LAPACK 3.3.1
Linear Algebra PACKage

dgehrd.f

Go to the documentation of this file.
00001       SUBROUTINE DGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )
00002 *
00003 *  -- LAPACK routine (version 3.3.1)                                  --
00004 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00005 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00006 *  -- April 2009                                                      --
00007 *
00008 *     .. Scalar Arguments ..
00009       INTEGER            IHI, ILO, INFO, LDA, LWORK, N
00010 *     ..
00011 *     .. Array Arguments ..
00012       DOUBLE PRECISION  A( LDA, * ), TAU( * ), WORK( * )
00013 *     ..
00014 *
00015 *  Purpose
00016 *  =======
00017 *
00018 *  DGEHRD reduces a real general matrix A to upper Hessenberg form H by
00019 *  an orthogonal similarity transformation:  Q**T * A * Q = H .
00020 *
00021 *  Arguments
00022 *  =========
00023 *
00024 *  N       (input) INTEGER
00025 *          The order of the matrix A.  N >= 0.
00026 *
00027 *  ILO     (input) INTEGER
00028 *  IHI     (input) INTEGER
00029 *          It is assumed that A is already upper triangular in rows
00030 *          and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
00031 *          set by a previous call to DGEBAL; otherwise they should be
00032 *          set to 1 and N respectively. See Further Details.
00033 *          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
00034 *
00035 *  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
00036 *          On entry, the N-by-N general matrix to be reduced.
00037 *          On exit, the upper triangle and the first subdiagonal of A
00038 *          are overwritten with the upper Hessenberg matrix H, and the
00039 *          elements below the first subdiagonal, with the array TAU,
00040 *          represent the orthogonal matrix Q as a product of elementary
00041 *          reflectors. See Further Details.
00042 *
00043 *  LDA     (input) INTEGER
00044 *          The leading dimension of the array A.  LDA >= max(1,N).
00045 *
00046 *  TAU     (output) DOUBLE PRECISION array, dimension (N-1)
00047 *          The scalar factors of the elementary reflectors (see Further
00048 *          Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to
00049 *          zero.
00050 *
00051 *  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
00052 *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
00053 *
00054 *  LWORK   (input) INTEGER
00055 *          The length of the array WORK.  LWORK >= max(1,N).
00056 *          For optimum performance LWORK >= N*NB, where NB is the
00057 *          optimal blocksize.
00058 *
00059 *          If LWORK = -1, then a workspace query is assumed; the routine
00060 *          only calculates the optimal size of the WORK array, returns
00061 *          this value as the first entry of the WORK array, and no error
00062 *          message related to LWORK is issued by XERBLA.
00063 *
00064 *  INFO    (output) INTEGER
00065 *          = 0:  successful exit
00066 *          < 0:  if INFO = -i, the i-th argument had an illegal value.
00067 *
00068 *  Further Details
00069 *  ===============
00070 *
00071 *  The matrix Q is represented as a product of (ihi-ilo) elementary
00072 *  reflectors
00073 *
00074 *     Q = H(ilo) H(ilo+1) . . . H(ihi-1).
00075 *
00076 *  Each H(i) has the form
00077 *
00078 *     H(i) = I - tau * v * v**T
00079 *
00080 *  where tau is a real scalar, and v is a real vector with
00081 *  v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
00082 *  exit in A(i+2:ihi,i), and tau in TAU(i).
00083 *
00084 *  The contents of A are illustrated by the following example, with
00085 *  n = 7, ilo = 2 and ihi = 6:
00086 *
00087 *  on entry,                        on exit,
00088 *
00089 *  ( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
00090 *  (     a   a   a   a   a   a )    (      a   h   h   h   h   a )
00091 *  (     a   a   a   a   a   a )    (      h   h   h   h   h   h )
00092 *  (     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
00093 *  (     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
00094 *  (     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
00095 *  (                         a )    (                          a )
00096 *
00097 *  where a denotes an element of the original matrix A, h denotes a
00098 *  modified element of the upper Hessenberg matrix H, and vi denotes an
00099 *  element of the vector defining H(i).
00100 *
00101 *  This file is a slight modification of LAPACK-3.0's DGEHRD
00102 *  subroutine incorporating improvements proposed by Quintana-Orti and
00103 *  Van de Geijn (2006). (See DLAHR2.)
00104 *
00105 *  =====================================================================
00106 *
00107 *     .. Parameters ..
00108       INTEGER            NBMAX, LDT
00109       PARAMETER          ( NBMAX = 64, LDT = NBMAX+1 )
00110       DOUBLE PRECISION  ZERO, ONE
00111       PARAMETER          ( ZERO = 0.0D+0, 
00112      $                     ONE = 1.0D+0 )
00113 *     ..
00114 *     .. Local Scalars ..
00115       LOGICAL            LQUERY
00116       INTEGER            I, IB, IINFO, IWS, J, LDWORK, LWKOPT, NB,
00117      $                   NBMIN, NH, NX
00118       DOUBLE PRECISION  EI
00119 *     ..
00120 *     .. Local Arrays ..
00121       DOUBLE PRECISION  T( LDT, NBMAX )
00122 *     ..
00123 *     .. External Subroutines ..
00124       EXTERNAL           DAXPY, DGEHD2, DGEMM, DLAHR2, DLARFB, DTRMM,
00125      $                   XERBLA
00126 *     ..
00127 *     .. Intrinsic Functions ..
00128       INTRINSIC          MAX, MIN
00129 *     ..
00130 *     .. External Functions ..
00131       INTEGER            ILAENV
00132       EXTERNAL           ILAENV
00133 *     ..
00134 *     .. Executable Statements ..
00135 *
00136 *     Test the input parameters
00137 *
00138       INFO = 0
00139       NB = MIN( NBMAX, ILAENV( 1, 'DGEHRD', ' ', N, ILO, IHI, -1 ) )
00140       LWKOPT = N*NB
00141       WORK( 1 ) = LWKOPT
00142       LQUERY = ( LWORK.EQ.-1 )
00143       IF( N.LT.0 ) THEN
00144          INFO = -1
00145       ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
00146          INFO = -2
00147       ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
00148          INFO = -3
00149       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
00150          INFO = -5
00151       ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
00152          INFO = -8
00153       END IF
00154       IF( INFO.NE.0 ) THEN
00155          CALL XERBLA( 'DGEHRD', -INFO )
00156          RETURN
00157       ELSE IF( LQUERY ) THEN
00158          RETURN
00159       END IF
00160 *
00161 *     Set elements 1:ILO-1 and IHI:N-1 of TAU to zero
00162 *
00163       DO 10 I = 1, ILO - 1
00164          TAU( I ) = ZERO
00165    10 CONTINUE
00166       DO 20 I = MAX( 1, IHI ), N - 1
00167          TAU( I ) = ZERO
00168    20 CONTINUE
00169 *
00170 *     Quick return if possible
00171 *
00172       NH = IHI - ILO + 1
00173       IF( NH.LE.1 ) THEN
00174          WORK( 1 ) = 1
00175          RETURN
00176       END IF
00177 *
00178 *     Determine the block size
00179 *
00180       NB = MIN( NBMAX, ILAENV( 1, 'DGEHRD', ' ', N, ILO, IHI, -1 ) )
00181       NBMIN = 2
00182       IWS = 1
00183       IF( NB.GT.1 .AND. NB.LT.NH ) THEN
00184 *
00185 *        Determine when to cross over from blocked to unblocked code
00186 *        (last block is always handled by unblocked code)
00187 *
00188          NX = MAX( NB, ILAENV( 3, 'DGEHRD', ' ', N, ILO, IHI, -1 ) )
00189          IF( NX.LT.NH ) THEN
00190 *
00191 *           Determine if workspace is large enough for blocked code
00192 *
00193             IWS = N*NB
00194             IF( LWORK.LT.IWS ) THEN
00195 *
00196 *              Not enough workspace to use optimal NB:  determine the
00197 *              minimum value of NB, and reduce NB or force use of
00198 *              unblocked code
00199 *
00200                NBMIN = MAX( 2, ILAENV( 2, 'DGEHRD', ' ', N, ILO, IHI,
00201      $                 -1 ) )
00202                IF( LWORK.GE.N*NBMIN ) THEN
00203                   NB = LWORK / N
00204                ELSE
00205                   NB = 1
00206                END IF
00207             END IF
00208          END IF
00209       END IF
00210       LDWORK = N
00211 *
00212       IF( NB.LT.NBMIN .OR. NB.GE.NH ) THEN
00213 *
00214 *        Use unblocked code below
00215 *
00216          I = ILO
00217 *
00218       ELSE
00219 *
00220 *        Use blocked code
00221 *
00222          DO 40 I = ILO, IHI - 1 - NX, NB
00223             IB = MIN( NB, IHI-I )
00224 *
00225 *           Reduce columns i:i+ib-1 to Hessenberg form, returning the
00226 *           matrices V and T of the block reflector H = I - V*T*V**T
00227 *           which performs the reduction, and also the matrix Y = A*V*T
00228 *
00229             CALL DLAHR2( IHI, I, IB, A( 1, I ), LDA, TAU( I ), T, LDT,
00230      $                   WORK, LDWORK )
00231 *
00232 *           Apply the block reflector H to A(1:ihi,i+ib:ihi) from the
00233 *           right, computing  A := A - Y * V**T. V(i+ib,ib-1) must be set
00234 *           to 1
00235 *
00236             EI = A( I+IB, I+IB-1 )
00237             A( I+IB, I+IB-1 ) = ONE
00238             CALL DGEMM( 'No transpose', 'Transpose', 
00239      $                  IHI, IHI-I-IB+1,
00240      $                  IB, -ONE, WORK, LDWORK, A( I+IB, I ), LDA, ONE,
00241      $                  A( 1, I+IB ), LDA )
00242             A( I+IB, I+IB-1 ) = EI
00243 *
00244 *           Apply the block reflector H to A(1:i,i+1:i+ib-1) from the
00245 *           right
00246 *
00247             CALL DTRMM( 'Right', 'Lower', 'Transpose',
00248      $                  'Unit', I, IB-1,
00249      $                  ONE, A( I+1, I ), LDA, WORK, LDWORK )
00250             DO 30 J = 0, IB-2
00251                CALL DAXPY( I, -ONE, WORK( LDWORK*J+1 ), 1,
00252      $                     A( 1, I+J+1 ), 1 )
00253    30       CONTINUE
00254 *
00255 *           Apply the block reflector H to A(i+1:ihi,i+ib:n) from the
00256 *           left
00257 *
00258             CALL DLARFB( 'Left', 'Transpose', 'Forward',
00259      $                   'Columnwise',
00260      $                   IHI-I, N-I-IB+1, IB, A( I+1, I ), LDA, T, LDT,
00261      $                   A( I+1, I+IB ), LDA, WORK, LDWORK )
00262    40    CONTINUE
00263       END IF
00264 *
00265 *     Use unblocked code to reduce the rest of the matrix
00266 *
00267       CALL DGEHD2( N, I, IHI, A, LDA, TAU, WORK, IINFO )
00268       WORK( 1 ) = IWS
00269 *
00270       RETURN
00271 *
00272 *     End of DGEHRD
00273 *
00274       END
 All Files Functions