Annotation of rpl/lapack/lapack/dlahr2.f, revision 1.1.1.1

1.1       bertrand    1:       SUBROUTINE DLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
                      2: *
                      3: *  -- LAPACK auxiliary routine (version 3.2.1)                        --
                      4: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
                      5: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
                      6: *  -- April 2009                                                      --
                      7: *
                      8: *     .. Scalar Arguments ..
                      9:       INTEGER            K, LDA, LDT, LDY, N, NB
                     10: *     ..
                     11: *     .. Array Arguments ..
                     12:       DOUBLE PRECISION  A( LDA, * ), T( LDT, NB ), TAU( NB ),
                     13:      $                   Y( LDY, NB )
                     14: *     ..
                     15: *
                     16: *  Purpose
                     17: *  =======
                     18: *
                     19: *  DLAHR2 reduces the first NB columns of A real general n-BY-(n-k+1)
                     20: *  matrix A so that elements below the k-th subdiagonal are zero. The
                     21: *  reduction is performed by an orthogonal similarity transformation
                     22: *  Q' * A * Q. The routine returns the matrices V and T which determine
                     23: *  Q as a block reflector I - V*T*V', and also the matrix Y = A * V * T.
                     24: *
                     25: *  This is an auxiliary routine called by DGEHRD.
                     26: *
                     27: *  Arguments
                     28: *  =========
                     29: *
                     30: *  N       (input) INTEGER
                     31: *          The order of the matrix A.
                     32: *
                     33: *  K       (input) INTEGER
                     34: *          The offset for the reduction. Elements below the k-th
                     35: *          subdiagonal in the first NB columns are reduced to zero.
                     36: *          K < N.
                     37: *
                     38: *  NB      (input) INTEGER
                     39: *          The number of columns to be reduced.
                     40: *
                     41: *  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N-K+1)
                     42: *          On entry, the n-by-(n-k+1) general matrix A.
                     43: *          On exit, the elements on and above the k-th subdiagonal in
                     44: *          the first NB columns are overwritten with the corresponding
                     45: *          elements of the reduced matrix; the elements below the k-th
                     46: *          subdiagonal, with the array TAU, represent the matrix Q as a
                     47: *          product of elementary reflectors. The other columns of A are
                     48: *          unchanged. See Further Details.
                     49: *
                     50: *  LDA     (input) INTEGER
                     51: *          The leading dimension of the array A.  LDA >= max(1,N).
                     52: *
                     53: *  TAU     (output) DOUBLE PRECISION array, dimension (NB)
                     54: *          The scalar factors of the elementary reflectors. See Further
                     55: *          Details.
                     56: *
                     57: *  T       (output) DOUBLE PRECISION array, dimension (LDT,NB)
                     58: *          The upper triangular matrix T.
                     59: *
                     60: *  LDT     (input) INTEGER
                     61: *          The leading dimension of the array T.  LDT >= NB.
                     62: *
                     63: *  Y       (output) DOUBLE PRECISION array, dimension (LDY,NB)
                     64: *          The n-by-nb matrix Y.
                     65: *
                     66: *  LDY     (input) INTEGER
                     67: *          The leading dimension of the array Y. LDY >= N.
                     68: *
                     69: *  Further Details
                     70: *  ===============
                     71: *
                     72: *  The matrix Q is represented as a product of nb elementary reflectors
                     73: *
                     74: *     Q = H(1) H(2) . . . H(nb).
                     75: *
                     76: *  Each H(i) has the form
                     77: *
                     78: *     H(i) = I - tau * v * v'
                     79: *
                     80: *  where tau is a real scalar, and v is a real vector with
                     81: *  v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
                     82: *  A(i+k+1:n,i), and tau in TAU(i).
                     83: *
                     84: *  The elements of the vectors v together form the (n-k+1)-by-nb matrix
                     85: *  V which is needed, with T and Y, to apply the transformation to the
                     86: *  unreduced part of the matrix, using an update of the form:
                     87: *  A := (I - V*T*V') * (A - Y*V').
                     88: *
                     89: *  The contents of A on exit are illustrated by the following example
                     90: *  with n = 7, k = 3 and nb = 2:
                     91: *
                     92: *     ( a   a   a   a   a )
                     93: *     ( a   a   a   a   a )
                     94: *     ( a   a   a   a   a )
                     95: *     ( h   h   a   a   a )
                     96: *     ( v1  h   a   a   a )
                     97: *     ( v1  v2  a   a   a )
                     98: *     ( v1  v2  a   a   a )
                     99: *
                    100: *  where a denotes an element of the original matrix A, h denotes a
                    101: *  modified element of the upper Hessenberg matrix H, and vi denotes an
                    102: *  element of the vector defining H(i).
                    103: *
                    104: *  This subroutine is a slight modification of LAPACK-3.0's DLAHRD
                    105: *  incorporating improvements proposed by Quintana-Orti and Van de
                    106: *  Gejin. Note that the entries of A(1:K,2:NB) differ from those
                    107: *  returned by the original LAPACK-3.0's DLAHRD routine. (This
                    108: *  subroutine is not backward compatible with LAPACK-3.0's DLAHRD.)
                    109: *
                    110: *  References
                    111: *  ==========
                    112: *
                    113: *  Gregorio Quintana-Orti and Robert van de Geijn, "Improving the
                    114: *  performance of reduction to Hessenberg form," ACM Transactions on
                    115: *  Mathematical Software, 32(2):180-194, June 2006.
                    116: *
                    117: *  =====================================================================
                    118: *
                    119: *     .. Parameters ..
                    120:       DOUBLE PRECISION  ZERO, ONE
                    121:       PARAMETER          ( ZERO = 0.0D+0, 
                    122:      $                     ONE = 1.0D+0 )
                    123: *     ..
                    124: *     .. Local Scalars ..
                    125:       INTEGER            I
                    126:       DOUBLE PRECISION  EI
                    127: *     ..
                    128: *     .. External Subroutines ..
                    129:       EXTERNAL           DAXPY, DCOPY, DGEMM, DGEMV, DLACPY,
                    130:      $                   DLARFG, DSCAL, DTRMM, DTRMV
                    131: *     ..
                    132: *     .. Intrinsic Functions ..
                    133:       INTRINSIC          MIN
                    134: *     ..
                    135: *     .. Executable Statements ..
                    136: *
                    137: *     Quick return if possible
                    138: *
                    139:       IF( N.LE.1 )
                    140:      $   RETURN
                    141: *
                    142:       DO 10 I = 1, NB
                    143:          IF( I.GT.1 ) THEN
                    144: *
                    145: *           Update A(K+1:N,I)
                    146: *
                    147: *           Update I-th column of A - Y * V'
                    148: *
                    149:             CALL DGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE, Y(K+1,1), LDY,
                    150:      $                  A( K+I-1, 1 ), LDA, ONE, A( K+1, I ), 1 )
                    151: *
                    152: *           Apply I - V * T' * V' to this column (call it b) from the
                    153: *           left, using the last column of T as workspace
                    154: *
                    155: *           Let  V = ( V1 )   and   b = ( b1 )   (first I-1 rows)
                    156: *                    ( V2 )             ( b2 )
                    157: *
                    158: *           where V1 is unit lower triangular
                    159: *
                    160: *           w := V1' * b1
                    161: *
                    162:             CALL DCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 )
                    163:             CALL DTRMV( 'Lower', 'Transpose', 'UNIT', 
                    164:      $                  I-1, A( K+1, 1 ),
                    165:      $                  LDA, T( 1, NB ), 1 )
                    166: *
                    167: *           w := w + V2'*b2
                    168: *
                    169:             CALL DGEMV( 'Transpose', N-K-I+1, I-1, 
                    170:      $                  ONE, A( K+I, 1 ),
                    171:      $                  LDA, A( K+I, I ), 1, ONE, T( 1, NB ), 1 )
                    172: *
                    173: *           w := T'*w
                    174: *
                    175:             CALL DTRMV( 'Upper', 'Transpose', 'NON-UNIT', 
                    176:      $                  I-1, T, LDT,
                    177:      $                  T( 1, NB ), 1 )
                    178: *
                    179: *           b2 := b2 - V2*w
                    180: *
                    181:             CALL DGEMV( 'NO TRANSPOSE', N-K-I+1, I-1, -ONE, 
                    182:      $                  A( K+I, 1 ),
                    183:      $                  LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 )
                    184: *
                    185: *           b1 := b1 - V1*w
                    186: *
                    187:             CALL DTRMV( 'Lower', 'NO TRANSPOSE', 
                    188:      $                  'UNIT', I-1,
                    189:      $                  A( K+1, 1 ), LDA, T( 1, NB ), 1 )
                    190:             CALL DAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 )
                    191: *
                    192:             A( K+I-1, I-1 ) = EI
                    193:          END IF
                    194: *
                    195: *        Generate the elementary reflector H(I) to annihilate
                    196: *        A(K+I+1:N,I)
                    197: *
                    198:          CALL DLARFG( N-K-I+1, A( K+I, I ), A( MIN( K+I+1, N ), I ), 1,
                    199:      $                TAU( I ) )
                    200:          EI = A( K+I, I )
                    201:          A( K+I, I ) = ONE
                    202: *
                    203: *        Compute  Y(K+1:N,I)
                    204: *
                    205:          CALL DGEMV( 'NO TRANSPOSE', N-K, N-K-I+1, 
                    206:      $               ONE, A( K+1, I+1 ),
                    207:      $               LDA, A( K+I, I ), 1, ZERO, Y( K+1, I ), 1 )
                    208:          CALL DGEMV( 'Transpose', N-K-I+1, I-1, 
                    209:      $               ONE, A( K+I, 1 ), LDA,
                    210:      $               A( K+I, I ), 1, ZERO, T( 1, I ), 1 )
                    211:          CALL DGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE, 
                    212:      $               Y( K+1, 1 ), LDY,
                    213:      $               T( 1, I ), 1, ONE, Y( K+1, I ), 1 )
                    214:          CALL DSCAL( N-K, TAU( I ), Y( K+1, I ), 1 )
                    215: *
                    216: *        Compute T(1:I,I)
                    217: *
                    218:          CALL DSCAL( I-1, -TAU( I ), T( 1, I ), 1 )
                    219:          CALL DTRMV( 'Upper', 'No Transpose', 'NON-UNIT', 
                    220:      $               I-1, T, LDT,
                    221:      $               T( 1, I ), 1 )
                    222:          T( I, I ) = TAU( I )
                    223: *
                    224:    10 CONTINUE
                    225:       A( K+NB, NB ) = EI
                    226: *
                    227: *     Compute Y(1:K,1:NB)
                    228: *
                    229:       CALL DLACPY( 'ALL', K, NB, A( 1, 2 ), LDA, Y, LDY )
                    230:       CALL DTRMM( 'RIGHT', 'Lower', 'NO TRANSPOSE', 
                    231:      $            'UNIT', K, NB,
                    232:      $            ONE, A( K+1, 1 ), LDA, Y, LDY )
                    233:       IF( N.GT.K+NB )
                    234:      $   CALL DGEMM( 'NO TRANSPOSE', 'NO TRANSPOSE', K, 
                    235:      $               NB, N-K-NB, ONE,
                    236:      $               A( 1, 2+NB ), LDA, A( K+1+NB, 1 ), LDA, ONE, Y,
                    237:      $               LDY )
                    238:       CALL DTRMM( 'RIGHT', 'Upper', 'NO TRANSPOSE', 
                    239:      $            'NON-UNIT', K, NB,
                    240:      $            ONE, T, LDT, Y, LDY )
                    241: *
                    242:       RETURN
                    243: *
                    244: *     End of DLAHR2
                    245: *
                    246:       END

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