*> \brief \b SLAED8 used by SSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SLAED8 + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO, * CUTPNT, Z, DLAMDA, Q2, LDQ2, W, PERM, GIVPTR, * GIVCOL, GIVNUM, INDXP, INDX, INFO ) * * .. Scalar Arguments .. * INTEGER CUTPNT, GIVPTR, ICOMPQ, INFO, K, LDQ, LDQ2, N, * \$ QSIZ * REAL RHO * .. * .. Array Arguments .. * INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ), * \$ INDXQ( * ), PERM( * ) * REAL D( * ), DLAMDA( * ), GIVNUM( 2, * ), * \$ Q( LDQ, * ), Q2( LDQ2, * ), W( * ), Z( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SLAED8 merges the two sets of eigenvalues together into a single *> sorted set. Then it tries to deflate the size of the problem. *> There are two ways in which deflation can occur: when two or more *> eigenvalues are close together or if there is a tiny element in the *> Z vector. For each such occurrence the order of the related secular *> equation problem is reduced by one. *> \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[out] K *> \verbatim *> K is INTEGER *> The number of non-deflated eigenvalues, and the order of the *> related secular equation. *> \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,out] D *> \verbatim *> D is REAL array, dimension (N) *> On entry, the eigenvalues of the two submatrices to be *> combined. On exit, the trailing (N-K) updated eigenvalues *> (those which were deflated) sorted into increasing order. *> \endverbatim *> *> \param[in,out] Q *> \verbatim *> Q is REAL array, dimension (LDQ,N) *> If ICOMPQ = 0, Q is not referenced. Otherwise, *> on entry, Q contains the eigenvectors of the partially solved *> system which has been previously updated in matrix *> multiplies with other partially solved eigensystems. *> On exit, Q contains the trailing (N-K) updated eigenvectors *> (those which were deflated) in its last N-K columns. *> \endverbatim *> *> \param[in] LDQ *> \verbatim *> LDQ is INTEGER *> The leading dimension of the array Q. LDQ >= max(1,N). *> \endverbatim *> *> \param[in] INDXQ *> \verbatim *> INDXQ is INTEGER array, dimension (N) *> The permutation which separately sorts the two sub-problems *> in D into ascending order. Note that elements in the second *> half of this permutation must first have CUTPNT added to *> their values in order to be accurate. *> \endverbatim *> *> \param[in,out] RHO *> \verbatim *> RHO is REAL *> On entry, the off-diagonal element associated with the rank-1 *> cut which originally split the two submatrices which are now *> being recombined. *> On exit, RHO has been modified to the value required by *> SLAED3. *> \endverbatim *> *> \param[in] CUTPNT *> \verbatim *> CUTPNT is INTEGER *> The location of the last eigenvalue in the leading *> sub-matrix. min(1,N) <= CUTPNT <= N. *> \endverbatim *> *> \param[in] Z *> \verbatim *> Z is REAL array, dimension (N) *> On entry, Z contains the updating vector (the last row of *> the first sub-eigenvector matrix and the first row of the *> second sub-eigenvector matrix). *> On exit, the contents of Z are destroyed by the updating *> process. *> \endverbatim *> *> \param[out] DLAMDA *> \verbatim *> DLAMDA is REAL array, dimension (N) *> A copy of the first K eigenvalues which will be used by *> SLAED3 to form the secular equation. *> \endverbatim *> *> \param[out] Q2 *> \verbatim *> Q2 is REAL array, dimension (LDQ2,N) *> If ICOMPQ = 0, Q2 is not referenced. Otherwise, *> a copy of the first K eigenvectors which will be used by *> SLAED7 in a matrix multiply (SGEMM) to update the new *> eigenvectors. *> \endverbatim *> *> \param[in] LDQ2 *> \verbatim *> LDQ2 is INTEGER *> The leading dimension of the array Q2. LDQ2 >= max(1,N). *> \endverbatim *> *> \param[out] W *> \verbatim *> W is REAL array, dimension (N) *> The first k values of the final deflation-altered z-vector and *> will be passed to SLAED3. *> \endverbatim *> *> \param[out] PERM *> \verbatim *> PERM is INTEGER array, dimension (N) *> The permutations (from deflation and sorting) to be applied *> to each eigenblock. *> \endverbatim *> *> \param[out] GIVPTR *> \verbatim *> GIVPTR is INTEGER *> The number of Givens rotations which took place in this *> subproblem. *> \endverbatim *> *> \param[out] GIVCOL *> \verbatim *> GIVCOL is INTEGER array, dimension (2, N) *> Each pair of numbers indicates a pair of columns to take place *> in a Givens rotation. *> \endverbatim *> *> \param[out] GIVNUM *> \verbatim *> GIVNUM is REAL array, dimension (2, N) *> Each number indicates the S value to be used in the *> corresponding Givens rotation. *> \endverbatim *> *> \param[out] INDXP *> \verbatim *> INDXP is INTEGER array, dimension (N) *> The permutation used to place deflated values of D at the end *> of the array. INDXP(1:K) points to the nondeflated D-values *> and INDXP(K+1:N) points to the deflated eigenvalues. *> \endverbatim *> *> \param[out] INDX *> \verbatim *> INDX is INTEGER array, dimension (N) *> The permutation used to sort the contents of D into ascending *> order. *> \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 auxOTHERcomputational * *> \par Contributors: * ================== *> *> Jeff Rutter, Computer Science Division, University of California *> at Berkeley, USA * * ===================================================================== SUBROUTINE SLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO, \$ CUTPNT, Z, DLAMDA, Q2, LDQ2, W, PERM, GIVPTR, \$ GIVCOL, GIVNUM, INDXP, INDX, 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, GIVPTR, ICOMPQ, INFO, K, LDQ, LDQ2, N, \$ QSIZ REAL RHO * .. * .. Array Arguments .. INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ), \$ INDXQ( * ), PERM( * ) REAL D( * ), DLAMDA( * ), GIVNUM( 2, * ), \$ Q( LDQ, * ), Q2( LDQ2, * ), W( * ), Z( * ) * .. * * ===================================================================== * * .. Parameters .. REAL MONE, ZERO, ONE, TWO, EIGHT PARAMETER ( MONE = -1.0E0, ZERO = 0.0E0, ONE = 1.0E0, \$ TWO = 2.0E0, EIGHT = 8.0E0 ) * .. * .. Local Scalars .. * INTEGER I, IMAX, J, JLAM, JMAX, JP, K2, N1, N1P1, N2 REAL C, EPS, S, T, TAU, TOL * .. * .. External Functions .. INTEGER ISAMAX REAL SLAMCH, SLAPY2 EXTERNAL ISAMAX, SLAMCH, SLAPY2 * .. * .. External Subroutines .. EXTERNAL SCOPY, SLACPY, SLAMRG, SROT, SSCAL, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, SQRT * .. * .. 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 = -3 ELSE IF( ICOMPQ.EQ.1 .AND. QSIZ.LT.N ) THEN INFO = -4 ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( CUTPNT.LT.MIN( 1, N ) .OR. CUTPNT.GT.N ) THEN INFO = -10 ELSE IF( LDQ2.LT.MAX( 1, N ) ) THEN INFO = -14 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'SLAED8', -INFO ) RETURN END IF * * Need to initialize GIVPTR to O here in case of quick exit * to prevent an unspecified code behavior (usually sigfault) * when IWORK array on entry to *stedc is not zeroed * (or at least some IWORK entries which used in *laed7 for GIVPTR). * GIVPTR = 0 * * Quick return if possible * IF( N.EQ.0 ) \$ RETURN * N1 = CUTPNT N2 = N - N1 N1P1 = N1 + 1 * IF( RHO.LT.ZERO ) THEN CALL SSCAL( N2, MONE, Z( N1P1 ), 1 ) END IF * * Normalize z so that norm(z) = 1 * T = ONE / SQRT( TWO ) DO 10 J = 1, N INDX( J ) = J 10 CONTINUE CALL SSCAL( N, T, Z, 1 ) RHO = ABS( TWO*RHO ) * * Sort the eigenvalues into increasing order * DO 20 I = CUTPNT + 1, N INDXQ( I ) = INDXQ( I ) + CUTPNT 20 CONTINUE DO 30 I = 1, N DLAMDA( I ) = D( INDXQ( I ) ) W( I ) = Z( INDXQ( I ) ) 30 CONTINUE I = 1 J = CUTPNT + 1 CALL SLAMRG( N1, N2, DLAMDA, 1, 1, INDX ) DO 40 I = 1, N D( I ) = DLAMDA( INDX( I ) ) Z( I ) = W( INDX( I ) ) 40 CONTINUE * * Calculate the allowable deflation tolerance * IMAX = ISAMAX( N, Z, 1 ) JMAX = ISAMAX( N, D, 1 ) EPS = SLAMCH( 'Epsilon' ) TOL = EIGHT*EPS*ABS( D( JMAX ) ) * * If the rank-1 modifier is small enough, no more needs to be done * except to reorganize Q so that its columns correspond with the * elements in D. * IF( RHO*ABS( Z( IMAX ) ).LE.TOL ) THEN K = 0 IF( ICOMPQ.EQ.0 ) THEN DO 50 J = 1, N PERM( J ) = INDXQ( INDX( J ) ) 50 CONTINUE ELSE DO 60 J = 1, N PERM( J ) = INDXQ( INDX( J ) ) CALL SCOPY( QSIZ, Q( 1, PERM( J ) ), 1, Q2( 1, J ), 1 ) 60 CONTINUE CALL SLACPY( 'A', QSIZ, N, Q2( 1, 1 ), LDQ2, Q( 1, 1 ), \$ LDQ ) END IF RETURN END IF * * If there are multiple eigenvalues then the problem deflates. Here * the number of equal eigenvalues are found. As each equal * eigenvalue is found, an elementary reflector is computed to rotate * the corresponding eigensubspace so that the corresponding * components of Z are zero in this new basis. * K = 0 K2 = N + 1 DO 70 J = 1, N IF( RHO*ABS( Z( J ) ).LE.TOL ) THEN * * Deflate due to small z component. * K2 = K2 - 1 INDXP( K2 ) = J IF( J.EQ.N ) \$ GO TO 110 ELSE JLAM = J GO TO 80 END IF 70 CONTINUE 80 CONTINUE J = J + 1 IF( J.GT.N ) \$ GO TO 100 IF( RHO*ABS( Z( J ) ).LE.TOL ) THEN * * Deflate due to small z component. * K2 = K2 - 1 INDXP( K2 ) = J ELSE * * Check if eigenvalues are close enough to allow deflation. * S = Z( JLAM ) C = Z( J ) * * Find sqrt(a**2+b**2) without overflow or * destructive underflow. * TAU = SLAPY2( C, S ) T = D( J ) - D( JLAM ) C = C / TAU S = -S / TAU IF( ABS( T*C*S ).LE.TOL ) THEN * * Deflation is possible. * Z( J ) = TAU Z( JLAM ) = ZERO * * Record the appropriate Givens rotation * GIVPTR = GIVPTR + 1 GIVCOL( 1, GIVPTR ) = INDXQ( INDX( JLAM ) ) GIVCOL( 2, GIVPTR ) = INDXQ( INDX( J ) ) GIVNUM( 1, GIVPTR ) = C GIVNUM( 2, GIVPTR ) = S IF( ICOMPQ.EQ.1 ) THEN CALL SROT( QSIZ, Q( 1, INDXQ( INDX( JLAM ) ) ), 1, \$ Q( 1, INDXQ( INDX( J ) ) ), 1, C, S ) END IF T = D( JLAM )*C*C + D( J )*S*S D( J ) = D( JLAM )*S*S + D( J )*C*C D( JLAM ) = T K2 = K2 - 1 I = 1 90 CONTINUE IF( K2+I.LE.N ) THEN IF( D( JLAM ).LT.D( INDXP( K2+I ) ) ) THEN INDXP( K2+I-1 ) = INDXP( K2+I ) INDXP( K2+I ) = JLAM I = I + 1 GO TO 90 ELSE INDXP( K2+I-1 ) = JLAM END IF ELSE INDXP( K2+I-1 ) = JLAM END IF JLAM = J ELSE K = K + 1 W( K ) = Z( JLAM ) DLAMDA( K ) = D( JLAM ) INDXP( K ) = JLAM JLAM = J END IF END IF GO TO 80 100 CONTINUE * * Record the last eigenvalue. * K = K + 1 W( K ) = Z( JLAM ) DLAMDA( K ) = D( JLAM ) INDXP( K ) = JLAM * 110 CONTINUE * * Sort the eigenvalues and corresponding eigenvectors into DLAMDA * and Q2 respectively. The eigenvalues/vectors which were not * deflated go into the first K slots of DLAMDA and Q2 respectively, * while those which were deflated go into the last N - K slots. * IF( ICOMPQ.EQ.0 ) THEN DO 120 J = 1, N JP = INDXP( J ) DLAMDA( J ) = D( JP ) PERM( J ) = INDXQ( INDX( JP ) ) 120 CONTINUE ELSE DO 130 J = 1, N JP = INDXP( J ) DLAMDA( J ) = D( JP ) PERM( J ) = INDXQ( INDX( JP ) ) CALL SCOPY( QSIZ, Q( 1, PERM( J ) ), 1, Q2( 1, J ), 1 ) 130 CONTINUE END IF * * The deflated eigenvalues and their corresponding vectors go back * into the last N - K slots of D and Q respectively. * IF( K.LT.N ) THEN IF( ICOMPQ.EQ.0 ) THEN CALL SCOPY( N-K, DLAMDA( K+1 ), 1, D( K+1 ), 1 ) ELSE CALL SCOPY( N-K, DLAMDA( K+1 ), 1, D( K+1 ), 1 ) CALL SLACPY( 'A', QSIZ, N-K, Q2( 1, K+1 ), LDQ2, \$ Q( 1, K+1 ), LDQ ) END IF END IF * RETURN * * End of SLAED8 * END