SUBROUTINE SLAED2( K, N, D, Q, LDQ, INDXQ, RHO, CUTPNT, Z, DLAMDA, $ Q2, LDQ2, INDXC, W, INDXP, INDX, COLTYP, INFO ) * * -- LAPACK routine (version 2.0) -- * Univ. of Tennessee, Oak Ridge National Lab, Argonne National Lab, * Courant Institute, NAG Ltd., and Rice University * September 30, 1994 * * .. Scalar Arguments .. INTEGER CUTPNT, INFO, K, LDQ, LDQ2, N REAL RHO * .. * .. Array Arguments .. INTEGER COLTYP( * ), INDX( * ), INDXC( * ), INDXP( * ), $ INDXQ( * ) REAL D( * ), DLAMDA( * ), Q( LDQ, * ), $ Q2( LDQ2, * ), W( * ), Z( * ) * .. * * Purpose * ======= * * SLAED2 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 entry in the * Z vector. For each such occurrence the order of the related secular * equation problem is reduced by one. * * Arguments * ========= * * K (output) INTEGER * The number of non-deflated eigenvalues, and the order of the * related secular equation. 0 <= K <=N. * * N (input) INTEGER * The dimension of the symmetric tridiagonal matrix. N >= 0. * * D (input/output) REAL 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. * * Q (input/output) REAL array, dimension (LDQ, N) * On entry, Q contains the eigenvectors of two submatrices in * the two square blocks with corners at (1,1), (CUTPNT,CUTPNT) * and (CUTPNT+1, CUTPNT+1), (N,N). * 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). * * INDXQ (input/output) 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. Destroyed on exit. * * RHO (input/output) 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. * * CUTPNT (input) INTEGER * The location of the last eigenvalue in the leading sub-matrix. * min(1,N) <= CUTPNT <= N. * * Z (input) 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 have been destroyed by the updating * process. * * DLAMDA (output) REAL array, dimension (N) * A copy of the first K eigenvalues which will be used by * SLAED3 to form the secular equation. * * Q2 (output) REAL array, dimension (LDQ2, N) * A copy of the first K eigenvectors which will be used by * SLAED3 in a matrix multiply (SGEMM) to solve for the new * eigenvectors. Q2 is arranged into three blocks. The * first block contains non-zero elements only at and above * CUTPNT, the second contains non-zero elements only below * CUTPNT, and the third is dense. * * LDQ2 (input) INTEGER * The leading dimension of the array Q2. LDQ2 >= max(1,N). * * INDXC (output) INTEGER array, dimension (N) * The permutation used to arrange the columns of the deflated * Q matrix into three groups: the first group contains non-zero * elements only at and above CUTPNT, the second contains * non-zero elements only below CUTPNT, and the third is dense. * * W (output) REAL array, dimension (N) * The first k values of the final deflation-altered z-vector * which will be passed to SLAED3. * * INDXP (workspace) 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. * * INDX (workspace) INTEGER array, dimension (N) * The permutation used to sort the contents of D into ascending * order. * * COLTYP (workspace/output) INTEGER array, dimension (N) * During execution, a label which will indicate which of the * following types a column in the Q2 matrix is: * 1 : non-zero in the upper half only; * 2 : non-zero in the lower half only; * 3 : dense; * 4 : deflated. * On exit, COLTYP(i) is the number of columns of type i, * for i=1 to 4 only. * * INFO (output) INTEGER * = 0: successful exit. * < 0: if INFO = -i, the i-th argument had an illegal value. * * ===================================================================== * * .. Parameters .. REAL MONE, ZERO, ONE, TWO, EIGHT PARAMETER ( MONE = -1.0E0, ZERO = 0.0E0, ONE = 1.0E0, $ TWO = 2.0E0, EIGHT = 8.0E0 ) * .. * .. Local Arrays .. INTEGER CTOT( 4 ), PSM( 4 ) * .. * .. Local Scalars .. INTEGER CT, 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( N.LT.0 ) THEN INFO = -2 ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( MIN( 1, N ).GT.CUTPNT .OR. N.LT.CUTPNT ) THEN INFO = -8 ELSE IF( LDQ2.LT.MAX( 1, N ) ) THEN INFO = -12 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'SLAED2', -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 SSCAL( N2, MONE, Z( N1P1 ), 1 ) END IF * * Normalize z so that norm(z) = 1. Since z is the concatenation of * two normalized vectors, norm2(z) = sqrt(2). * T = ONE / SQRT( TWO ) DO 10 J = 1, N INDX( J ) = J 10 CONTINUE CALL SSCAL( N, T, Z, 1 ) * * RHO = ABS( norm(z)**2 * RHO ) * RHO = ABS( TWO*RHO ) * DO 20 I = 1, CUTPNT COLTYP( I ) = 1 20 CONTINUE DO 30 I = CUTPNT + 1, N COLTYP( I ) = 2 30 CONTINUE * * Sort the eigenvalues into increasing order * DO 40 I = CUTPNT + 1, N INDXQ( I ) = INDXQ( I ) + CUTPNT 40 CONTINUE * * re-integrate the deflated parts from the last pass * DO 50 I = 1, N DLAMDA( I ) = D( INDXQ( I ) ) W( I ) = Z( INDXQ( I ) ) INDXC( I ) = COLTYP( INDXQ( I ) ) 50 CONTINUE CALL SLAMRG( N1, N2, DLAMDA, 1, 1, INDX ) DO 60 I = 1, N D( I ) = DLAMDA( INDX( I ) ) Z( I ) = W( INDX( I ) ) COLTYP( I ) = INDXC( INDX( I ) ) 60 CONTINUE * * Calculate the allowable deflation tolerance * IMAX = ISAMAX( N, Z, 1 ) JMAX = ISAMAX( N, D, 1 ) EPS = SLAMCH( 'Epsilon' ) TOL = EIGHT*EPS*MAX( ABS( D( JMAX ) ), ABS( Z( IMAX ) ) ) * * 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 70 J = 1, N CALL SCOPY( N, Q( 1, INDXQ( INDX( J ) ) ), 1, Q2( 1, J ), $ 1 ) 70 CONTINUE CALL SLACPY( 'A', N, N, Q2, LDQ2, Q, LDQ ) GO TO 180 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 80 J = 1, N IF( RHO*ABS( Z( J ) ).LE.TOL ) THEN * * Deflate due to small z component. * K2 = K2 - 1 INDXP( K2 ) = J COLTYP( J ) = 4 IF( J.EQ.N ) $ GO TO 120 ELSE JLAM = J GO TO 90 END IF 80 CONTINUE 90 CONTINUE J = J + 1 IF( J.GT.N ) $ GO TO 110 IF( RHO*ABS( Z( J ) ).LE.TOL ) THEN * * Deflate due to small z component. * K2 = K2 - 1 INDXP( K2 ) = J COLTYP( J ) = 4 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 IF( COLTYP( J ).NE.COLTYP( JLAM ) ) $ COLTYP( J ) = 3 COLTYP( JLAM ) = 4 CALL SROT( N, Q( 1, INDXQ( INDX( JLAM ) ) ), 1, $ Q( 1, INDXQ( INDX( J ) ) ), 1, C, S ) T = D( JLAM )*C**2 + D( J )*S**2 D( J ) = D( JLAM )*S**2 + D( J )*C**2 D( JLAM ) = T K2 = K2 - 1 I = 1 100 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 100 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 90 110 CONTINUE * * Record the last eigenvalue. * K = K + 1 W( K ) = Z( JLAM ) DLAMDA( K ) = D( JLAM ) INDXP( K ) = JLAM * 120 CONTINUE * * Count up the total number of the various types of columns, then * form a permutation which positions the four column types into * four uniform groups (although one or more of these groups may be * empty). * DO 130 J = 1, 4 CTOT( J ) = 0 130 CONTINUE DO 140 J = 1, N CT = COLTYP( J ) CTOT( CT ) = CTOT( CT ) + 1 140 CONTINUE * * PSM(*) = Position in SubMatrix (of types 1 through 4) * PSM( 1 ) = 1 PSM( 2 ) = 1 + CTOT( 1 ) PSM( 3 ) = PSM( 2 ) + CTOT( 2 ) PSM( 4 ) = PSM( 3 ) + CTOT( 3 ) * * Fill out the INDXC array so that the permutation which it induces * will place all type-1 columns first, all type-2 columns next, * then all type-3's, and finally all type-4's. * DO 150 J = 1, N JP = INDXP( J ) CT = COLTYP( JP ) INDXC( PSM( CT ) ) = J PSM( CT ) = PSM( CT ) + 1 150 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 160 J = 1, N JP = INDXP( J ) DLAMDA( J ) = D( JP ) CALL SCOPY( N, Q( 1, INDXQ( INDX( INDXP( INDXC( J ) ) ) ) ), 1, $ Q2( 1, J ), 1 ) 160 CONTINUE * * The deflated eigenvalues and their corresponding vectors go back * into the last N - K slots of D and Q respectively. * CALL SCOPY( N-K, DLAMDA( K+1 ), 1, D( K+1 ), 1 ) CALL SLACPY( 'A', N, N-K, Q2( 1, K+1 ), LDQ2, Q( 1, K+1 ), LDQ ) * * Copy CTOT into COLTYP for referencing in SLAED3. * DO 170 J = 1, 4 COLTYP( J ) = CTOT( J ) 170 CONTINUE * 180 CONTINUE RETURN * * End of SLAED2 * END