root / src / lapack / double / dlasq2.f @ 7
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SUBROUTINE DLASQ2( N, Z, INFO ) |
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* |
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* -- LAPACK routine (version 3.2) -- |
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* |
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* -- Contributed by Osni Marques of the Lawrence Berkeley National -- |
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* -- Laboratory and Beresford Parlett of the Univ. of California at -- |
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* -- Berkeley -- |
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* -- November 2008 -- |
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* |
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* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
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* |
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* .. Scalar Arguments .. |
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INTEGER INFO, N |
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* .. |
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* .. Array Arguments .. |
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DOUBLE PRECISION Z( * ) |
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* .. |
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* |
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* Purpose |
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* ======= |
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* |
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* DLASQ2 computes all the eigenvalues of the symmetric positive |
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* definite tridiagonal matrix associated with the qd array Z to high |
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* relative accuracy are computed to high relative accuracy, in the |
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* absence of denormalization, underflow and overflow. |
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* |
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* To see the relation of Z to the tridiagonal matrix, let L be a |
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* unit lower bidiagonal matrix with subdiagonals Z(2,4,6,,..) and |
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* let U be an upper bidiagonal matrix with 1's above and diagonal |
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* Z(1,3,5,,..). The tridiagonal is L*U or, if you prefer, the |
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* symmetric tridiagonal to which it is similar. |
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* |
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* Note : DLASQ2 defines a logical variable, IEEE, which is true |
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* on machines which follow ieee-754 floating-point standard in their |
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* handling of infinities and NaNs, and false otherwise. This variable |
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* is passed to DLASQ3. |
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* |
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* Arguments |
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* ========= |
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* |
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* N (input) INTEGER |
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* The number of rows and columns in the matrix. N >= 0. |
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* |
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* Z (input/output) DOUBLE PRECISION array, dimension ( 4*N ) |
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* On entry Z holds the qd array. On exit, entries 1 to N hold |
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* the eigenvalues in decreasing order, Z( 2*N+1 ) holds the |
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* trace, and Z( 2*N+2 ) holds the sum of the eigenvalues. If |
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* N > 2, then Z( 2*N+3 ) holds the iteration count, Z( 2*N+4 ) |
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* holds NDIVS/NIN^2, and Z( 2*N+5 ) holds the percentage of |
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* shifts that failed. |
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* |
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* INFO (output) INTEGER |
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* = 0: successful exit |
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* < 0: if the i-th argument is a scalar and had an illegal |
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* value, then INFO = -i, if the i-th argument is an |
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* array and the j-entry had an illegal value, then |
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* INFO = -(i*100+j) |
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* > 0: the algorithm failed |
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* = 1, a split was marked by a positive value in E |
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* = 2, current block of Z not diagonalized after 30*N |
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* iterations (in inner while loop) |
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* = 3, termination criterion of outer while loop not met |
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* (program created more than N unreduced blocks) |
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* |
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* Further Details |
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* =============== |
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* Local Variables: I0:N0 defines a current unreduced segment of Z. |
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* The shifts are accumulated in SIGMA. Iteration count is in ITER. |
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* Ping-pong is controlled by PP (alternates between 0 and 1). |
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* |
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* ===================================================================== |
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* |
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* .. Parameters .. |
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DOUBLE PRECISION CBIAS |
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PARAMETER ( CBIAS = 1.50D0 ) |
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DOUBLE PRECISION ZERO, HALF, ONE, TWO, FOUR, HUNDRD |
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PARAMETER ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0, |
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$ TWO = 2.0D0, FOUR = 4.0D0, HUNDRD = 100.0D0 ) |
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* .. |
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* .. Local Scalars .. |
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LOGICAL IEEE |
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INTEGER I0, I4, IINFO, IPN4, ITER, IWHILA, IWHILB, K, |
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$ KMIN, N0, NBIG, NDIV, NFAIL, PP, SPLT, TTYPE |
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DOUBLE PRECISION D, DEE, DEEMIN, DESIG, DMIN, DMIN1, DMIN2, DN, |
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$ DN1, DN2, E, EMAX, EMIN, EPS, G, OLDEMN, QMAX, |
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$ QMIN, S, SAFMIN, SIGMA, T, TAU, TEMP, TOL, |
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$ TOL2, TRACE, ZMAX |
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* .. |
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* .. External Subroutines .. |
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EXTERNAL DLASQ3, DLASRT, XERBLA |
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* .. |
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* .. External Functions .. |
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INTEGER ILAENV |
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DOUBLE PRECISION DLAMCH |
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EXTERNAL DLAMCH, ILAENV |
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* .. |
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* .. Intrinsic Functions .. |
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INTRINSIC ABS, DBLE, MAX, MIN, SQRT |
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* .. |
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* .. Executable Statements .. |
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* |
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* Test the input arguments. |
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* (in case DLASQ2 is not called by DLASQ1) |
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* |
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INFO = 0 |
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EPS = DLAMCH( 'Precision' ) |
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SAFMIN = DLAMCH( 'Safe minimum' ) |
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TOL = EPS*HUNDRD |
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TOL2 = TOL**2 |
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* |
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IF( N.LT.0 ) THEN |
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INFO = -1 |
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CALL XERBLA( 'DLASQ2', 1 ) |
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RETURN |
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ELSE IF( N.EQ.0 ) THEN |
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RETURN |
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ELSE IF( N.EQ.1 ) THEN |
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* |
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* 1-by-1 case. |
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* |
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IF( Z( 1 ).LT.ZERO ) THEN |
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INFO = -201 |
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CALL XERBLA( 'DLASQ2', 2 ) |
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END IF |
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RETURN |
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ELSE IF( N.EQ.2 ) THEN |
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* |
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* 2-by-2 case. |
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* |
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IF( Z( 2 ).LT.ZERO .OR. Z( 3 ).LT.ZERO ) THEN |
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INFO = -2 |
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CALL XERBLA( 'DLASQ2', 2 ) |
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RETURN |
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ELSE IF( Z( 3 ).GT.Z( 1 ) ) THEN |
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D = Z( 3 ) |
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Z( 3 ) = Z( 1 ) |
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Z( 1 ) = D |
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END IF |
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Z( 5 ) = Z( 1 ) + Z( 2 ) + Z( 3 ) |
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IF( Z( 2 ).GT.Z( 3 )*TOL2 ) THEN |
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T = HALF*( ( Z( 1 )-Z( 3 ) )+Z( 2 ) ) |
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S = Z( 3 )*( Z( 2 ) / T ) |
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IF( S.LE.T ) THEN |
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S = Z( 3 )*( Z( 2 ) / ( T*( ONE+SQRT( ONE+S / T ) ) ) ) |
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ELSE |
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S = Z( 3 )*( Z( 2 ) / ( T+SQRT( T )*SQRT( T+S ) ) ) |
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END IF |
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T = Z( 1 ) + ( S+Z( 2 ) ) |
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Z( 3 ) = Z( 3 )*( Z( 1 ) / T ) |
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Z( 1 ) = T |
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END IF |
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Z( 2 ) = Z( 3 ) |
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Z( 6 ) = Z( 2 ) + Z( 1 ) |
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RETURN |
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END IF |
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* |
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* Check for negative data and compute sums of q's and e's. |
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* |
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Z( 2*N ) = ZERO |
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EMIN = Z( 2 ) |
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QMAX = ZERO |
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ZMAX = ZERO |
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D = ZERO |
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E = ZERO |
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* |
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DO 10 K = 1, 2*( N-1 ), 2 |
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IF( Z( K ).LT.ZERO ) THEN |
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INFO = -( 200+K ) |
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CALL XERBLA( 'DLASQ2', 2 ) |
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RETURN |
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ELSE IF( Z( K+1 ).LT.ZERO ) THEN |
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INFO = -( 200+K+1 ) |
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CALL XERBLA( 'DLASQ2', 2 ) |
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RETURN |
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END IF |
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D = D + Z( K ) |
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E = E + Z( K+1 ) |
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QMAX = MAX( QMAX, Z( K ) ) |
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EMIN = MIN( EMIN, Z( K+1 ) ) |
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ZMAX = MAX( QMAX, ZMAX, Z( K+1 ) ) |
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10 CONTINUE |
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IF( Z( 2*N-1 ).LT.ZERO ) THEN |
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INFO = -( 200+2*N-1 ) |
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CALL XERBLA( 'DLASQ2', 2 ) |
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RETURN |
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END IF |
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D = D + Z( 2*N-1 ) |
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QMAX = MAX( QMAX, Z( 2*N-1 ) ) |
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ZMAX = MAX( QMAX, ZMAX ) |
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* |
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* Check for diagonality. |
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* |
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IF( E.EQ.ZERO ) THEN |
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DO 20 K = 2, N |
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Z( K ) = Z( 2*K-1 ) |
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20 CONTINUE |
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CALL DLASRT( 'D', N, Z, IINFO ) |
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Z( 2*N-1 ) = D |
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RETURN |
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END IF |
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* |
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TRACE = D + E |
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* |
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* Check for zero data. |
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* |
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IF( TRACE.EQ.ZERO ) THEN |
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Z( 2*N-1 ) = ZERO |
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RETURN |
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END IF |
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* |
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* Check whether the machine is IEEE conformable. |
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* |
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IEEE = ILAENV( 10, 'DLASQ2', 'N', 1, 2, 3, 4 ).EQ.1 .AND. |
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$ ILAENV( 11, 'DLASQ2', 'N', 1, 2, 3, 4 ).EQ.1 |
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* |
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* Rearrange data for locality: Z=(q1,qq1,e1,ee1,q2,qq2,e2,ee2,...). |
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* |
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DO 30 K = 2*N, 2, -2 |
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Z( 2*K ) = ZERO |
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Z( 2*K-1 ) = Z( K ) |
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Z( 2*K-2 ) = ZERO |
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Z( 2*K-3 ) = Z( K-1 ) |
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30 CONTINUE |
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* |
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I0 = 1 |
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N0 = N |
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* |
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* Reverse the qd-array, if warranted. |
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* |
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IF( CBIAS*Z( 4*I0-3 ).LT.Z( 4*N0-3 ) ) THEN |
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IPN4 = 4*( I0+N0 ) |
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DO 40 I4 = 4*I0, 2*( I0+N0-1 ), 4 |
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TEMP = Z( I4-3 ) |
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Z( I4-3 ) = Z( IPN4-I4-3 ) |
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Z( IPN4-I4-3 ) = TEMP |
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TEMP = Z( I4-1 ) |
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Z( I4-1 ) = Z( IPN4-I4-5 ) |
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Z( IPN4-I4-5 ) = TEMP |
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40 CONTINUE |
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END IF |
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* |
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* Initial split checking via dqd and Li's test. |
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* |
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PP = 0 |
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* |
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DO 80 K = 1, 2 |
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* |
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D = Z( 4*N0+PP-3 ) |
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DO 50 I4 = 4*( N0-1 ) + PP, 4*I0 + PP, -4 |
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IF( Z( I4-1 ).LE.TOL2*D ) THEN |
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Z( I4-1 ) = -ZERO |
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D = Z( I4-3 ) |
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ELSE |
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D = Z( I4-3 )*( D / ( D+Z( I4-1 ) ) ) |
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END IF |
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50 CONTINUE |
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* |
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* dqd maps Z to ZZ plus Li's test. |
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* |
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EMIN = Z( 4*I0+PP+1 ) |
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D = Z( 4*I0+PP-3 ) |
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DO 60 I4 = 4*I0 + PP, 4*( N0-1 ) + PP, 4 |
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Z( I4-2*PP-2 ) = D + Z( I4-1 ) |
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IF( Z( I4-1 ).LE.TOL2*D ) THEN |
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Z( I4-1 ) = -ZERO |
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Z( I4-2*PP-2 ) = D |
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Z( I4-2*PP ) = ZERO |
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D = Z( I4+1 ) |
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ELSE IF( SAFMIN*Z( I4+1 ).LT.Z( I4-2*PP-2 ) .AND. |
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$ SAFMIN*Z( I4-2*PP-2 ).LT.Z( I4+1 ) ) THEN |
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TEMP = Z( I4+1 ) / Z( I4-2*PP-2 ) |
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Z( I4-2*PP ) = Z( I4-1 )*TEMP |
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D = D*TEMP |
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ELSE |
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Z( I4-2*PP ) = Z( I4+1 )*( Z( I4-1 ) / Z( I4-2*PP-2 ) ) |
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D = Z( I4+1 )*( D / Z( I4-2*PP-2 ) ) |
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END IF |
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EMIN = MIN( EMIN, Z( I4-2*PP ) ) |
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60 CONTINUE |
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Z( 4*N0-PP-2 ) = D |
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* |
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* Now find qmax. |
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* |
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QMAX = Z( 4*I0-PP-2 ) |
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DO 70 I4 = 4*I0 - PP + 2, 4*N0 - PP - 2, 4 |
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QMAX = MAX( QMAX, Z( I4 ) ) |
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70 CONTINUE |
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* |
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* Prepare for the next iteration on K. |
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* |
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PP = 1 - PP |
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80 CONTINUE |
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* |
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* Initialise variables to pass to DLASQ3. |
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* |
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TTYPE = 0 |
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DMIN1 = ZERO |
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DMIN2 = ZERO |
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DN = ZERO |
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DN1 = ZERO |
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DN2 = ZERO |
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G = ZERO |
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TAU = ZERO |
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* |
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ITER = 2 |
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NFAIL = 0 |
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NDIV = 2*( N0-I0 ) |
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* |
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DO 160 IWHILA = 1, N + 1 |
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IF( N0.LT.1 ) |
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$ GO TO 170 |
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* |
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* While array unfinished do |
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* |
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* E(N0) holds the value of SIGMA when submatrix in I0:N0 |
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* splits from the rest of the array, but is negated. |
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* |
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DESIG = ZERO |
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IF( N0.EQ.N ) THEN |
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SIGMA = ZERO |
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ELSE |
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SIGMA = -Z( 4*N0-1 ) |
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END IF |
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IF( SIGMA.LT.ZERO ) THEN |
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INFO = 1 |
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RETURN |
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END IF |
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* |
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* Find last unreduced submatrix's top index I0, find QMAX and |
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* EMIN. Find Gershgorin-type bound if Q's much greater than E's. |
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* |
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EMAX = ZERO |
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IF( N0.GT.I0 ) THEN |
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EMIN = ABS( Z( 4*N0-5 ) ) |
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ELSE |
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EMIN = ZERO |
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END IF |
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QMIN = Z( 4*N0-3 ) |
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QMAX = QMIN |
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DO 90 I4 = 4*N0, 8, -4 |
| 342 |
IF( Z( I4-5 ).LE.ZERO ) |
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$ GO TO 100 |
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IF( QMIN.GE.FOUR*EMAX ) THEN |
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QMIN = MIN( QMIN, Z( I4-3 ) ) |
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EMAX = MAX( EMAX, Z( I4-5 ) ) |
| 347 |
END IF |
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QMAX = MAX( QMAX, Z( I4-7 )+Z( I4-5 ) ) |
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EMIN = MIN( EMIN, Z( I4-5 ) ) |
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90 CONTINUE |
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I4 = 4 |
| 352 |
* |
| 353 |
100 CONTINUE |
| 354 |
I0 = I4 / 4 |
| 355 |
PP = 0 |
| 356 |
* |
| 357 |
IF( N0-I0.GT.1 ) THEN |
| 358 |
DEE = Z( 4*I0-3 ) |
| 359 |
DEEMIN = DEE |
| 360 |
KMIN = I0 |
| 361 |
DO 110 I4 = 4*I0+1, 4*N0-3, 4 |
| 362 |
DEE = Z( I4 )*( DEE /( DEE+Z( I4-2 ) ) ) |
| 363 |
IF( DEE.LE.DEEMIN ) THEN |
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DEEMIN = DEE |
| 365 |
KMIN = ( I4+3 )/4 |
| 366 |
END IF |
| 367 |
110 CONTINUE |
| 368 |
IF( (KMIN-I0)*2.LT.N0-KMIN .AND. |
| 369 |
$ DEEMIN.LE.HALF*Z(4*N0-3) ) THEN |
| 370 |
IPN4 = 4*( I0+N0 ) |
| 371 |
PP = 2 |
| 372 |
DO 120 I4 = 4*I0, 2*( I0+N0-1 ), 4 |
| 373 |
TEMP = Z( I4-3 ) |
| 374 |
Z( I4-3 ) = Z( IPN4-I4-3 ) |
| 375 |
Z( IPN4-I4-3 ) = TEMP |
| 376 |
TEMP = Z( I4-2 ) |
| 377 |
Z( I4-2 ) = Z( IPN4-I4-2 ) |
| 378 |
Z( IPN4-I4-2 ) = TEMP |
| 379 |
TEMP = Z( I4-1 ) |
| 380 |
Z( I4-1 ) = Z( IPN4-I4-5 ) |
| 381 |
Z( IPN4-I4-5 ) = TEMP |
| 382 |
TEMP = Z( I4 ) |
| 383 |
Z( I4 ) = Z( IPN4-I4-4 ) |
| 384 |
Z( IPN4-I4-4 ) = TEMP |
| 385 |
120 CONTINUE |
| 386 |
END IF |
| 387 |
END IF |
| 388 |
* |
| 389 |
* Put -(initial shift) into DMIN. |
| 390 |
* |
| 391 |
DMIN = -MAX( ZERO, QMIN-TWO*SQRT( QMIN )*SQRT( EMAX ) ) |
| 392 |
* |
| 393 |
* Now I0:N0 is unreduced. |
| 394 |
* PP = 0 for ping, PP = 1 for pong. |
| 395 |
* PP = 2 indicates that flipping was applied to the Z array and |
| 396 |
* and that the tests for deflation upon entry in DLASQ3 |
| 397 |
* should not be performed. |
| 398 |
* |
| 399 |
NBIG = 30*( N0-I0+1 ) |
| 400 |
DO 140 IWHILB = 1, NBIG |
| 401 |
IF( I0.GT.N0 ) |
| 402 |
$ GO TO 150 |
| 403 |
* |
| 404 |
* While submatrix unfinished take a good dqds step. |
| 405 |
* |
| 406 |
CALL DLASQ3( I0, N0, Z, PP, DMIN, SIGMA, DESIG, QMAX, NFAIL, |
| 407 |
$ ITER, NDIV, IEEE, TTYPE, DMIN1, DMIN2, DN, DN1, |
| 408 |
$ DN2, G, TAU ) |
| 409 |
* |
| 410 |
PP = 1 - PP |
| 411 |
* |
| 412 |
* When EMIN is very small check for splits. |
| 413 |
* |
| 414 |
IF( PP.EQ.0 .AND. N0-I0.GE.3 ) THEN |
| 415 |
IF( Z( 4*N0 ).LE.TOL2*QMAX .OR. |
| 416 |
$ Z( 4*N0-1 ).LE.TOL2*SIGMA ) THEN |
| 417 |
SPLT = I0 - 1 |
| 418 |
QMAX = Z( 4*I0-3 ) |
| 419 |
EMIN = Z( 4*I0-1 ) |
| 420 |
OLDEMN = Z( 4*I0 ) |
| 421 |
DO 130 I4 = 4*I0, 4*( N0-3 ), 4 |
| 422 |
IF( Z( I4 ).LE.TOL2*Z( I4-3 ) .OR. |
| 423 |
$ Z( I4-1 ).LE.TOL2*SIGMA ) THEN |
| 424 |
Z( I4-1 ) = -SIGMA |
| 425 |
SPLT = I4 / 4 |
| 426 |
QMAX = ZERO |
| 427 |
EMIN = Z( I4+3 ) |
| 428 |
OLDEMN = Z( I4+4 ) |
| 429 |
ELSE |
| 430 |
QMAX = MAX( QMAX, Z( I4+1 ) ) |
| 431 |
EMIN = MIN( EMIN, Z( I4-1 ) ) |
| 432 |
OLDEMN = MIN( OLDEMN, Z( I4 ) ) |
| 433 |
END IF |
| 434 |
130 CONTINUE |
| 435 |
Z( 4*N0-1 ) = EMIN |
| 436 |
Z( 4*N0 ) = OLDEMN |
| 437 |
I0 = SPLT + 1 |
| 438 |
END IF |
| 439 |
END IF |
| 440 |
* |
| 441 |
140 CONTINUE |
| 442 |
* |
| 443 |
INFO = 2 |
| 444 |
RETURN |
| 445 |
* |
| 446 |
* end IWHILB |
| 447 |
* |
| 448 |
150 CONTINUE |
| 449 |
* |
| 450 |
160 CONTINUE |
| 451 |
* |
| 452 |
INFO = 3 |
| 453 |
RETURN |
| 454 |
* |
| 455 |
* end IWHILA |
| 456 |
* |
| 457 |
170 CONTINUE |
| 458 |
* |
| 459 |
* Move q's to the front. |
| 460 |
* |
| 461 |
DO 180 K = 2, N |
| 462 |
Z( K ) = Z( 4*K-3 ) |
| 463 |
180 CONTINUE |
| 464 |
* |
| 465 |
* Sort and compute sum of eigenvalues. |
| 466 |
* |
| 467 |
CALL DLASRT( 'D', N, Z, IINFO ) |
| 468 |
* |
| 469 |
E = ZERO |
| 470 |
DO 190 K = N, 1, -1 |
| 471 |
E = E + Z( K ) |
| 472 |
190 CONTINUE |
| 473 |
* |
| 474 |
* Store trace, sum(eigenvalues) and information on performance. |
| 475 |
* |
| 476 |
Z( 2*N+1 ) = TRACE |
| 477 |
Z( 2*N+2 ) = E |
| 478 |
Z( 2*N+3 ) = DBLE( ITER ) |
| 479 |
Z( 2*N+4 ) = DBLE( NDIV ) / DBLE( N**2 ) |
| 480 |
Z( 2*N+5 ) = HUNDRD*NFAIL / DBLE( ITER ) |
| 481 |
RETURN |
| 482 |
* |
| 483 |
* End of DLASQ2 |
| 484 |
* |
| 485 |
END |