LAPACK 3.3.1
Linear Algebra PACKage

ztzrqf.f

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00001       SUBROUTINE ZTZRQF( M, N, A, LDA, TAU, 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, M, N
00010 *     ..
00011 *     .. Array Arguments ..
00012       COMPLEX*16         A( LDA, * ), TAU( * )
00013 *     ..
00014 *
00015 *  Purpose
00016 *  =======
00017 *
00018 *  This routine is deprecated and has been replaced by routine ZTZRZF.
00019 *
00020 *  ZTZRQF reduces the M-by-N ( M<=N ) complex upper trapezoidal matrix A
00021 *  to upper triangular form by means of unitary transformations.
00022 *
00023 *  The upper trapezoidal matrix A is factored as
00024 *
00025 *     A = ( R  0 ) * Z,
00026 *
00027 *  where Z is an N-by-N unitary matrix and R is an M-by-M upper
00028 *  triangular matrix.
00029 *
00030 *  Arguments
00031 *  =========
00032 *
00033 *  M       (input) INTEGER
00034 *          The number of rows of the matrix A.  M >= 0.
00035 *
00036 *  N       (input) INTEGER
00037 *          The number of columns of the matrix A.  N >= M.
00038 *
00039 *  A       (input/output) COMPLEX*16 array, dimension (LDA,N)
00040 *          On entry, the leading M-by-N upper trapezoidal part of the
00041 *          array A must contain the matrix to be factorized.
00042 *          On exit, the leading M-by-M upper triangular part of A
00043 *          contains the upper triangular matrix R, and elements M+1 to
00044 *          N of the first M rows of A, with the array TAU, represent the
00045 *          unitary matrix Z as a product of M elementary reflectors.
00046 *
00047 *  LDA     (input) INTEGER
00048 *          The leading dimension of the array A.  LDA >= max(1,M).
00049 *
00050 *  TAU     (output) COMPLEX*16 array, dimension (M)
00051 *          The scalar factors of the elementary reflectors.
00052 *
00053 *  INFO    (output) INTEGER
00054 *          = 0: successful exit
00055 *          < 0: if INFO = -i, the i-th argument had an illegal value
00056 *
00057 *  Further Details
00058 *  ===============
00059 *
00060 *  The  factorization is obtained by Householder's method.  The kth
00061 *  transformation matrix, Z( k ), whose conjugate transpose is used to
00062 *  introduce zeros into the (m - k + 1)th row of A, is given in the form
00063 *
00064 *     Z( k ) = ( I     0   ),
00065 *              ( 0  T( k ) )
00066 *
00067 *  where
00068 *
00069 *     T( k ) = I - tau*u( k )*u( k )**H,   u( k ) = (   1    ),
00070 *                                                   (   0    )
00071 *                                                   ( z( k ) )
00072 *
00073 *  tau is a scalar and z( k ) is an ( n - m ) element vector.
00074 *  tau and z( k ) are chosen to annihilate the elements of the kth row
00075 *  of X.
00076 *
00077 *  The scalar tau is returned in the kth element of TAU and the vector
00078 *  u( k ) in the kth row of A, such that the elements of z( k ) are
00079 *  in  a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in
00080 *  the upper triangular part of A.
00081 *
00082 *  Z is given by
00083 *
00084 *     Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).
00085 *
00086 * =====================================================================
00087 *
00088 *     .. Parameters ..
00089       COMPLEX*16         CONE, CZERO
00090       PARAMETER          ( CONE = ( 1.0D+0, 0.0D+0 ),
00091      $                   CZERO = ( 0.0D+0, 0.0D+0 ) )
00092 *     ..
00093 *     .. Local Scalars ..
00094       INTEGER            I, K, M1
00095       COMPLEX*16         ALPHA
00096 *     ..
00097 *     .. Intrinsic Functions ..
00098       INTRINSIC          DCONJG, MAX, MIN
00099 *     ..
00100 *     .. External Subroutines ..
00101       EXTERNAL           XERBLA, ZAXPY, ZCOPY, ZGEMV, ZGERC, ZLACGV,
00102      $                   ZLARFG
00103 *     ..
00104 *     .. Executable Statements ..
00105 *
00106 *     Test the input parameters.
00107 *
00108       INFO = 0
00109       IF( M.LT.0 ) THEN
00110          INFO = -1
00111       ELSE IF( N.LT.M ) THEN
00112          INFO = -2
00113       ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
00114          INFO = -4
00115       END IF
00116       IF( INFO.NE.0 ) THEN
00117          CALL XERBLA( 'ZTZRQF', -INFO )
00118          RETURN
00119       END IF
00120 *
00121 *     Perform the factorization.
00122 *
00123       IF( M.EQ.0 )
00124      $   RETURN
00125       IF( M.EQ.N ) THEN
00126          DO 10 I = 1, N
00127             TAU( I ) = CZERO
00128    10    CONTINUE
00129       ELSE
00130          M1 = MIN( M+1, N )
00131          DO 20 K = M, 1, -1
00132 *
00133 *           Use a Householder reflection to zero the kth row of A.
00134 *           First set up the reflection.
00135 *
00136             A( K, K ) = DCONJG( A( K, K ) )
00137             CALL ZLACGV( N-M, A( K, M1 ), LDA )
00138             ALPHA = A( K, K )
00139             CALL ZLARFG( N-M+1, ALPHA, A( K, M1 ), LDA, TAU( K ) )
00140             A( K, K ) = ALPHA
00141             TAU( K ) = DCONJG( TAU( K ) )
00142 *
00143             IF( TAU( K ).NE.CZERO .AND. K.GT.1 ) THEN
00144 *
00145 *              We now perform the operation  A := A*P( k )**H.
00146 *
00147 *              Use the first ( k - 1 ) elements of TAU to store  a( k ),
00148 *              where  a( k ) consists of the first ( k - 1 ) elements of
00149 *              the  kth column  of  A.  Also  let  B  denote  the  first
00150 *              ( k - 1 ) rows of the last ( n - m ) columns of A.
00151 *
00152                CALL ZCOPY( K-1, A( 1, K ), 1, TAU, 1 )
00153 *
00154 *              Form   w = a( k ) + B*z( k )  in TAU.
00155 *
00156                CALL ZGEMV( 'No transpose', K-1, N-M, CONE, A( 1, M1 ),
00157      $                     LDA, A( K, M1 ), LDA, CONE, TAU, 1 )
00158 *
00159 *              Now form  a( k ) := a( k ) - conjg(tau)*w
00160 *              and       B      := B      - conjg(tau)*w*z( k )**H.
00161 *
00162                CALL ZAXPY( K-1, -DCONJG( TAU( K ) ), TAU, 1, A( 1, K ),
00163      $                     1 )
00164                CALL ZGERC( K-1, N-M, -DCONJG( TAU( K ) ), TAU, 1,
00165      $                     A( K, M1 ), LDA, A( 1, M1 ), LDA )
00166             END IF
00167    20    CONTINUE
00168       END IF
00169 *
00170       RETURN
00171 *
00172 *     End of ZTZRQF
00173 *
00174       END
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