 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ dsbt21()

 subroutine dsbt21 ( character UPLO, integer N, integer KA, integer KS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision, dimension( ldu, * ) U, integer LDU, double precision, dimension( * ) WORK, double precision, dimension( 2 ) RESULT )

DSBT21

Purpose:
``` DSBT21  generally checks a decomposition of the form

A = U S U**T

where **T means transpose, A is symmetric banded, U is
orthogonal, and S is diagonal (if KS=0) or symmetric
tridiagonal (if KS=1).

Specifically:

RESULT(1) = | A - U S U**T | / ( |A| n ulp ) and
RESULT(2) = | I - U U**T | / ( n ulp )```
Parameters
 [in] UPLO ``` UPLO is CHARACTER If UPLO='U', the upper triangle of A and V will be used and the (strictly) lower triangle will not be referenced. If UPLO='L', the lower triangle of A and V will be used and the (strictly) upper triangle will not be referenced.``` [in] N ``` N is INTEGER The size of the matrix. If it is zero, DSBT21 does nothing. It must be at least zero.``` [in] KA ``` KA is INTEGER The bandwidth of the matrix A. It must be at least zero. If it is larger than N-1, then max( 0, N-1 ) will be used.``` [in] KS ``` KS is INTEGER The bandwidth of the matrix S. It may only be zero or one. If zero, then S is diagonal, and E is not referenced. If one, then S is symmetric tri-diagonal.``` [in] A ``` A is DOUBLE PRECISION array, dimension (LDA, N) The original (unfactored) matrix. It is assumed to be symmetric, and only the upper (UPLO='U') or only the lower (UPLO='L') will be referenced.``` [in] LDA ``` LDA is INTEGER The leading dimension of A. It must be at least 1 and at least min( KA, N-1 ).``` [in] D ``` D is DOUBLE PRECISION array, dimension (N) The diagonal of the (symmetric tri-) diagonal matrix S.``` [in] E ``` E is DOUBLE PRECISION array, dimension (N-1) The off-diagonal of the (symmetric tri-) diagonal matrix S. E(1) is the (1,2) and (2,1) element, E(2) is the (2,3) and (3,2) element, etc. Not referenced if KS=0.``` [in] U ``` U is DOUBLE PRECISION array, dimension (LDU, N) The orthogonal matrix in the decomposition, expressed as a dense matrix (i.e., not as a product of Householder transformations, Givens transformations, etc.)``` [in] LDU ``` LDU is INTEGER The leading dimension of U. LDU must be at least N and at least 1.``` [out] WORK ` WORK is DOUBLE PRECISION array, dimension (N**2+N)` [out] RESULT ``` RESULT is DOUBLE PRECISION array, dimension (2) The values computed by the two tests described above. The values are currently limited to 1/ulp, to avoid overflow.```

Definition at line 145 of file dsbt21.f.

147*
148* -- LAPACK test routine --
149* -- LAPACK is a software package provided by Univ. of Tennessee, --
150* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
151*
152* .. Scalar Arguments ..
153 CHARACTER UPLO
154 INTEGER KA, KS, LDA, LDU, N
155* ..
156* .. Array Arguments ..
157 DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), RESULT( 2 ),
158 \$ U( LDU, * ), WORK( * )
159* ..
160*
161* =====================================================================
162*
163* .. Parameters ..
164 DOUBLE PRECISION ZERO, ONE
165 parameter( zero = 0.0d0, one = 1.0d0 )
166* ..
167* .. Local Scalars ..
168 LOGICAL LOWER
169 CHARACTER CUPLO
170 INTEGER IKA, J, JC, JR, LW
171 DOUBLE PRECISION ANORM, ULP, UNFL, WNORM
172* ..
173* .. External Functions ..
174 LOGICAL LSAME
175 DOUBLE PRECISION DLAMCH, DLANGE, DLANSB, DLANSP
176 EXTERNAL lsame, dlamch, dlange, dlansb, dlansp
177* ..
178* .. External Subroutines ..
179 EXTERNAL dgemm, dspr, dspr2
180* ..
181* .. Intrinsic Functions ..
182 INTRINSIC dble, max, min
183* ..
184* .. Executable Statements ..
185*
186* Constants
187*
188 result( 1 ) = zero
189 result( 2 ) = zero
190 IF( n.LE.0 )
191 \$ RETURN
192*
193 ika = max( 0, min( n-1, ka ) )
194 lw = ( n*( n+1 ) ) / 2
195*
196 IF( lsame( uplo, 'U' ) ) THEN
197 lower = .false.
198 cuplo = 'U'
199 ELSE
200 lower = .true.
201 cuplo = 'L'
202 END IF
203*
204 unfl = dlamch( 'Safe minimum' )
205 ulp = dlamch( 'Epsilon' )*dlamch( 'Base' )
206*
207* Some Error Checks
208*
209* Do Test 1
210*
211* Norm of A:
212*
213 anorm = max( dlansb( '1', cuplo, n, ika, a, lda, work ), unfl )
214*
215* Compute error matrix: Error = A - U S U**T
216*
217* Copy A from SB to SP storage format.
218*
219 j = 0
220 DO 50 jc = 1, n
221 IF( lower ) THEN
222 DO 10 jr = 1, min( ika+1, n+1-jc )
223 j = j + 1
224 work( j ) = a( jr, jc )
225 10 CONTINUE
226 DO 20 jr = ika + 2, n + 1 - jc
227 j = j + 1
228 work( j ) = zero
229 20 CONTINUE
230 ELSE
231 DO 30 jr = ika + 2, jc
232 j = j + 1
233 work( j ) = zero
234 30 CONTINUE
235 DO 40 jr = min( ika, jc-1 ), 0, -1
236 j = j + 1
237 work( j ) = a( ika+1-jr, jc )
238 40 CONTINUE
239 END IF
240 50 CONTINUE
241*
242 DO 60 j = 1, n
243 CALL dspr( cuplo, n, -d( j ), u( 1, j ), 1, work )
244 60 CONTINUE
245*
246 IF( n.GT.1 .AND. ks.EQ.1 ) THEN
247 DO 70 j = 1, n - 1
248 CALL dspr2( cuplo, n, -e( j ), u( 1, j ), 1, u( 1, j+1 ), 1,
249 \$ work )
250 70 CONTINUE
251 END IF
252 wnorm = dlansp( '1', cuplo, n, work, work( lw+1 ) )
253*
254 IF( anorm.GT.wnorm ) THEN
255 result( 1 ) = ( wnorm / anorm ) / ( n*ulp )
256 ELSE
257 IF( anorm.LT.one ) THEN
258 result( 1 ) = ( min( wnorm, n*anorm ) / anorm ) / ( n*ulp )
259 ELSE
260 result( 1 ) = min( wnorm / anorm, dble( n ) ) / ( n*ulp )
261 END IF
262 END IF
263*
264* Do Test 2
265*
266* Compute U U**T - I
267*
268 CALL dgemm( 'N', 'C', n, n, n, one, u, ldu, u, ldu, zero, work,
269 \$ n )
270*
271 DO 80 j = 1, n
272 work( ( n+1 )*( j-1 )+1 ) = work( ( n+1 )*( j-1 )+1 ) - one
273 80 CONTINUE
274*
275 result( 2 ) = min( dlange( '1', n, n, work, n, work( n**2+1 ) ),
276 \$ dble( n ) ) / ( n*ulp )
277*
278 RETURN
279*
280* End of DSBT21
281*
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine dspr2(UPLO, N, ALPHA, X, INCX, Y, INCY, AP)
DSPR2
Definition: dspr2.f:142
subroutine dspr(UPLO, N, ALPHA, X, INCX, AP)
DSPR
Definition: dspr.f:127
subroutine dgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
DGEMM
Definition: dgemm.f:187
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
double precision function dlansp(NORM, UPLO, N, AP, WORK)
DLANSP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: dlansp.f:114
double precision function dlansb(NORM, UPLO, N, K, AB, LDAB, WORK)
DLANSB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: dlansb.f:129
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