 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ 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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