 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ dbdt04()

 subroutine dbdt04 ( character UPLO, integer N, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision, dimension( * ) S, integer NS, double precision, dimension( ldu, * ) U, integer LDU, double precision, dimension( ldvt, * ) VT, integer LDVT, double precision, dimension( * ) WORK, double precision RESID )

DBDT04

Purpose:
``` DBDT04 reconstructs a bidiagonal matrix B from its (partial) SVD:
S = U' * B * V
where U and V are orthogonal matrices and S is diagonal.

The test ratio to test the singular value decomposition is
RESID = norm( S - U' * B * V ) / ( n * norm(B) * EPS )
where VT = V' and EPS is the machine precision.```
Parameters
 [in] UPLO ``` UPLO is CHARACTER*1 Specifies whether the matrix B is upper or lower bidiagonal. = 'U': Upper bidiagonal = 'L': Lower bidiagonal``` [in] N ``` N is INTEGER The order of the matrix B.``` [in] D ``` D is DOUBLE PRECISION array, dimension (N) The n diagonal elements of the bidiagonal matrix B.``` [in] E ``` E is DOUBLE PRECISION array, dimension (N-1) The (n-1) superdiagonal elements of the bidiagonal matrix B if UPLO = 'U', or the (n-1) subdiagonal elements of B if UPLO = 'L'.``` [in] S ``` S is DOUBLE PRECISION array, dimension (NS) The singular values from the (partial) SVD of B, sorted in decreasing order.``` [in] NS ``` NS is INTEGER The number of singular values/vectors from the (partial) SVD of B.``` [in] U ``` U is DOUBLE PRECISION array, dimension (LDU,NS) The n by ns orthogonal matrix U in S = U' * B * V.``` [in] LDU ``` LDU is INTEGER The leading dimension of the array U. LDU >= max(1,N)``` [in] VT ``` VT is DOUBLE PRECISION array, dimension (LDVT,N) The n by ns orthogonal matrix V in S = U' * B * V.``` [in] LDVT ``` LDVT is INTEGER The leading dimension of the array VT.``` [out] WORK ` WORK is DOUBLE PRECISION array, dimension (2*N)` [out] RESID ``` RESID is DOUBLE PRECISION The test ratio: norm(S - U' * B * V) / ( n * norm(B) * EPS )```

Definition at line 129 of file dbdt04.f.

131*
132* -- LAPACK test routine --
133* -- LAPACK is a software package provided by Univ. of Tennessee, --
134* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
135*
136* .. Scalar Arguments ..
137 CHARACTER UPLO
138 INTEGER LDU, LDVT, N, NS
139 DOUBLE PRECISION RESID
140* ..
141* .. Array Arguments ..
142 DOUBLE PRECISION D( * ), E( * ), S( * ), U( LDU, * ),
143 \$ VT( LDVT, * ), WORK( * )
144* ..
145*
146* ======================================================================
147*
148* .. Parameters ..
149 DOUBLE PRECISION ZERO, ONE
150 parameter( zero = 0.0d+0, one = 1.0d+0 )
151* ..
152* .. Local Scalars ..
153 INTEGER I, J, K
154 DOUBLE PRECISION BNORM, EPS
155* ..
156* .. External Functions ..
157 LOGICAL LSAME
158 INTEGER IDAMAX
159 DOUBLE PRECISION DASUM, DLAMCH
160 EXTERNAL lsame, idamax, dasum, dlamch
161* ..
162* .. External Subroutines ..
163 EXTERNAL dgemm
164* ..
165* .. Intrinsic Functions ..
166 INTRINSIC abs, dble, max, min
167* ..
168* .. Executable Statements ..
169*
170* Quick return if possible.
171*
172 resid = zero
173 IF( n.LE.0 .OR. ns.LE.0 )
174 \$ RETURN
175*
176 eps = dlamch( 'Precision' )
177*
178* Compute S - U' * B * V.
179*
180 bnorm = zero
181*
182 IF( lsame( uplo, 'U' ) ) THEN
183*
184* B is upper bidiagonal.
185*
186 k = 0
187 DO 20 i = 1, ns
188 DO 10 j = 1, n-1
189 k = k + 1
190 work( k ) = d( j )*vt( i, j ) + e( j )*vt( i, j+1 )
191 10 CONTINUE
192 k = k + 1
193 work( k ) = d( n )*vt( i, n )
194 20 CONTINUE
195 bnorm = abs( d( 1 ) )
196 DO 30 i = 2, n
197 bnorm = max( bnorm, abs( d( i ) )+abs( e( i-1 ) ) )
198 30 CONTINUE
199 ELSE
200*
201* B is lower bidiagonal.
202*
203 k = 0
204 DO 50 i = 1, ns
205 k = k + 1
206 work( k ) = d( 1 )*vt( i, 1 )
207 DO 40 j = 1, n-1
208 k = k + 1
209 work( k ) = e( j )*vt( i, j ) + d( j+1 )*vt( i, j+1 )
210 40 CONTINUE
211 50 CONTINUE
212 bnorm = abs( d( n ) )
213 DO 60 i = 1, n-1
214 bnorm = max( bnorm, abs( d( i ) )+abs( e( i ) ) )
215 60 CONTINUE
216 END IF
217*
218 CALL dgemm( 'T', 'N', ns, ns, n, -one, u, ldu, work( 1 ),
219 \$ n, zero, work( 1+n*ns ), ns )
220*
221* norm(S - U' * B * V)
222*
223 k = n*ns
224 DO 70 i = 1, ns
225 work( k+i ) = work( k+i ) + s( i )
226 resid = max( resid, dasum( ns, work( k+1 ), 1 ) )
227 k = k + ns
228 70 CONTINUE
229*
230 IF( bnorm.LE.zero ) THEN
231 IF( resid.NE.zero )
232 \$ resid = one / eps
233 ELSE
234 IF( bnorm.GE.resid ) THEN
235 resid = ( resid / bnorm ) / ( dble( n )*eps )
236 ELSE
237 IF( bnorm.LT.one ) THEN
238 resid = ( min( resid, dble( n )*bnorm ) / bnorm ) /
239 \$ ( dble( n )*eps )
240 ELSE
241 resid = min( resid / bnorm, dble( n ) ) /
242 \$ ( dble( n )*eps )
243 END IF
244 END IF
245 END IF
246*
247 RETURN
248*
249* End of DBDT04
250*
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
integer function idamax(N, DX, INCX)
IDAMAX
Definition: idamax.f:71
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
double precision function dasum(N, DX, INCX)
DASUM
Definition: dasum.f:71
subroutine dgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
DGEMM
Definition: dgemm.f:187
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