1: *> \brief \b DGETSQRHRT
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DGETSQRHRT + dependencies
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11: *> [TGZ]</a>
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13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgetsqrhrt.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DGETSQRHRT( M, N, MB1, NB1, NB2, A, LDA, T, LDT, WORK,
22: * $ LWORK, INFO )
23: * IMPLICIT NONE
24: *
25: * .. Scalar Arguments ..
26: * INTEGER INFO, LDA, LDT, LWORK, M, N, NB1, NB2, MB1
27: * ..
28: * .. Array Arguments ..
29: * DOUBLE PRECISION A( LDA, * ), T( LDT, * ), WORK( * )
30: * ..
31: *
32: *
33: *> \par Purpose:
34: * =============
35: *>
36: *> \verbatim
37: *>
38: *> DGETSQRHRT computes a NB2-sized column blocked QR-factorization
39: *> of a real M-by-N matrix A with M >= N,
40: *>
41: *> A = Q * R.
42: *>
43: *> The routine uses internally a NB1-sized column blocked and MB1-sized
44: *> row blocked TSQR-factorization and perfors the reconstruction
45: *> of the Householder vectors from the TSQR output. The routine also
46: *> converts the R_tsqr factor from the TSQR-factorization output into
47: *> the R factor that corresponds to the Householder QR-factorization,
48: *>
49: *> A = Q_tsqr * R_tsqr = Q * R.
50: *>
51: *> The output Q and R factors are stored in the same format as in DGEQRT
52: *> (Q is in blocked compact WY-representation). See the documentation
53: *> of DGEQRT for more details on the format.
54: *> \endverbatim
55: *
56: * Arguments:
57: * ==========
58: *
59: *> \param[in] M
60: *> \verbatim
61: *> M is INTEGER
62: *> The number of rows of the matrix A. M >= 0.
63: *> \endverbatim
64: *>
65: *> \param[in] N
66: *> \verbatim
67: *> N is INTEGER
68: *> The number of columns of the matrix A. M >= N >= 0.
69: *> \endverbatim
70: *>
71: *> \param[in] MB1
72: *> \verbatim
73: *> MB1 is INTEGER
74: *> The row block size to be used in the blocked TSQR.
75: *> MB1 > N.
76: *> \endverbatim
77: *>
78: *> \param[in] NB1
79: *> \verbatim
80: *> NB1 is INTEGER
81: *> The column block size to be used in the blocked TSQR.
82: *> N >= NB1 >= 1.
83: *> \endverbatim
84: *>
85: *> \param[in] NB2
86: *> \verbatim
87: *> NB2 is INTEGER
88: *> The block size to be used in the blocked QR that is
89: *> output. NB2 >= 1.
90: *> \endverbatim
91: *>
92: *> \param[in,out] A
93: *> \verbatim
94: *> A is DOUBLE PRECISION array, dimension (LDA,N)
95: *>
96: *> On entry: an M-by-N matrix A.
97: *>
98: *> On exit:
99: *> a) the elements on and above the diagonal
100: *> of the array contain the N-by-N upper-triangular
101: *> matrix R corresponding to the Householder QR;
102: *> b) the elements below the diagonal represent Q by
103: *> the columns of blocked V (compact WY-representation).
104: *> \endverbatim
105: *>
106: *> \param[in] LDA
107: *> \verbatim
108: *> LDA is INTEGER
109: *> The leading dimension of the array A. LDA >= max(1,M).
110: *> \endverbatim
111: *>
112: *> \param[out] T
113: *> \verbatim
114: *> T is DOUBLE PRECISION array, dimension (LDT,N))
115: *> The upper triangular block reflectors stored in compact form
116: *> as a sequence of upper triangular blocks.
117: *> \endverbatim
118: *>
119: *> \param[in] LDT
120: *> \verbatim
121: *> LDT is INTEGER
122: *> The leading dimension of the array T. LDT >= NB2.
123: *> \endverbatim
124: *>
125: *> \param[out] WORK
126: *> \verbatim
127: *> (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
128: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
129: *> \endverbatim
130: *>
131: *> \param[in] LWORK
132: *> \verbatim
133: *> The dimension of the array WORK.
134: *> LWORK >= MAX( LWT + LW1, MAX( LWT+N*N+LW2, LWT+N*N+N ) ),
135: *> where
136: *> NUM_ALL_ROW_BLOCKS = CEIL((M-N)/(MB1-N)),
137: *> NB1LOCAL = MIN(NB1,N).
138: *> LWT = NUM_ALL_ROW_BLOCKS * N * NB1LOCAL,
139: *> LW1 = NB1LOCAL * N,
140: *> LW2 = NB1LOCAL * MAX( NB1LOCAL, ( N - NB1LOCAL ) ),
141: *> If LWORK = -1, then a workspace query is assumed.
142: *> The routine only calculates the optimal size of the WORK
143: *> array, returns this value as the first entry of the WORK
144: *> array, and no error message related to LWORK is issued
145: *> by XERBLA.
146: *> \endverbatim
147: *>
148: *> \param[out] INFO
149: *> \verbatim
150: *> INFO is INTEGER
151: *> = 0: successful exit
152: *> < 0: if INFO = -i, the i-th argument had an illegal value
153: *> \endverbatim
154: *
155: * Authors:
156: * ========
157: *
158: *> \author Univ. of Tennessee
159: *> \author Univ. of California Berkeley
160: *> \author Univ. of Colorado Denver
161: *> \author NAG Ltd.
162: *
163: *> \ingroup doubleOTHERcomputational
164: *
165: *> \par Contributors:
166: * ==================
167: *>
168: *> \verbatim
169: *>
170: *> November 2020, Igor Kozachenko,
171: *> Computer Science Division,
172: *> University of California, Berkeley
173: *>
174: *> \endverbatim
175: *>
176: * =====================================================================
177: SUBROUTINE DGETSQRHRT( M, N, MB1, NB1, NB2, A, LDA, T, LDT, WORK,
178: $ LWORK, INFO )
179: IMPLICIT NONE
180: *
181: * -- LAPACK computational routine --
182: * -- LAPACK is a software package provided by Univ. of Tennessee, --
183: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
184: *
185: * .. Scalar Arguments ..
186: INTEGER INFO, LDA, LDT, LWORK, M, N, NB1, NB2, MB1
187: * ..
188: * .. Array Arguments ..
189: DOUBLE PRECISION A( LDA, * ), T( LDT, * ), WORK( * )
190: * ..
191: *
192: * =====================================================================
193: *
194: * .. Parameters ..
195: DOUBLE PRECISION ONE
196: PARAMETER ( ONE = 1.0D+0 )
197: * ..
198: * .. Local Scalars ..
199: LOGICAL LQUERY
200: INTEGER I, IINFO, J, LW1, LW2, LWT, LDWT, LWORKOPT,
201: $ NB1LOCAL, NB2LOCAL, NUM_ALL_ROW_BLOCKS
202: * ..
203: * .. External Subroutines ..
204: EXTERNAL DCOPY, DLATSQR, DORGTSQR_ROW, DORHR_COL,
205: $ XERBLA
206: * ..
207: * .. Intrinsic Functions ..
208: INTRINSIC CEILING, DBLE, MAX, MIN
209: * ..
210: * .. Executable Statements ..
211: *
212: * Test the input arguments
213: *
214: INFO = 0
215: LQUERY = LWORK.EQ.-1
216: IF( M.LT.0 ) THEN
217: INFO = -1
218: ELSE IF( N.LT.0 .OR. M.LT.N ) THEN
219: INFO = -2
220: ELSE IF( MB1.LE.N ) THEN
221: INFO = -3
222: ELSE IF( NB1.LT.1 ) THEN
223: INFO = -4
224: ELSE IF( NB2.LT.1 ) THEN
225: INFO = -5
226: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
227: INFO = -7
228: ELSE IF( LDT.LT.MAX( 1, MIN( NB2, N ) ) ) THEN
229: INFO = -9
230: ELSE
231: *
232: * Test the input LWORK for the dimension of the array WORK.
233: * This workspace is used to store array:
234: * a) Matrix T and WORK for DLATSQR;
235: * b) N-by-N upper-triangular factor R_tsqr;
236: * c) Matrix T and array WORK for DORGTSQR_ROW;
237: * d) Diagonal D for DORHR_COL.
238: *
239: IF( LWORK.LT.N*N+1 .AND. .NOT.LQUERY ) THEN
240: INFO = -11
241: ELSE
242: *
243: * Set block size for column blocks
244: *
245: NB1LOCAL = MIN( NB1, N )
246: *
247: NUM_ALL_ROW_BLOCKS = MAX( 1,
248: $ CEILING( DBLE( M - N ) / DBLE( MB1 - N ) ) )
249: *
250: * Length and leading dimension of WORK array to place
251: * T array in TSQR.
252: *
253: LWT = NUM_ALL_ROW_BLOCKS * N * NB1LOCAL
254:
255: LDWT = NB1LOCAL
256: *
257: * Length of TSQR work array
258: *
259: LW1 = NB1LOCAL * N
260: *
261: * Length of DORGTSQR_ROW work array.
262: *
263: LW2 = NB1LOCAL * MAX( NB1LOCAL, ( N - NB1LOCAL ) )
264: *
265: LWORKOPT = MAX( LWT + LW1, MAX( LWT+N*N+LW2, LWT+N*N+N ) )
266: *
267: IF( ( LWORK.LT.MAX( 1, LWORKOPT ) ).AND.(.NOT.LQUERY) ) THEN
268: INFO = -11
269: END IF
270: *
271: END IF
272: END IF
273: *
274: * Handle error in the input parameters and return workspace query.
275: *
276: IF( INFO.NE.0 ) THEN
277: CALL XERBLA( 'DGETSQRHRT', -INFO )
278: RETURN
279: ELSE IF ( LQUERY ) THEN
280: WORK( 1 ) = DBLE( LWORKOPT )
281: RETURN
282: END IF
283: *
284: * Quick return if possible
285: *
286: IF( MIN( M, N ).EQ.0 ) THEN
287: WORK( 1 ) = DBLE( LWORKOPT )
288: RETURN
289: END IF
290: *
291: NB2LOCAL = MIN( NB2, N )
292: *
293: *
294: * (1) Perform TSQR-factorization of the M-by-N matrix A.
295: *
296: CALL DLATSQR( M, N, MB1, NB1LOCAL, A, LDA, WORK, LDWT,
297: $ WORK(LWT+1), LW1, IINFO )
298: *
299: * (2) Copy the factor R_tsqr stored in the upper-triangular part
300: * of A into the square matrix in the work array
301: * WORK(LWT+1:LWT+N*N) column-by-column.
302: *
303: DO J = 1, N
304: CALL DCOPY( J, A( 1, J ), 1, WORK( LWT + N*(J-1)+1 ), 1 )
305: END DO
306: *
307: * (3) Generate a M-by-N matrix Q with orthonormal columns from
308: * the result stored below the diagonal in the array A in place.
309: *
310:
311: CALL DORGTSQR_ROW( M, N, MB1, NB1LOCAL, A, LDA, WORK, LDWT,
312: $ WORK( LWT+N*N+1 ), LW2, IINFO )
313: *
314: * (4) Perform the reconstruction of Householder vectors from
315: * the matrix Q (stored in A) in place.
316: *
317: CALL DORHR_COL( M, N, NB2LOCAL, A, LDA, T, LDT,
318: $ WORK( LWT+N*N+1 ), IINFO )
319: *
320: * (5) Copy the factor R_tsqr stored in the square matrix in the
321: * work array WORK(LWT+1:LWT+N*N) into the upper-triangular
322: * part of A.
323: *
324: * (6) Compute from R_tsqr the factor R_hr corresponding to
325: * the reconstructed Householder vectors, i.e. R_hr = S * R_tsqr.
326: * This multiplication by the sign matrix S on the left means
327: * changing the sign of I-th row of the matrix R_tsqr according
328: * to sign of the I-th diagonal element DIAG(I) of the matrix S.
329: * DIAG is stored in WORK( LWT+N*N+1 ) from the DORHR_COL output.
330: *
331: * (5) and (6) can be combined in a single loop, so the rows in A
332: * are accessed only once.
333: *
334: DO I = 1, N
335: IF( WORK( LWT+N*N+I ).EQ.-ONE ) THEN
336: DO J = I, N
337: A( I, J ) = -ONE * WORK( LWT+N*(J-1)+I )
338: END DO
339: ELSE
340: CALL DCOPY( N-I+1, WORK(LWT+N*(I-1)+I), N, A( I, I ), LDA )
341: END IF
342: END DO
343: *
344: WORK( 1 ) = DBLE( LWORKOPT )
345: RETURN
346: *
347: * End of DGETSQRHRT
348: *
349: END
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