*> \brief \b DLAED7 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 dense. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download DLAED7 + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE DLAED7( ICOMPQ, N, QSIZ, TLVLS, CURLVL, CURPBM, D, Q, * LDQ, INDXQ, RHO, CUTPNT, QSTORE, QPTR, PRMPTR, * PERM, GIVPTR, GIVCOL, GIVNUM, WORK, IWORK, * INFO ) * * .. Scalar Arguments .. * INTEGER CURLVL, CURPBM, CUTPNT, ICOMPQ, INFO, LDQ, N, * $ QSIZ, TLVLS * DOUBLE PRECISION RHO * .. * .. Array Arguments .. * INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ), * $ IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * ) * DOUBLE PRECISION D( * ), GIVNUM( 2, * ), Q( LDQ, * ), * $ QSTORE( * ), WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> DLAED7 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 optionally eigenvectors of a dense symmetric matrix *> that has been reduced to tridiagonal form. DLAED1 handles *> the case in which all eigenvalues and eigenvectors of a symmetric *> tridiagonal matrix are desired. *> *> T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out) *> *> where Z = Q**Tu, 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 occurence the dimension of the *> secular equation problem is reduced by one. This stage is *> performed by the routine DLAED8. *> *> The second stage consists of calculating the updated *> eigenvalues. This is done by finding the roots of the secular *> equation via the routine DLAED4 (as called by DLAED9). *> 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] ICOMPQ *> \verbatim *> ICOMPQ is INTEGER *> = 0: Compute eigenvalues only. *> = 1: Compute eigenvectors of original dense symmetric matrix *> also. On entry, Q contains the orthogonal matrix used *> to reduce the original matrix to tridiagonal form. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The dimension of the symmetric tridiagonal matrix. N >= 0. *> \endverbatim *> *> \param[in] QSIZ *> \verbatim *> QSIZ is INTEGER *> The dimension of the orthogonal matrix used to reduce *> the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1. *> \endverbatim *> *> \param[in] TLVLS *> \verbatim *> TLVLS is INTEGER *> The total number of merging levels in the overall divide and *> conquer tree. *> \endverbatim *> *> \param[in] CURLVL *> \verbatim *> CURLVL is INTEGER *> The current level in the overall merge routine, *> 0 <= CURLVL <= TLVLS. *> \endverbatim *> *> \param[in] CURPBM *> \verbatim *> CURPBM is INTEGER *> The current problem in the current level in the overall *> merge routine (counting from upper left to lower right). *> \endverbatim *> *> \param[in,out] D *> \verbatim *> D is DOUBLE PRECISION 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 DOUBLE PRECISION 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[out] INDXQ *> \verbatim *> INDXQ is INTEGER array, dimension (N) *> The permutation which will reintegrate the subproblem just *> solved back into sorted order, i.e., D( INDXQ( I = 1, N ) ) *> will be in ascending order. *> \endverbatim *> *> \param[in] RHO *> \verbatim *> RHO is DOUBLE PRECISION *> The subdiagonal element used to create the rank-1 *> modification. *> \endverbatim *> *> \param[in] CUTPNT *> \verbatim *> CUTPNT is INTEGER *> Contains the location of the last eigenvalue in the leading *> sub-matrix. min(1,N) <= CUTPNT <= N. *> \endverbatim *> *> \param[in,out] QSTORE *> \verbatim *> QSTORE is DOUBLE PRECISION array, dimension (N**2+1) *> Stores eigenvectors of submatrices encountered during *> divide and conquer, packed together. QPTR points to *> beginning of the submatrices. *> \endverbatim *> *> \param[in,out] QPTR *> \verbatim *> QPTR is INTEGER array, dimension (N+2) *> List of indices pointing to beginning of submatrices stored *> in QSTORE. The submatrices are numbered starting at the *> bottom left of the divide and conquer tree, from left to *> right and bottom to top. *> \endverbatim *> *> \param[in] PRMPTR *> \verbatim *> PRMPTR is INTEGER array, dimension (N lg N) *> Contains a list of pointers which indicate where in PERM a *> level's permutation is stored. PRMPTR(i+1) - PRMPTR(i) *> indicates the size of the permutation and also the size of *> the full, non-deflated problem. *> \endverbatim *> *> \param[in] PERM *> \verbatim *> PERM is INTEGER array, dimension (N lg N) *> Contains the permutations (from deflation and sorting) to be *> applied to each eigenblock. *> \endverbatim *> *> \param[in] GIVPTR *> \verbatim *> GIVPTR is INTEGER array, dimension (N lg N) *> Contains a list of pointers which indicate where in GIVCOL a *> level's Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i) *> indicates the number of Givens rotations. *> \endverbatim *> *> \param[in] GIVCOL *> \verbatim *> GIVCOL is INTEGER array, dimension (2, N lg N) *> Each pair of numbers indicates a pair of columns to take place *> in a Givens rotation. *> \endverbatim *> *> \param[in] GIVNUM *> \verbatim *> GIVNUM is DOUBLE PRECISION array, dimension (2, N lg N) *> Each number indicates the S value to be used in the *> corresponding Givens rotation. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is DOUBLE PRECISION array, dimension (3*N+2*QSIZ*N) *> \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. * *> \date November 2015 * *> \ingroup auxOTHERcomputational * *> \par Contributors: * ================== *> *> Jeff Rutter, Computer Science Division, University of California *> at Berkeley, USA * * ===================================================================== SUBROUTINE DLAED7( ICOMPQ, N, QSIZ, TLVLS, CURLVL, CURPBM, D, Q, $ LDQ, INDXQ, RHO, CUTPNT, QSTORE, QPTR, PRMPTR, $ PERM, GIVPTR, GIVCOL, GIVNUM, WORK, IWORK, $ INFO ) * * -- LAPACK computational routine (version 3.6.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * November 2015 * * .. Scalar Arguments .. INTEGER CURLVL, CURPBM, CUTPNT, ICOMPQ, INFO, LDQ, N, $ QSIZ, TLVLS DOUBLE PRECISION RHO * .. * .. Array Arguments .. INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ), $ IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * ) DOUBLE PRECISION D( * ), GIVNUM( 2, * ), Q( LDQ, * ), $ QSTORE( * ), WORK( * ) * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0 ) * .. * .. Local Scalars .. INTEGER COLTYP, CURR, I, IDLMDA, INDX, INDXC, INDXP, $ IQ2, IS, IW, IZ, K, LDQ2, N1, N2, PTR * .. * .. External Subroutines .. EXTERNAL DGEMM, DLAED8, DLAED9, DLAEDA, DLAMRG, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 * IF( ICOMPQ.LT.0 .OR. ICOMPQ.GT.1 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( ICOMPQ.EQ.1 .AND. QSIZ.LT.N ) THEN INFO = -3 ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN INFO = -9 ELSE IF( MIN( 1, N ).GT.CUTPNT .OR. N.LT.CUTPNT ) THEN INFO = -12 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLAED7', -INFO ) RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) $ RETURN * * The following values are for bookkeeping purposes only. They are * integer pointers which indicate the portion of the workspace * used by a particular array in DLAED8 and DLAED9. * IF( ICOMPQ.EQ.1 ) THEN LDQ2 = QSIZ ELSE LDQ2 = N END IF * IZ = 1 IDLMDA = IZ + N IW = IDLMDA + N IQ2 = IW + N IS = IQ2 + N*LDQ2 * 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. * PTR = 1 + 2**TLVLS DO 10 I = 1, CURLVL - 1 PTR = PTR + 2**( TLVLS-I ) 10 CONTINUE CURR = PTR + CURPBM CALL DLAEDA( N, TLVLS, CURLVL, CURPBM, PRMPTR, PERM, GIVPTR, $ GIVCOL, GIVNUM, QSTORE, QPTR, WORK( IZ ), $ WORK( IZ+N ), INFO ) * * When solving the final problem, we no longer need the stored data, * so we will overwrite the data from this level onto the previously * used storage space. * IF( CURLVL.EQ.TLVLS ) THEN QPTR( CURR ) = 1 PRMPTR( CURR ) = 1 GIVPTR( CURR ) = 1 END IF * * Sort and Deflate eigenvalues. * CALL DLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO, CUTPNT, $ WORK( IZ ), WORK( IDLMDA ), WORK( IQ2 ), LDQ2, $ WORK( IW ), PERM( PRMPTR( CURR ) ), GIVPTR( CURR+1 ), $ GIVCOL( 1, GIVPTR( CURR ) ), $ GIVNUM( 1, GIVPTR( CURR ) ), IWORK( INDXP ), $ IWORK( INDX ), INFO ) PRMPTR( CURR+1 ) = PRMPTR( CURR ) + N GIVPTR( CURR+1 ) = GIVPTR( CURR+1 ) + GIVPTR( CURR ) * * Solve Secular Equation. * IF( K.NE.0 ) THEN CALL DLAED9( K, 1, K, N, D, WORK( IS ), K, RHO, WORK( IDLMDA ), $ WORK( IW ), QSTORE( QPTR( CURR ) ), K, INFO ) IF( INFO.NE.0 ) $ GO TO 30 IF( ICOMPQ.EQ.1 ) THEN CALL DGEMM( 'N', 'N', QSIZ, K, K, ONE, WORK( IQ2 ), LDQ2, $ QSTORE( QPTR( CURR ) ), K, ZERO, Q, LDQ ) END IF QPTR( CURR+1 ) = QPTR( CURR ) + K**2 * * Prepare the INDXQ sorting permutation. * N1 = K N2 = N - K CALL DLAMRG( N1, N2, D, 1, -1, INDXQ ) ELSE QPTR( CURR+1 ) = QPTR( CURR ) DO 20 I = 1, N INDXQ( I ) = I 20 CONTINUE END IF * 30 CONTINUE RETURN * * End of DLAED7 * END