574 lines
15 KiB
Fortran
574 lines
15 KiB
Fortran
*> \brief \b CSTEQR
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*
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* =========== DOCUMENTATION ===========
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*
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* Online html documentation available at
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* http://www.netlib.org/lapack/explore-html/
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*
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*> \htmlonly
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*> Download CSTEQR + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/csteqr.f">
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*> [TGZ]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/csteqr.f">
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*> [ZIP]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/csteqr.f">
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*> [TXT]</a>
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*> \endhtmlonly
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*
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* Definition:
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* ===========
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*
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* SUBROUTINE CSTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
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*
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* .. Scalar Arguments ..
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* CHARACTER COMPZ
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* INTEGER INFO, LDZ, N
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* ..
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* .. Array Arguments ..
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* REAL D( * ), E( * ), WORK( * )
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* COMPLEX Z( LDZ, * )
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* ..
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*
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*
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*> \par Purpose:
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* =============
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*>
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*> \verbatim
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*>
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*> CSTEQR computes all eigenvalues and, optionally, eigenvectors of a
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*> symmetric tridiagonal matrix using the implicit QL or QR method.
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*> The eigenvectors of a full or band complex Hermitian matrix can also
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*> be found if CHETRD or CHPTRD or CHBTRD has been used to reduce this
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*> matrix to tridiagonal form.
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*> \endverbatim
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*
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* Arguments:
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* ==========
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*
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*> \param[in] COMPZ
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*> \verbatim
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*> COMPZ is CHARACTER*1
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*> = 'N': Compute eigenvalues only.
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*> = 'V': Compute eigenvalues and eigenvectors of the original
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*> Hermitian matrix. On entry, Z must contain the
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*> unitary matrix used to reduce the original matrix
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*> to tridiagonal form.
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*> = 'I': Compute eigenvalues and eigenvectors of the
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*> tridiagonal matrix. Z is initialized to the identity
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*> matrix.
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*> \endverbatim
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*>
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*> \param[in] N
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*> \verbatim
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*> N is INTEGER
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*> The order of the matrix. N >= 0.
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*> \endverbatim
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*>
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*> \param[in,out] D
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*> \verbatim
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*> D is REAL array, dimension (N)
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*> On entry, the diagonal elements of the tridiagonal matrix.
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*> On exit, if INFO = 0, the eigenvalues in ascending order.
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*> \endverbatim
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*>
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*> \param[in,out] E
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*> \verbatim
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*> E is REAL array, dimension (N-1)
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*> On entry, the (n-1) subdiagonal elements of the tridiagonal
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*> matrix.
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*> On exit, E has been destroyed.
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*> \endverbatim
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*>
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*> \param[in,out] Z
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*> \verbatim
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*> Z is COMPLEX array, dimension (LDZ, N)
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*> On entry, if COMPZ = 'V', then Z contains the unitary
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*> matrix used in the reduction to tridiagonal form.
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*> On exit, if INFO = 0, then if COMPZ = 'V', Z contains the
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*> orthonormal eigenvectors of the original Hermitian matrix,
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*> and if COMPZ = 'I', Z contains the orthonormal eigenvectors
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*> of the symmetric tridiagonal matrix.
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*> If COMPZ = 'N', then Z is not referenced.
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*> \endverbatim
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*>
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*> \param[in] LDZ
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*> \verbatim
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*> LDZ is INTEGER
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*> The leading dimension of the array Z. LDZ >= 1, and if
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*> eigenvectors are desired, then LDZ >= max(1,N).
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*> \endverbatim
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*>
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*> \param[out] WORK
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*> \verbatim
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*> WORK is REAL array, dimension (max(1,2*N-2))
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*> If COMPZ = 'N', then WORK is not referenced.
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*> \endverbatim
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*>
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*> \param[out] INFO
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*> \verbatim
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*> INFO is INTEGER
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*> = 0: successful exit
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*> < 0: if INFO = -i, the i-th argument had an illegal value
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*> > 0: the algorithm has failed to find all the eigenvalues in
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*> a total of 30*N iterations; if INFO = i, then i
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*> elements of E have not converged to zero; on exit, D
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*> and E contain the elements of a symmetric tridiagonal
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*> matrix which is unitarily similar to the original
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*> matrix.
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*> \endverbatim
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*
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* Authors:
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* ========
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*
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*> \author Univ. of Tennessee
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*> \author Univ. of California Berkeley
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*> \author Univ. of Colorado Denver
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*> \author NAG Ltd.
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*
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*> \ingroup complexOTHERcomputational
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*
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* =====================================================================
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SUBROUTINE CSTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
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*
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* -- LAPACK computational routine --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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*
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* .. Scalar Arguments ..
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CHARACTER COMPZ
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INTEGER INFO, LDZ, N
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* ..
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* .. Array Arguments ..
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REAL D( * ), E( * ), WORK( * )
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COMPLEX Z( LDZ, * )
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* ..
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*
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* =====================================================================
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*
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* .. Parameters ..
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REAL ZERO, ONE, TWO, THREE
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PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0, TWO = 2.0E0,
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$ THREE = 3.0E0 )
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COMPLEX CZERO, CONE
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PARAMETER ( CZERO = ( 0.0E0, 0.0E0 ),
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$ CONE = ( 1.0E0, 0.0E0 ) )
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INTEGER MAXIT
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PARAMETER ( MAXIT = 30 )
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* ..
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* .. Local Scalars ..
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INTEGER I, ICOMPZ, II, ISCALE, J, JTOT, K, L, L1, LEND,
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$ LENDM1, LENDP1, LENDSV, LM1, LSV, M, MM, MM1,
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$ NM1, NMAXIT
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REAL ANORM, B, C, EPS, EPS2, F, G, P, R, RT1, RT2,
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$ S, SAFMAX, SAFMIN, SSFMAX, SSFMIN, TST
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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REAL SLAMCH, SLANST, SLAPY2
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EXTERNAL LSAME, SLAMCH, SLANST, SLAPY2
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* ..
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* .. External Subroutines ..
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EXTERNAL CLASET, CLASR, CSWAP, SLAE2, SLAEV2, SLARTG,
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$ SLASCL, SLASRT, XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, MAX, SIGN, SQRT
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* ..
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* .. Executable Statements ..
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*
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* Test the input parameters.
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*
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INFO = 0
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*
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IF( LSAME( COMPZ, 'N' ) ) THEN
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ICOMPZ = 0
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ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
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ICOMPZ = 1
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ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
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ICOMPZ = 2
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ELSE
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ICOMPZ = -1
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END IF
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IF( ICOMPZ.LT.0 ) THEN
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INFO = -1
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ELSE IF( N.LT.0 ) THEN
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INFO = -2
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ELSE IF( ( LDZ.LT.1 ) .OR. ( ICOMPZ.GT.0 .AND. LDZ.LT.MAX( 1,
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$ N ) ) ) THEN
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INFO = -6
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'CSTEQR', -INFO )
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RETURN
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END IF
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*
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* Quick return if possible
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*
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IF( N.EQ.0 )
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$ RETURN
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*
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IF( N.EQ.1 ) THEN
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IF( ICOMPZ.EQ.2 )
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$ Z( 1, 1 ) = CONE
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RETURN
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END IF
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*
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* Determine the unit roundoff and over/underflow thresholds.
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*
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EPS = SLAMCH( 'E' )
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EPS2 = EPS**2
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SAFMIN = SLAMCH( 'S' )
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SAFMAX = ONE / SAFMIN
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SSFMAX = SQRT( SAFMAX ) / THREE
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SSFMIN = SQRT( SAFMIN ) / EPS2
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*
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* Compute the eigenvalues and eigenvectors of the tridiagonal
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* matrix.
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*
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IF( ICOMPZ.EQ.2 )
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$ CALL CLASET( 'Full', N, N, CZERO, CONE, Z, LDZ )
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*
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NMAXIT = N*MAXIT
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JTOT = 0
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*
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* Determine where the matrix splits and choose QL or QR iteration
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* for each block, according to whether top or bottom diagonal
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* element is smaller.
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*
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L1 = 1
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NM1 = N - 1
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*
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10 CONTINUE
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IF( L1.GT.N )
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$ GO TO 160
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IF( L1.GT.1 )
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$ E( L1-1 ) = ZERO
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IF( L1.LE.NM1 ) THEN
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DO 20 M = L1, NM1
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TST = ABS( E( M ) )
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IF( TST.EQ.ZERO )
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$ GO TO 30
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IF( TST.LE.( SQRT( ABS( D( M ) ) )*SQRT( ABS( D( M+
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$ 1 ) ) ) )*EPS ) THEN
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E( M ) = ZERO
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GO TO 30
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END IF
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20 CONTINUE
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END IF
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M = N
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*
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30 CONTINUE
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L = L1
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LSV = L
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LEND = M
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LENDSV = LEND
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L1 = M + 1
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IF( LEND.EQ.L )
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$ GO TO 10
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*
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* Scale submatrix in rows and columns L to LEND
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*
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ANORM = SLANST( 'I', LEND-L+1, D( L ), E( L ) )
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ISCALE = 0
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IF( ANORM.EQ.ZERO )
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$ GO TO 10
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IF( ANORM.GT.SSFMAX ) THEN
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ISCALE = 1
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CALL SLASCL( 'G', 0, 0, ANORM, SSFMAX, LEND-L+1, 1, D( L ), N,
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$ INFO )
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CALL SLASCL( 'G', 0, 0, ANORM, SSFMAX, LEND-L, 1, E( L ), N,
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$ INFO )
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ELSE IF( ANORM.LT.SSFMIN ) THEN
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ISCALE = 2
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CALL SLASCL( 'G', 0, 0, ANORM, SSFMIN, LEND-L+1, 1, D( L ), N,
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$ INFO )
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CALL SLASCL( 'G', 0, 0, ANORM, SSFMIN, LEND-L, 1, E( L ), N,
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$ INFO )
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END IF
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*
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* Choose between QL and QR iteration
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*
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IF( ABS( D( LEND ) ).LT.ABS( D( L ) ) ) THEN
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LEND = LSV
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L = LENDSV
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END IF
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*
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IF( LEND.GT.L ) THEN
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*
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* QL Iteration
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*
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* Look for small subdiagonal element.
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*
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40 CONTINUE
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IF( L.NE.LEND ) THEN
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LENDM1 = LEND - 1
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DO 50 M = L, LENDM1
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TST = ABS( E( M ) )**2
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IF( TST.LE.( EPS2*ABS( D( M ) ) )*ABS( D( M+1 ) )+
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$ SAFMIN )GO TO 60
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50 CONTINUE
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END IF
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*
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M = LEND
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*
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60 CONTINUE
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IF( M.LT.LEND )
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$ E( M ) = ZERO
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P = D( L )
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IF( M.EQ.L )
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$ GO TO 80
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*
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* If remaining matrix is 2-by-2, use SLAE2 or SLAEV2
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* to compute its eigensystem.
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*
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IF( M.EQ.L+1 ) THEN
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IF( ICOMPZ.GT.0 ) THEN
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CALL SLAEV2( D( L ), E( L ), D( L+1 ), RT1, RT2, C, S )
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WORK( L ) = C
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WORK( N-1+L ) = S
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CALL CLASR( 'R', 'V', 'B', N, 2, WORK( L ),
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$ WORK( N-1+L ), Z( 1, L ), LDZ )
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ELSE
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CALL SLAE2( D( L ), E( L ), D( L+1 ), RT1, RT2 )
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END IF
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D( L ) = RT1
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D( L+1 ) = RT2
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E( L ) = ZERO
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L = L + 2
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IF( L.LE.LEND )
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$ GO TO 40
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GO TO 140
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END IF
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*
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IF( JTOT.EQ.NMAXIT )
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$ GO TO 140
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JTOT = JTOT + 1
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*
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* Form shift.
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*
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G = ( D( L+1 )-P ) / ( TWO*E( L ) )
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R = SLAPY2( G, ONE )
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G = D( M ) - P + ( E( L ) / ( G+SIGN( R, G ) ) )
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*
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S = ONE
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C = ONE
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P = ZERO
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*
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* Inner loop
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*
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MM1 = M - 1
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DO 70 I = MM1, L, -1
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F = S*E( I )
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B = C*E( I )
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CALL SLARTG( G, F, C, S, R )
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IF( I.NE.M-1 )
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$ E( I+1 ) = R
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G = D( I+1 ) - P
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R = ( D( I )-G )*S + TWO*C*B
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P = S*R
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D( I+1 ) = G + P
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G = C*R - B
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*
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* If eigenvectors are desired, then save rotations.
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*
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IF( ICOMPZ.GT.0 ) THEN
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WORK( I ) = C
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WORK( N-1+I ) = -S
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END IF
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*
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70 CONTINUE
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*
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* If eigenvectors are desired, then apply saved rotations.
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*
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IF( ICOMPZ.GT.0 ) THEN
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MM = M - L + 1
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CALL CLASR( 'R', 'V', 'B', N, MM, WORK( L ), WORK( N-1+L ),
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$ Z( 1, L ), LDZ )
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END IF
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*
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D( L ) = D( L ) - P
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E( L ) = G
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GO TO 40
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*
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* Eigenvalue found.
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*
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80 CONTINUE
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D( L ) = P
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*
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L = L + 1
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IF( L.LE.LEND )
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$ GO TO 40
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GO TO 140
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*
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ELSE
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*
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* QR Iteration
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*
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* Look for small superdiagonal element.
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*
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90 CONTINUE
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IF( L.NE.LEND ) THEN
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LENDP1 = LEND + 1
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DO 100 M = L, LENDP1, -1
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TST = ABS( E( M-1 ) )**2
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IF( TST.LE.( EPS2*ABS( D( M ) ) )*ABS( D( M-1 ) )+
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$ SAFMIN )GO TO 110
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100 CONTINUE
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END IF
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*
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M = LEND
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*
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110 CONTINUE
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IF( M.GT.LEND )
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$ E( M-1 ) = ZERO
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P = D( L )
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IF( M.EQ.L )
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$ GO TO 130
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*
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* If remaining matrix is 2-by-2, use SLAE2 or SLAEV2
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* to compute its eigensystem.
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*
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IF( M.EQ.L-1 ) THEN
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IF( ICOMPZ.GT.0 ) THEN
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CALL SLAEV2( D( L-1 ), E( L-1 ), D( L ), RT1, RT2, C, S )
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WORK( M ) = C
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WORK( N-1+M ) = S
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CALL CLASR( 'R', 'V', 'F', N, 2, WORK( M ),
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$ WORK( N-1+M ), Z( 1, L-1 ), LDZ )
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ELSE
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CALL SLAE2( D( L-1 ), E( L-1 ), D( L ), RT1, RT2 )
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END IF
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D( L-1 ) = RT1
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D( L ) = RT2
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E( L-1 ) = ZERO
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L = L - 2
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IF( L.GE.LEND )
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$ GO TO 90
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GO TO 140
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END IF
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*
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IF( JTOT.EQ.NMAXIT )
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$ GO TO 140
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JTOT = JTOT + 1
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*
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* Form shift.
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*
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G = ( D( L-1 )-P ) / ( TWO*E( L-1 ) )
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R = SLAPY2( G, ONE )
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G = D( M ) - P + ( E( L-1 ) / ( G+SIGN( R, G ) ) )
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*
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S = ONE
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C = ONE
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P = ZERO
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*
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* Inner loop
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*
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LM1 = L - 1
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DO 120 I = M, LM1
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F = S*E( I )
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B = C*E( I )
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CALL SLARTG( G, F, C, S, R )
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IF( I.NE.M )
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$ E( I-1 ) = R
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G = D( I ) - P
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R = ( D( I+1 )-G )*S + TWO*C*B
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P = S*R
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D( I ) = G + P
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G = C*R - B
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*
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* If eigenvectors are desired, then save rotations.
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*
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IF( ICOMPZ.GT.0 ) THEN
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WORK( I ) = C
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WORK( N-1+I ) = S
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END IF
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*
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120 CONTINUE
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*
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* If eigenvectors are desired, then apply saved rotations.
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*
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IF( ICOMPZ.GT.0 ) THEN
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MM = L - M + 1
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CALL CLASR( 'R', 'V', 'F', N, MM, WORK( M ), WORK( N-1+M ),
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$ Z( 1, M ), LDZ )
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END IF
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*
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D( L ) = D( L ) - P
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E( LM1 ) = G
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GO TO 90
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*
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* Eigenvalue found.
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*
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130 CONTINUE
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D( L ) = P
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*
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L = L - 1
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IF( L.GE.LEND )
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$ GO TO 90
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GO TO 140
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*
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END IF
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*
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* Undo scaling if necessary
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*
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140 CONTINUE
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IF( ISCALE.EQ.1 ) THEN
|
|
CALL SLASCL( 'G', 0, 0, SSFMAX, ANORM, LENDSV-LSV+1, 1,
|
|
$ D( LSV ), N, INFO )
|
|
CALL SLASCL( 'G', 0, 0, SSFMAX, ANORM, LENDSV-LSV, 1, E( LSV ),
|
|
$ N, INFO )
|
|
ELSE IF( ISCALE.EQ.2 ) THEN
|
|
CALL SLASCL( 'G', 0, 0, SSFMIN, ANORM, LENDSV-LSV+1, 1,
|
|
$ D( LSV ), N, INFO )
|
|
CALL SLASCL( 'G', 0, 0, SSFMIN, ANORM, LENDSV-LSV, 1, E( LSV ),
|
|
$ N, INFO )
|
|
END IF
|
|
*
|
|
* Check for no convergence to an eigenvalue after a total
|
|
* of N*MAXIT iterations.
|
|
*
|
|
IF( JTOT.EQ.NMAXIT ) THEN
|
|
DO 150 I = 1, N - 1
|
|
IF( E( I ).NE.ZERO )
|
|
$ INFO = INFO + 1
|
|
150 CONTINUE
|
|
RETURN
|
|
END IF
|
|
GO TO 10
|
|
*
|
|
* Order eigenvalues and eigenvectors.
|
|
*
|
|
160 CONTINUE
|
|
IF( ICOMPZ.EQ.0 ) THEN
|
|
*
|
|
* Use Quick Sort
|
|
*
|
|
CALL SLASRT( 'I', N, D, INFO )
|
|
*
|
|
ELSE
|
|
*
|
|
* Use Selection Sort to minimize swaps of eigenvectors
|
|
*
|
|
DO 180 II = 2, N
|
|
I = II - 1
|
|
K = I
|
|
P = D( I )
|
|
DO 170 J = II, N
|
|
IF( D( J ).LT.P ) THEN
|
|
K = J
|
|
P = D( J )
|
|
END IF
|
|
170 CONTINUE
|
|
IF( K.NE.I ) THEN
|
|
D( K ) = D( I )
|
|
D( I ) = P
|
|
CALL CSWAP( N, Z( 1, I ), 1, Z( 1, K ), 1 )
|
|
END IF
|
|
180 CONTINUE
|
|
END IF
|
|
RETURN
|
|
*
|
|
* End of CSTEQR
|
|
*
|
|
END
|