SUBROUTINE ZTZRZF( M, N, A, LDA, TAU, WORK, LWORK, INFO ) * * -- LAPACK routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. INTEGER INFO, LDA, LWORK, M, N * .. * .. Array Arguments .. COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * ) * .. * * Purpose * ======= * * ZTZRZF reduces the M-by-N ( M<=N ) complex upper trapezoidal matrix A * to upper triangular form by means of unitary transformations. * * The upper trapezoidal matrix A is factored as * * A = ( R 0 ) * Z, * * where Z is an N-by-N unitary matrix and R is an M-by-M upper * triangular matrix. * * Arguments * ========= * * M (input) INTEGER * The number of rows of the matrix A. M >= 0. * * N (input) INTEGER * The number of columns of the matrix A. N >= M. * * A (input/output) COMPLEX*16 array, dimension (LDA,N) * On entry, the leading M-by-N upper trapezoidal part of the * array A must contain the matrix to be factorized. * On exit, the leading M-by-M upper triangular part of A * contains the upper triangular matrix R, and elements M+1 to * N of the first M rows of A, with the array TAU, represent the * unitary matrix Z as a product of M elementary reflectors. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,M). * * TAU (output) COMPLEX*16 array, dimension (M) * The scalar factors of the elementary reflectors. * * WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK)) * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. * * LWORK (input) INTEGER * The dimension of the array WORK. LWORK >= max(1,M). * For optimum performance LWORK >= M*NB, where NB is * the optimal blocksize. * * If LWORK = -1, then a workspace query is assumed; the routine * only calculates the optimal size of the WORK array, returns * this value as the first entry of the WORK array, and no error * message related to LWORK is issued by XERBLA. * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * * Further Details * =============== * * Based on contributions by * A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA * * The factorization is obtained by Householder's method. The kth * transformation matrix, Z( k ), which is used to introduce zeros into * the ( m - k + 1 )th row of A, is given in the form * * Z( k ) = ( I 0 ), * ( 0 T( k ) ) * * where * * T( k ) = I - tau*u( k )*u( k )', u( k ) = ( 1 ), * ( 0 ) * ( z( k ) ) * * tau is a scalar and z( k ) is an ( n - m ) element vector. * tau and z( k ) are chosen to annihilate the elements of the kth row * of X. * * The scalar tau is returned in the kth element of TAU and the vector * u( k ) in the kth row of A, such that the elements of z( k ) are * in a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in * the upper triangular part of A. * * Z is given by * * Z = Z( 1 ) * Z( 2 ) * ... * Z( m ). * * ===================================================================== * * .. Parameters .. COMPLEX*16 ZERO PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ) ) * .. * .. Local Scalars .. LOGICAL LQUERY INTEGER I, IB, IWS, KI, KK, LDWORK, LWKOPT, M1, MU, NB, \$ NBMIN, NX * .. * .. External Subroutines .. EXTERNAL XERBLA, ZLARZB, ZLARZT, ZLATRZ * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN * .. * .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV * .. * .. Executable Statements .. * * Test the input arguments * INFO = 0 LQUERY = ( LWORK.EQ.-1 ) IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.M ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -4 END IF * IF( INFO.EQ.0 ) THEN IF( M.EQ.0 .OR. M.EQ.N ) THEN LWKOPT = 1 ELSE * * Determine the block size. * NB = ILAENV( 1, 'ZGERQF', ' ', M, N, -1, -1 ) LWKOPT = M*NB END IF WORK( 1 ) = LWKOPT * IF( LWORK.LT.MAX( 1, M ) .AND. .NOT.LQUERY ) THEN INFO = -7 END IF END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZTZRZF', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible * IF( M.EQ.0 ) THEN RETURN ELSE IF( M.EQ.N ) THEN DO 10 I = 1, N TAU( I ) = ZERO 10 CONTINUE RETURN END IF * NBMIN = 2 NX = 1 IWS = M IF( NB.GT.1 .AND. NB.LT.M ) THEN * * Determine when to cross over from blocked to unblocked code. * NX = MAX( 0, ILAENV( 3, 'ZGERQF', ' ', M, N, -1, -1 ) ) IF( NX.LT.M ) THEN * * Determine if workspace is large enough for blocked code. * LDWORK = M IWS = LDWORK*NB IF( LWORK.LT.IWS ) THEN * * Not enough workspace to use optimal NB: reduce NB and * determine the minimum value of NB. * NB = LWORK / LDWORK NBMIN = MAX( 2, ILAENV( 2, 'ZGERQF', ' ', M, N, -1, \$ -1 ) ) END IF END IF END IF * IF( NB.GE.NBMIN .AND. NB.LT.M .AND. NX.LT.M ) THEN * * Use blocked code initially. * The last kk rows are handled by the block method. * M1 = MIN( M+1, N ) KI = ( ( M-NX-1 ) / NB )*NB KK = MIN( M, KI+NB ) * DO 20 I = M - KK + KI + 1, M - KK + 1, -NB IB = MIN( M-I+1, NB ) * * Compute the TZ factorization of the current block * A(i:i+ib-1,i:n) * CALL ZLATRZ( IB, N-I+1, N-M, A( I, I ), LDA, TAU( I ), \$ WORK ) IF( I.GT.1 ) THEN * * Form the triangular factor of the block reflector * H = H(i+ib-1) . . . H(i+1) H(i) * CALL ZLARZT( 'Backward', 'Rowwise', N-M, IB, A( I, M1 ), \$ LDA, TAU( I ), WORK, LDWORK ) * * Apply H to A(1:i-1,i:n) from the right * CALL ZLARZB( 'Right', 'No transpose', 'Backward', \$ 'Rowwise', I-1, N-I+1, IB, N-M, A( I, M1 ), \$ LDA, WORK, LDWORK, A( 1, I ), LDA, \$ WORK( IB+1 ), LDWORK ) END IF 20 CONTINUE MU = I + NB - 1 ELSE MU = M END IF * * Use unblocked code to factor the last or only block * IF( MU.GT.0 ) \$ CALL ZLATRZ( MU, N, N-M, A, LDA, TAU, WORK ) * WORK( 1 ) = LWKOPT * RETURN * * End of ZTZRZF * END