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Mise à jour de lapack vers la version 3.3.0.
1: SUBROUTINE DGELS( TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK, 2: $ INFO ) 3: * 4: * -- LAPACK driver routine (version 3.2) -- 5: * -- LAPACK is a software package provided by Univ. of Tennessee, -- 6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 7: * November 2006 8: * 9: * .. Scalar Arguments .. 10: CHARACTER TRANS 11: INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS 12: * .. 13: * .. Array Arguments .. 14: DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * ) 15: * .. 16: * 17: * Purpose 18: * ======= 19: * 20: * DGELS solves overdetermined or underdetermined real linear systems 21: * involving an M-by-N matrix A, or its transpose, using a QR or LQ 22: * factorization of A. It is assumed that A has full rank. 23: * 24: * The following options are provided: 25: * 26: * 1. If TRANS = 'N' and m >= n: find the least squares solution of 27: * an overdetermined system, i.e., solve the least squares problem 28: * minimize || B - A*X ||. 29: * 30: * 2. If TRANS = 'N' and m < n: find the minimum norm solution of 31: * an underdetermined system A * X = B. 32: * 33: * 3. If TRANS = 'T' and m >= n: find the minimum norm solution of 34: * an undetermined system A**T * X = B. 35: * 36: * 4. If TRANS = 'T' and m < n: find the least squares solution of 37: * an overdetermined system, i.e., solve the least squares problem 38: * minimize || B - A**T * X ||. 39: * 40: * Several right hand side vectors b and solution vectors x can be 41: * handled in a single call; they are stored as the columns of the 42: * M-by-NRHS right hand side matrix B and the N-by-NRHS solution 43: * matrix X. 44: * 45: * Arguments 46: * ========= 47: * 48: * TRANS (input) CHARACTER*1 49: * = 'N': the linear system involves A; 50: * = 'T': the linear system involves A**T. 51: * 52: * M (input) INTEGER 53: * The number of rows of the matrix A. M >= 0. 54: * 55: * N (input) INTEGER 56: * The number of columns of the matrix A. N >= 0. 57: * 58: * NRHS (input) INTEGER 59: * The number of right hand sides, i.e., the number of 60: * columns of the matrices B and X. NRHS >=0. 61: * 62: * A (input/output) DOUBLE PRECISION array, dimension (LDA,N) 63: * On entry, the M-by-N matrix A. 64: * On exit, 65: * if M >= N, A is overwritten by details of its QR 66: * factorization as returned by DGEQRF; 67: * if M < N, A is overwritten by details of its LQ 68: * factorization as returned by DGELQF. 69: * 70: * LDA (input) INTEGER 71: * The leading dimension of the array A. LDA >= max(1,M). 72: * 73: * B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) 74: * On entry, the matrix B of right hand side vectors, stored 75: * columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS 76: * if TRANS = 'T'. 77: * On exit, if INFO = 0, B is overwritten by the solution 78: * vectors, stored columnwise: 79: * if TRANS = 'N' and m >= n, rows 1 to n of B contain the least 80: * squares solution vectors; the residual sum of squares for the 81: * solution in each column is given by the sum of squares of 82: * elements N+1 to M in that column; 83: * if TRANS = 'N' and m < n, rows 1 to N of B contain the 84: * minimum norm solution vectors; 85: * if TRANS = 'T' and m >= n, rows 1 to M of B contain the 86: * minimum norm solution vectors; 87: * if TRANS = 'T' and m < n, rows 1 to M of B contain the 88: * least squares solution vectors; the residual sum of squares 89: * for the solution in each column is given by the sum of 90: * squares of elements M+1 to N in that column. 91: * 92: * LDB (input) INTEGER 93: * The leading dimension of the array B. LDB >= MAX(1,M,N). 94: * 95: * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) 96: * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 97: * 98: * LWORK (input) INTEGER 99: * The dimension of the array WORK. 100: * LWORK >= max( 1, MN + max( MN, NRHS ) ). 101: * For optimal performance, 102: * LWORK >= max( 1, MN + max( MN, NRHS )*NB ). 103: * where MN = min(M,N) and NB is the optimum block size. 104: * 105: * If LWORK = -1, then a workspace query is assumed; the routine 106: * only calculates the optimal size of the WORK array, returns 107: * this value as the first entry of the WORK array, and no error 108: * message related to LWORK is issued by XERBLA. 109: * 110: * INFO (output) INTEGER 111: * = 0: successful exit 112: * < 0: if INFO = -i, the i-th argument had an illegal value 113: * > 0: if INFO = i, the i-th diagonal element of the 114: * triangular factor of A is zero, so that A does not have 115: * full rank; the least squares solution could not be 116: * computed. 117: * 118: * ===================================================================== 119: * 120: * .. Parameters .. 121: DOUBLE PRECISION ZERO, ONE 122: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) 123: * .. 124: * .. Local Scalars .. 125: LOGICAL LQUERY, TPSD 126: INTEGER BROW, I, IASCL, IBSCL, J, MN, NB, SCLLEN, WSIZE 127: DOUBLE PRECISION ANRM, BIGNUM, BNRM, SMLNUM 128: * .. 129: * .. Local Arrays .. 130: DOUBLE PRECISION RWORK( 1 ) 131: * .. 132: * .. External Functions .. 133: LOGICAL LSAME 134: INTEGER ILAENV 135: DOUBLE PRECISION DLAMCH, DLANGE 136: EXTERNAL LSAME, ILAENV, DLABAD, DLAMCH, DLANGE 137: * .. 138: * .. External Subroutines .. 139: EXTERNAL DGELQF, DGEQRF, DLASCL, DLASET, DORMLQ, DORMQR, 140: $ DTRTRS, XERBLA 141: * .. 142: * .. Intrinsic Functions .. 143: INTRINSIC DBLE, MAX, MIN 144: * .. 145: * .. Executable Statements .. 146: * 147: * Test the input arguments. 148: * 149: INFO = 0 150: MN = MIN( M, N ) 151: LQUERY = ( LWORK.EQ.-1 ) 152: IF( .NOT.( LSAME( TRANS, 'N' ) .OR. LSAME( TRANS, 'T' ) ) ) THEN 153: INFO = -1 154: ELSE IF( M.LT.0 ) THEN 155: INFO = -2 156: ELSE IF( N.LT.0 ) THEN 157: INFO = -3 158: ELSE IF( NRHS.LT.0 ) THEN 159: INFO = -4 160: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN 161: INFO = -6 162: ELSE IF( LDB.LT.MAX( 1, M, N ) ) THEN 163: INFO = -8 164: ELSE IF( LWORK.LT.MAX( 1, MN+MAX( MN, NRHS ) ) .AND. .NOT.LQUERY ) 165: $ THEN 166: INFO = -10 167: END IF 168: * 169: * Figure out optimal block size 170: * 171: IF( INFO.EQ.0 .OR. INFO.EQ.-10 ) THEN 172: * 173: TPSD = .TRUE. 174: IF( LSAME( TRANS, 'N' ) ) 175: $ TPSD = .FALSE. 176: * 177: IF( M.GE.N ) THEN 178: NB = ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 ) 179: IF( TPSD ) THEN 180: NB = MAX( NB, ILAENV( 1, 'DORMQR', 'LN', M, NRHS, N, 181: $ -1 ) ) 182: ELSE 183: NB = MAX( NB, ILAENV( 1, 'DORMQR', 'LT', M, NRHS, N, 184: $ -1 ) ) 185: END IF 186: ELSE 187: NB = ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 ) 188: IF( TPSD ) THEN 189: NB = MAX( NB, ILAENV( 1, 'DORMLQ', 'LT', N, NRHS, M, 190: $ -1 ) ) 191: ELSE 192: NB = MAX( NB, ILAENV( 1, 'DORMLQ', 'LN', N, NRHS, M, 193: $ -1 ) ) 194: END IF 195: END IF 196: * 197: WSIZE = MAX( 1, MN+MAX( MN, NRHS )*NB ) 198: WORK( 1 ) = DBLE( WSIZE ) 199: * 200: END IF 201: * 202: IF( INFO.NE.0 ) THEN 203: CALL XERBLA( 'DGELS ', -INFO ) 204: RETURN 205: ELSE IF( LQUERY ) THEN 206: RETURN 207: END IF 208: * 209: * Quick return if possible 210: * 211: IF( MIN( M, N, NRHS ).EQ.0 ) THEN 212: CALL DLASET( 'Full', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB ) 213: RETURN 214: END IF 215: * 216: * Get machine parameters 217: * 218: SMLNUM = DLAMCH( 'S' ) / DLAMCH( 'P' ) 219: BIGNUM = ONE / SMLNUM 220: CALL DLABAD( SMLNUM, BIGNUM ) 221: * 222: * Scale A, B if max element outside range [SMLNUM,BIGNUM] 223: * 224: ANRM = DLANGE( 'M', M, N, A, LDA, RWORK ) 225: IASCL = 0 226: IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN 227: * 228: * Scale matrix norm up to SMLNUM 229: * 230: CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO ) 231: IASCL = 1 232: ELSE IF( ANRM.GT.BIGNUM ) THEN 233: * 234: * Scale matrix norm down to BIGNUM 235: * 236: CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO ) 237: IASCL = 2 238: ELSE IF( ANRM.EQ.ZERO ) THEN 239: * 240: * Matrix all zero. Return zero solution. 241: * 242: CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB ) 243: GO TO 50 244: END IF 245: * 246: BROW = M 247: IF( TPSD ) 248: $ BROW = N 249: BNRM = DLANGE( 'M', BROW, NRHS, B, LDB, RWORK ) 250: IBSCL = 0 251: IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN 252: * 253: * Scale matrix norm up to SMLNUM 254: * 255: CALL DLASCL( 'G', 0, 0, BNRM, SMLNUM, BROW, NRHS, B, LDB, 256: $ INFO ) 257: IBSCL = 1 258: ELSE IF( BNRM.GT.BIGNUM ) THEN 259: * 260: * Scale matrix norm down to BIGNUM 261: * 262: CALL DLASCL( 'G', 0, 0, BNRM, BIGNUM, BROW, NRHS, B, LDB, 263: $ INFO ) 264: IBSCL = 2 265: END IF 266: * 267: IF( M.GE.N ) THEN 268: * 269: * compute QR factorization of A 270: * 271: CALL DGEQRF( M, N, A, LDA, WORK( 1 ), WORK( MN+1 ), LWORK-MN, 272: $ INFO ) 273: * 274: * workspace at least N, optimally N*NB 275: * 276: IF( .NOT.TPSD ) THEN 277: * 278: * Least-Squares Problem min || A * X - B || 279: * 280: * B(1:M,1:NRHS) := Q' * B(1:M,1:NRHS) 281: * 282: CALL DORMQR( 'Left', 'Transpose', M, NRHS, N, A, LDA, 283: $ WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN, 284: $ INFO ) 285: * 286: * workspace at least NRHS, optimally NRHS*NB 287: * 288: * B(1:N,1:NRHS) := inv(R) * B(1:N,1:NRHS) 289: * 290: CALL DTRTRS( 'Upper', 'No transpose', 'Non-unit', N, NRHS, 291: $ A, LDA, B, LDB, INFO ) 292: * 293: IF( INFO.GT.0 ) THEN 294: RETURN 295: END IF 296: * 297: SCLLEN = N 298: * 299: ELSE 300: * 301: * Overdetermined system of equations A' * X = B 302: * 303: * B(1:N,1:NRHS) := inv(R') * B(1:N,1:NRHS) 304: * 305: CALL DTRTRS( 'Upper', 'Transpose', 'Non-unit', N, NRHS, 306: $ A, LDA, B, LDB, INFO ) 307: * 308: IF( INFO.GT.0 ) THEN 309: RETURN 310: END IF 311: * 312: * B(N+1:M,1:NRHS) = ZERO 313: * 314: DO 20 J = 1, NRHS 315: DO 10 I = N + 1, M 316: B( I, J ) = ZERO 317: 10 CONTINUE 318: 20 CONTINUE 319: * 320: * B(1:M,1:NRHS) := Q(1:N,:) * B(1:N,1:NRHS) 321: * 322: CALL DORMQR( 'Left', 'No transpose', M, NRHS, N, A, LDA, 323: $ WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN, 324: $ INFO ) 325: * 326: * workspace at least NRHS, optimally NRHS*NB 327: * 328: SCLLEN = M 329: * 330: END IF 331: * 332: ELSE 333: * 334: * Compute LQ factorization of A 335: * 336: CALL DGELQF( M, N, A, LDA, WORK( 1 ), WORK( MN+1 ), LWORK-MN, 337: $ INFO ) 338: * 339: * workspace at least M, optimally M*NB. 340: * 341: IF( .NOT.TPSD ) THEN 342: * 343: * underdetermined system of equations A * X = B 344: * 345: * B(1:M,1:NRHS) := inv(L) * B(1:M,1:NRHS) 346: * 347: CALL DTRTRS( 'Lower', 'No transpose', 'Non-unit', M, NRHS, 348: $ A, LDA, B, LDB, INFO ) 349: * 350: IF( INFO.GT.0 ) THEN 351: RETURN 352: END IF 353: * 354: * B(M+1:N,1:NRHS) = 0 355: * 356: DO 40 J = 1, NRHS 357: DO 30 I = M + 1, N 358: B( I, J ) = ZERO 359: 30 CONTINUE 360: 40 CONTINUE 361: * 362: * B(1:N,1:NRHS) := Q(1:N,:)' * B(1:M,1:NRHS) 363: * 364: CALL DORMLQ( 'Left', 'Transpose', N, NRHS, M, A, LDA, 365: $ WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN, 366: $ INFO ) 367: * 368: * workspace at least NRHS, optimally NRHS*NB 369: * 370: SCLLEN = N 371: * 372: ELSE 373: * 374: * overdetermined system min || A' * X - B || 375: * 376: * B(1:N,1:NRHS) := Q * B(1:N,1:NRHS) 377: * 378: CALL DORMLQ( 'Left', 'No transpose', N, NRHS, M, A, LDA, 379: $ WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN, 380: $ INFO ) 381: * 382: * workspace at least NRHS, optimally NRHS*NB 383: * 384: * B(1:M,1:NRHS) := inv(L') * B(1:M,1:NRHS) 385: * 386: CALL DTRTRS( 'Lower', 'Transpose', 'Non-unit', M, NRHS, 387: $ A, LDA, B, LDB, INFO ) 388: * 389: IF( INFO.GT.0 ) THEN 390: RETURN 391: END IF 392: * 393: SCLLEN = M 394: * 395: END IF 396: * 397: END IF 398: * 399: * Undo scaling 400: * 401: IF( IASCL.EQ.1 ) THEN 402: CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, SCLLEN, NRHS, B, LDB, 403: $ INFO ) 404: ELSE IF( IASCL.EQ.2 ) THEN 405: CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, SCLLEN, NRHS, B, LDB, 406: $ INFO ) 407: END IF 408: IF( IBSCL.EQ.1 ) THEN 409: CALL DLASCL( 'G', 0, 0, SMLNUM, BNRM, SCLLEN, NRHS, B, LDB, 410: $ INFO ) 411: ELSE IF( IBSCL.EQ.2 ) THEN 412: CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, SCLLEN, NRHS, B, LDB, 413: $ INFO ) 414: END IF 415: * 416: 50 CONTINUE 417: WORK( 1 ) = DBLE( WSIZE ) 418: * 419: RETURN 420: * 421: * End of DGELS 422: * 423: END