LAPACK 3.3.1
Linear Algebra PACKage

dgeqrf.f

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00001       SUBROUTINE DGEQRF( M, N, 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 2011                                                      --
00007 *
00008 *     .. Scalar Arguments ..
00009       INTEGER            INFO, LDA, LWORK, M, N
00010 *     ..
00011 *     .. Array Arguments ..
00012       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
00013 *     ..
00014 *
00015 *  Purpose
00016 *  =======
00017 *
00018 *  DGEQRF computes a QR factorization of a real M-by-N matrix A:
00019 *  A = Q * R.
00020 *
00021 *  Arguments
00022 *  =========
00023 *
00024 *  M       (input) INTEGER
00025 *          The number of rows of the matrix A.  M >= 0.
00026 *
00027 *  N       (input) INTEGER
00028 *          The number of columns of the matrix A.  N >= 0.
00029 *
00030 *  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
00031 *          On entry, the M-by-N matrix A.
00032 *          On exit, the elements on and above the diagonal of the array
00033 *          contain the min(M,N)-by-N upper trapezoidal matrix R (R is
00034 *          upper triangular if m >= n); the elements below the diagonal,
00035 *          with the array TAU, represent the orthogonal matrix Q as a
00036 *          product of min(m,n) elementary reflectors (see Further
00037 *          Details).
00038 *
00039 *  LDA     (input) INTEGER
00040 *          The leading dimension of the array A.  LDA >= max(1,M).
00041 *
00042 *  TAU     (output) DOUBLE PRECISION array, dimension (min(M,N))
00043 *          The scalar factors of the elementary reflectors (see Further
00044 *          Details).
00045 *
00046 *  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
00047 *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
00048 *
00049 *  LWORK   (input) INTEGER
00050 *          The dimension of the array WORK.  LWORK >= max(1,N).
00051 *          For optimum performance LWORK >= N*NB, where NB is
00052 *          the optimal blocksize.
00053 *
00054 *          If LWORK = -1, then a workspace query is assumed; the routine
00055 *          only calculates the optimal size of the WORK array, returns
00056 *          this value as the first entry of the WORK array, and no error
00057 *          message related to LWORK is issued by XERBLA.
00058 *
00059 *  INFO    (output) INTEGER
00060 *          = 0:  successful exit
00061 *          < 0:  if INFO = -i, the i-th argument had an illegal value
00062 *
00063 *  Further Details
00064 *  ===============
00065 *
00066 *  The matrix Q is represented as a product of elementary reflectors
00067 *
00068 *     Q = H(1) H(2) . . . H(k), where k = min(m,n).
00069 *
00070 *  Each H(i) has the form
00071 *
00072 *     H(i) = I - tau * v * v**T
00073 *
00074 *  where tau is a real scalar, and v is a real vector with
00075 *  v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
00076 *  and tau in TAU(i).
00077 *
00078 *  =====================================================================
00079 *
00080 *     .. Local Scalars ..
00081       LOGICAL            LQUERY
00082       INTEGER            I, IB, IINFO, IWS, K, LDWORK, LWKOPT, NB,
00083      $                   NBMIN, NX
00084 *     ..
00085 *     .. External Subroutines ..
00086       EXTERNAL           DGEQR2, DLARFB, DLARFT, XERBLA
00087 *     ..
00088 *     .. Intrinsic Functions ..
00089       INTRINSIC          MAX, MIN
00090 *     ..
00091 *     .. External Functions ..
00092       INTEGER            ILAENV
00093       EXTERNAL           ILAENV
00094 *     ..
00095 *     .. Executable Statements ..
00096 *
00097 *     Test the input arguments
00098 *
00099       INFO = 0
00100       NB = ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
00101       LWKOPT = N*NB
00102       WORK( 1 ) = LWKOPT
00103       LQUERY = ( LWORK.EQ.-1 )
00104       IF( M.LT.0 ) THEN
00105          INFO = -1
00106       ELSE IF( N.LT.0 ) THEN
00107          INFO = -2
00108       ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
00109          INFO = -4
00110       ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
00111          INFO = -7
00112       END IF
00113       IF( INFO.NE.0 ) THEN
00114          CALL XERBLA( 'DGEQRF', -INFO )
00115          RETURN
00116       ELSE IF( LQUERY ) THEN
00117          RETURN
00118       END IF
00119 *
00120 *     Quick return if possible
00121 *
00122       K = MIN( M, N )
00123       IF( K.EQ.0 ) THEN
00124          WORK( 1 ) = 1
00125          RETURN
00126       END IF
00127 *
00128       NBMIN = 2
00129       NX = 0
00130       IWS = N
00131       IF( NB.GT.1 .AND. NB.LT.K ) THEN
00132 *
00133 *        Determine when to cross over from blocked to unblocked code.
00134 *
00135          NX = MAX( 0, ILAENV( 3, 'DGEQRF', ' ', M, N, -1, -1 ) )
00136          IF( NX.LT.K ) THEN
00137 *
00138 *           Determine if workspace is large enough for blocked code.
00139 *
00140             LDWORK = N
00141             IWS = LDWORK*NB
00142             IF( LWORK.LT.IWS ) THEN
00143 *
00144 *              Not enough workspace to use optimal NB:  reduce NB and
00145 *              determine the minimum value of NB.
00146 *
00147                NB = LWORK / LDWORK
00148                NBMIN = MAX( 2, ILAENV( 2, 'DGEQRF', ' ', M, N, -1,
00149      $                 -1 ) )
00150             END IF
00151          END IF
00152       END IF
00153 *
00154       IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
00155 *
00156 *        Use blocked code initially
00157 *
00158          DO 10 I = 1, K - NX, NB
00159             IB = MIN( K-I+1, NB )
00160 *
00161 *           Compute the QR factorization of the current block
00162 *           A(i:m,i:i+ib-1)
00163 *
00164             CALL DGEQR2( M-I+1, IB, A( I, I ), LDA, TAU( I ), WORK,
00165      $                   IINFO )
00166             IF( I+IB.LE.N ) THEN
00167 *
00168 *              Form the triangular factor of the block reflector
00169 *              H = H(i) H(i+1) . . . H(i+ib-1)
00170 *
00171                CALL DLARFT( 'Forward', 'Columnwise', M-I+1, IB,
00172      $                      A( I, I ), LDA, TAU( I ), WORK, LDWORK )
00173 *
00174 *              Apply H**T to A(i:m,i+ib:n) from the left
00175 *
00176                CALL DLARFB( 'Left', 'Transpose', 'Forward',
00177      $                      'Columnwise', M-I+1, N-I-IB+1, IB,
00178      $                      A( I, I ), LDA, WORK, LDWORK, A( I, I+IB ),
00179      $                      LDA, WORK( IB+1 ), LDWORK )
00180             END IF
00181    10    CONTINUE
00182       ELSE
00183          I = 1
00184       END IF
00185 *
00186 *     Use unblocked code to factor the last or only block.
00187 *
00188       IF( I.LE.K )
00189      $   CALL DGEQR2( M-I+1, N-I+1, A( I, I ), LDA, TAU( I ), WORK,
00190      $                IINFO )
00191 *
00192       WORK( 1 ) = IWS
00193       RETURN
00194 *
00195 *     End of DGEQRF
00196 *
00197       END
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