LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ stzrqf()

 subroutine stzrqf ( integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, integer INFO )

STZRQF

Purpose:
``` This routine is deprecated and has been replaced by routine STZRZF.

STZRQF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A
to upper triangular form by means of orthogonal transformations.

The upper trapezoidal matrix A is factored as

A = ( R  0 ) * Z,

where Z is an N-by-N orthogonal matrix and R is an M-by-M upper
triangular matrix.```
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 >= M.``` [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 M+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] INFO ``` INFO is 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 ), 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 ( 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 ).```

Definition at line 137 of file stzrqf.f.

138*
139* -- LAPACK computational routine --
140* -- LAPACK is a software package provided by Univ. of Tennessee, --
141* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
142*
143* .. Scalar Arguments ..
144 INTEGER INFO, LDA, M, N
145* ..
146* .. Array Arguments ..
147 REAL A( LDA, * ), TAU( * )
148* ..
149*
150* =====================================================================
151*
152* .. Parameters ..
153 REAL ONE, ZERO
154 parameter( one = 1.0e+0, zero = 0.0e+0 )
155* ..
156* .. Local Scalars ..
157 INTEGER I, K, M1
158* ..
159* .. Intrinsic Functions ..
160 INTRINSIC max, min
161* ..
162* .. External Subroutines ..
163 EXTERNAL saxpy, scopy, sgemv, sger, slarfg, xerbla
164* ..
165* .. Executable Statements ..
166*
167* Test the input parameters.
168*
169 info = 0
170 IF( m.LT.0 ) THEN
171 info = -1
172 ELSE IF( n.LT.m ) THEN
173 info = -2
174 ELSE IF( lda.LT.max( 1, m ) ) THEN
175 info = -4
176 END IF
177 IF( info.NE.0 ) THEN
178 CALL xerbla( 'STZRQF', -info )
179 RETURN
180 END IF
181*
182* Perform the factorization.
183*
184 IF( m.EQ.0 )
185 \$ RETURN
186 IF( m.EQ.n ) THEN
187 DO 10 i = 1, n
188 tau( i ) = zero
189 10 CONTINUE
190 ELSE
191 m1 = min( m+1, n )
192 DO 20 k = m, 1, -1
193*
194* Use a Householder reflection to zero the kth row of A.
195* First set up the reflection.
196*
197 CALL slarfg( n-m+1, a( k, k ), a( k, m1 ), lda, tau( k ) )
198*
199 IF( ( tau( k ).NE.zero ) .AND. ( k.GT.1 ) ) THEN
200*
201* We now perform the operation A := A*P( k ).
202*
203* Use the first ( k - 1 ) elements of TAU to store a( k ),
204* where a( k ) consists of the first ( k - 1 ) elements of
205* the kth column of A. Also let B denote the first
206* ( k - 1 ) rows of the last ( n - m ) columns of A.
207*
208 CALL scopy( k-1, a( 1, k ), 1, tau, 1 )
209*
210* Form w = a( k ) + B*z( k ) in TAU.
211*
212 CALL sgemv( 'No transpose', k-1, n-m, one, a( 1, m1 ),
213 \$ lda, a( k, m1 ), lda, one, tau, 1 )
214*
215* Now form a( k ) := a( k ) - tau*w
216* and B := B - tau*w*z( k )**T.
217*
218 CALL saxpy( k-1, -tau( k ), tau, 1, a( 1, k ), 1 )
219 CALL sger( k-1, n-m, -tau( k ), tau, 1, a( k, m1 ), lda,
220 \$ a( 1, m1 ), lda )
221 END IF
222 20 CONTINUE
223 END IF
224*
225 RETURN
226*
227* End of STZRQF
228*
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine slarfg(N, ALPHA, X, INCX, TAU)
SLARFG generates an elementary reflector (Householder matrix).
Definition: slarfg.f:106
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
subroutine saxpy(N, SA, SX, INCX, SY, INCY)
SAXPY
Definition: saxpy.f:89
subroutine sger(M, N, ALPHA, X, INCX, Y, INCY, A, LDA)
SGER
Definition: sger.f:130
subroutine sgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SGEMV
Definition: sgemv.f:156
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