Annotation of rpl/lapack/lapack/dlaqps.f, revision 1.1

1.1     ! bertrand    1:       SUBROUTINE DLAQPS( M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1,
        !             2:      $                   VN2, AUXV, F, LDF )
        !             3: *
        !             4: *  -- LAPACK auxiliary 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            KB, LDA, LDF, M, N, NB, OFFSET
        !            11: *     ..
        !            12: *     .. Array Arguments ..
        !            13:       INTEGER            JPVT( * )
        !            14:       DOUBLE PRECISION   A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * ),
        !            15:      $                   VN1( * ), VN2( * )
        !            16: *     ..
        !            17: *
        !            18: *  Purpose
        !            19: *  =======
        !            20: *
        !            21: *  DLAQPS computes a step of QR factorization with column pivoting
        !            22: *  of a real M-by-N matrix A by using Blas-3.  It tries to factorize
        !            23: *  NB columns from A starting from the row OFFSET+1, and updates all
        !            24: *  of the matrix with Blas-3 xGEMM.
        !            25: *
        !            26: *  In some cases, due to catastrophic cancellations, it cannot
        !            27: *  factorize NB columns.  Hence, the actual number of factorized
        !            28: *  columns is returned in KB.
        !            29: *
        !            30: *  Block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.
        !            31: *
        !            32: *  Arguments
        !            33: *  =========
        !            34: *
        !            35: *  M       (input) INTEGER
        !            36: *          The number of rows of the matrix A. M >= 0.
        !            37: *
        !            38: *  N       (input) INTEGER
        !            39: *          The number of columns of the matrix A. N >= 0
        !            40: *
        !            41: *  OFFSET  (input) INTEGER
        !            42: *          The number of rows of A that have been factorized in
        !            43: *          previous steps.
        !            44: *
        !            45: *  NB      (input) INTEGER
        !            46: *          The number of columns to factorize.
        !            47: *
        !            48: *  KB      (output) INTEGER
        !            49: *          The number of columns actually factorized.
        !            50: *
        !            51: *  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
        !            52: *          On entry, the M-by-N matrix A.
        !            53: *          On exit, block A(OFFSET+1:M,1:KB) is the triangular
        !            54: *          factor obtained and block A(1:OFFSET,1:N) has been
        !            55: *          accordingly pivoted, but no factorized.
        !            56: *          The rest of the matrix, block A(OFFSET+1:M,KB+1:N) has
        !            57: *          been updated.
        !            58: *
        !            59: *  LDA     (input) INTEGER
        !            60: *          The leading dimension of the array A. LDA >= max(1,M).
        !            61: *
        !            62: *  JPVT    (input/output) INTEGER array, dimension (N)
        !            63: *          JPVT(I) = K <==> Column K of the full matrix A has been
        !            64: *          permuted into position I in AP.
        !            65: *
        !            66: *  TAU     (output) DOUBLE PRECISION array, dimension (KB)
        !            67: *          The scalar factors of the elementary reflectors.
        !            68: *
        !            69: *  VN1     (input/output) DOUBLE PRECISION array, dimension (N)
        !            70: *          The vector with the partial column norms.
        !            71: *
        !            72: *  VN2     (input/output) DOUBLE PRECISION array, dimension (N)
        !            73: *          The vector with the exact column norms.
        !            74: *
        !            75: *  AUXV    (input/output) DOUBLE PRECISION array, dimension (NB)
        !            76: *          Auxiliar vector.
        !            77: *
        !            78: *  F       (input/output) DOUBLE PRECISION array, dimension (LDF,NB)
        !            79: *          Matrix F' = L*Y'*A.
        !            80: *
        !            81: *  LDF     (input) INTEGER
        !            82: *          The leading dimension of the array F. LDF >= max(1,N).
        !            83: *
        !            84: *  Further Details
        !            85: *  ===============
        !            86: *
        !            87: *  Based on contributions by
        !            88: *    G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
        !            89: *    X. Sun, Computer Science Dept., Duke University, USA
        !            90: *
        !            91: *  Partial column norm updating strategy modified by
        !            92: *    Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
        !            93: *    University of Zagreb, Croatia.
        !            94: *    June 2006.
        !            95: *  For more details see LAPACK Working Note 176.
        !            96: *  =====================================================================
        !            97: *
        !            98: *     .. Parameters ..
        !            99:       DOUBLE PRECISION   ZERO, ONE
        !           100:       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
        !           101: *     ..
        !           102: *     .. Local Scalars ..
        !           103:       INTEGER            ITEMP, J, K, LASTRK, LSTICC, PVT, RK
        !           104:       DOUBLE PRECISION   AKK, TEMP, TEMP2, TOL3Z
        !           105: *     ..
        !           106: *     .. External Subroutines ..
        !           107:       EXTERNAL           DGEMM, DGEMV, DLARFP, DSWAP
        !           108: *     ..
        !           109: *     .. Intrinsic Functions ..
        !           110:       INTRINSIC          ABS, DBLE, MAX, MIN, NINT, SQRT
        !           111: *     ..
        !           112: *     .. External Functions ..
        !           113:       INTEGER            IDAMAX
        !           114:       DOUBLE PRECISION   DLAMCH, DNRM2
        !           115:       EXTERNAL           IDAMAX, DLAMCH, DNRM2
        !           116: *     ..
        !           117: *     .. Executable Statements ..
        !           118: *
        !           119:       LASTRK = MIN( M, N+OFFSET )
        !           120:       LSTICC = 0
        !           121:       K = 0
        !           122:       TOL3Z = SQRT(DLAMCH('Epsilon'))
        !           123: *
        !           124: *     Beginning of while loop.
        !           125: *
        !           126:    10 CONTINUE
        !           127:       IF( ( K.LT.NB ) .AND. ( LSTICC.EQ.0 ) ) THEN
        !           128:          K = K + 1
        !           129:          RK = OFFSET + K
        !           130: *
        !           131: *        Determine ith pivot column and swap if necessary
        !           132: *
        !           133:          PVT = ( K-1 ) + IDAMAX( N-K+1, VN1( K ), 1 )
        !           134:          IF( PVT.NE.K ) THEN
        !           135:             CALL DSWAP( M, A( 1, PVT ), 1, A( 1, K ), 1 )
        !           136:             CALL DSWAP( K-1, F( PVT, 1 ), LDF, F( K, 1 ), LDF )
        !           137:             ITEMP = JPVT( PVT )
        !           138:             JPVT( PVT ) = JPVT( K )
        !           139:             JPVT( K ) = ITEMP
        !           140:             VN1( PVT ) = VN1( K )
        !           141:             VN2( PVT ) = VN2( K )
        !           142:          END IF
        !           143: *
        !           144: *        Apply previous Householder reflectors to column K:
        !           145: *        A(RK:M,K) := A(RK:M,K) - A(RK:M,1:K-1)*F(K,1:K-1)'.
        !           146: *
        !           147:          IF( K.GT.1 ) THEN
        !           148:             CALL DGEMV( 'No transpose', M-RK+1, K-1, -ONE, A( RK, 1 ),
        !           149:      $                  LDA, F( K, 1 ), LDF, ONE, A( RK, K ), 1 )
        !           150:          END IF
        !           151: *
        !           152: *        Generate elementary reflector H(k).
        !           153: *
        !           154:          IF( RK.LT.M ) THEN
        !           155:             CALL DLARFP( M-RK+1, A( RK, K ), A( RK+1, K ), 1, TAU( K ) )
        !           156:          ELSE
        !           157:             CALL DLARFP( 1, A( RK, K ), A( RK, K ), 1, TAU( K ) )
        !           158:          END IF
        !           159: *
        !           160:          AKK = A( RK, K )
        !           161:          A( RK, K ) = ONE
        !           162: *
        !           163: *        Compute Kth column of F:
        !           164: *
        !           165: *        Compute  F(K+1:N,K) := tau(K)*A(RK:M,K+1:N)'*A(RK:M,K).
        !           166: *
        !           167:          IF( K.LT.N ) THEN
        !           168:             CALL DGEMV( 'Transpose', M-RK+1, N-K, TAU( K ),
        !           169:      $                  A( RK, K+1 ), LDA, A( RK, K ), 1, ZERO,
        !           170:      $                  F( K+1, K ), 1 )
        !           171:          END IF
        !           172: *
        !           173: *        Padding F(1:K,K) with zeros.
        !           174: *
        !           175:          DO 20 J = 1, K
        !           176:             F( J, K ) = ZERO
        !           177:    20    CONTINUE
        !           178: *
        !           179: *        Incremental updating of F:
        !           180: *        F(1:N,K) := F(1:N,K) - tau(K)*F(1:N,1:K-1)*A(RK:M,1:K-1)'
        !           181: *                    *A(RK:M,K).
        !           182: *
        !           183:          IF( K.GT.1 ) THEN
        !           184:             CALL DGEMV( 'Transpose', M-RK+1, K-1, -TAU( K ), A( RK, 1 ),
        !           185:      $                  LDA, A( RK, K ), 1, ZERO, AUXV( 1 ), 1 )
        !           186: *
        !           187:             CALL DGEMV( 'No transpose', N, K-1, ONE, F( 1, 1 ), LDF,
        !           188:      $                  AUXV( 1 ), 1, ONE, F( 1, K ), 1 )
        !           189:          END IF
        !           190: *
        !           191: *        Update the current row of A:
        !           192: *        A(RK,K+1:N) := A(RK,K+1:N) - A(RK,1:K)*F(K+1:N,1:K)'.
        !           193: *
        !           194:          IF( K.LT.N ) THEN
        !           195:             CALL DGEMV( 'No transpose', N-K, K, -ONE, F( K+1, 1 ), LDF,
        !           196:      $                  A( RK, 1 ), LDA, ONE, A( RK, K+1 ), LDA )
        !           197:          END IF
        !           198: *
        !           199: *        Update partial column norms.
        !           200: *
        !           201:          IF( RK.LT.LASTRK ) THEN
        !           202:             DO 30 J = K + 1, N
        !           203:                IF( VN1( J ).NE.ZERO ) THEN
        !           204: *
        !           205: *                 NOTE: The following 4 lines follow from the analysis in
        !           206: *                 Lapack Working Note 176.
        !           207: *
        !           208:                   TEMP = ABS( A( RK, J ) ) / VN1( J )
        !           209:                   TEMP = MAX( ZERO, ( ONE+TEMP )*( ONE-TEMP ) )
        !           210:                   TEMP2 = TEMP*( VN1( J ) / VN2( J ) )**2
        !           211:                   IF( TEMP2 .LE. TOL3Z ) THEN
        !           212:                      VN2( J ) = DBLE( LSTICC )
        !           213:                      LSTICC = J
        !           214:                   ELSE
        !           215:                      VN1( J ) = VN1( J )*SQRT( TEMP )
        !           216:                   END IF
        !           217:                END IF
        !           218:    30       CONTINUE
        !           219:          END IF
        !           220: *
        !           221:          A( RK, K ) = AKK
        !           222: *
        !           223: *        End of while loop.
        !           224: *
        !           225:          GO TO 10
        !           226:       END IF
        !           227:       KB = K
        !           228:       RK = OFFSET + KB
        !           229: *
        !           230: *     Apply the block reflector to the rest of the matrix:
        !           231: *     A(OFFSET+KB+1:M,KB+1:N) := A(OFFSET+KB+1:M,KB+1:N) -
        !           232: *                         A(OFFSET+KB+1:M,1:KB)*F(KB+1:N,1:KB)'.
        !           233: *
        !           234:       IF( KB.LT.MIN( N, M-OFFSET ) ) THEN
        !           235:          CALL DGEMM( 'No transpose', 'Transpose', M-RK, N-KB, KB, -ONE,
        !           236:      $               A( RK+1, 1 ), LDA, F( KB+1, 1 ), LDF, ONE,
        !           237:      $               A( RK+1, KB+1 ), LDA )
        !           238:       END IF
        !           239: *
        !           240: *     Recomputation of difficult columns.
        !           241: *
        !           242:    40 CONTINUE
        !           243:       IF( LSTICC.GT.0 ) THEN
        !           244:          ITEMP = NINT( VN2( LSTICC ) )
        !           245:          VN1( LSTICC ) = DNRM2( M-RK, A( RK+1, LSTICC ), 1 )
        !           246: *
        !           247: *        NOTE: The computation of VN1( LSTICC ) relies on the fact that 
        !           248: *        SNRM2 does not fail on vectors with norm below the value of
        !           249: *        SQRT(DLAMCH('S')) 
        !           250: *
        !           251:          VN2( LSTICC ) = VN1( LSTICC )
        !           252:          LSTICC = ITEMP
        !           253:          GO TO 40
        !           254:       END IF
        !           255: *
        !           256:       RETURN
        !           257: *
        !           258: *     End of DLAQPS
        !           259: *
        !           260:       END

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