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Annotation of rpl/lapack/lapack/dlatrz.f, revision 1.6

1.1       bertrand    1:       SUBROUTINE DLATRZ( M, N, L, A, LDA, TAU, WORK )
                      2: *
1.5       bertrand    3: *  -- LAPACK routine (version 3.2.2) --
1.1       bertrand    4: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
                      5: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.5       bertrand    6: *     June 2010
1.1       bertrand    7: *
                      8: *     .. Scalar Arguments ..
                      9:       INTEGER            L, LDA, M, N
                     10: *     ..
                     11: *     .. Array Arguments ..
                     12:       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
                     13: *     ..
                     14: *
                     15: *  Purpose
                     16: *  =======
                     17: *
                     18: *  DLATRZ factors the M-by-(M+L) real upper trapezoidal matrix
                     19: *  [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R  0 ) * Z, by means
                     20: *  of orthogonal transformations.  Z is an (M+L)-by-(M+L) orthogonal
                     21: *  matrix and, R and A1 are M-by-M upper triangular matrices.
                     22: *
                     23: *  Arguments
                     24: *  =========
                     25: *
                     26: *  M       (input) INTEGER
                     27: *          The number of rows of the matrix A.  M >= 0.
                     28: *
                     29: *  N       (input) INTEGER
                     30: *          The number of columns of the matrix A.  N >= 0.
                     31: *
                     32: *  L       (input) INTEGER
                     33: *          The number of columns of the matrix A containing the
                     34: *          meaningful part of the Householder vectors. N-M >= L >= 0.
                     35: *
                     36: *  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
                     37: *          On entry, the leading M-by-N upper trapezoidal part of the
                     38: *          array A must contain the matrix to be factorized.
                     39: *          On exit, the leading M-by-M upper triangular part of A
                     40: *          contains the upper triangular matrix R, and elements N-L+1 to
                     41: *          N of the first M rows of A, with the array TAU, represent the
                     42: *          orthogonal matrix Z as a product of M elementary reflectors.
                     43: *
                     44: *  LDA     (input) INTEGER
                     45: *          The leading dimension of the array A.  LDA >= max(1,M).
                     46: *
                     47: *  TAU     (output) DOUBLE PRECISION array, dimension (M)
                     48: *          The scalar factors of the elementary reflectors.
                     49: *
                     50: *  WORK    (workspace) DOUBLE PRECISION array, dimension (M)
                     51: *
                     52: *  Further Details
                     53: *  ===============
                     54: *
                     55: *  Based on contributions by
                     56: *    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
                     57: *
                     58: *  The factorization is obtained by Householder's method.  The kth
                     59: *  transformation matrix, Z( k ), which is used to introduce zeros into
                     60: *  the ( m - k + 1 )th row of A, is given in the form
                     61: *
                     62: *     Z( k ) = ( I     0   ),
                     63: *              ( 0  T( k ) )
                     64: *
                     65: *  where
                     66: *
                     67: *     T( k ) = I - tau*u( k )*u( k )',   u( k ) = (   1    ),
                     68: *                                                 (   0    )
                     69: *                                                 ( z( k ) )
                     70: *
                     71: *  tau is a scalar and z( k ) is an l element vector. tau and z( k )
                     72: *  are chosen to annihilate the elements of the kth row of A2.
                     73: *
                     74: *  The scalar tau is returned in the kth element of TAU and the vector
                     75: *  u( k ) in the kth row of A2, such that the elements of z( k ) are
                     76: *  in  a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in
                     77: *  the upper triangular part of A1.
                     78: *
                     79: *  Z is given by
                     80: *
                     81: *     Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).
                     82: *
                     83: *  =====================================================================
                     84: *
                     85: *     .. Parameters ..
                     86:       DOUBLE PRECISION   ZERO
                     87:       PARAMETER          ( ZERO = 0.0D+0 )
                     88: *     ..
                     89: *     .. Local Scalars ..
                     90:       INTEGER            I
                     91: *     ..
                     92: *     .. External Subroutines ..
1.5       bertrand   93:       EXTERNAL           DLARFG, DLARZ
1.1       bertrand   94: *     ..
                     95: *     .. Executable Statements ..
                     96: *
                     97: *     Test the input arguments
                     98: *
                     99: *     Quick return if possible
                    100: *
                    101:       IF( M.EQ.0 ) THEN
                    102:          RETURN
                    103:       ELSE IF( M.EQ.N ) THEN
                    104:          DO 10 I = 1, N
                    105:             TAU( I ) = ZERO
                    106:    10    CONTINUE
                    107:          RETURN
                    108:       END IF
                    109: *
                    110:       DO 20 I = M, 1, -1
                    111: *
                    112: *        Generate elementary reflector H(i) to annihilate
                    113: *        [ A(i,i) A(i,n-l+1:n) ]
                    114: *
1.5       bertrand  115:          CALL DLARFG( L+1, A( I, I ), A( I, N-L+1 ), LDA, TAU( I ) )
1.1       bertrand  116: *
                    117: *        Apply H(i) to A(1:i-1,i:n) from the right
                    118: *
                    119:          CALL DLARZ( 'Right', I-1, N-I+1, L, A( I, N-L+1 ), LDA,
                    120:      $               TAU( I ), A( 1, I ), LDA, WORK )
                    121: *
                    122:    20 CONTINUE
                    123: *
                    124:       RETURN
                    125: *
                    126: *     End of DLATRZ
                    127: *
                    128:       END