 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ sbdt03()

 subroutine sbdt03 ( character UPLO, integer N, integer KD, real, dimension( * ) D, real, dimension( * ) E, real, dimension( ldu, * ) U, integer LDU, real, dimension( * ) S, real, dimension( ldvt, * ) VT, integer LDVT, real, dimension( * ) WORK, real RESID )

SBDT03

Purpose:
``` SBDT03 reconstructs a bidiagonal matrix B from its 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( B - U * S * VT ) / ( 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] KD ``` KD is INTEGER The bandwidth of the bidiagonal matrix B. If KD = 1, the matrix B is bidiagonal, and if KD = 0, B is diagonal and E is not referenced. If KD is greater than 1, it is assumed to be 1, and if KD is less than 0, it is assumed to be 0.``` [in] D ``` D is REAL array, dimension (N) The n diagonal elements of the bidiagonal matrix B.``` [in] E ``` E is REAL 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] U ``` U is REAL array, dimension (LDU,N) The n by n orthogonal matrix U in the reduction B = U'*A*P.``` [in] LDU ``` LDU is INTEGER The leading dimension of the array U. LDU >= max(1,N)``` [in] S ``` S is REAL array, dimension (N) The singular values from the SVD of B, sorted in decreasing order.``` [in] VT ``` VT is REAL array, dimension (LDVT,N) The n by n orthogonal matrix V' in the reduction B = U * S * V'.``` [in] LDVT ``` LDVT is INTEGER The leading dimension of the array VT.``` [out] WORK ` WORK is REAL array, dimension (2*N)` [out] RESID ``` RESID is REAL The test ratio: norm(B - U * S * V') / ( n * norm(A) * EPS )```

Definition at line 133 of file sbdt03.f.

135*
136* -- LAPACK test routine --
137* -- LAPACK is a software package provided by Univ. of Tennessee, --
138* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
139*
140* .. Scalar Arguments ..
141 CHARACTER UPLO
142 INTEGER KD, LDU, LDVT, N
143 REAL RESID
144* ..
145* .. Array Arguments ..
146 REAL D( * ), E( * ), S( * ), U( LDU, * ),
147 \$ VT( LDVT, * ), WORK( * )
148* ..
149*
150* ======================================================================
151*
152* .. Parameters ..
153 REAL ZERO, ONE
154 parameter( zero = 0.0e+0, one = 1.0e+0 )
155* ..
156* .. Local Scalars ..
157 INTEGER I, J
158 REAL BNORM, EPS
159* ..
160* .. External Functions ..
161 LOGICAL LSAME
162 INTEGER ISAMAX
163 REAL SASUM, SLAMCH
164 EXTERNAL lsame, isamax, sasum, slamch
165* ..
166* .. External Subroutines ..
167 EXTERNAL sgemv
168* ..
169* .. Intrinsic Functions ..
170 INTRINSIC abs, max, min, real
171* ..
172* .. Executable Statements ..
173*
174* Quick return if possible
175*
176 resid = zero
177 IF( n.LE.0 )
178 \$ RETURN
179*
180* Compute B - U * S * V' one column at a time.
181*
182 bnorm = zero
183 IF( kd.GE.1 ) THEN
184*
185* B is bidiagonal.
186*
187 IF( lsame( uplo, 'U' ) ) THEN
188*
189* B is upper bidiagonal.
190*
191 DO 20 j = 1, n
192 DO 10 i = 1, n
193 work( n+i ) = s( i )*vt( i, j )
194 10 CONTINUE
195 CALL sgemv( 'No transpose', n, n, -one, u, ldu,
196 \$ work( n+1 ), 1, zero, work, 1 )
197 work( j ) = work( j ) + d( j )
198 IF( j.GT.1 ) THEN
199 work( j-1 ) = work( j-1 ) + e( j-1 )
200 bnorm = max( bnorm, abs( d( j ) )+abs( e( j-1 ) ) )
201 ELSE
202 bnorm = max( bnorm, abs( d( j ) ) )
203 END IF
204 resid = max( resid, sasum( n, work, 1 ) )
205 20 CONTINUE
206 ELSE
207*
208* B is lower bidiagonal.
209*
210 DO 40 j = 1, n
211 DO 30 i = 1, n
212 work( n+i ) = s( i )*vt( i, j )
213 30 CONTINUE
214 CALL sgemv( 'No transpose', n, n, -one, u, ldu,
215 \$ work( n+1 ), 1, zero, work, 1 )
216 work( j ) = work( j ) + d( j )
217 IF( j.LT.n ) THEN
218 work( j+1 ) = work( j+1 ) + e( j )
219 bnorm = max( bnorm, abs( d( j ) )+abs( e( j ) ) )
220 ELSE
221 bnorm = max( bnorm, abs( d( j ) ) )
222 END IF
223 resid = max( resid, sasum( n, work, 1 ) )
224 40 CONTINUE
225 END IF
226 ELSE
227*
228* B is diagonal.
229*
230 DO 60 j = 1, n
231 DO 50 i = 1, n
232 work( n+i ) = s( i )*vt( i, j )
233 50 CONTINUE
234 CALL sgemv( 'No transpose', n, n, -one, u, ldu, work( n+1 ),
235 \$ 1, zero, work, 1 )
236 work( j ) = work( j ) + d( j )
237 resid = max( resid, sasum( n, work, 1 ) )
238 60 CONTINUE
239 j = isamax( n, d, 1 )
240 bnorm = abs( d( j ) )
241 END IF
242*
243* Compute norm(B - U * S * V') / ( n * norm(B) * EPS )
244*
245 eps = slamch( 'Precision' )
246*
247 IF( bnorm.LE.zero ) THEN
248 IF( resid.NE.zero )
249 \$ resid = one / eps
250 ELSE
251 IF( bnorm.GE.resid ) THEN
252 resid = ( resid / bnorm ) / ( real( n )*eps )
253 ELSE
254 IF( bnorm.LT.one ) THEN
255 resid = ( min( resid, real( n )*bnorm ) / bnorm ) /
256 \$ ( real( n )*eps )
257 ELSE
258 resid = min( resid / bnorm, real( n ) ) /
259 \$ ( real( n )*eps )
260 END IF
261 END IF
262 END IF
263*
264 RETURN
265*
266* End of SBDT03
267*
integer function isamax(N, SX, INCX)
ISAMAX
Definition: isamax.f:71
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
real function sasum(N, SX, INCX)
SASUM
Definition: sasum.f:72
subroutine sgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SGEMV
Definition: sgemv.f:156
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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