*> \brief \b SLASQ2 computes all the eigenvalues of the symmetric positive definite tridiagonal matrix associated with the qd Array Z to high relative accuracy. Used by sbdsqr and sstegr. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SLASQ2 + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SLASQ2( N, Z, INFO ) * * .. Scalar Arguments .. * INTEGER INFO, N * .. * .. Array Arguments .. * REAL Z( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SLASQ2 computes all the eigenvalues of the symmetric positive *> definite tridiagonal matrix associated with the qd array Z to high *> relative accuracy are computed to high relative accuracy, in the *> absence of denormalization, underflow and overflow. *> *> To see the relation of Z to the tridiagonal matrix, let L be a *> unit lower bidiagonal matrix with subdiagonals Z(2,4,6,,..) and *> let U be an upper bidiagonal matrix with 1's above and diagonal *> Z(1,3,5,,..). The tridiagonal is L*U or, if you prefer, the *> symmetric tridiagonal to which it is similar. *> *> Note : SLASQ2 defines a logical variable, IEEE, which is true *> on machines which follow ieee-754 floating-point standard in their *> handling of infinities and NaNs, and false otherwise. This variable *> is passed to SLASQ3. *> \endverbatim * * Arguments: * ========== * *> \param[in] N *> \verbatim *> N is INTEGER *> The number of rows and columns in the matrix. N >= 0. *> \endverbatim *> *> \param[in,out] Z *> \verbatim *> Z is REAL array, dimension ( 4*N ) *> On entry Z holds the qd array. On exit, entries 1 to N hold *> the eigenvalues in decreasing order, Z( 2*N+1 ) holds the *> trace, and Z( 2*N+2 ) holds the sum of the eigenvalues. If *> N > 2, then Z( 2*N+3 ) holds the iteration count, Z( 2*N+4 ) *> holds NDIVS/NIN^2, and Z( 2*N+5 ) holds the percentage of *> shifts that failed. *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: successful exit *> < 0: if the i-th argument is a scalar and had an illegal *> value, then INFO = -i, if the i-th argument is an *> array and the j-entry had an illegal value, then *> INFO = -(i*100+j) *> > 0: the algorithm failed *> = 1, a split was marked by a positive value in E *> = 2, current block of Z not diagonalized after 100*N *> iterations (in inner while loop). On exit Z holds *> a qd array with the same eigenvalues as the given Z. *> = 3, termination criterion of outer while loop not met *> (program created more than N unreduced blocks) *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup auxOTHERcomputational * *> \par Further Details: * ===================== *> *> \verbatim *> *> Local Variables: I0:N0 defines a current unreduced segment of Z. *> The shifts are accumulated in SIGMA. Iteration count is in ITER. *> Ping-pong is controlled by PP (alternates between 0 and 1). *> \endverbatim *> * ===================================================================== SUBROUTINE SLASQ2( N, Z, 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 INFO, N * .. * .. Array Arguments .. REAL Z( * ) * .. * * ===================================================================== * * .. Parameters .. REAL CBIAS PARAMETER ( CBIAS = 1.50E0 ) REAL ZERO, HALF, ONE, TWO, FOUR, HUNDRD PARAMETER ( ZERO = 0.0E0, HALF = 0.5E0, ONE = 1.0E0, \$ TWO = 2.0E0, FOUR = 4.0E0, HUNDRD = 100.0E0 ) * .. * .. Local Scalars .. LOGICAL IEEE INTEGER I0, I4, IINFO, IPN4, ITER, IWHILA, IWHILB, K, \$ KMIN, N0, NBIG, NDIV, NFAIL, PP, SPLT, TTYPE, \$ I1, N1 REAL D, DEE, DEEMIN, DESIG, DMIN, DMIN1, DMIN2, DN, \$ DN1, DN2, E, EMAX, EMIN, EPS, G, OLDEMN, QMAX, \$ QMIN, S, SAFMIN, SIGMA, T, TAU, TEMP, TOL, \$ TOL2, TRACE, ZMAX, TEMPE, TEMPQ * .. * .. External Subroutines .. EXTERNAL SLASQ3, SLASRT, XERBLA * .. * .. External Functions .. REAL SLAMCH EXTERNAL SLAMCH * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, REAL, SQRT * .. * .. Executable Statements .. * * Test the input arguments. * (in case SLASQ2 is not called by SLASQ1) * INFO = 0 EPS = SLAMCH( 'Precision' ) SAFMIN = SLAMCH( 'Safe minimum' ) TOL = EPS*HUNDRD TOL2 = TOL**2 * IF( N.LT.0 ) THEN INFO = -1 CALL XERBLA( 'SLASQ2', 1 ) RETURN ELSE IF( N.EQ.0 ) THEN RETURN ELSE IF( N.EQ.1 ) THEN * * 1-by-1 case. * IF( Z( 1 ).LT.ZERO ) THEN INFO = -201 CALL XERBLA( 'SLASQ2', 2 ) END IF RETURN ELSE IF( N.EQ.2 ) THEN * * 2-by-2 case. * IF( Z( 1 ).LT.ZERO ) THEN INFO = -201 CALL XERBLA( 'SLASQ2', 2 ) RETURN ELSE IF( Z( 2 ).LT.ZERO ) THEN INFO = -202 CALL XERBLA( 'SLASQ2', 2 ) RETURN ELSE IF( Z( 3 ).LT.ZERO ) THEN INFO = -203 CALL XERBLA( 'SLASQ2', 2 ) RETURN ELSE IF( Z( 3 ).GT.Z( 1 ) ) THEN D = Z( 3 ) Z( 3 ) = Z( 1 ) Z( 1 ) = D END IF Z( 5 ) = Z( 1 ) + Z( 2 ) + Z( 3 ) IF( Z( 2 ).GT.Z( 3 )*TOL2 ) THEN T = HALF*( ( Z( 1 )-Z( 3 ) )+Z( 2 ) ) S = Z( 3 )*( Z( 2 ) / T ) IF( S.LE.T ) THEN S = Z( 3 )*( Z( 2 ) / ( T*( ONE+SQRT( ONE+S / T ) ) ) ) ELSE S = Z( 3 )*( Z( 2 ) / ( T+SQRT( T )*SQRT( T+S ) ) ) END IF T = Z( 1 ) + ( S+Z( 2 ) ) Z( 3 ) = Z( 3 )*( Z( 1 ) / T ) Z( 1 ) = T END IF Z( 2 ) = Z( 3 ) Z( 6 ) = Z( 2 ) + Z( 1 ) RETURN END IF * * Check for negative data and compute sums of q's and e's. * Z( 2*N ) = ZERO EMIN = Z( 2 ) QMAX = ZERO ZMAX = ZERO D = ZERO E = ZERO * DO 10 K = 1, 2*( N-1 ), 2 IF( Z( K ).LT.ZERO ) THEN INFO = -( 200+K ) CALL XERBLA( 'SLASQ2', 2 ) RETURN ELSE IF( Z( K+1 ).LT.ZERO ) THEN INFO = -( 200+K+1 ) CALL XERBLA( 'SLASQ2', 2 ) RETURN END IF D = D + Z( K ) E = E + Z( K+1 ) QMAX = MAX( QMAX, Z( K ) ) EMIN = MIN( EMIN, Z( K+1 ) ) ZMAX = MAX( QMAX, ZMAX, Z( K+1 ) ) 10 CONTINUE IF( Z( 2*N-1 ).LT.ZERO ) THEN INFO = -( 200+2*N-1 ) CALL XERBLA( 'SLASQ2', 2 ) RETURN END IF D = D + Z( 2*N-1 ) QMAX = MAX( QMAX, Z( 2*N-1 ) ) ZMAX = MAX( QMAX, ZMAX ) * * Check for diagonality. * IF( E.EQ.ZERO ) THEN DO 20 K = 2, N Z( K ) = Z( 2*K-1 ) 20 CONTINUE CALL SLASRT( 'D', N, Z, IINFO ) Z( 2*N-1 ) = D RETURN END IF * TRACE = D + E * * Check for zero data. * IF( TRACE.EQ.ZERO ) THEN Z( 2*N-1 ) = ZERO RETURN END IF * * Check whether the machine is IEEE conformable. * * IEEE = ( ILAENV( 10, 'SLASQ2', 'N', 1, 2, 3, 4 ).EQ.1 ) * * [11/15/2008] The case IEEE=.TRUE. has a problem in single precision with * some the test matrices of type 16. The double precision code is fine. * IEEE = .FALSE. * * Rearrange data for locality: Z=(q1,qq1,e1,ee1,q2,qq2,e2,ee2,...). * DO 30 K = 2*N, 2, -2 Z( 2*K ) = ZERO Z( 2*K-1 ) = Z( K ) Z( 2*K-2 ) = ZERO Z( 2*K-3 ) = Z( K-1 ) 30 CONTINUE * I0 = 1 N0 = N * * Reverse the qd-array, if warranted. * IF( CBIAS*Z( 4*I0-3 ).LT.Z( 4*N0-3 ) ) THEN IPN4 = 4*( I0+N0 ) DO 40 I4 = 4*I0, 2*( I0+N0-1 ), 4 TEMP = Z( I4-3 ) Z( I4-3 ) = Z( IPN4-I4-3 ) Z( IPN4-I4-3 ) = TEMP TEMP = Z( I4-1 ) Z( I4-1 ) = Z( IPN4-I4-5 ) Z( IPN4-I4-5 ) = TEMP 40 CONTINUE END IF * * Initial split checking via dqd and Li's test. * PP = 0 * DO 80 K = 1, 2 * D = Z( 4*N0+PP-3 ) DO 50 I4 = 4*( N0-1 ) + PP, 4*I0 + PP, -4 IF( Z( I4-1 ).LE.TOL2*D ) THEN Z( I4-1 ) = -ZERO D = Z( I4-3 ) ELSE D = Z( I4-3 )*( D / ( D+Z( I4-1 ) ) ) END IF 50 CONTINUE * * dqd maps Z to ZZ plus Li's test. * EMIN = Z( 4*I0+PP+1 ) D = Z( 4*I0+PP-3 ) DO 60 I4 = 4*I0 + PP, 4*( N0-1 ) + PP, 4 Z( I4-2*PP-2 ) = D + Z( I4-1 ) IF( Z( I4-1 ).LE.TOL2*D ) THEN Z( I4-1 ) = -ZERO Z( I4-2*PP-2 ) = D Z( I4-2*PP ) = ZERO D = Z( I4+1 ) ELSE IF( SAFMIN*Z( I4+1 ).LT.Z( I4-2*PP-2 ) .AND. \$ SAFMIN*Z( I4-2*PP-2 ).LT.Z( I4+1 ) ) THEN TEMP = Z( I4+1 ) / Z( I4-2*PP-2 ) Z( I4-2*PP ) = Z( I4-1 )*TEMP D = D*TEMP ELSE Z( I4-2*PP ) = Z( I4+1 )*( Z( I4-1 ) / Z( I4-2*PP-2 ) ) D = Z( I4+1 )*( D / Z( I4-2*PP-2 ) ) END IF EMIN = MIN( EMIN, Z( I4-2*PP ) ) 60 CONTINUE Z( 4*N0-PP-2 ) = D * * Now find qmax. * QMAX = Z( 4*I0-PP-2 ) DO 70 I4 = 4*I0 - PP + 2, 4*N0 - PP - 2, 4 QMAX = MAX( QMAX, Z( I4 ) ) 70 CONTINUE * * Prepare for the next iteration on K. * PP = 1 - PP 80 CONTINUE * * Initialise variables to pass to SLASQ3. * TTYPE = 0 DMIN1 = ZERO DMIN2 = ZERO DN = ZERO DN1 = ZERO DN2 = ZERO G = ZERO TAU = ZERO * ITER = 2 NFAIL = 0 NDIV = 2*( N0-I0 ) * DO 160 IWHILA = 1, N + 1 IF( N0.LT.1 ) \$ GO TO 170 * * While array unfinished do * * E(N0) holds the value of SIGMA when submatrix in I0:N0 * splits from the rest of the array, but is negated. * DESIG = ZERO IF( N0.EQ.N ) THEN SIGMA = ZERO ELSE SIGMA = -Z( 4*N0-1 ) END IF IF( SIGMA.LT.ZERO ) THEN INFO = 1 RETURN END IF * * Find last unreduced submatrix's top index I0, find QMAX and * EMIN. Find Gershgorin-type bound if Q's much greater than E's. * EMAX = ZERO IF( N0.GT.I0 ) THEN EMIN = ABS( Z( 4*N0-5 ) ) ELSE EMIN = ZERO END IF QMIN = Z( 4*N0-3 ) QMAX = QMIN DO 90 I4 = 4*N0, 8, -4 IF( Z( I4-5 ).LE.ZERO ) \$ GO TO 100 IF( QMIN.GE.FOUR*EMAX ) THEN QMIN = MIN( QMIN, Z( I4-3 ) ) EMAX = MAX( EMAX, Z( I4-5 ) ) END IF QMAX = MAX( QMAX, Z( I4-7 )+Z( I4-5 ) ) EMIN = MIN( EMIN, Z( I4-5 ) ) 90 CONTINUE I4 = 4 * 100 CONTINUE I0 = I4 / 4 PP = 0 * IF( N0-I0.GT.1 ) THEN DEE = Z( 4*I0-3 ) DEEMIN = DEE KMIN = I0 DO 110 I4 = 4*I0+1, 4*N0-3, 4 DEE = Z( I4 )*( DEE /( DEE+Z( I4-2 ) ) ) IF( DEE.LE.DEEMIN ) THEN DEEMIN = DEE KMIN = ( I4+3 )/4 END IF 110 CONTINUE IF( (KMIN-I0)*2.LT.N0-KMIN .AND. \$ DEEMIN.LE.HALF*Z(4*N0-3) ) THEN IPN4 = 4*( I0+N0 ) PP = 2 DO 120 I4 = 4*I0, 2*( I0+N0-1 ), 4 TEMP = Z( I4-3 ) Z( I4-3 ) = Z( IPN4-I4-3 ) Z( IPN4-I4-3 ) = TEMP TEMP = Z( I4-2 ) Z( I4-2 ) = Z( IPN4-I4-2 ) Z( IPN4-I4-2 ) = TEMP TEMP = Z( I4-1 ) Z( I4-1 ) = Z( IPN4-I4-5 ) Z( IPN4-I4-5 ) = TEMP TEMP = Z( I4 ) Z( I4 ) = Z( IPN4-I4-4 ) Z( IPN4-I4-4 ) = TEMP 120 CONTINUE END IF END IF * * Put -(initial shift) into DMIN. * DMIN = -MAX( ZERO, QMIN-TWO*SQRT( QMIN )*SQRT( EMAX ) ) * * Now I0:N0 is unreduced. * PP = 0 for ping, PP = 1 for pong. * PP = 2 indicates that flipping was applied to the Z array and * and that the tests for deflation upon entry in SLASQ3 * should not be performed. * NBIG = 100*( N0-I0+1 ) DO 140 IWHILB = 1, NBIG IF( I0.GT.N0 ) \$ GO TO 150 * * While submatrix unfinished take a good dqds step. * CALL SLASQ3( I0, N0, Z, PP, DMIN, SIGMA, DESIG, QMAX, NFAIL, \$ ITER, NDIV, IEEE, TTYPE, DMIN1, DMIN2, DN, DN1, \$ DN2, G, TAU ) * PP = 1 - PP * * When EMIN is very small check for splits. * IF( PP.EQ.0 .AND. N0-I0.GE.3 ) THEN IF( Z( 4*N0 ).LE.TOL2*QMAX .OR. \$ Z( 4*N0-1 ).LE.TOL2*SIGMA ) THEN SPLT = I0 - 1 QMAX = Z( 4*I0-3 ) EMIN = Z( 4*I0-1 ) OLDEMN = Z( 4*I0 ) DO 130 I4 = 4*I0, 4*( N0-3 ), 4 IF( Z( I4 ).LE.TOL2*Z( I4-3 ) .OR. \$ Z( I4-1 ).LE.TOL2*SIGMA ) THEN Z( I4-1 ) = -SIGMA SPLT = I4 / 4 QMAX = ZERO EMIN = Z( I4+3 ) OLDEMN = Z( I4+4 ) ELSE QMAX = MAX( QMAX, Z( I4+1 ) ) EMIN = MIN( EMIN, Z( I4-1 ) ) OLDEMN = MIN( OLDEMN, Z( I4 ) ) END IF 130 CONTINUE Z( 4*N0-1 ) = EMIN Z( 4*N0 ) = OLDEMN I0 = SPLT + 1 END IF END IF * 140 CONTINUE * INFO = 2 * * Maximum number of iterations exceeded, restore the shift * SIGMA and place the new d's and e's in a qd array. * This might need to be done for several blocks * I1 = I0 N1 = N0 145 CONTINUE TEMPQ = Z( 4*I0-3 ) Z( 4*I0-3 ) = Z( 4*I0-3 ) + SIGMA DO K = I0+1, N0 TEMPE = Z( 4*K-5 ) Z( 4*K-5 ) = Z( 4*K-5 ) * (TEMPQ / Z( 4*K-7 )) TEMPQ = Z( 4*K-3 ) Z( 4*K-3 ) = Z( 4*K-3 ) + SIGMA + TEMPE - Z( 4*K-5 ) END DO * * Prepare to do this on the previous block if there is one * IF( I1.GT.1 ) THEN N1 = I1-1 DO WHILE( ( I1.GE.2 ) .AND. ( Z(4*I1-5).GE.ZERO ) ) I1 = I1 - 1 END DO IF( I1.GE.1 ) THEN SIGMA = -Z(4*N1-1) GO TO 145 END IF END IF DO K = 1, N Z( 2*K-1 ) = Z( 4*K-3 ) * * Only the block 1..N0 is unfinished. The rest of the e's * must be essentially zero, although sometimes other data * has been stored in them. * IF( K.LT.N0 ) THEN Z( 2*K ) = Z( 4*K-1 ) ELSE Z( 2*K ) = 0 END IF END DO RETURN * * end IWHILB * 150 CONTINUE * 160 CONTINUE * INFO = 3 RETURN * * end IWHILA * 170 CONTINUE * * Move q's to the front. * DO 180 K = 2, N Z( K ) = Z( 4*K-3 ) 180 CONTINUE * * Sort and compute sum of eigenvalues. * CALL SLASRT( 'D', N, Z, IINFO ) * E = ZERO DO 190 K = N, 1, -1 E = E + Z( K ) 190 CONTINUE * * Store trace, sum(eigenvalues) and information on performance. * Z( 2*N+1 ) = TRACE Z( 2*N+2 ) = E Z( 2*N+3 ) = REAL( ITER ) Z( 2*N+4 ) = REAL( NDIV ) / REAL( N**2 ) Z( 2*N+5 ) = HUNDRD*NFAIL / REAL( ITER ) RETURN * * End of SLASQ2 * END