SUBROUTINE SLAED3( K, N, N1, D, Q, LDQ, RHO, DLAMDA, Q2, INDX, $ CTOT, W, S, INFO ) * * -- LAPACK routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. INTEGER INFO, K, LDQ, N, N1 REAL RHO * .. * .. Array Arguments .. INTEGER CTOT( * ), INDX( * ) REAL D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ), $ S( * ), W( * ) * .. * * Purpose * ======= * * SLAED3 finds the roots of the secular equation, as defined by the * values in D, W, and RHO, between 1 and K. It makes the * appropriate calls to SLAED4 and then updates the eigenvectors by * multiplying the matrix of eigenvectors of the pair of eigensystems * being combined by the matrix of eigenvectors of the K-by-K system * which is solved here. * * This code makes very mild assumptions about floating point * arithmetic. It will work on machines with a guard digit in * add/subtract, or on those binary machines without guard digits * which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. * It could conceivably fail on hexadecimal or decimal machines * without guard digits, but we know of none. * * Arguments * ========= * * K (input) INTEGER * The number of terms in the rational function to be solved by * SLAED4. K >= 0. * * N (input) INTEGER * The number of rows and columns in the Q matrix. * N >= K (deflation may result in N>K). * * N1 (input) INTEGER * The location of the last eigenvalue in the leading submatrix. * min(1,N) <= N1 <= N/2. * * D (output) REAL array, dimension (N) * D(I) contains the updated eigenvalues for * 1 <= I <= K. * * Q (output) REAL array, dimension (LDQ,N) * Initially the first K columns are used as workspace. * On output the columns 1 to K contain * the updated eigenvectors. * * LDQ (input) INTEGER * The leading dimension of the array Q. LDQ >= max(1,N). * * RHO (input) REAL * The value of the parameter in the rank one update equation. * RHO >= 0 required. * * DLAMDA (input/output) REAL array, dimension (K) * The first K elements of this array contain the old roots * of the deflated updating problem. These are the poles * of the secular equation. May be changed on output by * having lowest order bit set to zero on Cray X-MP, Cray Y-MP, * Cray-2, or Cray C-90, as described above. * * Q2 (input) REAL array, dimension (LDQ2, N) * The first K columns of this matrix contain the non-deflated * eigenvectors for the split problem. * * INDX (input) INTEGER array, dimension (N) * The permutation used to arrange the columns of the deflated * Q matrix into three groups (see SLAED2). * The rows of the eigenvectors found by SLAED4 must be likewise * permuted before the matrix multiply can take place. * * CTOT (input) INTEGER array, dimension (4) * A count of the total number of the various types of columns * in Q, as described in INDX. The fourth column type is any * column which has been deflated. * * W (input/output) REAL array, dimension (K) * The first K elements of this array contain the components * of the deflation-adjusted updating vector. Destroyed on * output. * * S (workspace) REAL array, dimension (N1 + 1)*K * Will contain the eigenvectors of the repaired matrix which * will be multiplied by the previously accumulated eigenvectors * to update the system. * * LDS (input) INTEGER * The leading dimension of S. LDS >= max(1,K). * * INFO (output) 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 * * Further Details * =============== * * Based on contributions by * Jeff Rutter, Computer Science Division, University of California * at Berkeley, USA * Modified by Francoise Tisseur, University of Tennessee. * * ===================================================================== * * .. Parameters .. REAL ONE, ZERO PARAMETER ( ONE = 1.0E0, ZERO = 0.0E0 ) * .. * .. Local Scalars .. INTEGER I, II, IQ2, J, N12, N2, N23 REAL TEMP * .. * .. External Functions .. REAL SLAMC3, SNRM2 EXTERNAL SLAMC3, SNRM2 * .. * .. External Subroutines .. EXTERNAL SCOPY, SGEMM, SLACPY, SLAED4, SLASET, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX, SIGN, SQRT * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 * IF( K.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.K ) THEN INFO = -2 ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN INFO = -6 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'SLAED3', -INFO ) RETURN END IF * * Quick return if possible * IF( K.EQ.0 ) $ RETURN * * Modify values DLAMDA(i) to make sure all DLAMDA(i)-DLAMDA(j) can * be computed with high relative accuracy (barring over/underflow). * This is a problem on machines without a guard digit in * add/subtract (Cray XMP, Cray YMP, Cray C 90 and Cray 2). * The following code replaces DLAMDA(I) by 2*DLAMDA(I)-DLAMDA(I), * which on any of these machines zeros out the bottommost * bit of DLAMDA(I) if it is 1; this makes the subsequent * subtractions DLAMDA(I)-DLAMDA(J) unproblematic when cancellation * occurs. On binary machines with a guard digit (almost all * machines) it does not change DLAMDA(I) at all. On hexadecimal * and decimal machines with a guard digit, it slightly * changes the bottommost bits of DLAMDA(I). It does not account * for hexadecimal or decimal machines without guard digits * (we know of none). We use a subroutine call to compute * 2*DLAMBDA(I) to prevent optimizing compilers from eliminating * this code. * DO 10 I = 1, K DLAMDA( I ) = SLAMC3( DLAMDA( I ), DLAMDA( I ) ) - DLAMDA( I ) 10 CONTINUE * DO 20 J = 1, K CALL SLAED4( K, J, DLAMDA, W, Q( 1, J ), RHO, D( J ), INFO ) * * If the zero finder fails, the computation is terminated. * IF( INFO.NE.0 ) $ GO TO 120 20 CONTINUE * IF( K.EQ.1 ) $ GO TO 110 IF( K.EQ.2 ) THEN DO 30 J = 1, K W( 1 ) = Q( 1, J ) W( 2 ) = Q( 2, J ) II = INDX( 1 ) Q( 1, J ) = W( II ) II = INDX( 2 ) Q( 2, J ) = W( II ) 30 CONTINUE GO TO 110 END IF * * Compute updated W. * CALL SCOPY( K, W, 1, S, 1 ) * * Initialize W(I) = Q(I,I) * CALL SCOPY( K, Q, LDQ+1, W, 1 ) DO 60 J = 1, K DO 40 I = 1, J - 1 W( I ) = W( I )*( Q( I, J ) / ( DLAMDA( I )-DLAMDA( J ) ) ) 40 CONTINUE DO 50 I = J + 1, K W( I ) = W( I )*( Q( I, J ) / ( DLAMDA( I )-DLAMDA( J ) ) ) 50 CONTINUE 60 CONTINUE DO 70 I = 1, K W( I ) = SIGN( SQRT( -W( I ) ), S( I ) ) 70 CONTINUE * * Compute eigenvectors of the modified rank-1 modification. * DO 100 J = 1, K DO 80 I = 1, K S( I ) = W( I ) / Q( I, J ) 80 CONTINUE TEMP = SNRM2( K, S, 1 ) DO 90 I = 1, K II = INDX( I ) Q( I, J ) = S( II ) / TEMP 90 CONTINUE 100 CONTINUE * * Compute the updated eigenvectors. * 110 CONTINUE * N2 = N - N1 N12 = CTOT( 1 ) + CTOT( 2 ) N23 = CTOT( 2 ) + CTOT( 3 ) * CALL SLACPY( 'A', N23, K, Q( CTOT( 1 )+1, 1 ), LDQ, S, N23 ) IQ2 = N1*N12 + 1 IF( N23.NE.0 ) THEN CALL SGEMM( 'N', 'N', N2, K, N23, ONE, Q2( IQ2 ), N2, S, N23, $ ZERO, Q( N1+1, 1 ), LDQ ) ELSE CALL SLASET( 'A', N2, K, ZERO, ZERO, Q( N1+1, 1 ), LDQ ) END IF * CALL SLACPY( 'A', N12, K, Q, LDQ, S, N12 ) IF( N12.NE.0 ) THEN CALL SGEMM( 'N', 'N', N1, K, N12, ONE, Q2, N1, S, N12, ZERO, Q, $ LDQ ) ELSE CALL SLASET( 'A', N1, K, ZERO, ZERO, Q( 1, 1 ), LDQ ) END IF * * 120 CONTINUE RETURN * * End of SLAED3 * END