 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ dqrt14()

 double precision function dqrt14 ( character TRANS, integer M, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldx, * ) X, integer LDX, double precision, dimension( lwork ) WORK, integer LWORK )

DQRT14

Purpose:
``` DQRT14 checks whether X is in the row space of A or A'.  It does so
by scaling both X and A such that their norms are in the range
[sqrt(eps), 1/sqrt(eps)], then computing a QR factorization of [A,X]
(if TRANS = 'T') or an LQ factorization of [A',X]' (if TRANS = 'N'),
and returning the norm of the trailing triangle, scaled by
MAX(M,N,NRHS)*eps.```
Parameters
 [in] TRANS ``` TRANS is CHARACTER*1 = 'N': No transpose, check for X in the row space of A = 'T': Transpose, check for X in the row space of A'.``` [in] M ``` M is INTEGER The number of rows of the matrix A.``` [in] N ``` N is INTEGER The number of columns of the matrix A.``` [in] NRHS ``` NRHS is INTEGER The number of right hand sides, i.e., the number of columns of X.``` [in] A ``` A is DOUBLE PRECISION array, dimension (LDA,N) The M-by-N matrix A.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A.``` [in] X ``` X is DOUBLE PRECISION array, dimension (LDX,NRHS) If TRANS = 'N', the N-by-NRHS matrix X. IF TRANS = 'T', the M-by-NRHS matrix X.``` [in] LDX ``` LDX is INTEGER The leading dimension of the array X.``` [out] WORK ` WORK is DOUBLE PRECISION array dimension (LWORK)` [in] LWORK ``` LWORK is INTEGER length of workspace array required If TRANS = 'N', LWORK >= (M+NRHS)*(N+2); if TRANS = 'T', LWORK >= (N+NRHS)*(M+2).```

Definition at line 114 of file dqrt14.f.

116 *
117 * -- LAPACK test routine --
118 * -- LAPACK is a software package provided by Univ. of Tennessee, --
119 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
120 *
121 * .. Scalar Arguments ..
122  CHARACTER TRANS
123  INTEGER LDA, LDX, LWORK, M, N, NRHS
124 * ..
125 * .. Array Arguments ..
126  DOUBLE PRECISION A( LDA, * ), WORK( LWORK ), X( LDX, * )
127 * ..
128 *
129 * =====================================================================
130 *
131 * .. Parameters ..
132  DOUBLE PRECISION ZERO, ONE
133  parameter( zero = 0.0d0, one = 1.0d0 )
134 * ..
135 * .. Local Scalars ..
136  LOGICAL TPSD
137  INTEGER I, INFO, J, LDWORK
138  DOUBLE PRECISION ANRM, ERR, XNRM
139 * ..
140 * .. Local Arrays ..
141  DOUBLE PRECISION RWORK( 1 )
142 * ..
143 * .. External Functions ..
144  LOGICAL LSAME
145  DOUBLE PRECISION DLAMCH, DLANGE
146  EXTERNAL lsame, dlamch, dlange
147 * ..
148 * .. External Subroutines ..
149  EXTERNAL dgelq2, dgeqr2, dlacpy, dlascl, xerbla
150 * ..
151 * .. Intrinsic Functions ..
152  INTRINSIC abs, dble, max, min
153 * ..
154 * .. Executable Statements ..
155 *
156  dqrt14 = zero
157  IF( lsame( trans, 'N' ) ) THEN
158  ldwork = m + nrhs
159  tpsd = .false.
160  IF( lwork.LT.( m+nrhs )*( n+2 ) ) THEN
161  CALL xerbla( 'DQRT14', 10 )
162  RETURN
163  ELSE IF( n.LE.0 .OR. nrhs.LE.0 ) THEN
164  RETURN
165  END IF
166  ELSE IF( lsame( trans, 'T' ) ) THEN
167  ldwork = m
168  tpsd = .true.
169  IF( lwork.LT.( n+nrhs )*( m+2 ) ) THEN
170  CALL xerbla( 'DQRT14', 10 )
171  RETURN
172  ELSE IF( m.LE.0 .OR. nrhs.LE.0 ) THEN
173  RETURN
174  END IF
175  ELSE
176  CALL xerbla( 'DQRT14', 1 )
177  RETURN
178  END IF
179 *
180 * Copy and scale A
181 *
182  CALL dlacpy( 'All', m, n, a, lda, work, ldwork )
183  anrm = dlange( 'M', m, n, work, ldwork, rwork )
184  IF( anrm.NE.zero )
185  \$ CALL dlascl( 'G', 0, 0, anrm, one, m, n, work, ldwork, info )
186 *
187 * Copy X or X' into the right place and scale it
188 *
189  IF( tpsd ) THEN
190 *
191 * Copy X into columns n+1:n+nrhs of work
192 *
193  CALL dlacpy( 'All', m, nrhs, x, ldx, work( n*ldwork+1 ),
194  \$ ldwork )
195  xnrm = dlange( 'M', m, nrhs, work( n*ldwork+1 ), ldwork,
196  \$ rwork )
197  IF( xnrm.NE.zero )
198  \$ CALL dlascl( 'G', 0, 0, xnrm, one, m, nrhs,
199  \$ work( n*ldwork+1 ), ldwork, info )
200 *
201 * Compute QR factorization of X
202 *
203  CALL dgeqr2( m, n+nrhs, work, ldwork,
204  \$ work( ldwork*( n+nrhs )+1 ),
205  \$ work( ldwork*( n+nrhs )+min( m, n+nrhs )+1 ),
206  \$ info )
207 *
208 * Compute largest entry in upper triangle of
209 * work(n+1:m,n+1:n+nrhs)
210 *
211  err = zero
212  DO 20 j = n + 1, n + nrhs
213  DO 10 i = n + 1, min( m, j )
214  err = max( err, abs( work( i+( j-1 )*m ) ) )
215  10 CONTINUE
216  20 CONTINUE
217 *
218  ELSE
219 *
220 * Copy X' into rows m+1:m+nrhs of work
221 *
222  DO 40 i = 1, n
223  DO 30 j = 1, nrhs
224  work( m+j+( i-1 )*ldwork ) = x( i, j )
225  30 CONTINUE
226  40 CONTINUE
227 *
228  xnrm = dlange( 'M', nrhs, n, work( m+1 ), ldwork, rwork )
229  IF( xnrm.NE.zero )
230  \$ CALL dlascl( 'G', 0, 0, xnrm, one, nrhs, n, work( m+1 ),
231  \$ ldwork, info )
232 *
233 * Compute LQ factorization of work
234 *
235  CALL dgelq2( ldwork, n, work, ldwork, work( ldwork*n+1 ),
236  \$ work( ldwork*( n+1 )+1 ), info )
237 *
238 * Compute largest entry in lower triangle in
239 * work(m+1:m+nrhs,m+1:n)
240 *
241  err = zero
242  DO 60 j = m + 1, n
243  DO 50 i = j, ldwork
244  err = max( err, abs( work( i+( j-1 )*ldwork ) ) )
245  50 CONTINUE
246  60 CONTINUE
247 *
248  END IF
249 *
250  dqrt14 = err / ( dble( max( m, n, nrhs ) )*dlamch( 'Epsilon' ) )
251 *
252  RETURN
253 *
254 * End of DQRT14
255 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine dlascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
DLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: dlascl.f:143
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:103
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
double precision function dqrt14(TRANS, M, N, NRHS, A, LDA, X, LDX, WORK, LWORK)
DQRT14
Definition: dqrt14.f:116
double precision function dlange(NORM, M, N, A, LDA, WORK)
DLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: dlange.f:114
subroutine dgelq2(M, N, A, LDA, TAU, WORK, INFO)
DGELQ2 computes the LQ factorization of a general rectangular matrix using an unblocked algorithm.
Definition: dgelq2.f:129
subroutine dgeqr2(M, N, A, LDA, TAU, WORK, INFO)
DGEQR2 computes the QR factorization of a general rectangular matrix using an unblocked algorithm.
Definition: dgeqr2.f:130
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