SUBROUTINE DTZRQF( M, N, A, LDA, TAU, INFO ) * * -- LAPACK routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. INTEGER INFO, LDA, M, N * .. * .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), TAU( * ) * .. * * Purpose * ======= * * This routine is deprecated and has been replaced by routine DTZRZF. * * DTZRQF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A * to upper triangular form by means of orthogonal transformations. * * The upper trapezoidal matrix A is factored as * * A = ( R 0 ) * Z, * * where Z is an N-by-N orthogonal 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) DOUBLE PRECISION 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 * orthogonal 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) DOUBLE PRECISION array, dimension (M) * The scalar factors of the elementary reflectors. * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * * Further Details * =============== * * 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 .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) * .. * .. Local Scalars .. INTEGER I, K, M1 * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN * .. * .. External Subroutines .. EXTERNAL DAXPY, DCOPY, DGEMV, DGER, DLARFG, XERBLA * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 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.NE.0 ) THEN CALL XERBLA( 'DTZRQF', -INFO ) RETURN END IF * * Perform the factorization. * IF( M.EQ.0 ) $ RETURN IF( M.EQ.N ) THEN DO 10 I = 1, N TAU( I ) = ZERO 10 CONTINUE ELSE M1 = MIN( M+1, N ) DO 20 K = M, 1, -1 * * Use a Householder reflection to zero the kth row of A. * First set up the reflection. * CALL DLARFG( N-M+1, A( K, K ), A( K, M1 ), LDA, TAU( K ) ) * IF( ( TAU( K ).NE.ZERO ) .AND. ( K.GT.1 ) ) THEN * * We now perform the operation A := A*P( k ). * * Use the first ( k - 1 ) elements of TAU to store a( k ), * where a( k ) consists of the first ( k - 1 ) elements of * the kth column of A. Also let B denote the first * ( k - 1 ) rows of the last ( n - m ) columns of A. * CALL DCOPY( K-1, A( 1, K ), 1, TAU, 1 ) * * Form w = a( k ) + B*z( k ) in TAU. * CALL DGEMV( 'No transpose', K-1, N-M, ONE, A( 1, M1 ), $ LDA, A( K, M1 ), LDA, ONE, TAU, 1 ) * * Now form a( k ) := a( k ) - tau*w * and B := B - tau*w*z( k )'. * CALL DAXPY( K-1, -TAU( K ), TAU, 1, A( 1, K ), 1 ) CALL DGER( K-1, N-M, -TAU( K ), TAU, 1, A( K, M1 ), LDA, $ A( 1, M1 ), LDA ) END IF 20 CONTINUE END IF * RETURN * * End of DTZRQF * END