*> \brief \b ZSYCON_3 * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download ZSYCON_3 + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE ZSYCON_3( UPLO, N, A, LDA, E, IPIV, ANORM, RCOND, * WORK, INFO ) * * .. Scalar Arguments .. * CHARACTER UPLO * INTEGER INFO, LDA, N * DOUBLE PRECISION ANORM, RCOND * .. * .. Array Arguments .. * INTEGER IPIV( * ) * COMPLEX*16 A( LDA, * ), E ( * ), WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> ZSYCON_3 estimates the reciprocal of the condition number (in the *> 1-norm) of a complex symmetric matrix A using the factorization *> computed by ZSYTRF_RK or ZSYTRF_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 ZSYTRS_3. *> \endverbatim * * Arguments: * ========== * *> \param[in] UPLO *> \verbatim *> 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). *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrix A. N >= 0. *> \endverbatim *> *> \param[in] A *> \verbatim *> A is COMPLEX*16 array, dimension (LDA,N) *> Diagonal of the block diagonal matrix D and factors U or L *> as computed by ZSYTRF_RK and ZSYTRF_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. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= max(1,N). *> \endverbatim *> *> \param[in] E *> \verbatim *> E is COMPLEX*16 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. *> \endverbatim *> *> \param[in] IPIV *> \verbatim *> IPIV is INTEGER array, dimension (N) *> Details of the interchanges and the block structure of D *> as determined by ZSYTRF_RK or ZSYTRF_BK. *> \endverbatim *> *> \param[in] ANORM *> \verbatim *> ANORM is DOUBLE PRECISION *> The 1-norm of the original matrix A. *> \endverbatim *> *> \param[out] RCOND *> \verbatim *> 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. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is COMPLEX*16 array, dimension (2*N) *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: successful exit *> < 0: if INFO = -i, the i-th argument had an illegal value *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup complex16SYcomputational * *> \par Contributors: * ================== *> \verbatim *> *> 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 *> *> \endverbatim * * ===================================================================== SUBROUTINE ZSYCON_3( UPLO, N, A, LDA, E, IPIV, ANORM, RCOND, \$ WORK, INFO ) * * -- LAPACK computational routine -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * * .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDA, N DOUBLE PRECISION ANORM, RCOND * .. * .. Array Arguments .. INTEGER IPIV( * ) COMPLEX*16 A( LDA, * ), E( * ), WORK( * ) * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) COMPLEX*16 CZERO PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ) ) * .. * .. Local Scalars .. LOGICAL UPPER INTEGER I, KASE DOUBLE PRECISION AINVNM * .. * .. Local Arrays .. INTEGER ISAVE( 3 ) * .. * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. External Subroutines .. EXTERNAL ZLACN2, ZSYTRS_3, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -4 ELSE IF( ANORM.LT.ZERO ) THEN INFO = -7 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZSYCON_3', -INFO ) RETURN END IF * * Quick return if possible * RCOND = ZERO IF( N.EQ.0 ) THEN RCOND = ONE RETURN ELSE IF( ANORM.LE.ZERO ) THEN RETURN END IF * * Check that the diagonal matrix D is nonsingular. * IF( UPPER ) THEN * * Upper triangular storage: examine D from bottom to top * DO I = N, 1, -1 IF( IPIV( I ).GT.0 .AND. A( I, I ).EQ.CZERO ) \$ RETURN END DO ELSE * * Lower triangular storage: examine D from top to bottom. * DO I = 1, N IF( IPIV( I ).GT.0 .AND. A( I, I ).EQ.CZERO ) \$ RETURN END DO END IF * * Estimate the 1-norm of the inverse. * KASE = 0 30 CONTINUE CALL ZLACN2( N, WORK( N+1 ), WORK, AINVNM, KASE, ISAVE ) IF( KASE.NE.0 ) THEN * * Multiply by inv(L*D*L**T) or inv(U*D*U**T). * CALL ZSYTRS_3( UPLO, N, 1, A, LDA, E, IPIV, WORK, N, INFO ) GO TO 30 END IF * * Compute the estimate of the reciprocal condition number. * IF( AINVNM.NE.ZERO ) \$ RCOND = ( ONE / AINVNM ) / ANORM * RETURN * * End of ZSYCON_3 * END