USAGE: tau, info, a = NumRu::Lapack.ztzrqf( a, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE ZTZRQF( M, N, A, LDA, TAU, INFO ) * Purpose * ======= * * This routine is deprecated and has been replaced by routine ZTZRZF. * * 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,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. * * 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 ). * * ===================================================================== *go to the page top

USAGE: tau, work, info, a = NumRu::Lapack.ztzrzf( a, [:lwork => lwork, :usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE ZTZRZF( M, N, A, LDA, TAU, WORK, LWORK, INFO ) * 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 ). * * ===================================================================== *go to the page top

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