LAPACK: SRC/zlarft.f Source File

Go to the documentation of this file.
00001       SUBROUTINE ZLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
00002 *
00003 *  -- LAPACK auxiliary routine (version 3.2) --
00004 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00005 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00006 *     November 2006
00007 *
00008 *     .. Scalar Arguments ..
00009       CHARACTER          DIRECT, STOREV
00010       INTEGER            K, LDT, LDV, N
00011 *     ..
00012 *     .. Array Arguments ..
00013       COMPLEX*16         T( LDT, * ), TAU( * ), V( LDV, * )
00014 *     ..
00015 *
00016 *  Purpose
00017 *  =======
00018 *
00019 *  ZLARFT forms the triangular factor T of a complex block reflector H
00020 *  of order n, which is defined as a product of k elementary reflectors.
00021 *
00022 *  If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;
00023 *
00024 *  If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular.
00025 *
00026 *  If STOREV = 'C', the vector which defines the elementary reflector
00027 *  H(i) is stored in the i-th column of the array V, and
00028 *
00029 *     H  =  I - V * T * V'
00030 *
00031 *  If STOREV = 'R', the vector which defines the elementary reflector
00032 *  H(i) is stored in the i-th row of the array V, and
00033 *
00034 *     H  =  I - V' * T * V
00035 *
00036 *  Arguments
00037 *  =========
00038 *
00039 *  DIRECT  (input) CHARACTER*1
00040 *          Specifies the order in which the elementary reflectors are
00041 *          multiplied to form the block reflector:
00042 *          = 'F': H = H(1) H(2) . . . H(k) (Forward)
00043 *          = 'B': H = H(k) . . . H(2) H(1) (Backward)
00044 *
00045 *  STOREV  (input) CHARACTER*1
00046 *          Specifies how the vectors which define the elementary
00047 *          reflectors are stored (see also Further Details):
00048 *          = 'C': columnwise
00049 *          = 'R': rowwise
00050 *
00051 *  N       (input) INTEGER
00052 *          The order of the block reflector H. N >= 0.
00053 *
00054 *  K       (input) INTEGER
00055 *          The order of the triangular factor T (= the number of
00056 *          elementary reflectors). K >= 1.
00057 *
00058 *  V       (input/output) COMPLEX*16 array, dimension
00059 *                               (LDV,K) if STOREV = 'C'
00060 *                               (LDV,N) if STOREV = 'R'
00061 *          The matrix V. See further details.
00062 *
00063 *  LDV     (input) INTEGER
00064 *          The leading dimension of the array V.
00065 *          If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.
00066 *
00067 *  TAU     (input) COMPLEX*16 array, dimension (K)
00068 *          TAU(i) must contain the scalar factor of the elementary
00069 *          reflector H(i).
00070 *
00071 *  T       (output) COMPLEX*16 array, dimension (LDT,K)
00072 *          The k by k triangular factor T of the block reflector.
00073 *          If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is
00074 *          lower triangular. The rest of the array is not used.
00075 *
00076 *  LDT     (input) INTEGER
00077 *          The leading dimension of the array T. LDT >= K.
00078 *
00079 *  Further Details
00080 *  ===============
00081 *
00082 *  The shape of the matrix V and the storage of the vectors which define
00083 *  the H(i) is best illustrated by the following example with n = 5 and
00084 *  k = 3. The elements equal to 1 are not stored; the corresponding
00085 *  array elements are modified but restored on exit. The rest of the
00086 *  array is not used.
00087 *
00088 *  DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':
00089 *
00090 *               V = (  1       )                 V = (  1 v1 v1 v1 v1 )
00091 *                   ( v1  1    )                     (     1 v2 v2 v2 )
00092 *                   ( v1 v2  1 )                     (        1 v3 v3 )
00093 *                   ( v1 v2 v3 )
00094 *                   ( v1 v2 v3 )
00095 *
00096 *  DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':
00097 *
00098 *               V = ( v1 v2 v3 )                 V = ( v1 v1  1       )
00099 *                   ( v1 v2 v3 )                     ( v2 v2 v2  1    )
00100 *                   (  1 v2 v3 )                     ( v3 v3 v3 v3  1 )
00101 *                   (     1 v3 )
00102 *                   (        1 )
00103 *
00104 *  =====================================================================
00105 *
00106 *     .. Parameters ..
00107       COMPLEX*16         ONE, ZERO
00108       PARAMETER          ( ONE = ( 1.0D+0, 0.0D+0 ),
00109      $                   ZERO = ( 0.0D+0, 0.0D+0 ) )
00110 *     ..
00111 *     .. Local Scalars ..
00112       INTEGER            I, J, PREVLASTV, LASTV
00113       COMPLEX*16         VII
00114 *     ..
00115 *     .. External Subroutines ..
00116       EXTERNAL           ZGEMV, ZLACGV, ZTRMV
00117 *     ..
00118 *     .. External Functions ..
00119       LOGICAL            LSAME
00120       EXTERNAL           LSAME
00121 *     ..
00122 *     .. Executable Statements ..
00123 *
00124 *     Quick return if possible
00125 *
00126       IF( N.EQ.0 )
00127      $   RETURN
00128 *
00129       IF( LSAME( DIRECT, 'F' ) ) THEN
00130          PREVLASTV = N
00131          DO 20 I = 1, K
00132             PREVLASTV = MAX( PREVLASTV, I )
00133             IF( TAU( I ).EQ.ZERO ) THEN
00134 *
00135 *              H(i)  =  I
00136 *
00137                DO 10 J = 1, I
00138                   T( J, I ) = ZERO
00139    10          CONTINUE
00140             ELSE
00141 *
00142 *              general case
00143 *
00144                VII = V( I, I )
00145                V( I, I ) = ONE
00146                IF( LSAME( STOREV, 'C' ) ) THEN
00147 !                 Skip any trailing zeros.
00148                   DO LASTV = N, I+1, -1
00149                      IF( V( LASTV, I ).NE.ZERO ) EXIT
00150                   END DO
00151                   J = MIN( LASTV, PREVLASTV )
00152 *
00153 *                 T(1:i-1,i) := - tau(i) * V(i:j,1:i-1)' * V(i:j,i)
00154 *
00155                   CALL ZGEMV( 'Conjugate transpose', J-I+1, I-1,
00156      $                        -TAU( I ), V( I, 1 ), LDV, V( I, I ), 1,
00157      $                        ZERO, T( 1, I ), 1 )
00158                ELSE
00159 !                 Skip any trailing zeros.
00160                   DO LASTV = N, I+1, -1
00161                      IF( V( I, LASTV ).NE.ZERO ) EXIT
00162                   END DO
00163                   J = MIN( LASTV, PREVLASTV )
00164 *
00165 *                 T(1:i-1,i) := - tau(i) * V(1:i-1,i:j) * V(i,i:j)'
00166 *
00167                   IF( I.LT.J )
00168      $               CALL ZLACGV( J-I, V( I, I+1 ), LDV )
00169                   CALL ZGEMV( 'No transpose', I-1, J-I+1, -TAU( I ),
00170      $                        V( 1, I ), LDV, V( I, I ), LDV, ZERO,
00171      $                        T( 1, I ), 1 )
00172                   IF( I.LT.J )
00173      $               CALL ZLACGV( J-I, V( I, I+1 ), LDV )
00174                END IF
00175                V( I, I ) = VII
00176 *
00177 *              T(1:i-1,i) := T(1:i-1,1:i-1) * T(1:i-1,i)
00178 *
00179                CALL ZTRMV( 'Upper', 'No transpose', 'Non-unit', I-1, T,
00180      $                     LDT, T( 1, I ), 1 )
00181                T( I, I ) = TAU( I )
00182                IF( I.GT.1 ) THEN
00183                   PREVLASTV = MAX( PREVLASTV, LASTV )
00184                ELSE
00185                   PREVLASTV = LASTV
00186                END IF
00187              END IF
00188    20    CONTINUE
00189       ELSE
00190          PREVLASTV = 1
00191          DO 40 I = K, 1, -1
00192             IF( TAU( I ).EQ.ZERO ) THEN
00193 *
00194 *              H(i)  =  I
00195 *
00196                DO 30 J = I, K
00197                   T( J, I ) = ZERO
00198    30          CONTINUE
00199             ELSE
00200 *
00201 *              general case
00202 *
00203                IF( I.LT.K ) THEN
00204                   IF( LSAME( STOREV, 'C' ) ) THEN
00205                      VII = V( N-K+I, I )
00206                      V( N-K+I, I ) = ONE
00207 !                    Skip any leading zeros.
00208                      DO LASTV = 1, I-1
00209                         IF( V( LASTV, I ).NE.ZERO ) EXIT
00210                      END DO
00211                      J = MAX( LASTV, PREVLASTV )
00212 *
00213 *                    T(i+1:k,i) :=
00214 *                            - tau(i) * V(j:n-k+i,i+1:k)' * V(j:n-k+i,i)
00215 *
00216                      CALL ZGEMV( 'Conjugate transpose', N-K+I-J+1, K-I,
00217      $                           -TAU( I ), V( J, I+1 ), LDV, V( J, I ),
00218      $                           1, ZERO, T( I+1, I ), 1 )
00219                      V( N-K+I, I ) = VII
00220                   ELSE
00221                      VII = V( I, N-K+I )
00222                      V( I, N-K+I ) = ONE
00223 !                    Skip any leading zeros.
00224                      DO LASTV = 1, I-1
00225                         IF( V( I, LASTV ).NE.ZERO ) EXIT
00226                      END DO
00227                      J = MAX( LASTV, PREVLASTV )
00228 *
00229 *                    T(i+1:k,i) :=
00230 *                            - tau(i) * V(i+1:k,j:n-k+i) * V(i,j:n-k+i)'
00231 *
00232                      CALL ZLACGV( N-K+I-1-J+1, V( I, J ), LDV )
00233                      CALL ZGEMV( 'No transpose', K-I, N-K+I-J+1,
00234      $                    -TAU( I ), V( I+1, J ), LDV, V( I, J ), LDV,
00235      $                    ZERO, T( I+1, I ), 1 )
00236                      CALL ZLACGV( N-K+I-1-J+1, V( I, J ), LDV )
00237                      V( I, N-K+I ) = VII
00238                   END IF
00239 *
00240 *                 T(i+1:k,i) := T(i+1:k,i+1:k) * T(i+1:k,i)
00241 *
00242                   CALL ZTRMV( 'Lower', 'No transpose', 'Non-unit', K-I,
00243      $                        T( I+1, I+1 ), LDT, T( I+1, I ), 1 )
00244                   IF( I.GT.1 ) THEN
00245                      PREVLASTV = MIN( PREVLASTV, LASTV )
00246                   ELSE
00247                      PREVLASTV = LASTV
00248                   END IF
00249                END IF
00250                T( I, I ) = TAU( I )
00251             END IF
00252    40    CONTINUE
00253       END IF
00254       RETURN
00255 *
00256 *     End of ZLARFT
00257 *
00258       END