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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 |
SUBROUTINE DGEQRF ( M, N, A, LDA, TAU, WORK, LWORK, INFO )
* * -- LAPACK routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * March 2008 * * .. Scalar Arguments .. INTEGER INFO, LDA, LWORK, M, N * .. * .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) * .. * * Purpose * ======= * * DGEQRF computes a QR factorization of a real M-by-N matrix A: * A = Q * R. * * This is the left-looking Level 3 BLAS version of the algorithm. * * 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. * * A (input/output) DOUBLE PRECISION array, dimension (LDA,N) * On entry, the M-by-N matrix A. * On exit, the elements on and above the diagonal of the array * contain the min(M,N)-by-N upper trapezoidal matrix R (R is * upper triangular if m >= n); the elements below the diagonal, * with the array TAU, represent the orthogonal matrix Q as a * product of min(m,n) elementary reflectors (see Further * Details). * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,M). * * TAU (output) DOUBLE PRECISION array, dimension (min(M,N)) * The scalar factors of the elementary reflectors (see Further * Details). * * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. * * LWORK (input) INTEGER * * The dimension of the array WORK. The dimension can be divided into three parts. * * 1) The part for the triangular factor T. If the very last T is not bigger * than any of the rest, then this part is NB x ceiling(K/NB), otherwise, * NB x (K-NT), where K = min(M,N) and NT is the dimension of the very last T * * 2) The part for the very last T when T is bigger than any of the rest T. * The size of this part is NT x NT, where NT = K - ceiling ((K-NX)/NB) x NB, * where K = min(M,N), NX is calculated by * NX = MAX( 0, ILAENV( 3, 'DGEQRF', ' ', M, N, -1, -1 ) ) * * 3) The part for dlarfb is of size max((N-M)*K, (N-M)*NB, K*NB, NB*NB) * * So LWORK = part1 + part2 + part3 * * If LWORK = -1, then a workspace query is assumed; the routine * only calculates the optimal size of the WORK array, returns * this value as the first entry of the WORK array, and no error * message related to LWORK is issued by XERBLA. * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * * Further Details * =============== * * The matrix Q is represented as a product of elementary reflectors * * Q = H(1) H(2) . . . H(k), where k = min(m,n). * * Each H(i) has the form * * H(i) = I - tau * v * v' * * where tau is a real scalar, and v is a real vector with * v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), * and tau in TAU(i). * * ===================================================================== * * .. Local Scalars .. LOGICAL LQUERY INTEGER I, IB, IINFO, IWS, J, K, LWKOPT, NB, $ NBMIN, NX, LBWORK, NT, LLWORK * .. * .. External Subroutines .. EXTERNAL DGEQR2, DLARFB, DLARFT, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN * .. * .. External Functions .. INTEGER ILAENV REAL SCEIL EXTERNAL ILAENV, SCEIL * .. * .. Executable Statements .. INFO = 0 NBMIN = 2 NX = 0 IWS = N K = MIN( M, N ) NB = ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 ) IF( NB.GT.1 .AND. NB.LT.K ) THEN * * Determine when to cross over from blocked to unblocked code. * NX = MAX( 0, ILAENV( 3, 'DGEQRF', ' ', M, N, -1, -1 ) ) END IF * * Get NT, the size of the very last T, which is the left-over from in-between K-NX and K to K, eg.: * * NB=3 2NB=6 K=10 * | | | * 1--2--3--4--5--6--7--8--9--10 * | \________/ * K-NX=5 NT=4 * * So here 4 x 4 is the last T stored in the workspace * NT = K-SCEIL(REAL(K-NX)/REAL(NB))*NB * * optimal workspace = space for dlarfb + space for normal T's + space for the last T * LLWORK = MAX (MAX((N-M)*K, (N-M)*NB), MAX(K*NB, NB*NB)) LLWORK = SCEIL(REAL(LLWORK)/REAL(NB)) IF ( NT.GT.NB ) THEN LBWORK = K-NT * * Optimal workspace for dlarfb = MAX(1,N)*NT * LWKOPT = (LBWORK+LLWORK)*NB WORK( 1 ) = (LWKOPT+NT*NT) ELSE LBWORK = SCEIL(REAL(K)/REAL(NB))*NB LWKOPT = (LBWORK+LLWORK-NB)*NB WORK( 1 ) = LWKOPT END IF * * Test the input arguments * LQUERY = ( LWORK.EQ.-1 ) IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -4 ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN INFO = -7 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGEQRF', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible * IF( K.EQ.0 ) THEN WORK( 1 ) = 1 RETURN END IF * IF( NB.GT.1 .AND. NB.LT.K ) THEN IF( NX.LT.K ) THEN * * Determine if workspace is large enough for blocked code. * IF ( NT.LE.NB ) THEN IWS = (LBWORK+LLWORK-NB)*NB ELSE IWS = (LBWORK+LLWORK)*NB+NT*NT END IF IF( LWORK.LT.IWS ) THEN * * Not enough workspace to use optimal NB: reduce NB and * determine the minimum value of NB. * IF ( NT.LE.NB ) THEN NB = LWORK / (LLWORK+(LBWORK-NB)) ELSE NB = (LWORK-NT*NT)/(LBWORK+LLWORK) END IF NBMIN = MAX( 2, ILAENV( 2, 'DGEQRF', ' ', M, N, -1, $ -1 ) ) END IF END IF END IF * IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN * * Use blocked code initially * DO 10 I = 1, K - NX, NB IB = MIN( K-I+1, NB ) * * Update the current column using old T's * DO 20 J = 1, I - NB, NB * * Apply H' to A(J:M,I:I+IB-1) from the left * CALL DLARFB( 'Left', 'Transpose', 'Forward', $ 'Columnwise', M-J+1, IB, NB, $ A( J, J ), LDA, WORK(J), LBWORK, $ A( J, I ), LDA, WORK(LBWORK*NB+NT*NT+1), $ IB) 20 CONTINUE * * Compute the QR factorization of the current block * A(I:M,I:I+IB-1) * CALL DGEQR2( M-I+1, IB, A( I, I ), LDA, TAU( I ), $ WORK(LBWORK*NB+NT*NT+1), IINFO ) IF( I+IB.LE.N ) THEN * * Form the triangular factor of the block reflector * H = H(i) H(i+1) . . . H(i+ib-1) * CALL DLARFT( 'Forward', 'Columnwise', M-I+1, IB, $ A( I, I ), LDA, TAU( I ), $ WORK(I), LBWORK ) * END IF 10 CONTINUE ELSE I = 1 END IF * * Use unblocked code to factor the last or only block. * IF( I.LE.K ) THEN IF ( I .NE. 1 ) THEN DO 30 J = 1, I - NB, NB * * Apply H' to A(J:M,I:K) from the left * CALL DLARFB( 'Left', 'Transpose', 'Forward', $ 'Columnwise', M-J+1, K-I+1, NB, $ A( J, J ), LDA, WORK(J), LBWORK, $ A( J, I ), LDA, WORK(LBWORK*NB+NT*NT+1), $ K-I+1) 30 CONTINUE CALL DGEQR2( M-I+1, K-I+1, A( I, I ), LDA, TAU( I ), $ WORK(LBWORK*NB+NT*NT+1),IINFO ) ELSE * * Use unblocked code to factor the last or only block. * CALL DGEQR2( M-I+1, N-I+1, A( I, I ), LDA, TAU( I ), $ WORK,IINFO ) END IF END IF * * Apply update to the column M+1:N when N > M * IF ( M.LT.N .AND. I.NE.1) THEN * * Form the last triangular factor of the block reflector * H = H(i) H(i+1) . . . H(i+ib-1) * IF ( NT .LE. NB ) THEN CALL DLARFT( 'Forward', 'Columnwise', M-I+1, K-I+1, $ A( I, I ), LDA, TAU( I ), WORK(I), LBWORK ) ELSE CALL DLARFT( 'Forward', 'Columnwise', M-I+1, K-I+1, $ A( I, I ), LDA, TAU( I ), $ WORK(LBWORK*NB+1), NT ) END IF * * Apply H' to A(1:M,M+1:N) from the left * DO 40 J = 1, K-NX, NB IB = MIN( K-J+1, NB ) CALL DLARFB( 'Left', 'Transpose', 'Forward', $ 'Columnwise', M-J+1, N-M, IB, $ A( J, J ), LDA, WORK(J), LBWORK, $ A( J, M+1 ), LDA, WORK(LBWORK*NB+NT*NT+1), $ N-M) 40 CONTINUE IF ( NT.LE.NB ) THEN CALL DLARFB( 'Left', 'Transpose', 'Forward', $ 'Columnwise', M-J+1, N-M, K-J+1, $ A( J, J ), LDA, WORK(J), LBWORK, $ A( J, M+1 ), LDA, WORK(LBWORK*NB+NT*NT+1), $ N-M) ELSE CALL DLARFB( 'Left', 'Transpose', 'Forward', $ 'Columnwise', M-J+1, N-M, K-J+1, $ A( J, J ), LDA, $ WORK(LBWORK*NB+1), $ NT, A( J, M+1 ), LDA, WORK(LBWORK*NB+NT*NT+1), $ N-M) END IF END IF WORK( 1 ) = IWS RETURN * * End of DGEQRF * END |