1: *> \brief \b ZLAQPS computes a step of QR factorization with column pivoting of a real m-by-n matrix A by using BLAS level 3.
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZLAQPS + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlaqps.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlaqps.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlaqps.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZLAQPS( M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1,
22: * VN2, AUXV, F, LDF )
23: *
24: * .. Scalar Arguments ..
25: * INTEGER KB, LDA, LDF, M, N, NB, OFFSET
26: * ..
27: * .. Array Arguments ..
28: * INTEGER JPVT( * )
29: * DOUBLE PRECISION VN1( * ), VN2( * )
30: * COMPLEX*16 A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * )
31: * ..
32: *
33: *
34: *> \par Purpose:
35: * =============
36: *>
37: *> \verbatim
38: *>
39: *> ZLAQPS computes a step of QR factorization with column pivoting
40: *> of a complex M-by-N matrix A by using Blas-3. It tries to factorize
41: *> NB columns from A starting from the row OFFSET+1, and updates all
42: *> of the matrix with Blas-3 xGEMM.
43: *>
44: *> In some cases, due to catastrophic cancellations, it cannot
45: *> factorize NB columns. Hence, the actual number of factorized
46: *> columns is returned in KB.
47: *>
48: *> Block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.
49: *> \endverbatim
50: *
51: * Arguments:
52: * ==========
53: *
54: *> \param[in] M
55: *> \verbatim
56: *> M is INTEGER
57: *> The number of rows of the matrix A. M >= 0.
58: *> \endverbatim
59: *>
60: *> \param[in] N
61: *> \verbatim
62: *> N is INTEGER
63: *> The number of columns of the matrix A. N >= 0
64: *> \endverbatim
65: *>
66: *> \param[in] OFFSET
67: *> \verbatim
68: *> OFFSET is INTEGER
69: *> The number of rows of A that have been factorized in
70: *> previous steps.
71: *> \endverbatim
72: *>
73: *> \param[in] NB
74: *> \verbatim
75: *> NB is INTEGER
76: *> The number of columns to factorize.
77: *> \endverbatim
78: *>
79: *> \param[out] KB
80: *> \verbatim
81: *> KB is INTEGER
82: *> The number of columns actually factorized.
83: *> \endverbatim
84: *>
85: *> \param[in,out] A
86: *> \verbatim
87: *> A is COMPLEX*16 array, dimension (LDA,N)
88: *> On entry, the M-by-N matrix A.
89: *> On exit, block A(OFFSET+1:M,1:KB) is the triangular
90: *> factor obtained and block A(1:OFFSET,1:N) has been
91: *> accordingly pivoted, but no factorized.
92: *> The rest of the matrix, block A(OFFSET+1:M,KB+1:N) has
93: *> been updated.
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[in,out] JPVT
103: *> \verbatim
104: *> JPVT is INTEGER array, dimension (N)
105: *> JPVT(I) = K <==> Column K of the full matrix A has been
106: *> permuted into position I in AP.
107: *> \endverbatim
108: *>
109: *> \param[out] TAU
110: *> \verbatim
111: *> TAU is COMPLEX*16 array, dimension (KB)
112: *> The scalar factors of the elementary reflectors.
113: *> \endverbatim
114: *>
115: *> \param[in,out] VN1
116: *> \verbatim
117: *> VN1 is DOUBLE PRECISION array, dimension (N)
118: *> The vector with the partial column norms.
119: *> \endverbatim
120: *>
121: *> \param[in,out] VN2
122: *> \verbatim
123: *> VN2 is DOUBLE PRECISION array, dimension (N)
124: *> The vector with the exact column norms.
125: *> \endverbatim
126: *>
127: *> \param[in,out] AUXV
128: *> \verbatim
129: *> AUXV is COMPLEX*16 array, dimension (NB)
130: *> Auxiliary vector.
131: *> \endverbatim
132: *>
133: *> \param[in,out] F
134: *> \verbatim
135: *> F is COMPLEX*16 array, dimension (LDF,NB)
136: *> Matrix F**H = L * Y**H * A.
137: *> \endverbatim
138: *>
139: *> \param[in] LDF
140: *> \verbatim
141: *> LDF is INTEGER
142: *> The leading dimension of the array F. LDF >= max(1,N).
143: *> \endverbatim
144: *
145: * Authors:
146: * ========
147: *
148: *> \author Univ. of Tennessee
149: *> \author Univ. of California Berkeley
150: *> \author Univ. of Colorado Denver
151: *> \author NAG Ltd.
152: *
153: *> \ingroup complex16OTHERauxiliary
154: *
155: *> \par Contributors:
156: * ==================
157: *>
158: *> G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
159: *> X. Sun, Computer Science Dept., Duke University, USA
160: *> \n
161: *> Partial column norm updating strategy modified on April 2011
162: *> Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
163: *> University of Zagreb, Croatia.
164: *
165: *> \par References:
166: * ================
167: *>
168: *> LAPACK Working Note 176
169: *
170: *> \htmlonly
171: *> <a href="http://www.netlib.org/lapack/lawnspdf/lawn176.pdf">[PDF]</a>
172: *> \endhtmlonly
173: *
174: * =====================================================================
175: SUBROUTINE ZLAQPS( M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1,
176: $ VN2, AUXV, F, LDF )
177: *
178: * -- LAPACK auxiliary routine --
179: * -- LAPACK is a software package provided by Univ. of Tennessee, --
180: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
181: *
182: * .. Scalar Arguments ..
183: INTEGER KB, LDA, LDF, M, N, NB, OFFSET
184: * ..
185: * .. Array Arguments ..
186: INTEGER JPVT( * )
187: DOUBLE PRECISION VN1( * ), VN2( * )
188: COMPLEX*16 A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * )
189: * ..
190: *
191: * =====================================================================
192: *
193: * .. Parameters ..
194: DOUBLE PRECISION ZERO, ONE
195: COMPLEX*16 CZERO, CONE
196: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0,
197: $ CZERO = ( 0.0D+0, 0.0D+0 ),
198: $ CONE = ( 1.0D+0, 0.0D+0 ) )
199: * ..
200: * .. Local Scalars ..
201: INTEGER ITEMP, J, K, LASTRK, LSTICC, PVT, RK
202: DOUBLE PRECISION TEMP, TEMP2, TOL3Z
203: COMPLEX*16 AKK
204: * ..
205: * .. External Subroutines ..
206: EXTERNAL ZGEMM, ZGEMV, ZLARFG, ZSWAP
207: * ..
208: * .. Intrinsic Functions ..
209: INTRINSIC ABS, DBLE, DCONJG, MAX, MIN, NINT, SQRT
210: * ..
211: * .. External Functions ..
212: INTEGER IDAMAX
213: DOUBLE PRECISION DLAMCH, DZNRM2
214: EXTERNAL IDAMAX, DLAMCH, DZNRM2
215: * ..
216: * .. Executable Statements ..
217: *
218: LASTRK = MIN( M, N+OFFSET )
219: LSTICC = 0
220: K = 0
221: TOL3Z = SQRT(DLAMCH('Epsilon'))
222: *
223: * Beginning of while loop.
224: *
225: 10 CONTINUE
226: IF( ( K.LT.NB ) .AND. ( LSTICC.EQ.0 ) ) THEN
227: K = K + 1
228: RK = OFFSET + K
229: *
230: * Determine ith pivot column and swap if necessary
231: *
232: PVT = ( K-1 ) + IDAMAX( N-K+1, VN1( K ), 1 )
233: IF( PVT.NE.K ) THEN
234: CALL ZSWAP( M, A( 1, PVT ), 1, A( 1, K ), 1 )
235: CALL ZSWAP( K-1, F( PVT, 1 ), LDF, F( K, 1 ), LDF )
236: ITEMP = JPVT( PVT )
237: JPVT( PVT ) = JPVT( K )
238: JPVT( K ) = ITEMP
239: VN1( PVT ) = VN1( K )
240: VN2( PVT ) = VN2( K )
241: END IF
242: *
243: * Apply previous Householder reflectors to column K:
244: * A(RK:M,K) := A(RK:M,K) - A(RK:M,1:K-1)*F(K,1:K-1)**H.
245: *
246: IF( K.GT.1 ) THEN
247: DO 20 J = 1, K - 1
248: F( K, J ) = DCONJG( F( K, J ) )
249: 20 CONTINUE
250: CALL ZGEMV( 'No transpose', M-RK+1, K-1, -CONE, A( RK, 1 ),
251: $ LDA, F( K, 1 ), LDF, CONE, A( RK, K ), 1 )
252: DO 30 J = 1, K - 1
253: F( K, J ) = DCONJG( F( K, J ) )
254: 30 CONTINUE
255: END IF
256: *
257: * Generate elementary reflector H(k).
258: *
259: IF( RK.LT.M ) THEN
260: CALL ZLARFG( M-RK+1, A( RK, K ), A( RK+1, K ), 1, TAU( K ) )
261: ELSE
262: CALL ZLARFG( 1, A( RK, K ), A( RK, K ), 1, TAU( K ) )
263: END IF
264: *
265: AKK = A( RK, K )
266: A( RK, K ) = CONE
267: *
268: * Compute Kth column of F:
269: *
270: * Compute F(K+1:N,K) := tau(K)*A(RK:M,K+1:N)**H*A(RK:M,K).
271: *
272: IF( K.LT.N ) THEN
273: CALL ZGEMV( 'Conjugate transpose', M-RK+1, N-K, TAU( K ),
274: $ A( RK, K+1 ), LDA, A( RK, K ), 1, CZERO,
275: $ F( K+1, K ), 1 )
276: END IF
277: *
278: * Padding F(1:K,K) with zeros.
279: *
280: DO 40 J = 1, K
281: F( J, K ) = CZERO
282: 40 CONTINUE
283: *
284: * Incremental updating of F:
285: * F(1:N,K) := F(1:N,K) - tau(K)*F(1:N,1:K-1)*A(RK:M,1:K-1)**H
286: * *A(RK:M,K).
287: *
288: IF( K.GT.1 ) THEN
289: CALL ZGEMV( 'Conjugate transpose', M-RK+1, K-1, -TAU( K ),
290: $ A( RK, 1 ), LDA, A( RK, K ), 1, CZERO,
291: $ AUXV( 1 ), 1 )
292: *
293: CALL ZGEMV( 'No transpose', N, K-1, CONE, F( 1, 1 ), LDF,
294: $ AUXV( 1 ), 1, CONE, F( 1, K ), 1 )
295: END IF
296: *
297: * Update the current row of A:
298: * A(RK,K+1:N) := A(RK,K+1:N) - A(RK,1:K)*F(K+1:N,1:K)**H.
299: *
300: IF( K.LT.N ) THEN
301: CALL ZGEMM( 'No transpose', 'Conjugate transpose', 1, N-K,
302: $ K, -CONE, A( RK, 1 ), LDA, F( K+1, 1 ), LDF,
303: $ CONE, A( RK, K+1 ), LDA )
304: END IF
305: *
306: * Update partial column norms.
307: *
308: IF( RK.LT.LASTRK ) THEN
309: DO 50 J = K + 1, N
310: IF( VN1( J ).NE.ZERO ) THEN
311: *
312: * NOTE: The following 4 lines follow from the analysis in
313: * Lapack Working Note 176.
314: *
315: TEMP = ABS( A( RK, J ) ) / VN1( J )
316: TEMP = MAX( ZERO, ( ONE+TEMP )*( ONE-TEMP ) )
317: TEMP2 = TEMP*( VN1( J ) / VN2( J ) )**2
318: IF( TEMP2 .LE. TOL3Z ) THEN
319: VN2( J ) = DBLE( LSTICC )
320: LSTICC = J
321: ELSE
322: VN1( J ) = VN1( J )*SQRT( TEMP )
323: END IF
324: END IF
325: 50 CONTINUE
326: END IF
327: *
328: A( RK, K ) = AKK
329: *
330: * End of while loop.
331: *
332: GO TO 10
333: END IF
334: KB = K
335: RK = OFFSET + KB
336: *
337: * Apply the block reflector to the rest of the matrix:
338: * A(OFFSET+KB+1:M,KB+1:N) := A(OFFSET+KB+1:M,KB+1:N) -
339: * A(OFFSET+KB+1:M,1:KB)*F(KB+1:N,1:KB)**H.
340: *
341: IF( KB.LT.MIN( N, M-OFFSET ) ) THEN
342: CALL ZGEMM( 'No transpose', 'Conjugate transpose', M-RK, N-KB,
343: $ KB, -CONE, A( RK+1, 1 ), LDA, F( KB+1, 1 ), LDF,
344: $ CONE, A( RK+1, KB+1 ), LDA )
345: END IF
346: *
347: * Recomputation of difficult columns.
348: *
349: 60 CONTINUE
350: IF( LSTICC.GT.0 ) THEN
351: ITEMP = NINT( VN2( LSTICC ) )
352: VN1( LSTICC ) = DZNRM2( M-RK, A( RK+1, LSTICC ), 1 )
353: *
354: * NOTE: The computation of VN1( LSTICC ) relies on the fact that
355: * SNRM2 does not fail on vectors with norm below the value of
356: * SQRT(DLAMCH('S'))
357: *
358: VN2( LSTICC ) = VN1( LSTICC )
359: LSTICC = ITEMP
360: GO TO 60
361: END IF
362: *
363: RETURN
364: *
365: * End of ZLAQPS
366: *
367: END
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