Annotation of rpl/lapack/lapack/dlahrd.f, revision 1.15

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: *>
1.15    ! bertrand   37: *> This routine is deprecated and has been replaced by routine DLAHR2.
        !            38: *>
1.9       bertrand   39: *> DLAHRD reduces the first NB columns of a real general n-by-(n-k+1)
                     40: *> matrix A so that elements below the k-th subdiagonal are zero. The
                     41: *> reduction is performed by an orthogonal similarity transformation
                     42: *> Q**T * A * Q. The routine returns the matrices V and T which determine
                     43: *> Q as a block reflector I - V*T*V**T, and also the matrix Y = A * V * T.
                     44: *> \endverbatim
                     45: *
                     46: *  Arguments:
                     47: *  ==========
                     48: *
                     49: *> \param[in] N
                     50: *> \verbatim
                     51: *>          N is INTEGER
                     52: *>          The order of the matrix A.
                     53: *> \endverbatim
                     54: *>
                     55: *> \param[in] K
                     56: *> \verbatim
                     57: *>          K is INTEGER
                     58: *>          The offset for the reduction. Elements below the k-th
                     59: *>          subdiagonal in the first NB columns are reduced to zero.
                     60: *> \endverbatim
                     61: *>
                     62: *> \param[in] NB
                     63: *> \verbatim
                     64: *>          NB is INTEGER
                     65: *>          The number of columns to be reduced.
                     66: *> \endverbatim
                     67: *>
                     68: *> \param[in,out] A
                     69: *> \verbatim
                     70: *>          A is DOUBLE PRECISION array, dimension (LDA,N-K+1)
                     71: *>          On entry, the n-by-(n-k+1) general matrix A.
                     72: *>          On exit, the elements on and above the k-th subdiagonal in
                     73: *>          the first NB columns are overwritten with the corresponding
                     74: *>          elements of the reduced matrix; the elements below the k-th
                     75: *>          subdiagonal, with the array TAU, represent the matrix Q as a
                     76: *>          product of elementary reflectors. The other columns of A are
                     77: *>          unchanged. See Further Details.
                     78: *> \endverbatim
                     79: *>
                     80: *> \param[in] LDA
                     81: *> \verbatim
                     82: *>          LDA is INTEGER
                     83: *>          The leading dimension of the array A.  LDA >= max(1,N).
                     84: *> \endverbatim
                     85: *>
                     86: *> \param[out] TAU
                     87: *> \verbatim
                     88: *>          TAU is DOUBLE PRECISION array, dimension (NB)
                     89: *>          The scalar factors of the elementary reflectors. See Further
                     90: *>          Details.
                     91: *> \endverbatim
                     92: *>
                     93: *> \param[out] T
                     94: *> \verbatim
                     95: *>          T is DOUBLE PRECISION array, dimension (LDT,NB)
                     96: *>          The upper triangular matrix T.
                     97: *> \endverbatim
                     98: *>
                     99: *> \param[in] LDT
                    100: *> \verbatim
                    101: *>          LDT is INTEGER
                    102: *>          The leading dimension of the array T.  LDT >= NB.
                    103: *> \endverbatim
                    104: *>
                    105: *> \param[out] Y
                    106: *> \verbatim
                    107: *>          Y is DOUBLE PRECISION array, dimension (LDY,NB)
                    108: *>          The n-by-nb matrix Y.
                    109: *> \endverbatim
                    110: *>
                    111: *> \param[in] LDY
                    112: *> \verbatim
                    113: *>          LDY is INTEGER
                    114: *>          The leading dimension of the array Y. LDY >= N.
                    115: *> \endverbatim
                    116: *
                    117: *  Authors:
                    118: *  ========
                    119: *
                    120: *> \author Univ. of Tennessee 
                    121: *> \author Univ. of California Berkeley 
                    122: *> \author Univ. of Colorado Denver 
                    123: *> \author NAG Ltd. 
                    124: *
1.15    ! bertrand  125: *> \date November 2015
1.9       bertrand  126: *
                    127: *> \ingroup doubleOTHERauxiliary
                    128: *
                    129: *> \par Further Details:
                    130: *  =====================
                    131: *>
                    132: *> \verbatim
                    133: *>
                    134: *>  The matrix Q is represented as a product of nb elementary reflectors
                    135: *>
                    136: *>     Q = H(1) H(2) . . . H(nb).
                    137: *>
                    138: *>  Each H(i) has the form
                    139: *>
                    140: *>     H(i) = I - tau * v * v**T
                    141: *>
                    142: *>  where tau is a real scalar, and v is a real vector with
                    143: *>  v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
                    144: *>  A(i+k+1:n,i), and tau in TAU(i).
                    145: *>
                    146: *>  The elements of the vectors v together form the (n-k+1)-by-nb matrix
                    147: *>  V which is needed, with T and Y, to apply the transformation to the
                    148: *>  unreduced part of the matrix, using an update of the form:
                    149: *>  A := (I - V*T*V**T) * (A - Y*V**T).
                    150: *>
                    151: *>  The contents of A on exit are illustrated by the following example
                    152: *>  with n = 7, k = 3 and nb = 2:
                    153: *>
                    154: *>     ( a   h   a   a   a )
                    155: *>     ( a   h   a   a   a )
                    156: *>     ( a   h   a   a   a )
                    157: *>     ( h   h   a   a   a )
                    158: *>     ( v1  h   a   a   a )
                    159: *>     ( v1  v2  a   a   a )
                    160: *>     ( v1  v2  a   a   a )
                    161: *>
                    162: *>  where a denotes an element of the original matrix A, h denotes a
                    163: *>  modified element of the upper Hessenberg matrix H, and vi denotes an
                    164: *>  element of the vector defining H(i).
                    165: *> \endverbatim
                    166: *>
                    167: *  =====================================================================
1.1       bertrand  168:       SUBROUTINE DLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
                    169: *
1.15    ! bertrand  170: *  -- LAPACK auxiliary routine (version 3.6.0) --
1.1       bertrand  171: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
                    172: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.15    ! bertrand  173: *     November 2015
1.1       bertrand  174: *
                    175: *     .. Scalar Arguments ..
                    176:       INTEGER            K, LDA, LDT, LDY, N, NB
                    177: *     ..
                    178: *     .. Array Arguments ..
                    179:       DOUBLE PRECISION   A( LDA, * ), T( LDT, NB ), TAU( NB ),
                    180:      $                   Y( LDY, NB )
                    181: *     ..
                    182: *
                    183: *  =====================================================================
                    184: *
                    185: *     .. Parameters ..
                    186:       DOUBLE PRECISION   ZERO, ONE
                    187:       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
                    188: *     ..
                    189: *     .. Local Scalars ..
                    190:       INTEGER            I
                    191:       DOUBLE PRECISION   EI
                    192: *     ..
                    193: *     .. External Subroutines ..
                    194:       EXTERNAL           DAXPY, DCOPY, DGEMV, DLARFG, DSCAL, DTRMV
                    195: *     ..
                    196: *     .. Intrinsic Functions ..
                    197:       INTRINSIC          MIN
                    198: *     ..
                    199: *     .. Executable Statements ..
                    200: *
                    201: *     Quick return if possible
                    202: *
                    203:       IF( N.LE.1 )
                    204:      $   RETURN
                    205: *
                    206:       DO 10 I = 1, NB
                    207:          IF( I.GT.1 ) THEN
                    208: *
                    209: *           Update A(1:n,i)
                    210: *
1.8       bertrand  211: *           Compute i-th column of A - Y * V**T
1.1       bertrand  212: *
                    213:             CALL DGEMV( 'No transpose', N, I-1, -ONE, Y, LDY,
                    214:      $                  A( K+I-1, 1 ), LDA, ONE, A( 1, I ), 1 )
                    215: *
1.8       bertrand  216: *           Apply I - V * T**T * V**T to this column (call it b) from the
1.1       bertrand  217: *           left, using the last column of T as workspace
                    218: *
                    219: *           Let  V = ( V1 )   and   b = ( b1 )   (first I-1 rows)
                    220: *                    ( V2 )             ( b2 )
                    221: *
                    222: *           where V1 is unit lower triangular
                    223: *
1.8       bertrand  224: *           w := V1**T * b1
1.1       bertrand  225: *
                    226:             CALL DCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 )
                    227:             CALL DTRMV( 'Lower', 'Transpose', 'Unit', I-1, A( K+1, 1 ),
                    228:      $                  LDA, T( 1, NB ), 1 )
                    229: *
1.8       bertrand  230: *           w := w + V2**T *b2
1.1       bertrand  231: *
                    232:             CALL DGEMV( 'Transpose', N-K-I+1, I-1, ONE, A( K+I, 1 ),
                    233:      $                  LDA, A( K+I, I ), 1, ONE, T( 1, NB ), 1 )
                    234: *
1.8       bertrand  235: *           w := T**T *w
1.1       bertrand  236: *
                    237:             CALL DTRMV( 'Upper', 'Transpose', 'Non-unit', I-1, T, LDT,
                    238:      $                  T( 1, NB ), 1 )
                    239: *
                    240: *           b2 := b2 - V2*w
                    241: *
                    242:             CALL DGEMV( 'No transpose', N-K-I+1, I-1, -ONE, A( K+I, 1 ),
                    243:      $                  LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 )
                    244: *
                    245: *           b1 := b1 - V1*w
                    246: *
                    247:             CALL DTRMV( 'Lower', 'No transpose', 'Unit', I-1,
                    248:      $                  A( K+1, 1 ), LDA, T( 1, NB ), 1 )
                    249:             CALL DAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 )
                    250: *
                    251:             A( K+I-1, I-1 ) = EI
                    252:          END IF
                    253: *
                    254: *        Generate the elementary reflector H(i) to annihilate
                    255: *        A(k+i+1:n,i)
                    256: *
                    257:          CALL DLARFG( N-K-I+1, A( K+I, I ), A( MIN( K+I+1, N ), I ), 1,
                    258:      $                TAU( I ) )
                    259:          EI = A( K+I, I )
                    260:          A( K+I, I ) = ONE
                    261: *
                    262: *        Compute  Y(1:n,i)
                    263: *
                    264:          CALL DGEMV( 'No transpose', N, N-K-I+1, ONE, A( 1, I+1 ), LDA,
                    265:      $               A( K+I, I ), 1, ZERO, Y( 1, I ), 1 )
                    266:          CALL DGEMV( 'Transpose', N-K-I+1, I-1, ONE, A( K+I, 1 ), LDA,
                    267:      $               A( K+I, I ), 1, ZERO, T( 1, I ), 1 )
                    268:          CALL DGEMV( 'No transpose', N, I-1, -ONE, Y, LDY, T( 1, I ), 1,
                    269:      $               ONE, Y( 1, I ), 1 )
                    270:          CALL DSCAL( N, TAU( I ), Y( 1, I ), 1 )
                    271: *
                    272: *        Compute T(1:i,i)
                    273: *
                    274:          CALL DSCAL( I-1, -TAU( I ), T( 1, I ), 1 )
                    275:          CALL DTRMV( 'Upper', 'No transpose', 'Non-unit', I-1, T, LDT,
                    276:      $               T( 1, I ), 1 )
                    277:          T( I, I ) = TAU( I )
                    278: *
                    279:    10 CONTINUE
                    280:       A( K+NB, NB ) = EI
                    281: *
                    282:       RETURN
                    283: *
                    284: *     End of DLAHRD
                    285: *
                    286:       END

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