LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 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 )```
Date
November 2011

Definition at line 137 of file sbdt03.f.

137 *
138 * -- LAPACK test routine (version 3.4.0) --
139 * -- LAPACK is a software package provided by Univ. of Tennessee, --
140 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
141 * November 2011
142 *
143 * .. Scalar Arguments ..
144  CHARACTER uplo
145  INTEGER kd, ldu, ldvt, n
146  REAL resid
147 * ..
148 * .. Array Arguments ..
149  REAL d( * ), e( * ), s( * ), u( ldu, * ),
150  \$ vt( ldvt, * ), work( * )
151 * ..
152 *
153 * ======================================================================
154 *
155 * .. Parameters ..
156  REAL zero, one
157  parameter ( zero = 0.0e+0, one = 1.0e+0 )
158 * ..
159 * .. Local Scalars ..
160  INTEGER i, j
161  REAL bnorm, eps
162 * ..
163 * .. External Functions ..
164  LOGICAL lsame
165  INTEGER isamax
166  REAL sasum, slamch
167  EXTERNAL lsame, isamax, sasum, slamch
168 * ..
169 * .. External Subroutines ..
170  EXTERNAL sgemv
171 * ..
172 * .. Intrinsic Functions ..
173  INTRINSIC abs, max, min, real
174 * ..
175 * .. Executable Statements ..
176 *
177 * Quick return if possible
178 *
179  resid = zero
180  IF( n.LE.0 )
181  \$ RETURN
182 *
183 * Compute B - U * S * V' one column at a time.
184 *
185  bnorm = zero
186  IF( kd.GE.1 ) THEN
187 *
188 * B is bidiagonal.
189 *
190  IF( lsame( uplo, 'U' ) ) THEN
191 *
192 * B is upper bidiagonal.
193 *
194  DO 20 j = 1, n
195  DO 10 i = 1, n
196  work( n+i ) = s( i )*vt( i, j )
197  10 CONTINUE
198  CALL sgemv( 'No transpose', n, n, -one, u, ldu,
199  \$ work( n+1 ), 1, zero, work, 1 )
200  work( j ) = work( j ) + d( j )
201  IF( j.GT.1 ) THEN
202  work( j-1 ) = work( j-1 ) + e( j-1 )
203  bnorm = max( bnorm, abs( d( j ) )+abs( e( j-1 ) ) )
204  ELSE
205  bnorm = max( bnorm, abs( d( j ) ) )
206  END IF
207  resid = max( resid, sasum( n, work, 1 ) )
208  20 CONTINUE
209  ELSE
210 *
211 * B is lower bidiagonal.
212 *
213  DO 40 j = 1, n
214  DO 30 i = 1, n
215  work( n+i ) = s( i )*vt( i, j )
216  30 CONTINUE
217  CALL sgemv( 'No transpose', n, n, -one, u, ldu,
218  \$ work( n+1 ), 1, zero, work, 1 )
219  work( j ) = work( j ) + d( j )
220  IF( j.LT.n ) THEN
221  work( j+1 ) = work( j+1 ) + e( j )
222  bnorm = max( bnorm, abs( d( j ) )+abs( e( j ) ) )
223  ELSE
224  bnorm = max( bnorm, abs( d( j ) ) )
225  END IF
226  resid = max( resid, sasum( n, work, 1 ) )
227  40 CONTINUE
228  END IF
229  ELSE
230 *
231 * B is diagonal.
232 *
233  DO 60 j = 1, n
234  DO 50 i = 1, n
235  work( n+i ) = s( i )*vt( i, j )
236  50 CONTINUE
237  CALL sgemv( 'No transpose', n, n, -one, u, ldu, work( n+1 ),
238  \$ 1, zero, work, 1 )
239  work( j ) = work( j ) + d( j )
240  resid = max( resid, sasum( n, work, 1 ) )
241  60 CONTINUE
242  j = isamax( n, d, 1 )
243  bnorm = abs( d( j ) )
244  END IF
245 *
246 * Compute norm(B - U * S * V') / ( n * norm(B) * EPS )
247 *
248  eps = slamch( 'Precision' )
249 *
250  IF( bnorm.LE.zero ) THEN
251  IF( resid.NE.zero )
252  \$ resid = one / eps
253  ELSE
254  IF( bnorm.GE.resid ) THEN
255  resid = ( resid / bnorm ) / ( REAL( n )*eps )
256  ELSE
257  IF( bnorm.LT.one ) THEN
258  resid = ( min( resid, REAL( n )*bnorm ) / bnorm ) /
259  \$ ( REAL( n )*eps )
260  ELSE
261  resid = min( resid / bnorm, REAL( N ) ) /
262  \$ ( REAL( n )*eps )
263  END IF
264  END IF
265  END IF
266 *
267  RETURN
268 *
269 * End of SBDT03
270 *
integer function isamax(N, SX, INCX)
ISAMAX
Definition: isamax.f:53
subroutine sgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SGEMV
Definition: sgemv.f:158
real function sasum(N, SX, INCX)
SASUM
Definition: sasum.f:54
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55

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