Annotation of rpl/lapack/lapack/zlahr2.f, revision 1.19

1.12      bertrand    1: *> \brief \b ZLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified 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: *
1.16      bertrand    5: * Online html documentation available at
                      6: *            http://www.netlib.org/lapack/explore-html/
1.9       bertrand    7: *
                      8: *> \htmlonly
1.16      bertrand    9: *> Download ZLAHR2 + dependencies
                     10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlahr2.f">
                     11: *> [TGZ]</a>
                     12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlahr2.f">
                     13: *> [ZIP]</a>
                     14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlahr2.f">
1.9       bertrand   15: *> [TXT]</a>
1.16      bertrand   16: *> \endhtmlonly
1.9       bertrand   17: *
                     18: *  Definition:
                     19: *  ===========
                     20: *
                     21: *       SUBROUTINE ZLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
1.16      bertrand   22: *
1.9       bertrand   23: *       .. Scalar Arguments ..
                     24: *       INTEGER            K, LDA, LDT, LDY, N, NB
                     25: *       ..
                     26: *       .. Array Arguments ..
                     27: *       COMPLEX*16        A( LDA, * ), T( LDT, NB ), TAU( NB ),
                     28: *      $                   Y( LDY, NB )
                     29: *       ..
1.16      bertrand   30: *
1.9       bertrand   31: *
                     32: *> \par Purpose:
                     33: *  =============
                     34: *>
                     35: *> \verbatim
                     36: *>
                     37: *> ZLAHR2 reduces the first NB columns of A complex 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 unitary similarity transformation
                     40: *> Q**H * A * Q. The routine returns the matrices V and T which determine
                     41: *> Q as a block reflector I - V*T*V**H, and also the matrix Y = A * V * T.
                     42: *>
                     43: *> This is an auxiliary routine called by ZGEHRD.
                     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: *>          K < N.
                     61: *> \endverbatim
                     62: *>
                     63: *> \param[in] NB
                     64: *> \verbatim
                     65: *>          NB is INTEGER
                     66: *>          The number of columns to be reduced.
                     67: *> \endverbatim
                     68: *>
                     69: *> \param[in,out] A
                     70: *> \verbatim
                     71: *>          A is COMPLEX*16 array, dimension (LDA,N-K+1)
                     72: *>          On entry, the n-by-(n-k+1) general matrix A.
                     73: *>          On exit, the elements on and above the k-th subdiagonal in
                     74: *>          the first NB columns are overwritten with the corresponding
                     75: *>          elements of the reduced matrix; the elements below the k-th
                     76: *>          subdiagonal, with the array TAU, represent the matrix Q as a
                     77: *>          product of elementary reflectors. The other columns of A are
                     78: *>          unchanged. See Further Details.
                     79: *> \endverbatim
                     80: *>
                     81: *> \param[in] LDA
                     82: *> \verbatim
                     83: *>          LDA is INTEGER
                     84: *>          The leading dimension of the array A.  LDA >= max(1,N).
                     85: *> \endverbatim
                     86: *>
                     87: *> \param[out] TAU
                     88: *> \verbatim
                     89: *>          TAU is COMPLEX*16 array, dimension (NB)
                     90: *>          The scalar factors of the elementary reflectors. See Further
                     91: *>          Details.
                     92: *> \endverbatim
                     93: *>
                     94: *> \param[out] T
                     95: *> \verbatim
                     96: *>          T is COMPLEX*16 array, dimension (LDT,NB)
                     97: *>          The upper triangular matrix T.
                     98: *> \endverbatim
                     99: *>
                    100: *> \param[in] LDT
                    101: *> \verbatim
                    102: *>          LDT is INTEGER
                    103: *>          The leading dimension of the array T.  LDT >= NB.
                    104: *> \endverbatim
                    105: *>
                    106: *> \param[out] Y
                    107: *> \verbatim
                    108: *>          Y is COMPLEX*16 array, dimension (LDY,NB)
                    109: *>          The n-by-nb matrix Y.
                    110: *> \endverbatim
                    111: *>
                    112: *> \param[in] LDY
                    113: *> \verbatim
                    114: *>          LDY is INTEGER
                    115: *>          The leading dimension of the array Y. LDY >= N.
                    116: *> \endverbatim
                    117: *
                    118: *  Authors:
                    119: *  ========
                    120: *
1.16      bertrand  121: *> \author Univ. of Tennessee
                    122: *> \author Univ. of California Berkeley
                    123: *> \author Univ. of Colorado Denver
                    124: *> \author NAG Ltd.
1.9       bertrand  125: *
                    126: *> \ingroup complex16OTHERauxiliary
                    127: *
                    128: *> \par Further Details:
                    129: *  =====================
                    130: *>
                    131: *> \verbatim
                    132: *>
                    133: *>  The matrix Q is represented as a product of nb elementary reflectors
                    134: *>
                    135: *>     Q = H(1) H(2) . . . H(nb).
                    136: *>
                    137: *>  Each H(i) has the form
                    138: *>
                    139: *>     H(i) = I - tau * v * v**H
                    140: *>
                    141: *>  where tau is a complex scalar, and v is a complex vector with
                    142: *>  v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
                    143: *>  A(i+k+1:n,i), and tau in TAU(i).
                    144: *>
                    145: *>  The elements of the vectors v together form the (n-k+1)-by-nb matrix
                    146: *>  V which is needed, with T and Y, to apply the transformation to the
                    147: *>  unreduced part of the matrix, using an update of the form:
                    148: *>  A := (I - V*T*V**H) * (A - Y*V**H).
                    149: *>
                    150: *>  The contents of A on exit are illustrated by the following example
                    151: *>  with n = 7, k = 3 and nb = 2:
                    152: *>
                    153: *>     ( a   a   a   a   a )
                    154: *>     ( a   a   a   a   a )
                    155: *>     ( a   a   a   a   a )
                    156: *>     ( h   h   a   a   a )
                    157: *>     ( v1  h   a   a   a )
                    158: *>     ( v1  v2  a   a   a )
                    159: *>     ( v1  v2  a   a   a )
                    160: *>
                    161: *>  where a denotes an element of the original matrix A, h denotes a
                    162: *>  modified element of the upper Hessenberg matrix H, and vi denotes an
                    163: *>  element of the vector defining H(i).
                    164: *>
1.19    ! bertrand  165: *>  This subroutine is a slight modification of LAPACK-3.0's ZLAHRD
1.9       bertrand  166: *>  incorporating improvements proposed by Quintana-Orti and Van de
                    167: *>  Gejin. Note that the entries of A(1:K,2:NB) differ from those
1.19    ! bertrand  168: *>  returned by the original LAPACK-3.0's ZLAHRD routine. (This
        !           169: *>  subroutine is not backward compatible with LAPACK-3.0's ZLAHRD.)
1.9       bertrand  170: *> \endverbatim
                    171: *
                    172: *> \par References:
                    173: *  ================
                    174: *>
                    175: *>  Gregorio Quintana-Orti and Robert van de Geijn, "Improving the
                    176: *>  performance of reduction to Hessenberg form," ACM Transactions on
                    177: *>  Mathematical Software, 32(2):180-194, June 2006.
                    178: *>
                    179: *  =====================================================================
1.1       bertrand  180:       SUBROUTINE ZLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
                    181: *
1.19    ! bertrand  182: *  -- LAPACK auxiliary routine --
1.1       bertrand  183: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
                    184: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
                    185: *
                    186: *     .. Scalar Arguments ..
                    187:       INTEGER            K, LDA, LDT, LDY, N, NB
                    188: *     ..
                    189: *     .. Array Arguments ..
                    190:       COMPLEX*16        A( LDA, * ), T( LDT, NB ), TAU( NB ),
                    191:      $                   Y( LDY, NB )
                    192: *     ..
                    193: *
                    194: *  =====================================================================
                    195: *
                    196: *     .. Parameters ..
                    197:       COMPLEX*16        ZERO, ONE
1.16      bertrand  198:       PARAMETER          ( ZERO = ( 0.0D+0, 0.0D+0 ),
1.1       bertrand  199:      $                     ONE = ( 1.0D+0, 0.0D+0 ) )
                    200: *     ..
                    201: *     .. Local Scalars ..
                    202:       INTEGER            I
                    203:       COMPLEX*16        EI
                    204: *     ..
                    205: *     .. External Subroutines ..
                    206:       EXTERNAL           ZAXPY, ZCOPY, ZGEMM, ZGEMV, ZLACPY,
                    207:      $                   ZLARFG, ZSCAL, ZTRMM, ZTRMV, ZLACGV
                    208: *     ..
                    209: *     .. Intrinsic Functions ..
                    210:       INTRINSIC          MIN
                    211: *     ..
                    212: *     .. Executable Statements ..
                    213: *
                    214: *     Quick return if possible
                    215: *
                    216:       IF( N.LE.1 )
                    217:      $   RETURN
                    218: *
                    219:       DO 10 I = 1, NB
                    220:          IF( I.GT.1 ) THEN
                    221: *
                    222: *           Update A(K+1:N,I)
                    223: *
1.8       bertrand  224: *           Update I-th column of A - Y * V**H
1.1       bertrand  225: *
1.16      bertrand  226:             CALL ZLACGV( I-1, A( K+I-1, 1 ), LDA )
1.1       bertrand  227:             CALL ZGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE, Y(K+1,1), LDY,
                    228:      $                  A( K+I-1, 1 ), LDA, ONE, A( K+1, I ), 1 )
1.16      bertrand  229:             CALL ZLACGV( I-1, A( K+I-1, 1 ), LDA )
1.1       bertrand  230: *
1.8       bertrand  231: *           Apply I - V * T**H * V**H to this column (call it b) from the
1.1       bertrand  232: *           left, using the last column of T as workspace
                    233: *
                    234: *           Let  V = ( V1 )   and   b = ( b1 )   (first I-1 rows)
                    235: *                    ( V2 )             ( b2 )
                    236: *
                    237: *           where V1 is unit lower triangular
                    238: *
1.8       bertrand  239: *           w := V1**H * b1
1.1       bertrand  240: *
                    241:             CALL ZCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 )
1.16      bertrand  242:             CALL ZTRMV( 'Lower', 'Conjugate transpose', 'UNIT',
1.1       bertrand  243:      $                  I-1, A( K+1, 1 ),
                    244:      $                  LDA, T( 1, NB ), 1 )
                    245: *
1.8       bertrand  246: *           w := w + V2**H * b2
1.1       bertrand  247: *
1.16      bertrand  248:             CALL ZGEMV( 'Conjugate transpose', N-K-I+1, I-1,
1.1       bertrand  249:      $                  ONE, A( K+I, 1 ),
                    250:      $                  LDA, A( K+I, I ), 1, ONE, T( 1, NB ), 1 )
                    251: *
1.8       bertrand  252: *           w := T**H * w
1.1       bertrand  253: *
1.16      bertrand  254:             CALL ZTRMV( 'Upper', 'Conjugate transpose', 'NON-UNIT',
1.1       bertrand  255:      $                  I-1, T, LDT,
                    256:      $                  T( 1, NB ), 1 )
                    257: *
                    258: *           b2 := b2 - V2*w
                    259: *
1.16      bertrand  260:             CALL ZGEMV( 'NO TRANSPOSE', N-K-I+1, I-1, -ONE,
1.1       bertrand  261:      $                  A( K+I, 1 ),
                    262:      $                  LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 )
                    263: *
                    264: *           b1 := b1 - V1*w
                    265: *
1.16      bertrand  266:             CALL ZTRMV( 'Lower', 'NO TRANSPOSE',
1.1       bertrand  267:      $                  'UNIT', I-1,
                    268:      $                  A( K+1, 1 ), LDA, T( 1, NB ), 1 )
                    269:             CALL ZAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 )
                    270: *
                    271:             A( K+I-1, I-1 ) = EI
                    272:          END IF
                    273: *
                    274: *        Generate the elementary reflector H(I) to annihilate
                    275: *        A(K+I+1:N,I)
                    276: *
                    277:          CALL ZLARFG( N-K-I+1, A( K+I, I ), A( MIN( K+I+1, N ), I ), 1,
                    278:      $                TAU( I ) )
                    279:          EI = A( K+I, I )
                    280:          A( K+I, I ) = ONE
                    281: *
                    282: *        Compute  Y(K+1:N,I)
                    283: *
1.16      bertrand  284:          CALL ZGEMV( 'NO TRANSPOSE', N-K, N-K-I+1,
1.1       bertrand  285:      $               ONE, A( K+1, I+1 ),
                    286:      $               LDA, A( K+I, I ), 1, ZERO, Y( K+1, I ), 1 )
1.16      bertrand  287:          CALL ZGEMV( 'Conjugate transpose', N-K-I+1, I-1,
1.1       bertrand  288:      $               ONE, A( K+I, 1 ), LDA,
                    289:      $               A( K+I, I ), 1, ZERO, T( 1, I ), 1 )
1.16      bertrand  290:          CALL ZGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE,
1.1       bertrand  291:      $               Y( K+1, 1 ), LDY,
                    292:      $               T( 1, I ), 1, ONE, Y( K+1, I ), 1 )
                    293:          CALL ZSCAL( N-K, TAU( I ), Y( K+1, I ), 1 )
                    294: *
                    295: *        Compute T(1:I,I)
                    296: *
                    297:          CALL ZSCAL( I-1, -TAU( I ), T( 1, I ), 1 )
1.16      bertrand  298:          CALL ZTRMV( 'Upper', 'No Transpose', 'NON-UNIT',
1.1       bertrand  299:      $               I-1, T, LDT,
                    300:      $               T( 1, I ), 1 )
                    301:          T( I, I ) = TAU( I )
                    302: *
                    303:    10 CONTINUE
                    304:       A( K+NB, NB ) = EI
                    305: *
                    306: *     Compute Y(1:K,1:NB)
                    307: *
                    308:       CALL ZLACPY( 'ALL', K, NB, A( 1, 2 ), LDA, Y, LDY )
1.16      bertrand  309:       CALL ZTRMM( 'RIGHT', 'Lower', 'NO TRANSPOSE',
1.1       bertrand  310:      $            'UNIT', K, NB,
                    311:      $            ONE, A( K+1, 1 ), LDA, Y, LDY )
                    312:       IF( N.GT.K+NB )
1.16      bertrand  313:      $   CALL ZGEMM( 'NO TRANSPOSE', 'NO TRANSPOSE', K,
1.1       bertrand  314:      $               NB, N-K-NB, ONE,
                    315:      $               A( 1, 2+NB ), LDA, A( K+1+NB, 1 ), LDA, ONE, Y,
                    316:      $               LDY )
1.16      bertrand  317:       CALL ZTRMM( 'RIGHT', 'Upper', 'NO TRANSPOSE',
1.1       bertrand  318:      $            'NON-UNIT', K, NB,
                    319:      $            ONE, T, LDT, Y, LDY )
                    320: *
                    321:       RETURN
                    322: *
                    323: *     End of ZLAHR2
                    324: *
                    325:       END

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