ZTZRQF(3F) ZTZRQF(3F)
ZTZRQF - reduce the M-by-N ( M<=N ) complex upper trapezoidal matrix A to
upper triangular form by means of unitary transformations
SUBROUTINE ZTZRQF( M, N, A, LDA, TAU, INFO )
INTEGER INFO, LDA, M, N
COMPLEX*16 A( LDA, * ), TAU( * )
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.
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
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ZTZRQF(3F) ZTZRQF(3F)
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 ).
ZTZRQF(3F) ZTZRQF(3F)
ZTZRQF - reduce the M-by-N ( M<=N ) complex upper trapezoidal matrix A to
upper triangular form by means of unitary transformations
SUBROUTINE ZTZRQF( M, N, A, LDA, TAU, INFO )
INTEGER INFO, LDA, M, N
COMPLEX*16 A( LDA, * ), TAU( * )
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.
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
Page 1
ZTZRQF(3F) ZTZRQF(3F)
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 ).
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