LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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dorhr_col.f
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1*> \brief \b DORHR_COL
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
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13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorhr_col.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE DORHR_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* DOUBLE PRECISION A( LDA, * ), D( * ), T( LDT, * )
28* ..
29*
30*> \par Purpose:
31* =============
32*>
33*> \verbatim
34*>
35*> DORHR_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 DGEQRT).
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 DOUBLE PRECISION 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*> DGEQRT). 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 DOUBLE PRECISION 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 DGEQRT).
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 DOUBLE PRECISION 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 unhr_col
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 dorhr_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 DOUBLE PRECISION A( LDA, * ), D( * ), T( LDT, * )
270* ..
271*
272* =====================================================================
273*
274* .. Parameters ..
275 DOUBLE PRECISION ONE, ZERO
276 parameter( one = 1.0d+0, zero = 0.0d+0 )
277* ..
278* .. Local Scalars ..
279 INTEGER I, IINFO, J, JB, JBTEMP1, JBTEMP2, JNB,
280 \$ NPLUSONE
281* ..
282* .. External Subroutines ..
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( 'DORHR_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 dlaorhr_col_getrfnp( n, n, a, lda, d, iinfo )
333*
334* (1-2) Solve for V2.
335*
336 IF( m.GT.n ) THEN
337 CALL dtrsm( '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 dcopy( 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 dscal( 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 DTRSM 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 DTRSM.
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 dtrsm( '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 DORHR_COL
437*
438 END
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine dcopy(n, dx, incx, dy, incy)
DCOPY
Definition dcopy.f:82
subroutine dlaorhr_col_getrfnp(m, n, a, lda, d, info)
DLAORHR_COL_GETRFNP
subroutine dscal(n, da, dx, incx)
DSCAL
Definition dscal.f:79
subroutine dtrsm(side, uplo, transa, diag, m, n, alpha, a, lda, b, ldb)
DTRSM
Definition dtrsm.f:181
subroutine dorhr_col(m, n, nb, a, lda, t, ldt, d, info)
DORHR_COL
Definition dorhr_col.f:259