LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ slatrz()

 subroutine slatrz ( integer M, integer N, integer L, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK )

SLATRZ factors an upper trapezoidal matrix by means of orthogonal transformations.

Purpose:
``` SLATRZ factors the M-by-(M+L) real upper trapezoidal matrix
[ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R  0 ) * Z, by means
of orthogonal transformations.  Z is an (M+L)-by-(M+L) orthogonal
matrix and, R and A1 are M-by-M upper triangular matrices.```
Parameters
 [in] M ``` M is INTEGER The number of rows of the matrix A. M >= 0.``` [in] N ``` N is INTEGER The number of columns of the matrix A. N >= 0.``` [in] L ``` L is INTEGER The number of columns of the matrix A containing the meaningful part of the Householder vectors. N-M >= L >= 0.``` [in,out] A ``` A is REAL 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 N-L+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.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).``` [out] TAU ``` TAU is REAL array, dimension (M) The scalar factors of the elementary reflectors.``` [out] WORK ` WORK is REAL array, dimension (M)`
Contributors:
A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
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 )**T,   u( k ) = (   1    ),
(   0    )
( z( k ) )

tau is a scalar and z( k ) is an l element vector. tau and z( k )
are chosen to annihilate the elements of the kth row of A2.

The scalar tau is returned in the kth element of TAU and the vector
u( k ) in the kth row of A2, such that the elements of z( k ) are
in  a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in
the upper triangular part of A1.

Z is given by

Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).```

Definition at line 139 of file slatrz.f.

140 *
141 * -- LAPACK computational routine --
142 * -- LAPACK is a software package provided by Univ. of Tennessee, --
143 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
144 *
145 * .. Scalar Arguments ..
146  INTEGER L, LDA, M, N
147 * ..
148 * .. Array Arguments ..
149  REAL A( LDA, * ), TAU( * ), WORK( * )
150 * ..
151 *
152 * =====================================================================
153 *
154 * .. Parameters ..
155  REAL ZERO
156  parameter( zero = 0.0e+0 )
157 * ..
158 * .. Local Scalars ..
159  INTEGER I
160 * ..
161 * .. External Subroutines ..
162  EXTERNAL slarfg, slarz
163 * ..
164 * .. Executable Statements ..
165 *
166 * Test the input arguments
167 *
168 * Quick return if possible
169 *
170  IF( m.EQ.0 ) THEN
171  RETURN
172  ELSE IF( m.EQ.n ) THEN
173  DO 10 i = 1, n
174  tau( i ) = zero
175  10 CONTINUE
176  RETURN
177  END IF
178 *
179  DO 20 i = m, 1, -1
180 *
181 * Generate elementary reflector H(i) to annihilate
182 * [ A(i,i) A(i,n-l+1:n) ]
183 *
184  CALL slarfg( l+1, a( i, i ), a( i, n-l+1 ), lda, tau( i ) )
185 *
186 * Apply H(i) to A(1:i-1,i:n) from the right
187 *
188  CALL slarz( 'Right', i-1, n-i+1, l, a( i, n-l+1 ), lda,
189  \$ tau( i ), a( 1, i ), lda, work )
190 *
191  20 CONTINUE
192 *
193  RETURN
194 *
195 * End of SLATRZ
196 *
subroutine slarfg(N, ALPHA, X, INCX, TAU)
SLARFG generates an elementary reflector (Householder matrix).
Definition: slarfg.f:106
subroutine slarz(SIDE, M, N, L, V, INCV, TAU, C, LDC, WORK)
SLARZ applies an elementary reflector (as returned by stzrzf) to a general matrix.
Definition: slarz.f:145
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