
65 *> of Householder column vector blocks in the array A and
66 *> corresponding block reflectors in the array T. NB >= 1.
67 *> (Note that if NB > N, then N is used instead of NB
68 *> as the column block size.)
69 *> \endverbatim
70 *>
71 *> \param[in,out] A
72 *> \verbatim
73 *> A is REAL array, dimension (LDA,N)
74 *>
75 *> On entry:
76 *>
77 *> The array A contains an M-by-N orthonormal matrix Q_in,
78 *> i.e the columns of A are orthogonal unit vectors.
79 *>
80 *> On exit:
81 *>
82 *> The elements below the diagonal of A represent the unit
83 *> lower-trapezoidal matrix V of Householder column vectors
84 *> V(i). The unit diagonal entries of V are not stored
85 *> (same format as the output below the diagonal in A from
86 *> SGEQRT). The matrix T and the matrix V stored on output
87 *> in A implicitly define Q_out.
88 *>
89 *> The elements above the diagonal contain the factor U
90 *> of the "modified" LU-decomposition:
91 *> Q_in - ( S ) = V * U
92 *> ( 0 )
93 *> where 0 is a (M-N)-by-(M-N) zero matrix.
94 *> \endverbatim
95 *>
96 *> \param[in] LDA
97 *> \verbatim
98 *> LDA is INTEGER
99 *> The leading dimension of the array A. LDA >= max(1,M).
100 *> \endverbatim
101 *>
102 *> \param[out] T
103 *> \verbatim
104 *> T is REAL array,
105 *> dimension (LDT, N)
106 *>
107 *> Let NOCB = Number_of_output_col_blocks
108 *> = CEIL(N/NB)
109 *>
110 *> On exit, T(1:NB, 1:N) contains NOCB upper-triangular
111 *> block reflectors used to define Q_out stored in compact
112 *> form as a sequence of upper-triangular NB-by-NB column
113 *> blocks (same format as the output T in SGEQRT).
114 *> The matrix T and the matrix V stored on output in A
115 *> implicitly define Q_out. NOTE: The lower triangles
116 *> below the upper-triangular blocks will be filled with
117 *> zeros. See Further Details.
118 *> \endverbatim
119 *>
120 *> \param[in] LDT
121 *> \verbatim
122 *> LDT is INTEGER
123 *> The leading dimension of the array T.
124 *> LDT >= max(1,min(NB,N)).
125 *> \endverbatim
126 *>
127 *> \param[out] D
128 *> \verbatim
129 *> D is REAL array, dimension min(M,N).
130 *> The elements can be only plus or minus one.
131 *>
132 *> D(i) is constructed as D(i) = -SIGN(Q_in_i(i,i)), where
133 *> 1 <= i <= min(M,N), and Q_in_i is Q_in after performing
134 *> i-1 steps of “modified” Gaussian elimination.
135 *> See Further Details.
136 *> \endverbatim
137 *>
138 *> \param[out] INFO
139 *> \verbatim
140 *> INFO is INTEGER
141 *> = 0: successful exit
142 *> < 0: if INFO = -i, the i-th argument had an illegal value
143 *> \endverbatim
144 *>
145 *> \par Further Details:
146 * =====================
147 *>
148 *> \verbatim
149 *>
150 *> The computed M-by-M orthogonal factor Q_out is defined implicitly as
151 *> a product of orthogonal matrices Q_out(i). Each Q_out(i) is stored in
152 *> the compact WY-representation format in the corresponding blocks of
153 *> matrices V (stored in A) and T.
154 *>
155 *> The M-by-N unit lower-trapezoidal matrix V stored in the M-by-N
156 *> matrix A contains the column vectors V(i) in NB-size column
157 *> blocks VB(j). For example, VB(1) contains the columns
158 *> V(1), V(2), ... V(NB). NOTE: The unit entries on
159 *> the diagonal of Y are not stored in A.
160 *>
161 *> The number of column blocks is
162 *>
163 *> NOCB = Number_of_output_col_blocks = CEIL(N/NB)
164 *>
165 *> where each block is of order NB except for the last block, which
166 *> is of order LAST_NB = N - (NOCB-1)*NB.
167 *>
168 *> For example, if M=6, N=5 and NB=2, the matrix V is
169 *>
170 *>
171 *> V = ( VB(1), VB(2), VB(3) ) =
172 *>
173 *> = ( 1 )
174 *> ( v21 1 )
175 *> ( v31 v32 1 )
176 *> ( v41 v42 v43 1 )
177 *> ( v51 v52 v53 v54 1 )
178 *> ( v61 v62 v63 v54 v65 )
179 *>
180 *>
181 *> For each of the column blocks VB(i), an upper-triangular block
182 *> reflector TB(i) is computed. These blocks are stored as
183 *> a sequence of upper-triangular column blocks in the NB-by-N
184 *> matrix T. The size of each TB(i) block is NB-by-NB, except
185 *> for the last block, whose size is LAST_NB-by-LAST_NB.
186 *>
187 *> For example, if M=6, N=5 and NB=2, the matrix T is
188 *>
189 *> T = ( TB(1), TB(2), TB(3) ) =
190 *>
191 *> = ( t11 t12 t13 t14 t15 )
192 *> ( t22 t24 )
193 *>
194 *>
195 *> The M-by-M factor Q_out is given as a product of NOCB
196 *> orthogonal M-by-M matrices Q_out(i).
197 *>
198 *> Q_out = Q_out(1) * Q_out(2) * ... * Q_out(NOCB),
199 *>
200 *> where each matrix Q_out(i) is given by the WY-representation
201 *> using corresponding blocks from the matrices V and T:
202 *>
203 *> Q_out(i) = I - VB(i) * TB(i) * (VB(i))**T,
204 *>
205 *> where I is the identity matrix. Here is the formula with matrix
206 *> dimensions:
207 *>
208 *> Q(i){M-by-M} = I{M-by-M} -
209 *> VB(i){M-by-INB} * TB(i){INB-by-INB} * (VB(i))**T {INB-by-M},
210 *>
211 *> where INB = NB, except for the last block NOCB
212 *> for which INB=LAST_NB.
213 *>
214 *> =====
215 *> NOTE:
216 *> =====
217 *>
218 *> If Q_in is the result of doing a QR factorization
219 *> B = Q_in * R_in, then:
220 *>
221 *> B = (Q_out*S) * R_in = Q_out * (S * R_in) = Q_out * R_out.
222 *>
223 *> So if one wants to interpret Q_out as the result
224 *> of the QR factorization of B, then the corresponding R_out
225 *> should be equal to R_out = S * R_in, i.e. some rows of R_in
226 *> should be multiplied by -1.
227 *>
228 *> For the details of the algorithm, see [1].
229 *>
230 *> [1] "Reconstructing Householder vectors from tall-skinny QR",
231 *> G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
232 *> E. Solomonik, J. Parallel Distrib. Comput.,
233 *> vol. 85, pp. 3-31, 2015.
234 *> \endverbatim
235 *>
236 * Authors:
237 * ========
238 *
239 *> \author Univ. of Tennessee
240 *> \author Univ. of California Berkeley
241 *> \author Univ. of Colorado Denver
242 *> \author NAG Ltd.
243 *
244 *> \ingroup singleOTHERcomputational
245 *
246 *> \par Contributors:
247 * ==================
248 *>
249 *> \verbatim
250 *>
251 *> November 2019, Igor Kozachenko,
252 *> Computer Science Division,
253 *> University of California, Berkeley
254 *>
255 *> \endverbatim
256 *
257 * =====================================================================
258  SUBROUTINE sorhr_col( M, N, NB, A, LDA, T, LDT, D, INFO )
259  IMPLICIT NONE
260 *
261 * -- LAPACK computational routine --
262 * -- LAPACK is a software package provided by Univ. of Tennessee, --
263 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
264 *
265 * .. Scalar Arguments ..
266  INTEGER INFO, LDA, LDT, M, N, NB
267 * ..
268 * .. Array Arguments ..
269  REAL A( LDA, * ), D( * ), T( LDT, * )
270 * ..
271 *
272 * =====================================================================
273 *
274 * .. Parameters ..
275  REAL ONE, ZERO
276  parameter( one = 1.0e+0, zero = 0.0e+0 )
277 * ..
278 * .. Local Scalars ..
279  INTEGER I, IINFO, J, JB, JBTEMP1, JBTEMP2, JNB,
280  \$ NPLUSONE
281 * ..
282 * .. External Subroutines ..
283  EXTERNAL scopy, slaorhr_col_getrfnp, sscal, strsm,
284  \$ xerbla
285 * ..
286 * .. Intrinsic Functions ..
287  INTRINSIC max, min
288 * ..
289 * .. Executable Statements ..
290 *
291 * Test the input parameters
292 *
293  info = 0
294  IF( m.LT.0 ) THEN
295  info = -1
296  ELSE IF( n.LT.0 .OR. n.GT.m ) THEN
297  info = -2
298  ELSE IF( nb.LT.1 ) THEN
299  info = -3
300  ELSE IF( lda.LT.max( 1, m ) ) THEN
301  info = -5
302  ELSE IF( ldt.LT.max( 1, min( nb, n ) ) ) THEN
303  info = -7
304  END IF
305 *
306 * Handle error in the input parameters.
307 *
308  IF( info.NE.0 ) THEN
309  CALL xerbla( 'SORHR_COL', -info )
310  RETURN
311  END IF
312 *
313 * Quick return if possible
314 *
315  IF( min( m, n ).EQ.0 ) THEN
316  RETURN
317  END IF
318 *
319 * On input, the M-by-N matrix A contains the orthogonal
320 * M-by-N matrix Q_in.
321 *
322 * (1) Compute the unit lower-trapezoidal V (ones on the diagonal
323 * are not stored) by performing the "modified" LU-decomposition.
324 *
325 * Q_in - ( S ) = V * U = ( V1 ) * U,
326 * ( 0 ) ( V2 )
327 *
328 * where 0 is an (M-N)-by-N zero matrix.
329 *
330 * (1-1) Factor V1 and U.
331
332  CALL slaorhr_col_getrfnp( n, n, a, lda, d, iinfo )
333 *
334 * (1-2) Solve for V2.
335 *
336  IF( m.GT.n ) THEN
337  CALL strsm( 'R', 'U', 'N', 'N', m-n, n, one, a, lda,
338  \$ a( n+1, 1 ), lda )
339  END IF
340 *
341 * (2) Reconstruct the block reflector T stored in T(1:NB, 1:N)
342 * as a sequence of upper-triangular blocks with NB-size column
343 * blocking.
344 *
345 * Loop over the column blocks of size NB of the array A(1:M,1:N)
346 * and the array T(1:NB,1:N), JB is the column index of a column
347 * block, JNB is the column block size at each step JB.
348 *
349  nplusone = n + 1
350  DO jb = 1, n, nb
351 *
352 * (2-0) Determine the column block size JNB.
353 *
354  jnb = min( nplusone-jb, nb )
355 *
356 * (2-1) Copy the upper-triangular part of the current JNB-by-JNB
357 * diagonal block U(JB) (of the N-by-N matrix U) stored
358 * in A(JB:JB+JNB-1,JB:JB+JNB-1) into the upper-triangular part
359 * of the current JNB-by-JNB block T(1:JNB,JB:JB+JNB-1)
360 * column-by-column, total JNB*(JNB+1)/2 elements.
361 *
362  jbtemp1 = jb - 1
363  DO j = jb, jb+jnb-1
364  CALL scopy( j-jbtemp1, a( jb, j ), 1, t( 1, j ), 1 )
365  END DO
366 *
367 * (2-2) Perform on the upper-triangular part of the current
368 * JNB-by-JNB diagonal block U(JB) (of the N-by-N matrix U) stored
369 * in T(1:JNB,JB:JB+JNB-1) the following operation in place:
370 * (-1)*U(JB)*S(JB), i.e the result will be stored in the upper-
371 * triangular part of T(1:JNB,JB:JB+JNB-1). This multiplication
372 * of the JNB-by-JNB diagonal block U(JB) by the JNB-by-JNB
373 * diagonal block S(JB) of the N-by-N sign matrix S from the
374 * right means changing the sign of each J-th column of the block
375 * U(JB) according to the sign of the diagonal element of the block
376 * S(JB), i.e. S(J,J) that is stored in the array element D(J).
377 *
378  DO j = jb, jb+jnb-1
379  IF( d( j ).EQ.one ) THEN
380  CALL sscal( j-jbtemp1, -one, t( 1, j ), 1 )
381  END IF
382  END DO
383 *
384 * (2-3) Perform the triangular solve for the current block
385 * matrix X(JB):
386 *
387 * X(JB) * (A(JB)**T) = B(JB), where:
388 *
389 * A(JB)**T is a JNB-by-JNB unit upper-triangular
390 * coefficient block, and A(JB)=V1(JB), which
391 * is a JNB-by-JNB unit lower-triangular block
392 * stored in A(JB:JB+JNB-1,JB:JB+JNB-1).
393 * The N-by-N matrix V1 is the upper part
394 * of the M-by-N lower-trapezoidal matrix V
395 * stored in A(1:M,1:N);
396 *
397 * B(JB) is a JNB-by-JNB upper-triangular right-hand
398 * side block, B(JB) = (-1)*U(JB)*S(JB), and
399 * B(JB) is stored in T(1:JNB,JB:JB+JNB-1);
400 *
401 * X(JB) is a JNB-by-JNB upper-triangular solution
402 * block, X(JB) is the upper-triangular block
403 * reflector T(JB), and X(JB) is stored
404 * in T(1:JNB,JB:JB+JNB-1).
405 *
406 * In other words, we perform the triangular solve for the
407 * upper-triangular block T(JB):
408 *
409 * T(JB) * (V1(JB)**T) = (-1)*U(JB)*S(JB).
410 *
411 * Even though the blocks X(JB) and B(JB) are upper-
412 * triangular, the routine STRSM will access all JNB**2
413 * elements of the square T(1:JNB,JB:JB+JNB-1). Therefore,
414 * we need to set to zero the elements of the block
415 * T(1:JNB,JB:JB+JNB-1) below the diagonal before the call
416 * to STRSM.
417 *
418 * (2-3a) Set the elements to zero.
419 *
420  jbtemp2 = jb - 2
421  DO j = jb, jb+jnb-2
422  DO i = j-jbtemp2, nb
423  t( i, j ) = zero
424  END DO
425  END DO
426 *
427 * (2-3b) Perform the triangular solve.
428 *
429  CALL strsm( 'R', 'L', 'T', 'U', jnb, jnb, one,
430  \$ a( jb, jb ), lda, t( 1, jb ), ldt )
431 *
432  END DO
433 *
434  RETURN
435 *
436 * End of SORHR_COL
437 *
438  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine slaorhr_col_getrfnp(M, N, A, LDA, D, INFO)
SLAORHR_COL_GETRFNP
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:79
subroutine strsm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
STRSM
Definition: strsm.f:181
subroutine sorhr_col(M, N, NB, A, LDA, T, LDT, D, INFO)
SORHR_COL
Definition: sorhr_col.f:259