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SUBROUTINE SLAED9( K, KSTART, KSTOP, N, D, Q, LDQ, RHO, DLAMDA, W,
$ S, LDS, INFO ) * * -- LAPACK routine (version 3.2) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * November 2006 * * .. Scalar Arguments .. INTEGER INFO, K, KSTART, KSTOP, LDQ, LDS, N REAL RHO * .. * .. Array Arguments .. REAL D( * ), DLAMDA( * ), Q( LDQ, * ), S( LDS, * ), $ W( * ) * .. * * Purpose * ======= * * SLAED9 finds the roots of the secular equation, as defined by the * values in D, Z, and RHO, between KSTART and KSTOP. It makes the * appropriate calls to SLAED4 and then stores the new matrix of * eigenvectors for use in calculating the next level of Z vectors. * * Arguments * ========= * * K (input) INTEGER * The number of terms in the rational function to be solved by * SLAED4. K >= 0. * * KSTART (input) INTEGER * KSTOP (input) INTEGER * The updated eigenvalues Lambda(I), KSTART <= I <= KSTOP * are to be computed. 1 <= KSTART <= KSTOP <= K. * * N (input) INTEGER * The number of rows and columns in the Q matrix. * N >= K (delation may result in N > K). * * D (output) REAL array, dimension (N) * D(I) contains the updated eigenvalues * for KSTART <= I <= KSTOP. * * Q (workspace) REAL array, dimension (LDQ,N) * * 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) 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. * * W (input) REAL array, dimension (K) * The first K elements of this array contain the components * of the deflation-adjusted updating vector. * * S (output) REAL array, dimension (LDS, K) * Will contain the eigenvectors of the repaired matrix which * will be stored for subsequent Z vector calculation and * 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 * * ===================================================================== * * .. Local Scalars .. INTEGER I, J REAL TEMP * .. * .. External Functions .. REAL SLAMC3, SNRM2 EXTERNAL SLAMC3, SNRM2 * .. * .. External Subroutines .. EXTERNAL SCOPY, SLAED4, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX, SIGN, SQRT * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 * IF( K.LT.0 ) THEN INFO = -1 ELSE IF( KSTART.LT.1 .OR. KSTART.GT.MAX( 1, K ) ) THEN INFO = -2 ELSE IF( MAX( 1, KSTOP ).LT.KSTART .OR. KSTOP.GT.MAX( 1, K ) ) $ THEN INFO = -3 ELSE IF( N.LT.K ) THEN INFO = -4 ELSE IF( LDQ.LT.MAX( 1, K ) ) THEN INFO = -7 ELSE IF( LDS.LT.MAX( 1, K ) ) THEN INFO = -12 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'SLAED9', -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, N DLAMDA( I ) = SLAMC3( DLAMDA( I ), DLAMDA( I ) ) - DLAMDA( I ) 10 CONTINUE * DO 20 J = KSTART, KSTOP 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 .OR. K.EQ.2 ) THEN DO 40 I = 1, K DO 30 J = 1, K S( J, I ) = Q( J, I ) 30 CONTINUE 40 CONTINUE GO TO 120 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 70 J = 1, K DO 50 I = 1, J - 1 W( I ) = W( I )*( Q( I, J ) / ( DLAMDA( I )-DLAMDA( J ) ) ) 50 CONTINUE DO 60 I = J + 1, K W( I ) = W( I )*( Q( I, J ) / ( DLAMDA( I )-DLAMDA( J ) ) ) 60 CONTINUE 70 CONTINUE DO 80 I = 1, K W( I ) = SIGN( SQRT( -W( I ) ), S( I, 1 ) ) 80 CONTINUE * * Compute eigenvectors of the modified rank-1 modification. * DO 110 J = 1, K DO 90 I = 1, K Q( I, J ) = W( I ) / Q( I, J ) 90 CONTINUE TEMP = SNRM2( K, Q( 1, J ), 1 ) DO 100 I = 1, K S( I, J ) = Q( I, J ) / TEMP 100 CONTINUE 110 CONTINUE * 120 CONTINUE RETURN * * End of SLAED9 * END |