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NAME ZTZRQF - reduce the M-by-N ( M<=N ) complex upper tra- pezoidal matrix A to upper triangular form by means of uni- tary transformations SYNOPSIS SUBROUTINE ZTZRQF( M, N, A, LDA, TAU, INFO ) INTEGER INFO, LDA, M, N COMPLEX*16 A( LDA, * ), TAU( * ) PURPOSE ZTZRQF 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,max(1,N)) On entry, the leading M-by-N upper trapezoidal part of the array A must contain the matrix to be factor- ized. 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 (max(1,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 ), whose conjugate transpose 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 ).