1DTZRZF(1) LAPACK routine (version 3.2) DTZRZF(1)
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6 DTZRZF - reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A to
7 upper triangular form by means of orthogonal transformations
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10 SUBROUTINE DTZRZF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
11 INTEGER INFO, LDA, LWORK, M, N
13 DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
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17 DTZRZF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A to
18 upper triangular form by means of orthogonal transformations. The
19 upper trapezoidal matrix A is factored as
20 A = ( R 0 ) * Z,
21 where Z is an N-by-N orthogonal matrix and R is an M-by-M upper trian‐
22 gular matrix.
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25 M (input) INTEGER
26 The number of rows of the matrix A. M >= 0.
27 N (input) INTEGER
29 The number of columns of the matrix A. N >= M.
30 A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
32 On entry, the leading M-by-N upper trapezoidal part of the
33 array A must contain the matrix to be factorized. On exit, the
34 leading M-by-M upper triangular part of A contains the upper
35 triangular matrix R, and elements M+1 to N of the first M rows
36 of A, with the array TAU, represent the orthogonal matrix Z as
37 a product of M elementary reflectors.
38 LDA (input) INTEGER
40 The leading dimension of the array A. LDA >= max(1,M).
41 TAU (output) DOUBLE PRECISION array, dimension (M)
43 The scalar factors of the elementary reflectors.
44 WORK (workspace/output) DOUBLE PRECISION array, dimension
46 (MAX(1,LWORK))
47 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
48 LWORK (input) INTEGER
50 The dimension of the array WORK. LWORK >= max(1,M). For opti‐
51 mum performance LWORK >= M*NB, where NB is the optimal block‐
52 size. If LWORK = -1, then a workspace query is assumed; the
53 routine only calculates the optimal size of the WORK array,
54 returns this value as the first entry of the WORK array, and no
55 error message related to LWORK is issued by XERBLA.
56 INFO (output) INTEGER
58 = 0: successful exit
59 < 0: if INFO = -i, the i-th argument had an illegal value
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62 Based on contributions by
63 A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
64 The factorization is obtained by Householder's method. The kth trans‐
65 formation matrix, Z( k ), which is used to introduce zeros into the ( m
66 - k + 1 )th row of A, is given in the form
67 Z( k ) = ( I 0 ),
68 ( 0 T( k ) )
69 where
70 T( k ) = I - tau*u( k )*u( k )', u( k ) = ( 1 ),
71 ( 0 )
72 ( z( k ) ) tau is a
73 scalar and z( k ) is an ( n - m ) element vector. tau and z( k ) are
74 chosen to annihilate the elements of the kth row of X.
75 The scalar tau is returned in the kth element of TAU and the vector u(
76 k ) in the kth row of A, such that the elements of z( k ) are in a( k,
77 m + 1 ), ..., a( k, n ). The elements of R are returned in the upper
78 triangular part of A.
79 Z is given by
80 Z = Z( 1 ) * Z( 2 ) * ... * Z( m ).
81 LAPACK routine (version 3.2) November 2008 DTZRZF(1)