Annotation of rpl/lapack/lapack/dlahrd.f, revision 1.12

1.12    ! bertrand    1: *> \brief \b DLAHRD reduces the first nb columns of a general rectangular matrix A so that elements below the k-th subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.
1.9       bertrand    2: *
                      3: *  =========== DOCUMENTATION ===========
                      4: *
                      5: * Online html documentation available at 
                      6: *            http://www.netlib.org/lapack/explore-html/ 
                      7: *
                      8: *> \htmlonly
                      9: *> Download DLAHRD + dependencies 
                     10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlahrd.f"> 
                     11: *> [TGZ]</a> 
                     12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlahrd.f"> 
                     13: *> [ZIP]</a> 
                     14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlahrd.f"> 
                     15: *> [TXT]</a>
                     16: *> \endhtmlonly 
                     17: *
                     18: *  Definition:
                     19: *  ===========
                     20: *
                     21: *       SUBROUTINE DLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
                     22: * 
                     23: *       .. Scalar Arguments ..
                     24: *       INTEGER            K, LDA, LDT, LDY, N, NB
                     25: *       ..
                     26: *       .. Array Arguments ..
                     27: *       DOUBLE PRECISION   A( LDA, * ), T( LDT, NB ), TAU( NB ),
                     28: *      $                   Y( LDY, NB )
                     29: *       ..
                     30: *  
                     31: *
                     32: *> \par Purpose:
                     33: *  =============
                     34: *>
                     35: *> \verbatim
                     36: *>
                     37: *> DLAHRD reduces the first NB columns of a real general n-by-(n-k+1)
                     38: *> matrix A so that elements below the k-th subdiagonal are zero. The
                     39: *> reduction is performed by an orthogonal similarity transformation
                     40: *> Q**T * A * Q. The routine returns the matrices V and T which determine
                     41: *> Q as a block reflector I - V*T*V**T, and also the matrix Y = A * V * T.
                     42: *>
                     43: *> This is an OBSOLETE auxiliary routine. 
                     44: *> This routine will be 'deprecated' in a  future release.
                     45: *> Please use the new routine DLAHR2 instead.
                     46: *> \endverbatim
                     47: *
                     48: *  Arguments:
                     49: *  ==========
                     50: *
                     51: *> \param[in] N
                     52: *> \verbatim
                     53: *>          N is INTEGER
                     54: *>          The order of the matrix A.
                     55: *> \endverbatim
                     56: *>
                     57: *> \param[in] K
                     58: *> \verbatim
                     59: *>          K is INTEGER
                     60: *>          The offset for the reduction. Elements below the k-th
                     61: *>          subdiagonal in the first NB columns are reduced to zero.
                     62: *> \endverbatim
                     63: *>
                     64: *> \param[in] NB
                     65: *> \verbatim
                     66: *>          NB is INTEGER
                     67: *>          The number of columns to be reduced.
                     68: *> \endverbatim
                     69: *>
                     70: *> \param[in,out] A
                     71: *> \verbatim
                     72: *>          A is DOUBLE PRECISION array, dimension (LDA,N-K+1)
                     73: *>          On entry, the n-by-(n-k+1) general matrix A.
                     74: *>          On exit, the elements on and above the k-th subdiagonal in
                     75: *>          the first NB columns are overwritten with the corresponding
                     76: *>          elements of the reduced matrix; the elements below the k-th
                     77: *>          subdiagonal, with the array TAU, represent the matrix Q as a
                     78: *>          product of elementary reflectors. The other columns of A are
                     79: *>          unchanged. See Further Details.
                     80: *> \endverbatim
                     81: *>
                     82: *> \param[in] LDA
                     83: *> \verbatim
                     84: *>          LDA is INTEGER
                     85: *>          The leading dimension of the array A.  LDA >= max(1,N).
                     86: *> \endverbatim
                     87: *>
                     88: *> \param[out] TAU
                     89: *> \verbatim
                     90: *>          TAU is DOUBLE PRECISION array, dimension (NB)
                     91: *>          The scalar factors of the elementary reflectors. See Further
                     92: *>          Details.
                     93: *> \endverbatim
                     94: *>
                     95: *> \param[out] T
                     96: *> \verbatim
                     97: *>          T is DOUBLE PRECISION array, dimension (LDT,NB)
                     98: *>          The upper triangular matrix T.
                     99: *> \endverbatim
                    100: *>
                    101: *> \param[in] LDT
                    102: *> \verbatim
                    103: *>          LDT is INTEGER
                    104: *>          The leading dimension of the array T.  LDT >= NB.
                    105: *> \endverbatim
                    106: *>
                    107: *> \param[out] Y
                    108: *> \verbatim
                    109: *>          Y is DOUBLE PRECISION array, dimension (LDY,NB)
                    110: *>          The n-by-nb matrix Y.
                    111: *> \endverbatim
                    112: *>
                    113: *> \param[in] LDY
                    114: *> \verbatim
                    115: *>          LDY is INTEGER
                    116: *>          The leading dimension of the array Y. LDY >= N.
                    117: *> \endverbatim
                    118: *
                    119: *  Authors:
                    120: *  ========
                    121: *
                    122: *> \author Univ. of Tennessee 
                    123: *> \author Univ. of California Berkeley 
                    124: *> \author Univ. of Colorado Denver 
                    125: *> \author NAG Ltd. 
                    126: *
1.12    ! bertrand  127: *> \date September 2012
1.9       bertrand  128: *
                    129: *> \ingroup doubleOTHERauxiliary
                    130: *
                    131: *> \par Further Details:
                    132: *  =====================
                    133: *>
                    134: *> \verbatim
                    135: *>
                    136: *>  The matrix Q is represented as a product of nb elementary reflectors
                    137: *>
                    138: *>     Q = H(1) H(2) . . . H(nb).
                    139: *>
                    140: *>  Each H(i) has the form
                    141: *>
                    142: *>     H(i) = I - tau * v * v**T
                    143: *>
                    144: *>  where tau is a real scalar, and v is a real vector with
                    145: *>  v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
                    146: *>  A(i+k+1:n,i), and tau in TAU(i).
                    147: *>
                    148: *>  The elements of the vectors v together form the (n-k+1)-by-nb matrix
                    149: *>  V which is needed, with T and Y, to apply the transformation to the
                    150: *>  unreduced part of the matrix, using an update of the form:
                    151: *>  A := (I - V*T*V**T) * (A - Y*V**T).
                    152: *>
                    153: *>  The contents of A on exit are illustrated by the following example
                    154: *>  with n = 7, k = 3 and nb = 2:
                    155: *>
                    156: *>     ( a   h   a   a   a )
                    157: *>     ( a   h   a   a   a )
                    158: *>     ( a   h   a   a   a )
                    159: *>     ( h   h   a   a   a )
                    160: *>     ( v1  h   a   a   a )
                    161: *>     ( v1  v2  a   a   a )
                    162: *>     ( v1  v2  a   a   a )
                    163: *>
                    164: *>  where a denotes an element of the original matrix A, h denotes a
                    165: *>  modified element of the upper Hessenberg matrix H, and vi denotes an
                    166: *>  element of the vector defining H(i).
                    167: *> \endverbatim
                    168: *>
                    169: *  =====================================================================
1.1       bertrand  170:       SUBROUTINE DLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
                    171: *
1.12    ! bertrand  172: *  -- LAPACK auxiliary routine (version 3.4.2) --
1.1       bertrand  173: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
                    174: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.12    ! bertrand  175: *     September 2012
1.1       bertrand  176: *
                    177: *     .. Scalar Arguments ..
                    178:       INTEGER            K, LDA, LDT, LDY, N, NB
                    179: *     ..
                    180: *     .. Array Arguments ..
                    181:       DOUBLE PRECISION   A( LDA, * ), T( LDT, NB ), TAU( NB ),
                    182:      $                   Y( LDY, NB )
                    183: *     ..
                    184: *
                    185: *  =====================================================================
                    186: *
                    187: *     .. Parameters ..
                    188:       DOUBLE PRECISION   ZERO, ONE
                    189:       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
                    190: *     ..
                    191: *     .. Local Scalars ..
                    192:       INTEGER            I
                    193:       DOUBLE PRECISION   EI
                    194: *     ..
                    195: *     .. External Subroutines ..
                    196:       EXTERNAL           DAXPY, DCOPY, DGEMV, DLARFG, DSCAL, DTRMV
                    197: *     ..
                    198: *     .. Intrinsic Functions ..
                    199:       INTRINSIC          MIN
                    200: *     ..
                    201: *     .. Executable Statements ..
                    202: *
                    203: *     Quick return if possible
                    204: *
                    205:       IF( N.LE.1 )
                    206:      $   RETURN
                    207: *
                    208:       DO 10 I = 1, NB
                    209:          IF( I.GT.1 ) THEN
                    210: *
                    211: *           Update A(1:n,i)
                    212: *
1.8       bertrand  213: *           Compute i-th column of A - Y * V**T
1.1       bertrand  214: *
                    215:             CALL DGEMV( 'No transpose', N, I-1, -ONE, Y, LDY,
                    216:      $                  A( K+I-1, 1 ), LDA, ONE, A( 1, I ), 1 )
                    217: *
1.8       bertrand  218: *           Apply I - V * T**T * V**T to this column (call it b) from the
1.1       bertrand  219: *           left, using the last column of T as workspace
                    220: *
                    221: *           Let  V = ( V1 )   and   b = ( b1 )   (first I-1 rows)
                    222: *                    ( V2 )             ( b2 )
                    223: *
                    224: *           where V1 is unit lower triangular
                    225: *
1.8       bertrand  226: *           w := V1**T * b1
1.1       bertrand  227: *
                    228:             CALL DCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 )
                    229:             CALL DTRMV( 'Lower', 'Transpose', 'Unit', I-1, A( K+1, 1 ),
                    230:      $                  LDA, T( 1, NB ), 1 )
                    231: *
1.8       bertrand  232: *           w := w + V2**T *b2
1.1       bertrand  233: *
                    234:             CALL DGEMV( 'Transpose', N-K-I+1, I-1, ONE, A( K+I, 1 ),
                    235:      $                  LDA, A( K+I, I ), 1, ONE, T( 1, NB ), 1 )
                    236: *
1.8       bertrand  237: *           w := T**T *w
1.1       bertrand  238: *
                    239:             CALL DTRMV( 'Upper', 'Transpose', 'Non-unit', I-1, T, LDT,
                    240:      $                  T( 1, NB ), 1 )
                    241: *
                    242: *           b2 := b2 - V2*w
                    243: *
                    244:             CALL DGEMV( 'No transpose', N-K-I+1, I-1, -ONE, A( K+I, 1 ),
                    245:      $                  LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 )
                    246: *
                    247: *           b1 := b1 - V1*w
                    248: *
                    249:             CALL DTRMV( 'Lower', 'No transpose', 'Unit', I-1,
                    250:      $                  A( K+1, 1 ), LDA, T( 1, NB ), 1 )
                    251:             CALL DAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 )
                    252: *
                    253:             A( K+I-1, I-1 ) = EI
                    254:          END IF
                    255: *
                    256: *        Generate the elementary reflector H(i) to annihilate
                    257: *        A(k+i+1:n,i)
                    258: *
                    259:          CALL DLARFG( N-K-I+1, A( K+I, I ), A( MIN( K+I+1, N ), I ), 1,
                    260:      $                TAU( I ) )
                    261:          EI = A( K+I, I )
                    262:          A( K+I, I ) = ONE
                    263: *
                    264: *        Compute  Y(1:n,i)
                    265: *
                    266:          CALL DGEMV( 'No transpose', N, N-K-I+1, ONE, A( 1, I+1 ), LDA,
                    267:      $               A( K+I, I ), 1, ZERO, Y( 1, I ), 1 )
                    268:          CALL DGEMV( 'Transpose', N-K-I+1, I-1, ONE, A( K+I, 1 ), LDA,
                    269:      $               A( K+I, I ), 1, ZERO, T( 1, I ), 1 )
                    270:          CALL DGEMV( 'No transpose', N, I-1, -ONE, Y, LDY, T( 1, I ), 1,
                    271:      $               ONE, Y( 1, I ), 1 )
                    272:          CALL DSCAL( N, TAU( I ), Y( 1, I ), 1 )
                    273: *
                    274: *        Compute T(1:i,i)
                    275: *
                    276:          CALL DSCAL( I-1, -TAU( I ), T( 1, I ), 1 )
                    277:          CALL DTRMV( 'Upper', 'No transpose', 'Non-unit', I-1, T, LDT,
                    278:      $               T( 1, I ), 1 )
                    279:          T( I, I ) = TAU( I )
                    280: *
                    281:    10 CONTINUE
                    282:       A( K+NB, NB ) = EI
                    283: *
                    284:       RETURN
                    285: *
                    286: *     End of DLAHRD
                    287: *
                    288:       END

CVSweb interface <joel.bertrand@systella.fr>