*> \brief \b CPTEQR * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download CPTEQR + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE CPTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO ) * * .. Scalar Arguments .. * CHARACTER COMPZ * INTEGER INFO, LDZ, N * .. * .. Array Arguments .. * REAL D( * ), E( * ), WORK( * ) * COMPLEX Z( LDZ, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> CPTEQR computes all eigenvalues and, optionally, eigenvectors of a *> symmetric positive definite tridiagonal matrix by first factoring the *> matrix using SPTTRF and then calling CBDSQR to compute the singular *> values of the bidiagonal factor. *> *> This routine computes the eigenvalues of the positive definite *> tridiagonal matrix to high relative accuracy. This means that if the *> eigenvalues range over many orders of magnitude in size, then the *> small eigenvalues and corresponding eigenvectors will be computed *> more accurately than, for example, with the standard QR method. *> *> The eigenvectors of a full or band positive definite Hermitian matrix *> can also be found if CHETRD, CHPTRD, or CHBTRD has been used to *> reduce this matrix to tridiagonal form. (The reduction to *> tridiagonal form, however, may preclude the possibility of obtaining *> high relative accuracy in the small eigenvalues of the original *> matrix, if these eigenvalues range over many orders of magnitude.) *> \endverbatim * * Arguments: * ========== * *> \param[in] COMPZ *> \verbatim *> COMPZ is CHARACTER*1 *> = 'N': Compute eigenvalues only. *> = 'V': Compute eigenvectors of original Hermitian *> matrix also. Array Z contains the unitary matrix *> used to reduce the original matrix to tridiagonal *> form. *> = 'I': Compute eigenvectors of tridiagonal matrix also. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrix. N >= 0. *> \endverbatim *> *> \param[in,out] D *> \verbatim *> D is REAL array, dimension (N) *> On entry, the n diagonal elements of the tridiagonal matrix. *> On normal exit, D contains the eigenvalues, in descending *> order. *> \endverbatim *> *> \param[in,out] E *> \verbatim *> E is REAL array, dimension (N-1) *> On entry, the (n-1) subdiagonal elements of the tridiagonal *> matrix. *> On exit, E has been destroyed. *> \endverbatim *> *> \param[in,out] Z *> \verbatim *> Z is COMPLEX array, dimension (LDZ, N) *> On entry, if COMPZ = 'V', the unitary matrix used in the *> reduction to tridiagonal form. *> On exit, if COMPZ = 'V', the orthonormal eigenvectors of the *> original Hermitian matrix; *> if COMPZ = 'I', the orthonormal eigenvectors of the *> tridiagonal matrix. *> If INFO > 0 on exit, Z contains the eigenvectors associated *> with only the stored eigenvalues. *> If COMPZ = 'N', then Z is not referenced. *> \endverbatim *> *> \param[in] LDZ *> \verbatim *> LDZ is INTEGER *> The leading dimension of the array Z. LDZ >= 1, and if *> COMPZ = 'V' or 'I', LDZ >= max(1,N). *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is REAL array, dimension (4*N) *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: successful exit. *> < 0: if INFO = -i, the i-th argument had an illegal value. *> > 0: if INFO = i, and i is: *> <= N the Cholesky factorization of the matrix could *> not be performed because the i-th principal minor *> was not positive definite. *> > N the SVD algorithm failed to converge; *> if INFO = N+i, i off-diagonal elements of the *> bidiagonal factor did not converge to zero. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup complexPTcomputational * * ===================================================================== SUBROUTINE CPTEQR( COMPZ, N, D, E, Z, LDZ, 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 COMPZ INTEGER INFO, LDZ, N * .. * .. Array Arguments .. REAL D( * ), E( * ), WORK( * ) COMPLEX Z( LDZ, * ) * .. * * ==================================================================== * * .. Parameters .. COMPLEX CZERO, CONE PARAMETER ( CZERO = ( 0.0E+0, 0.0E+0 ), \$ CONE = ( 1.0E+0, 0.0E+0 ) ) * .. * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. External Subroutines .. EXTERNAL CBDSQR, CLASET, SPTTRF, XERBLA * .. * .. Local Arrays .. COMPLEX C( 1, 1 ), VT( 1, 1 ) * .. * .. Local Scalars .. INTEGER I, ICOMPZ, NRU * .. * .. Intrinsic Functions .. INTRINSIC MAX, SQRT * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 * IF( LSAME( COMPZ, 'N' ) ) THEN ICOMPZ = 0 ELSE IF( LSAME( COMPZ, 'V' ) ) THEN ICOMPZ = 1 ELSE IF( LSAME( COMPZ, 'I' ) ) THEN ICOMPZ = 2 ELSE ICOMPZ = -1 END IF IF( ICOMPZ.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( ( LDZ.LT.1 ) .OR. ( ICOMPZ.GT.0 .AND. LDZ.LT.MAX( 1, \$ N ) ) ) THEN INFO = -6 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'CPTEQR', -INFO ) RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) \$ RETURN * IF( N.EQ.1 ) THEN IF( ICOMPZ.GT.0 ) \$ Z( 1, 1 ) = CONE RETURN END IF IF( ICOMPZ.EQ.2 ) \$ CALL CLASET( 'Full', N, N, CZERO, CONE, Z, LDZ ) * * Call SPTTRF to factor the matrix. * CALL SPTTRF( N, D, E, INFO ) IF( INFO.NE.0 ) \$ RETURN DO 10 I = 1, N D( I ) = SQRT( D( I ) ) 10 CONTINUE DO 20 I = 1, N - 1 E( I ) = E( I )*D( I ) 20 CONTINUE * * Call CBDSQR to compute the singular values/vectors of the * bidiagonal factor. * IF( ICOMPZ.GT.0 ) THEN NRU = N ELSE NRU = 0 END IF CALL CBDSQR( 'Lower', N, 0, NRU, 0, D, E, VT, 1, Z, LDZ, C, 1, \$ WORK, INFO ) * * Square the singular values. * IF( INFO.EQ.0 ) THEN DO 30 I = 1, N D( I ) = D( I )*D( I ) 30 CONTINUE ELSE INFO = N + INFO END IF * RETURN * * End of CPTEQR * END