SUBROUTINE ZLAED8( K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, Z, DLAMDA, $ Q2, LDQ2, W, INDXP, INDX, INDXQ, PERM, GIVPTR, $ GIVCOL, GIVNUM, INFO ) * * -- LAPACK routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. INTEGER CUTPNT, GIVPTR, INFO, K, LDQ, LDQ2, N, QSIZ DOUBLE PRECISION RHO * .. * .. Array Arguments .. INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ), $ INDXQ( * ), PERM( * ) DOUBLE PRECISION D( * ), DLAMDA( * ), GIVNUM( 2, * ), W( * ), $ Z( * ) COMPLEX*16 Q( LDQ, * ), Q2( LDQ2, * ) * .. * * Purpose * ======= * * ZLAED8 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. * * Arguments * ========= * * K (output) INTEGER * Contains the number of non-deflated eigenvalues. * This is the order of the related secular equation. * * N (input) INTEGER * The dimension of the symmetric tridiagonal matrix. N >= 0. * * QSIZ (input) INTEGER * The dimension of the unitary matrix used to reduce * the dense or band matrix to tridiagonal form. * QSIZ >= N if ICOMPQ = 1. * * Q (input/output) COMPLEX*16 array, dimension (LDQ,N) * 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. * * LDQ (input) INTEGER * The leading dimension of the array Q. LDQ >= max( 1, N ). * * D (input/output) DOUBLE PRECISION array, dimension (N) * On entry, D contains the eigenvalues of the two submatrices to * be combined. On exit, D contains the trailing (N-K) updated * eigenvalues (those which were deflated) sorted into increasing * order. * * RHO (input/output) DOUBLE PRECISION * Contains the off diagonal element associated with the rank-1 * cut which originally split the two submatrices which are now * being recombined. RHO is modified during the computation to * the value required by DLAED3. * * CUTPNT (input) INTEGER * Contains the location of the last eigenvalue in the leading * sub-matrix. MIN(1,N) <= CUTPNT <= N. * * Z (input) DOUBLE PRECISION array, dimension (N) * On input this vector contains the updating vector (the last * row of the first sub-eigenvector matrix and the first row of * the second sub-eigenvector matrix). The contents of Z are * destroyed during the updating process. * * DLAMDA (output) DOUBLE PRECISION array, dimension (N) * Contains a copy of the first K eigenvalues which will be used * by DLAED3 to form the secular equation. * * Q2 (output) COMPLEX*16 array, dimension (LDQ2,N) * If ICOMPQ = 0, Q2 is not referenced. Otherwise, * Contains a copy of the first K eigenvectors which will be used * by DLAED7 in a matrix multiply (DGEMM) to update the new * eigenvectors. * * LDQ2 (input) INTEGER * The leading dimension of the array Q2. LDQ2 >= max( 1, N ). * * W (output) DOUBLE PRECISION array, dimension (N) * This will hold the first k values of the final * deflation-altered z-vector and will be passed to DLAED3. * * INDXP (workspace) INTEGER array, dimension (N) * This will contain the permutation used to place deflated * values of D at the end of the array. On output INDXP(1:K) * points to the nondeflated D-values and INDXP(K+1:N) * points to the deflated eigenvalues. * * INDX (workspace) INTEGER array, dimension (N) * This will contain the permutation used to sort the contents of * D into ascending order. * * INDXQ (input) INTEGER array, dimension (N) * This contains 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. * * PERM (output) INTEGER array, dimension (N) * Contains the permutations (from deflation and sorting) to be * applied to each eigenblock. * * GIVPTR (output) INTEGER * Contains the number of Givens rotations which took place in * this subproblem. * * GIVCOL (output) INTEGER array, dimension (2, N) * Each pair of numbers indicates a pair of columns to take place * in a Givens rotation. * * GIVNUM (output) DOUBLE PRECISION array, dimension (2, N) * Each number indicates the S value to be used in the * corresponding Givens rotation. * * INFO (output) INTEGER * = 0: successful exit. * < 0: if INFO = -i, the i-th argument had an illegal value. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION MONE, ZERO, ONE, TWO, EIGHT PARAMETER ( MONE = -1.0D0, ZERO = 0.0D0, ONE = 1.0D0, $ TWO = 2.0D0, EIGHT = 8.0D0 ) * .. * .. Local Scalars .. INTEGER I, IMAX, J, JLAM, JMAX, JP, K2, N1, N1P1, N2 DOUBLE PRECISION C, EPS, S, T, TAU, TOL * .. * .. External Functions .. INTEGER IDAMAX DOUBLE PRECISION DLAMCH, DLAPY2 EXTERNAL IDAMAX, DLAMCH, DLAPY2 * .. * .. External Subroutines .. EXTERNAL DCOPY, DLAMRG, DSCAL, XERBLA, ZCOPY, ZDROT, $ ZLACPY * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, SQRT * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 * IF( N.LT.0 ) THEN INFO = -2 ELSE IF( QSIZ.LT.N ) THEN INFO = -3 ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( CUTPNT.LT.MIN( 1, N ) .OR. CUTPNT.GT.N ) THEN INFO = -8 ELSE IF( LDQ2.LT.MAX( 1, N ) ) THEN INFO = -12 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZLAED8', -INFO ) RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) $ RETURN * N1 = CUTPNT N2 = N - N1 N1P1 = N1 + 1 * IF( RHO.LT.ZERO ) THEN CALL DSCAL( 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 DSCAL( 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 DLAMRG( 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 = IDAMAX( N, Z, 1 ) JMAX = IDAMAX( N, D, 1 ) EPS = DLAMCH( '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 DO 50 J = 1, N PERM( J ) = INDXQ( INDX( J ) ) CALL ZCOPY( QSIZ, Q( 1, PERM( J ) ), 1, Q2( 1, J ), 1 ) 50 CONTINUE CALL ZLACPY( 'A', QSIZ, N, Q2( 1, 1 ), LDQ2, Q( 1, 1 ), LDQ ) 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 GIVPTR = 0 K2 = N + 1 DO 60 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 100 ELSE JLAM = J GO TO 70 END IF 60 CONTINUE 70 CONTINUE J = J + 1 IF( J.GT.N ) $ GO TO 90 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 = DLAPY2( 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 CALL ZDROT( QSIZ, Q( 1, INDXQ( INDX( JLAM ) ) ), 1, $ Q( 1, INDXQ( INDX( J ) ) ), 1, C, S ) 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 80 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 80 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 70 90 CONTINUE * * Record the last eigenvalue. * K = K + 1 W( K ) = Z( JLAM ) DLAMDA( K ) = D( JLAM ) INDXP( K ) = JLAM * 100 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. * DO 110 J = 1, N JP = INDXP( J ) DLAMDA( J ) = D( JP ) PERM( J ) = INDXQ( INDX( JP ) ) CALL ZCOPY( QSIZ, Q( 1, PERM( J ) ), 1, Q2( 1, J ), 1 ) 110 CONTINUE * * 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 CALL DCOPY( N-K, DLAMDA( K+1 ), 1, D( K+1 ), 1 ) CALL ZLACPY( 'A', QSIZ, N-K, Q2( 1, K+1 ), LDQ2, Q( 1, K+1 ), $ LDQ ) END IF * RETURN * * End of ZLAED8 * END