 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ dsycon_3()

 subroutine dsycon_3 ( character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) E, integer, dimension( * ) IPIV, double precision ANORM, double precision RCOND, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO )

DSYCON_3

Purpose:
``` DSYCON_3 estimates the reciprocal of the condition number (in the
1-norm) of a real symmetric matrix A using the factorization
computed by DSYTRF_RK or DSYTRF_BK:

A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),

where U (or L) is unit upper (or lower) triangular matrix,
U**T (or L**T) is the transpose of U (or L), P is a permutation
matrix, P**T is the transpose of P, and D is symmetric and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.

An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
This routine uses BLAS3 solver DSYTRS_3.```
Parameters
 [in] UPLO ``` UPLO is CHARACTER*1 Specifies whether the details of the factorization are stored as an upper or lower triangular matrix: = 'U': Upper triangular, form is A = P*U*D*(U**T)*(P**T); = 'L': Lower triangular, form is A = P*L*D*(L**T)*(P**T).``` [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in] A ``` A is DOUBLE PRECISION array, dimension (LDA,N) Diagonal of the block diagonal matrix D and factors U or L as computed by DSYTRF_RK and DSYTRF_BK: a) ONLY diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D should be provided on entry in array E), and b) If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [in] E ``` E is DOUBLE PRECISION array, dimension (N) On entry, contains the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced; If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced. NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is not referenced in both UPLO = 'U' or UPLO = 'L' cases.``` [in] IPIV ``` IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by DSYTRF_RK or DSYTRF_BK.``` [in] ANORM ``` ANORM is DOUBLE PRECISION The 1-norm of the original matrix A.``` [out] RCOND ``` RCOND is DOUBLE PRECISION The reciprocal of the condition number of the matrix A, computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an estimate of the 1-norm of inv(A) computed in this routine.``` [out] WORK ` WORK is DOUBLE PRECISION array, dimension (2*N)` [out] IWORK ` IWORK is INTEGER array, dimension (N)` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value```
Contributors:
```  June 2017,  Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester```

Definition at line 169 of file dsycon_3.f.

171*
172* -- LAPACK computational routine --
173* -- LAPACK is a software package provided by Univ. of Tennessee, --
174* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
175*
176* .. Scalar Arguments ..
177 CHARACTER UPLO
178 INTEGER INFO, LDA, N
179 DOUBLE PRECISION ANORM, RCOND
180* ..
181* .. Array Arguments ..
182 INTEGER IPIV( * ), IWORK( * )
183 DOUBLE PRECISION A( LDA, * ), E( * ), WORK( * )
184* ..
185*
186* =====================================================================
187*
188* .. Parameters ..
189 DOUBLE PRECISION ONE, ZERO
190 parameter( one = 1.0d+0, zero = 0.0d+0 )
191* ..
192* .. Local Scalars ..
193 LOGICAL UPPER
194 INTEGER I, KASE
195 DOUBLE PRECISION AINVNM
196* ..
197* .. Local Arrays ..
198 INTEGER ISAVE( 3 )
199* ..
200* .. External Functions ..
201 LOGICAL LSAME
202 EXTERNAL lsame
203* ..
204* .. External Subroutines ..
205 EXTERNAL dlacn2, dsytrs_3, xerbla
206* ..
207* .. Intrinsic Functions ..
208 INTRINSIC max
209* ..
210* .. Executable Statements ..
211*
212* Test the input parameters.
213*
214 info = 0
215 upper = lsame( uplo, 'U' )
216 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
217 info = -1
218 ELSE IF( n.LT.0 ) THEN
219 info = -2
220 ELSE IF( lda.LT.max( 1, n ) ) THEN
221 info = -4
222 ELSE IF( anorm.LT.zero ) THEN
223 info = -7
224 END IF
225 IF( info.NE.0 ) THEN
226 CALL xerbla( 'DSYCON_3', -info )
227 RETURN
228 END IF
229*
230* Quick return if possible
231*
232 rcond = zero
233 IF( n.EQ.0 ) THEN
234 rcond = one
235 RETURN
236 ELSE IF( anorm.LE.zero ) THEN
237 RETURN
238 END IF
239*
240* Check that the diagonal matrix D is nonsingular.
241*
242 IF( upper ) THEN
243*
244* Upper triangular storage: examine D from bottom to top
245*
246 DO i = n, 1, -1
247 IF( ipiv( i ).GT.0 .AND. a( i, i ).EQ.zero )
248 \$ RETURN
249 END DO
250 ELSE
251*
252* Lower triangular storage: examine D from top to bottom.
253*
254 DO i = 1, n
255 IF( ipiv( i ).GT.0 .AND. a( i, i ).EQ.zero )
256 \$ RETURN
257 END DO
258 END IF
259*
260* Estimate the 1-norm of the inverse.
261*
262 kase = 0
263 30 CONTINUE
264 CALL dlacn2( n, work( n+1 ), work, iwork, ainvnm, kase, isave )
265 IF( kase.NE.0 ) THEN
266*
267* Multiply by inv(L*D*L**T) or inv(U*D*U**T).
268*
269 CALL dsytrs_3( uplo, n, 1, a, lda, e, ipiv, work, n, info )
270 GO TO 30
271 END IF
272*
273* Compute the estimate of the reciprocal condition number.
274*
275 IF( ainvnm.NE.zero )
276 \$ rcond = ( one / ainvnm ) / anorm
277*
278 RETURN
279*
280* End of DSYCON_3
281*
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine dlacn2(N, V, X, ISGN, EST, KASE, ISAVE)
DLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vec...
Definition: dlacn2.f:136
subroutine dsytrs_3(UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB, INFO)
DSYTRS_3
Definition: dsytrs_3.f:165
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