Annotation of rpl/lapack/lapack/dgelsx.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
! 2: $ WORK, 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: INTEGER INFO, LDA, LDB, M, N, NRHS, RANK
! 11: DOUBLE PRECISION RCOND
! 12: * ..
! 13: * .. Array Arguments ..
! 14: INTEGER JPVT( * )
! 15: DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * )
! 16: * ..
! 17: *
! 18: * Purpose
! 19: * =======
! 20: *
! 21: * This routine is deprecated and has been replaced by routine DGELSY.
! 22: *
! 23: * DGELSX computes the minimum-norm solution to a real linear least
! 24: * squares problem:
! 25: * minimize || A * X - B ||
! 26: * using a complete orthogonal factorization of A. A is an M-by-N
! 27: * matrix which may be rank-deficient.
! 28: *
! 29: * Several right hand side vectors b and solution vectors x can be
! 30: * handled in a single call; they are stored as the columns of the
! 31: * M-by-NRHS right hand side matrix B and the N-by-NRHS solution
! 32: * matrix X.
! 33: *
! 34: * The routine first computes a QR factorization with column pivoting:
! 35: * A * P = Q * [ R11 R12 ]
! 36: * [ 0 R22 ]
! 37: * with R11 defined as the largest leading submatrix whose estimated
! 38: * condition number is less than 1/RCOND. The order of R11, RANK,
! 39: * is the effective rank of A.
! 40: *
! 41: * Then, R22 is considered to be negligible, and R12 is annihilated
! 42: * by orthogonal transformations from the right, arriving at the
! 43: * complete orthogonal factorization:
! 44: * A * P = Q * [ T11 0 ] * Z
! 45: * [ 0 0 ]
! 46: * The minimum-norm solution is then
! 47: * X = P * Z' [ inv(T11)*Q1'*B ]
! 48: * [ 0 ]
! 49: * where Q1 consists of the first RANK columns of Q.
! 50: *
! 51: * Arguments
! 52: * =========
! 53: *
! 54: * M (input) INTEGER
! 55: * The number of rows of the matrix A. M >= 0.
! 56: *
! 57: * N (input) INTEGER
! 58: * The number of columns of the matrix A. N >= 0.
! 59: *
! 60: * NRHS (input) INTEGER
! 61: * The number of right hand sides, i.e., the number of
! 62: * columns of matrices B and X. NRHS >= 0.
! 63: *
! 64: * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
! 65: * On entry, the M-by-N matrix A.
! 66: * On exit, A has been overwritten by details of its
! 67: * complete orthogonal factorization.
! 68: *
! 69: * LDA (input) INTEGER
! 70: * The leading dimension of the array A. LDA >= max(1,M).
! 71: *
! 72: * B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
! 73: * On entry, the M-by-NRHS right hand side matrix B.
! 74: * On exit, the N-by-NRHS solution matrix X.
! 75: * If m >= n and RANK = n, the residual sum-of-squares for
! 76: * the solution in the i-th column is given by the sum of
! 77: * squares of elements N+1:M in that column.
! 78: *
! 79: * LDB (input) INTEGER
! 80: * The leading dimension of the array B. LDB >= max(1,M,N).
! 81: *
! 82: * JPVT (input/output) INTEGER array, dimension (N)
! 83: * On entry, if JPVT(i) .ne. 0, the i-th column of A is an
! 84: * initial column, otherwise it is a free column. Before
! 85: * the QR factorization of A, all initial columns are
! 86: * permuted to the leading positions; only the remaining
! 87: * free columns are moved as a result of column pivoting
! 88: * during the factorization.
! 89: * On exit, if JPVT(i) = k, then the i-th column of A*P
! 90: * was the k-th column of A.
! 91: *
! 92: * RCOND (input) DOUBLE PRECISION
! 93: * RCOND is used to determine the effective rank of A, which
! 94: * is defined as the order of the largest leading triangular
! 95: * submatrix R11 in the QR factorization with pivoting of A,
! 96: * whose estimated condition number < 1/RCOND.
! 97: *
! 98: * RANK (output) INTEGER
! 99: * The effective rank of A, i.e., the order of the submatrix
! 100: * R11. This is the same as the order of the submatrix T11
! 101: * in the complete orthogonal factorization of A.
! 102: *
! 103: * WORK (workspace) DOUBLE PRECISION array, dimension
! 104: * (max( min(M,N)+3*N, 2*min(M,N)+NRHS )),
! 105: *
! 106: * INFO (output) INTEGER
! 107: * = 0: successful exit
! 108: * < 0: if INFO = -i, the i-th argument had an illegal value
! 109: *
! 110: * =====================================================================
! 111: *
! 112: * .. Parameters ..
! 113: INTEGER IMAX, IMIN
! 114: PARAMETER ( IMAX = 1, IMIN = 2 )
! 115: DOUBLE PRECISION ZERO, ONE, DONE, NTDONE
! 116: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, DONE = ZERO,
! 117: $ NTDONE = ONE )
! 118: * ..
! 119: * .. Local Scalars ..
! 120: INTEGER I, IASCL, IBSCL, ISMAX, ISMIN, J, K, MN
! 121: DOUBLE PRECISION ANRM, BIGNUM, BNRM, C1, C2, S1, S2, SMAX,
! 122: $ SMAXPR, SMIN, SMINPR, SMLNUM, T1, T2
! 123: * ..
! 124: * .. External Functions ..
! 125: DOUBLE PRECISION DLAMCH, DLANGE
! 126: EXTERNAL DLAMCH, DLANGE
! 127: * ..
! 128: * .. External Subroutines ..
! 129: EXTERNAL DGEQPF, DLAIC1, DLASCL, DLASET, DLATZM, DORM2R,
! 130: $ DTRSM, DTZRQF, XERBLA
! 131: * ..
! 132: * .. Intrinsic Functions ..
! 133: INTRINSIC ABS, MAX, MIN
! 134: * ..
! 135: * .. Executable Statements ..
! 136: *
! 137: MN = MIN( M, N )
! 138: ISMIN = MN + 1
! 139: ISMAX = 2*MN + 1
! 140: *
! 141: * Test the input arguments.
! 142: *
! 143: INFO = 0
! 144: IF( M.LT.0 ) THEN
! 145: INFO = -1
! 146: ELSE IF( N.LT.0 ) THEN
! 147: INFO = -2
! 148: ELSE IF( NRHS.LT.0 ) THEN
! 149: INFO = -3
! 150: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
! 151: INFO = -5
! 152: ELSE IF( LDB.LT.MAX( 1, M, N ) ) THEN
! 153: INFO = -7
! 154: END IF
! 155: *
! 156: IF( INFO.NE.0 ) THEN
! 157: CALL XERBLA( 'DGELSX', -INFO )
! 158: RETURN
! 159: END IF
! 160: *
! 161: * Quick return if possible
! 162: *
! 163: IF( MIN( M, N, NRHS ).EQ.0 ) THEN
! 164: RANK = 0
! 165: RETURN
! 166: END IF
! 167: *
! 168: * Get machine parameters
! 169: *
! 170: SMLNUM = DLAMCH( 'S' ) / DLAMCH( 'P' )
! 171: BIGNUM = ONE / SMLNUM
! 172: CALL DLABAD( SMLNUM, BIGNUM )
! 173: *
! 174: * Scale A, B if max elements outside range [SMLNUM,BIGNUM]
! 175: *
! 176: ANRM = DLANGE( 'M', M, N, A, LDA, WORK )
! 177: IASCL = 0
! 178: IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
! 179: *
! 180: * Scale matrix norm up to SMLNUM
! 181: *
! 182: CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
! 183: IASCL = 1
! 184: ELSE IF( ANRM.GT.BIGNUM ) THEN
! 185: *
! 186: * Scale matrix norm down to BIGNUM
! 187: *
! 188: CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
! 189: IASCL = 2
! 190: ELSE IF( ANRM.EQ.ZERO ) THEN
! 191: *
! 192: * Matrix all zero. Return zero solution.
! 193: *
! 194: CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
! 195: RANK = 0
! 196: GO TO 100
! 197: END IF
! 198: *
! 199: BNRM = DLANGE( 'M', M, NRHS, B, LDB, WORK )
! 200: IBSCL = 0
! 201: IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
! 202: *
! 203: * Scale matrix norm up to SMLNUM
! 204: *
! 205: CALL DLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO )
! 206: IBSCL = 1
! 207: ELSE IF( BNRM.GT.BIGNUM ) THEN
! 208: *
! 209: * Scale matrix norm down to BIGNUM
! 210: *
! 211: CALL DLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB, INFO )
! 212: IBSCL = 2
! 213: END IF
! 214: *
! 215: * Compute QR factorization with column pivoting of A:
! 216: * A * P = Q * R
! 217: *
! 218: CALL DGEQPF( M, N, A, LDA, JPVT, WORK( 1 ), WORK( MN+1 ), INFO )
! 219: *
! 220: * workspace 3*N. Details of Householder rotations stored
! 221: * in WORK(1:MN).
! 222: *
! 223: * Determine RANK using incremental condition estimation
! 224: *
! 225: WORK( ISMIN ) = ONE
! 226: WORK( ISMAX ) = ONE
! 227: SMAX = ABS( A( 1, 1 ) )
! 228: SMIN = SMAX
! 229: IF( ABS( A( 1, 1 ) ).EQ.ZERO ) THEN
! 230: RANK = 0
! 231: CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
! 232: GO TO 100
! 233: ELSE
! 234: RANK = 1
! 235: END IF
! 236: *
! 237: 10 CONTINUE
! 238: IF( RANK.LT.MN ) THEN
! 239: I = RANK + 1
! 240: CALL DLAIC1( IMIN, RANK, WORK( ISMIN ), SMIN, A( 1, I ),
! 241: $ A( I, I ), SMINPR, S1, C1 )
! 242: CALL DLAIC1( IMAX, RANK, WORK( ISMAX ), SMAX, A( 1, I ),
! 243: $ A( I, I ), SMAXPR, S2, C2 )
! 244: *
! 245: IF( SMAXPR*RCOND.LE.SMINPR ) THEN
! 246: DO 20 I = 1, RANK
! 247: WORK( ISMIN+I-1 ) = S1*WORK( ISMIN+I-1 )
! 248: WORK( ISMAX+I-1 ) = S2*WORK( ISMAX+I-1 )
! 249: 20 CONTINUE
! 250: WORK( ISMIN+RANK ) = C1
! 251: WORK( ISMAX+RANK ) = C2
! 252: SMIN = SMINPR
! 253: SMAX = SMAXPR
! 254: RANK = RANK + 1
! 255: GO TO 10
! 256: END IF
! 257: END IF
! 258: *
! 259: * Logically partition R = [ R11 R12 ]
! 260: * [ 0 R22 ]
! 261: * where R11 = R(1:RANK,1:RANK)
! 262: *
! 263: * [R11,R12] = [ T11, 0 ] * Y
! 264: *
! 265: IF( RANK.LT.N )
! 266: $ CALL DTZRQF( RANK, N, A, LDA, WORK( MN+1 ), INFO )
! 267: *
! 268: * Details of Householder rotations stored in WORK(MN+1:2*MN)
! 269: *
! 270: * B(1:M,1:NRHS) := Q' * B(1:M,1:NRHS)
! 271: *
! 272: CALL DORM2R( 'Left', 'Transpose', M, NRHS, MN, A, LDA, WORK( 1 ),
! 273: $ B, LDB, WORK( 2*MN+1 ), INFO )
! 274: *
! 275: * workspace NRHS
! 276: *
! 277: * B(1:RANK,1:NRHS) := inv(T11) * B(1:RANK,1:NRHS)
! 278: *
! 279: CALL DTRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', RANK,
! 280: $ NRHS, ONE, A, LDA, B, LDB )
! 281: *
! 282: DO 40 I = RANK + 1, N
! 283: DO 30 J = 1, NRHS
! 284: B( I, J ) = ZERO
! 285: 30 CONTINUE
! 286: 40 CONTINUE
! 287: *
! 288: * B(1:N,1:NRHS) := Y' * B(1:N,1:NRHS)
! 289: *
! 290: IF( RANK.LT.N ) THEN
! 291: DO 50 I = 1, RANK
! 292: CALL DLATZM( 'Left', N-RANK+1, NRHS, A( I, RANK+1 ), LDA,
! 293: $ WORK( MN+I ), B( I, 1 ), B( RANK+1, 1 ), LDB,
! 294: $ WORK( 2*MN+1 ) )
! 295: 50 CONTINUE
! 296: END IF
! 297: *
! 298: * workspace NRHS
! 299: *
! 300: * B(1:N,1:NRHS) := P * B(1:N,1:NRHS)
! 301: *
! 302: DO 90 J = 1, NRHS
! 303: DO 60 I = 1, N
! 304: WORK( 2*MN+I ) = NTDONE
! 305: 60 CONTINUE
! 306: DO 80 I = 1, N
! 307: IF( WORK( 2*MN+I ).EQ.NTDONE ) THEN
! 308: IF( JPVT( I ).NE.I ) THEN
! 309: K = I
! 310: T1 = B( K, J )
! 311: T2 = B( JPVT( K ), J )
! 312: 70 CONTINUE
! 313: B( JPVT( K ), J ) = T1
! 314: WORK( 2*MN+K ) = DONE
! 315: T1 = T2
! 316: K = JPVT( K )
! 317: T2 = B( JPVT( K ), J )
! 318: IF( JPVT( K ).NE.I )
! 319: $ GO TO 70
! 320: B( I, J ) = T1
! 321: WORK( 2*MN+K ) = DONE
! 322: END IF
! 323: END IF
! 324: 80 CONTINUE
! 325: 90 CONTINUE
! 326: *
! 327: * Undo scaling
! 328: *
! 329: IF( IASCL.EQ.1 ) THEN
! 330: CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO )
! 331: CALL DLASCL( 'U', 0, 0, SMLNUM, ANRM, RANK, RANK, A, LDA,
! 332: $ INFO )
! 333: ELSE IF( IASCL.EQ.2 ) THEN
! 334: CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO )
! 335: CALL DLASCL( 'U', 0, 0, BIGNUM, ANRM, RANK, RANK, A, LDA,
! 336: $ INFO )
! 337: END IF
! 338: IF( IBSCL.EQ.1 ) THEN
! 339: CALL DLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO )
! 340: ELSE IF( IBSCL.EQ.2 ) THEN
! 341: CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO )
! 342: END IF
! 343: *
! 344: 100 CONTINUE
! 345: *
! 346: RETURN
! 347: *
! 348: * End of DGELSX
! 349: *
! 350: END
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