|
1
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 |
SUBROUTINE ZLAQPS( M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1,
$ VN2, AUXV, F, LDF ) * * -- LAPACK auxiliary routine (version 3.3.1) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * -- April 2011 -- * * .. Scalar Arguments .. INTEGER KB, LDA, LDF, M, N, NB, OFFSET * .. * .. Array Arguments .. INTEGER JPVT( * ) DOUBLE PRECISION VN1( * ), VN2( * ) COMPLEX*16 A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * ) * .. * * Purpose * ======= * * ZLAQPS computes a step of QR factorization with column pivoting * of a complex M-by-N matrix A by using Blas-3. It tries to factorize * NB columns from A starting from the row OFFSET+1, and updates all * of the matrix with Blas-3 xGEMM. * * In some cases, due to catastrophic cancellations, it cannot * factorize NB columns. Hence, the actual number of factorized * columns is returned in KB. * * Block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized. * * Arguments * ========= * * M (input) INTEGER * The number of rows of the matrix A. M >= 0. * * N (input) INTEGER * The number of columns of the matrix A. N >= 0 * * OFFSET (input) INTEGER * The number of rows of A that have been factorized in * previous steps. * * NB (input) INTEGER * The number of columns to factorize. * * KB (output) INTEGER * The number of columns actually factorized. * * A (input/output) COMPLEX*16 array, dimension (LDA,N) * On entry, the M-by-N matrix A. * On exit, block A(OFFSET+1:M,1:KB) is the triangular * factor obtained and block A(1:OFFSET,1:N) has been * accordingly pivoted, but no factorized. * The rest of the matrix, block A(OFFSET+1:M,KB+1:N) has * been updated. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,M). * * JPVT (input/output) INTEGER array, dimension (N) * JPVT(I) = K <==> Column K of the full matrix A has been * permuted into position I in AP. * * TAU (output) COMPLEX*16 array, dimension (KB) * The scalar factors of the elementary reflectors. * * VN1 (input/output) DOUBLE PRECISION array, dimension (N) * The vector with the partial column norms. * * VN2 (input/output) DOUBLE PRECISION array, dimension (N) * The vector with the exact column norms. * * AUXV (input/output) COMPLEX*16 array, dimension (NB) * Auxiliar vector. * * F (input/output) COMPLEX*16 array, dimension (LDF,NB) * Matrix F**H = L * Y**H * A. * * LDF (input) INTEGER * The leading dimension of the array F. LDF >= max(1,N). * * Further Details * =============== * * Based on contributions by * G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain * X. Sun, Computer Science Dept., Duke University, USA * * Partial column norm updating strategy modified by * Z. Drmac and Z. Bujanovic, Dept. of Mathematics, * University of Zagreb, Croatia. * -- April 2011 -- * For more details see LAPACK Working Note 176. * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE COMPLEX*16 CZERO, CONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, $ CZERO = ( 0.0D+0, 0.0D+0 ), $ CONE = ( 1.0D+0, 0.0D+0 ) ) * .. * .. Local Scalars .. INTEGER ITEMP, J, K, LASTRK, LSTICC, PVT, RK DOUBLE PRECISION TEMP, TEMP2, TOL3Z COMPLEX*16 AKK * .. * .. External Subroutines .. EXTERNAL ZGEMM, ZGEMV, ZLARFG, ZSWAP * .. * .. Intrinsic Functions .. INTRINSIC ABS, DBLE, DCONJG, MAX, MIN, NINT, SQRT * .. * .. External Functions .. INTEGER IDAMAX DOUBLE PRECISION DLAMCH, DZNRM2 EXTERNAL IDAMAX, DLAMCH, DZNRM2 * .. * .. Executable Statements .. * LASTRK = MIN( M, N+OFFSET ) LSTICC = 0 K = 0 TOL3Z = SQRT(DLAMCH('Epsilon')) * * Beginning of while loop. * 10 CONTINUE IF( ( K.LT.NB ) .AND. ( LSTICC.EQ.0 ) ) THEN K = K + 1 RK = OFFSET + K * * Determine ith pivot column and swap if necessary * PVT = ( K-1 ) + IDAMAX( N-K+1, VN1( K ), 1 ) IF( PVT.NE.K ) THEN CALL ZSWAP( M, A( 1, PVT ), 1, A( 1, K ), 1 ) CALL ZSWAP( K-1, F( PVT, 1 ), LDF, F( K, 1 ), LDF ) ITEMP = JPVT( PVT ) JPVT( PVT ) = JPVT( K ) JPVT( K ) = ITEMP VN1( PVT ) = VN1( K ) VN2( PVT ) = VN2( K ) END IF * * Apply previous Householder reflectors to column K: * A(RK:M,K) := A(RK:M,K) - A(RK:M,1:K-1)*F(K,1:K-1)**H. * IF( K.GT.1 ) THEN DO 20 J = 1, K - 1 F( K, J ) = DCONJG( F( K, J ) ) 20 CONTINUE CALL ZGEMV( 'No transpose', M-RK+1, K-1, -CONE, A( RK, 1 ), $ LDA, F( K, 1 ), LDF, CONE, A( RK, K ), 1 ) DO 30 J = 1, K - 1 F( K, J ) = DCONJG( F( K, J ) ) 30 CONTINUE END IF * * Generate elementary reflector H(k). * IF( RK.LT.M ) THEN CALL ZLARFG( M-RK+1, A( RK, K ), A( RK+1, K ), 1, TAU( K ) ) ELSE CALL ZLARFG( 1, A( RK, K ), A( RK, K ), 1, TAU( K ) ) END IF * AKK = A( RK, K ) A( RK, K ) = CONE * * Compute Kth column of F: * * Compute F(K+1:N,K) := tau(K)*A(RK:M,K+1:N)**H*A(RK:M,K). * IF( K.LT.N ) THEN CALL ZGEMV( 'Conjugate transpose', M-RK+1, N-K, TAU( K ), $ A( RK, K+1 ), LDA, A( RK, K ), 1, CZERO, $ F( K+1, K ), 1 ) END IF * * Padding F(1:K,K) with zeros. * DO 40 J = 1, K F( J, K ) = CZERO 40 CONTINUE * * Incremental updating of F: * F(1:N,K) := F(1:N,K) - tau(K)*F(1:N,1:K-1)*A(RK:M,1:K-1)**H * *A(RK:M,K). * IF( K.GT.1 ) THEN CALL ZGEMV( 'Conjugate transpose', M-RK+1, K-1, -TAU( K ), $ A( RK, 1 ), LDA, A( RK, K ), 1, CZERO, $ AUXV( 1 ), 1 ) * CALL ZGEMV( 'No transpose', N, K-1, CONE, F( 1, 1 ), LDF, $ AUXV( 1 ), 1, CONE, F( 1, K ), 1 ) END IF * * Update the current row of A: * A(RK,K+1:N) := A(RK,K+1:N) - A(RK,1:K)*F(K+1:N,1:K)**H. * IF( K.LT.N ) THEN CALL ZGEMM( 'No transpose', 'Conjugate transpose', 1, N-K, $ K, -CONE, A( RK, 1 ), LDA, F( K+1, 1 ), LDF, $ CONE, A( RK, K+1 ), LDA ) END IF * * Update partial column norms. * IF( RK.LT.LASTRK ) THEN DO 50 J = K + 1, N IF( VN1( J ).NE.ZERO ) THEN * * NOTE: The following 4 lines follow from the analysis in * Lapack Working Note 176. * TEMP = ABS( A( RK, J ) ) / VN1( J ) TEMP = MAX( ZERO, ( ONE+TEMP )*( ONE-TEMP ) ) TEMP2 = TEMP*( VN1( J ) / VN2( J ) )**2 IF( TEMP2 .LE. TOL3Z ) THEN VN2( J ) = DBLE( LSTICC ) LSTICC = J ELSE VN1( J ) = VN1( J )*SQRT( TEMP ) END IF END IF 50 CONTINUE END IF * A( RK, K ) = AKK * * End of while loop. * GO TO 10 END IF KB = K RK = OFFSET + KB * * Apply the block reflector to the rest of the matrix: * A(OFFSET+KB+1:M,KB+1:N) := A(OFFSET+KB+1:M,KB+1:N) - * A(OFFSET+KB+1:M,1:KB)*F(KB+1:N,1:KB)**H. * IF( KB.LT.MIN( N, M-OFFSET ) ) THEN CALL ZGEMM( 'No transpose', 'Conjugate transpose', M-RK, N-KB, $ KB, -CONE, A( RK+1, 1 ), LDA, F( KB+1, 1 ), LDF, $ CONE, A( RK+1, KB+1 ), LDA ) END IF * * Recomputation of difficult columns. * 60 CONTINUE IF( LSTICC.GT.0 ) THEN ITEMP = NINT( VN2( LSTICC ) ) VN1( LSTICC ) = DZNRM2( M-RK, A( RK+1, LSTICC ), 1 ) * * NOTE: The computation of VN1( LSTICC ) relies on the fact that * SNRM2 does not fail on vectors with norm below the value of * SQRT(DLAMCH('S')) * VN2( LSTICC ) = VN1( LSTICC ) LSTICC = ITEMP GO TO 60 END IF * RETURN * * End of ZLAQPS * END |