Annotation of rpl/lapack/lapack/dgels.f, revision 1.1
1.1 ! bertrand 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
CVSweb interface <joel.bertrand@systella.fr>
![[BACK]](/cvsweb/icons/back.gif){kind=icon}
Return to dgels.f CVS log ![[TXT]](/cvsweb/icons/text.gif){kind=icon}
![[DIR]](/cvsweb/icons/dir.gif){kind=icon}
Up to [local] / rpl / lapack / lapack