*> \brief \b SLAED1 used by SSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is tridiagonal. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SLAED1 + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SLAED1( N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK, IWORK, * INFO ) * * .. Scalar Arguments .. * INTEGER CUTPNT, INFO, LDQ, N * REAL RHO * .. * .. Array Arguments .. * INTEGER INDXQ( * ), IWORK( * ) * REAL D( * ), Q( LDQ, * ), WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SLAED1 computes the updated eigensystem of a diagonal *> matrix after modification by a rank-one symmetric matrix. This *> routine is used only for the eigenproblem which requires all *> eigenvalues and eigenvectors of a tridiagonal matrix. SLAED7 handles *> the case in which eigenvalues only or eigenvalues and eigenvectors *> of a full symmetric matrix (which was reduced to tridiagonal form) *> are desired. *> *> T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out) *> *> where Z = Q**T*u, u is a vector of length N with ones in the *> CUTPNT and CUTPNT + 1 th elements and zeros elsewhere. *> *> The eigenvectors of the original matrix are stored in Q, and the *> eigenvalues are in D. The algorithm consists of three stages: *> *> The first stage consists of deflating the size of the problem *> when there are multiple eigenvalues or if there is a zero in *> the Z vector. For each such occurrence the dimension of the *> secular equation problem is reduced by one. This stage is *> performed by the routine SLAED2. *> *> The second stage consists of calculating the updated *> eigenvalues. This is done by finding the roots of the secular *> equation via the routine SLAED4 (as called by SLAED3). *> This routine also calculates the eigenvectors of the current *> problem. *> *> The final stage consists of computing the updated eigenvectors *> directly using the updated eigenvalues. The eigenvectors for *> the current problem are multiplied with the eigenvectors from *> the overall problem. *> \endverbatim * * Arguments: * ========== * *> \param[in] N *> \verbatim *> N is INTEGER *> The dimension of the symmetric tridiagonal matrix. N >= 0. *> \endverbatim *> *> \param[in,out] D *> \verbatim *> D is REAL array, dimension (N) *> On entry, the eigenvalues of the rank-1-perturbed matrix. *> On exit, the eigenvalues of the repaired matrix. *> \endverbatim *> *> \param[in,out] Q *> \verbatim *> Q is REAL array, dimension (LDQ,N) *> On entry, the eigenvectors of the rank-1-perturbed matrix. *> On exit, the eigenvectors of the repaired tridiagonal matrix. *> \endverbatim *> *> \param[in] LDQ *> \verbatim *> LDQ is INTEGER *> The leading dimension of the array Q. LDQ >= max(1,N). *> \endverbatim *> *> \param[in,out] INDXQ *> \verbatim *> INDXQ is INTEGER array, dimension (N) *> On entry, the permutation which separately sorts the two *> subproblems in D into ascending order. *> On exit, the permutation which will reintegrate the *> subproblems back into sorted order, *> i.e. D( INDXQ( I = 1, N ) ) will be in ascending order. *> \endverbatim *> *> \param[in] RHO *> \verbatim *> RHO is REAL *> The subdiagonal entry used to create the rank-1 modification. *> \endverbatim *> *> \param[in] CUTPNT *> \verbatim *> CUTPNT is INTEGER *> The location of the last eigenvalue in the leading sub-matrix. *> min(1,N) <= CUTPNT <= N/2. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is REAL array, dimension (4*N + N**2) *> \endverbatim *> *> \param[out] IWORK *> \verbatim *> IWORK is INTEGER 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 = 1, an eigenvalue did not converge *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup auxOTHERcomputational * *> \par Contributors: * ================== *> *> Jeff Rutter, Computer Science Division, University of California *> at Berkeley, USA \n *> Modified by Francoise Tisseur, University of Tennessee *> * ===================================================================== SUBROUTINE SLAED1( N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK, IWORK, \$ 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 .. INTEGER CUTPNT, INFO, LDQ, N REAL RHO * .. * .. Array Arguments .. INTEGER INDXQ( * ), IWORK( * ) REAL D( * ), Q( LDQ, * ), WORK( * ) * .. * * ===================================================================== * * .. Local Scalars .. INTEGER COLTYP, CPP1, I, IDLMDA, INDX, INDXC, INDXP, \$ IQ2, IS, IW, IZ, K, N1, N2 * .. * .. External Subroutines .. EXTERNAL SCOPY, SLAED2, SLAED3, SLAMRG, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 * IF( N.LT.0 ) THEN INFO = -1 ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN INFO = -4 ELSE IF( MIN( 1, N / 2 ).GT.CUTPNT .OR. ( N / 2 ).LT.CUTPNT ) THEN INFO = -7 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'SLAED1', -INFO ) RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) \$ RETURN * * The following values are integer pointers which indicate * the portion of the workspace * used by a particular array in SLAED2 and SLAED3. * IZ = 1 IDLMDA = IZ + N IW = IDLMDA + N IQ2 = IW + N * INDX = 1 INDXC = INDX + N COLTYP = INDXC + N INDXP = COLTYP + N * * * Form the z-vector which consists of the last row of Q_1 and the * first row of Q_2. * CALL SCOPY( CUTPNT, Q( CUTPNT, 1 ), LDQ, WORK( IZ ), 1 ) CPP1 = CUTPNT + 1 CALL SCOPY( N-CUTPNT, Q( CPP1, CPP1 ), LDQ, WORK( IZ+CUTPNT ), 1 ) * * Deflate eigenvalues. * CALL SLAED2( K, N, CUTPNT, D, Q, LDQ, INDXQ, RHO, WORK( IZ ), \$ WORK( IDLMDA ), WORK( IW ), WORK( IQ2 ), \$ IWORK( INDX ), IWORK( INDXC ), IWORK( INDXP ), \$ IWORK( COLTYP ), INFO ) * IF( INFO.NE.0 ) \$ GO TO 20 * * Solve Secular Equation. * IF( K.NE.0 ) THEN IS = ( IWORK( COLTYP )+IWORK( COLTYP+1 ) )*CUTPNT + \$ ( IWORK( COLTYP+1 )+IWORK( COLTYP+2 ) )*( N-CUTPNT ) + IQ2 CALL SLAED3( K, N, CUTPNT, D, Q, LDQ, RHO, WORK( IDLMDA ), \$ WORK( IQ2 ), IWORK( INDXC ), IWORK( COLTYP ), \$ WORK( IW ), WORK( IS ), INFO ) IF( INFO.NE.0 ) \$ GO TO 20 * * Prepare the INDXQ sorting permutation. * N1 = K N2 = N - K CALL SLAMRG( N1, N2, D, 1, -1, INDXQ ) ELSE DO 10 I = 1, N INDXQ( I ) = I 10 CONTINUE END IF * 20 CONTINUE RETURN * * End of SLAED1 * END