001:       SUBROUTINE SLAQPS( M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1,
002:      $                   VN2, AUXV, F, LDF )
003: *
004: *  -- LAPACK auxiliary routine (version 3.2) --
005: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
006: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
007: *     November 2006
008: *
009: *     .. Scalar Arguments ..
010:       INTEGER            KB, LDA, LDF, M, N, NB, OFFSET
011: *     ..
012: *     .. Array Arguments ..
013:       INTEGER            JPVT( * )
014:       REAL               A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * ),
015:      $                   VN1( * ), VN2( * )
016: *     ..
017: *
018: *  Purpose
019: *  =======
020: *
021: *  SLAQPS computes a step of QR factorization with column pivoting
022: *  of a real M-by-N matrix A by using Blas-3.  It tries to factorize
023: *  NB columns from A starting from the row OFFSET+1, and updates all
024: *  of the matrix with Blas-3 xGEMM.
025: *
026: *  In some cases, due to catastrophic cancellations, it cannot
027: *  factorize NB columns.  Hence, the actual number of factorized
028: *  columns is returned in KB.
029: *
030: *  Block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.
031: *
032: *  Arguments
033: *  =========
034: *
035: *  M       (input) INTEGER
036: *          The number of rows of the matrix A. M >= 0.
037: *
038: *  N       (input) INTEGER
039: *          The number of columns of the matrix A. N >= 0
040: *
041: *  OFFSET  (input) INTEGER
042: *          The number of rows of A that have been factorized in
043: *          previous steps.
044: *
045: *  NB      (input) INTEGER
046: *          The number of columns to factorize.
047: *
048: *  KB      (output) INTEGER
049: *          The number of columns actually factorized.
050: *
051: *  A       (input/output) REAL array, dimension (LDA,N)
052: *          On entry, the M-by-N matrix A.
053: *          On exit, block A(OFFSET+1:M,1:KB) is the triangular
054: *          factor obtained and block A(1:OFFSET,1:N) has been
055: *          accordingly pivoted, but no factorized.
056: *          The rest of the matrix, block A(OFFSET+1:M,KB+1:N) has
057: *          been updated.
058: *
059: *  LDA     (input) INTEGER
060: *          The leading dimension of the array A. LDA >= max(1,M).
061: *
062: *  JPVT    (input/output) INTEGER array, dimension (N)
063: *          JPVT(I) = K <==> Column K of the full matrix A has been
064: *          permuted into position I in AP.
065: *
066: *  TAU     (output) REAL array, dimension (KB)
067: *          The scalar factors of the elementary reflectors.
068: *
069: *  VN1     (input/output) REAL array, dimension (N)
070: *          The vector with the partial column norms.
071: *
072: *  VN2     (input/output) REAL array, dimension (N)
073: *          The vector with the exact column norms.
074: *
075: *  AUXV    (input/output) REAL array, dimension (NB)
076: *          Auxiliar vector.
077: *
078: *  F       (input/output) REAL array, dimension (LDF,NB)
079: *          Matrix F' = L*Y'*A.
080: *
081: *  LDF     (input) INTEGER
082: *          The leading dimension of the array F. LDF >= max(1,N).
083: *
084: *  Further Details
085: *  ===============
086: *
087: *  Based on contributions by
088: *    G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
089: *    X. Sun, Computer Science Dept., Duke University, USA
090: *
091: *  Partial column norm updating strategy modified by
092: *    Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
093: *    University of Zagreb, Croatia.
094: *    June 2006.
095: *  For more details see LAPACK Working Note 176.
096: *  =====================================================================
097: *
098: *     .. Parameters ..
099:       REAL               ZERO, ONE
100:       PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
101: *     ..
102: *     .. Local Scalars ..
103:       INTEGER            ITEMP, J, K, LASTRK, LSTICC, PVT, RK
104:       REAL               AKK, TEMP, TEMP2, TOL3Z
105: *     ..
106: *     .. External Subroutines ..
107:       EXTERNAL           SGEMM, SGEMV, SLARFP, SSWAP
108: *     ..
109: *     .. Intrinsic Functions ..
110:       INTRINSIC          ABS, MAX, MIN, NINT, REAL, SQRT
111: *     ..
112: *     .. External Functions ..
113:       INTEGER            ISAMAX
114:       REAL               SLAMCH, SNRM2
115:       EXTERNAL           ISAMAX, SLAMCH, SNRM2
116: *     ..
117: *     .. Executable Statements ..
118: *
119:       LASTRK = MIN( M, N+OFFSET )
120:       LSTICC = 0
121:       K = 0
122:       TOL3Z = SQRT(SLAMCH('Epsilon'))
123: *
124: *     Beginning of while loop.
125: *
126:    10 CONTINUE
127:       IF( ( K.LT.NB ) .AND. ( LSTICC.EQ.0 ) ) THEN
128:          K = K + 1
129:          RK = OFFSET + K
130: *
131: *        Determine ith pivot column and swap if necessary
132: *
133:          PVT = ( K-1 ) + ISAMAX( N-K+1, VN1( K ), 1 )
134:          IF( PVT.NE.K ) THEN
135:             CALL SSWAP( M, A( 1, PVT ), 1, A( 1, K ), 1 )
136:             CALL SSWAP( K-1, F( PVT, 1 ), LDF, F( K, 1 ), LDF )
137:             ITEMP = JPVT( PVT )
138:             JPVT( PVT ) = JPVT( K )
139:             JPVT( K ) = ITEMP
140:             VN1( PVT ) = VN1( K )
141:             VN2( PVT ) = VN2( K )
142:          END IF
143: *
144: *        Apply previous Householder reflectors to column K:
145: *        A(RK:M,K) := A(RK:M,K) - A(RK:M,1:K-1)*F(K,1:K-1)'.
146: *
147:          IF( K.GT.1 ) THEN
148:             CALL SGEMV( 'No transpose', M-RK+1, K-1, -ONE, A( RK, 1 ),
149:      $                  LDA, F( K, 1 ), LDF, ONE, A( RK, K ), 1 )
150:          END IF
151: *
152: *        Generate elementary reflector H(k).
153: *
154:          IF( RK.LT.M ) THEN
155:             CALL SLARFP( M-RK+1, A( RK, K ), A( RK+1, K ), 1, TAU( K ) )
156:          ELSE
157:             CALL SLARFP( 1, A( RK, K ), A( RK, K ), 1, TAU( K ) )
158:          END IF
159: *
160:          AKK = A( RK, K )
161:          A( RK, K ) = ONE
162: *
163: *        Compute Kth column of F:
164: *
165: *        Compute  F(K+1:N,K) := tau(K)*A(RK:M,K+1:N)'*A(RK:M,K).
166: *
167:          IF( K.LT.N ) THEN
168:             CALL SGEMV( 'Transpose', M-RK+1, N-K, TAU( K ),
169:      $                  A( RK, K+1 ), LDA, A( RK, K ), 1, ZERO,
170:      $                  F( K+1, K ), 1 )
171:          END IF
172: *
173: *        Padding F(1:K,K) with zeros.
174: *
175:          DO 20 J = 1, K
176:             F( J, K ) = ZERO
177:    20    CONTINUE
178: *
179: *        Incremental updating of F:
180: *        F(1:N,K) := F(1:N,K) - tau(K)*F(1:N,1:K-1)*A(RK:M,1:K-1)'
181: *                    *A(RK:M,K).
182: *
183:          IF( K.GT.1 ) THEN
184:             CALL SGEMV( 'Transpose', M-RK+1, K-1, -TAU( K ), A( RK, 1 ),
185:      $                  LDA, A( RK, K ), 1, ZERO, AUXV( 1 ), 1 )
186: *
187:             CALL SGEMV( 'No transpose', N, K-1, ONE, F( 1, 1 ), LDF,
188:      $                  AUXV( 1 ), 1, ONE, F( 1, K ), 1 )
189:          END IF
190: *
191: *        Update the current row of A:
192: *        A(RK,K+1:N) := A(RK,K+1:N) - A(RK,1:K)*F(K+1:N,1:K)'.
193: *
194:          IF( K.LT.N ) THEN
195:             CALL SGEMV( 'No transpose', N-K, K, -ONE, F( K+1, 1 ), LDF,
196:      $                  A( RK, 1 ), LDA, ONE, A( RK, K+1 ), LDA )
197:          END IF
198: *
199: *        Update partial column norms.
200: *
201:          IF( RK.LT.LASTRK ) THEN
202:             DO 30 J = K + 1, N
203:                IF( VN1( J ).NE.ZERO ) THEN
204: *
205: *                 NOTE: The following 4 lines follow from the analysis in
206: *                 Lapack Working Note 176.
207: *
208:                   TEMP = ABS( A( RK, J ) ) / VN1( J )
209:                   TEMP = MAX( ZERO, ( ONE+TEMP )*( ONE-TEMP ) )
210:                   TEMP2 = TEMP*( VN1( J ) / VN2( J ) )**2
211:                   IF( TEMP2 .LE. TOL3Z ) THEN
212:                      VN2( J ) = REAL( LSTICC )
213:                      LSTICC = J
214:                   ELSE
215:                      VN1( J ) = VN1( J )*SQRT( TEMP )
216:                   END IF
217:                END IF
218:    30       CONTINUE
219:          END IF
220: *
221:          A( RK, K ) = AKK
222: *
223: *        End of while loop.
224: *
225:          GO TO 10
226:       END IF
227:       KB = K
228:       RK = OFFSET + KB
229: *
230: *     Apply the block reflector to the rest of the matrix:
231: *     A(OFFSET+KB+1:M,KB+1:N) := A(OFFSET+KB+1:M,KB+1:N) -
232: *                         A(OFFSET+KB+1:M,1:KB)*F(KB+1:N,1:KB)'.
233: *
234:       IF( KB.LT.MIN( N, M-OFFSET ) ) THEN
235:          CALL SGEMM( 'No transpose', 'Transpose', M-RK, N-KB, KB, -ONE,
236:      $               A( RK+1, 1 ), LDA, F( KB+1, 1 ), LDF, ONE,
237:      $               A( RK+1, KB+1 ), LDA )
238:       END IF
239: *
240: *     Recomputation of difficult columns.
241: *
242:    40 CONTINUE
243:       IF( LSTICC.GT.0 ) THEN
244:          ITEMP = NINT( VN2( LSTICC ) )
245:          VN1( LSTICC ) = SNRM2( M-RK, A( RK+1, LSTICC ), 1 )
246: *
247: *        NOTE: The computation of VN1( LSTICC ) relies on the fact that 
248: *        SNRM2 does not fail on vectors with norm below the value of
249: *        SQRT(DLAMCH('S')) 
250: *
251:          VN2( LSTICC ) = VN1( LSTICC )
252:          LSTICC = ITEMP
253:          GO TO 40
254:       END IF
255: *
256:       RETURN
257: *
258: *     End of SLAQPS
259: *
260:       END
261: