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SUBROUTINE DLAMSH ( S, LDS, NBULGE, JBLK, H, LDH, N, ULP )
*
* -- ScaLAPACK auxiliary routine (version 1.7) --
* University of Tennessee, Knoxville, Oak Ridge National Laboratory,
* and University of California, Berkeley.
* May 1, 1997
*
* .. Scalar Arguments ..
INTEGER LDS, NBULGE, JBLK, LDH, N
DOUBLE PRECISION ULP
* ..
* .. Array Arguments ..
DOUBLE PRECISION S(LDS,*), H(LDH,*)
* ..
*
* Purpose
* =======
*
* DLAMSH sends multiple shifts through a small (single node) matrix to
* see how consecutive small subdiagonal elements are modified by
* subsequent shifts in an effort to maximize the number of bulges
* that can be sent through.
* DLAMSH should only be called when there are multiple shifts/bulges
* (NBULGE > 1) and the first shift is starting in the middle of an
* unreduced Hessenberg matrix because of two or more consecutive small
* subdiagonal elements.
*
* Arguments
* =========
*
* S (local input/output) DOUBLE PRECISION array, (LDS,*)
* On entry, the matrix of shifts. Only the 2x2 diagonal of S is
* referenced. It is assumed that S has JBLK double shifts
* (size 2).
* On exit, the data is rearranged in the best order for
* applying.
*
* LDS (local input) INTEGER
* On entry, the leading dimension of S. Unchanged on exit.
* 1 < NBULGE <= JBLK <= LDS/2
*
* NBULGE (local input/output) INTEGER
* On entry, the number of bulges to send through H ( >1 ).
* NBULGE should be less than the maximum determined (JBLK).
* 1 < NBULGE <= JBLK <= LDS/2
* On exit, the maximum number of bulges that can be sent
* through.
*
* JBLK (local input) INTEGER
* On entry, the number of shifts determined for S.
* Unchanged on exit.
*
* H (local input/output) DOUBLE PRECISION array (LDH,N)
* On entry, the local matrix to apply the shifts on.
* H should be aligned so that the starting row is 2.
* On exit, the data is destroyed.
*
* LDS (local input) INTEGER
* On entry, the leading dimension of S. Unchanged on exit.
*
* N (local input) INTEGER
* On entry, the size of H. If all the bulges are expected to
* go through, N should be at least 4*NBULGE+2.
* Otherwise, NBULGE may be reduced by this routine.
*
* ULP (local input) DOUBLE PRECISION
* On entry, machine precision
* Unchanged on exit.
*
* Implemented by: G. Henry, May 1, 1997
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, TEN
PARAMETER ( ZERO = 0.0D+0, TEN = 10.0D+0 )
* ..
* .. Local Scalars ..
INTEGER K, IBULGE, M, NR, J, IVAL, I
DOUBLE PRECISION H44, H33, H43H34, H11, H22, H21, H12, H44S,
$ H33S, V1, V2, V3, H00, H10, TST1, T1, T2, T3,
$ SUM, S1, DVAL
* ..
* .. Local Arrays ..
DOUBLE PRECISION V(3)
* ..
* .. External Subroutines ..
EXTERNAL DLARFG, DCOPY
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, ABS
* ..
* .. Executable Statements ..
*
M = 2
DO 10 IBULGE = 1, NBULGE
H44 = S(2*JBLK-2*IBULGE+2, 2*JBLK-2*IBULGE+2)
H33 = S(2*JBLK-2*IBULGE+1,2*JBLK-2*IBULGE+1)
H43H34 = S(2*JBLK-2*IBULGE+1,2*JBLK-2*IBULGE+2)*
$ S(2*JBLK-2*IBULGE+2, 2*JBLK-2*IBULGE+1)
H11 = H( M, M )
H22 = H( M+1, M+1 )
H21 = H( M+1, M )
H12 = H( M, M+1 )
H44S = H44 - H11
H33S = H33 - H11
V1 = ( H33S*H44S-H43H34 ) / H21 + H12
V2 = H22 - H11 - H33S - H44S
V3 = H( M+2, M+1 )
S1 = ABS( V1 ) + ABS( V2 ) + ABS( V3 )
V1 = V1 / S1
V2 = V2 / S1
V3 = V3 / S1
V( 1 ) = V1
V( 2 ) = V2
V( 3 ) = V3
H00 = H( M-1, M-1 )
H10 = H( M, M-1 )
TST1 = ABS( V1 )*( ABS( H00 )+ABS( H11 )+ABS( H22 ) )
IF( ABS( H10 )*( ABS( V2 )+ABS( V3 ) ).GT.ULP*TST1 ) THEN
* Find minimum
DVAL = (ABS(H10)*(ABS(V2)+ABS(V3))) / (ULP*TST1)
IVAL = IBULGE
DO 15 I = IBULGE+1, NBULGE
H44 = S(2*JBLK-2*I+2, 2*JBLK-2*I+2)
H33 = S(2*JBLK-2*I+1,2*JBLK-2*I+1)
H43H34 = S(2*JBLK-2*I+1,2*JBLK-2*I+2)*
$ S(2*JBLK-2*I+2, 2*JBLK-2*I+1)
H11 = H( M, M )
H22 = H( M+1, M+1 )
H21 = H( M+1, M )
H12 = H( M, M+1 )
H44S = H44 - H11
H33S = H33 - H11
V1 = ( H33S*H44S-H43H34 ) / H21 + H12
V2 = H22 - H11 - H33S - H44S
V3 = H( M+2, M+1 )
S1 = ABS( V1 ) + ABS( V2 ) + ABS( V3 )
V1 = V1 / S1
V2 = V2 / S1
V3 = V3 / S1
V( 1 ) = V1
V( 2 ) = V2
V( 3 ) = V3
H00 = H( M-1, M-1 )
H10 = H( M, M-1 )
TST1 = ABS( V1 )*( ABS( H00 )+ABS( H11 )+ABS( H22 ) )
IF ( (DVAL.GT.(ABS(H10)*(ABS(V2)+ABS(V3)))/(ULP*TST1))
$ .AND. ( DVAL .GT. 1.D0 ) ) THEN
DVAL = (ABS(H10)*(ABS(V2)+ABS(V3))) / (ULP*TST1)
IVAL = I
END IF
15 CONTINUE
IF ( (DVAL .LT. TEN) .AND. (IVAL .NE. IBULGE) ) THEN
H44 = S(2*JBLK-2*IVAL+2, 2*JBLK-2*IVAL+2)
H33 = S(2*JBLK-2*IVAL+1,2*JBLK-2*IVAL+1)
H43H34 = S(2*JBLK-2*IVAL+1,2*JBLK-2*IVAL+2)
H10 = S(2*JBLK-2*IVAL+2, 2*JBLK-2*IVAL+1)
S(2*JBLK-2*IVAL+2,2*JBLK-2*IVAL+2) =
$ S(2*JBLK-2*IBULGE+2,2*JBLK-2*IBULGE+2)
S(2*JBLK-2*IVAL+1,2*JBLK-2*IVAL+1) =
$ S(2*JBLK-2*IBULGE+1,2*JBLK-2*IBULGE+1)
S(2*JBLK-2*IVAL+1,2*JBLK-2*IVAL+2) =
$ S(2*JBLK-2*IBULGE+1,2*JBLK-2*IBULGE+2)
S(2*JBLK-2*IVAL+2, 2*JBLK-2*IVAL+1) =
$ S(2*JBLK-2*IBULGE+2, 2*JBLK-2*IBULGE+1)
S(2*JBLK-2*IBULGE+2, 2*JBLK-2*IBULGE+2) = H44
S(2*JBLK-2*IBULGE+1,2*JBLK-2*IBULGE+1) = H33
S(2*JBLK-2*IBULGE+1,2*JBLK-2*IBULGE+2) = H43H34
S(2*JBLK-2*IBULGE+2, 2*JBLK-2*IBULGE+1) = H10
END IF
H44 = S(2*JBLK-2*IBULGE+2, 2*JBLK-2*IBULGE+2)
H33 = S(2*JBLK-2*IBULGE+1,2*JBLK-2*IBULGE+1)
H43H34 = S(2*JBLK-2*IBULGE+1,2*JBLK-2*IBULGE+2)*
$ S(2*JBLK-2*IBULGE+2, 2*JBLK-2*IBULGE+1)
H11 = H( M, M )
H22 = H( M+1, M+1 )
H21 = H( M+1, M )
H12 = H( M, M+1 )
H44S = H44 - H11
H33S = H33 - H11
V1 = ( H33S*H44S-H43H34 ) / H21 + H12
V2 = H22 - H11 - H33S - H44S
V3 = H( M+2, M+1 )
S1 = ABS( V1 ) + ABS( V2 ) + ABS( V3 )
V1 = V1 / S1
V2 = V2 / S1
V3 = V3 / S1
V( 1 ) = V1
V( 2 ) = V2
V( 3 ) = V3
H00 = H( M-1, M-1 )
H10 = H( M, M-1 )
TST1 = ABS( V1 )*( ABS( H00 )+ABS( H11 )+ABS( H22 ) )
END IF
IF( ABS( H10 )*( ABS( V2 )+ABS( V3 ) ).GT.TEN*ULP*TST1 ) THEN
* IBULGE better not be 1 here or we have a bug!
NBULGE = MAX(IBULGE -1,1)
RETURN
END IF
DO 120 K = M, N - 1
NR = MIN( 3, N-K+1 )
IF( K.GT.M )
$ CALL DCOPY( NR, H( K, K-1 ), 1, V, 1 )
CALL DLARFG( NR, V( 1 ), V( 2 ), 1, T1 )
IF( K.GT.M ) THEN
H( K, K-1 ) = V( 1 )
H( K+1, K-1 ) = ZERO
IF( K.LT.N-1 )
$ H( K+2, K-1 ) = ZERO
ELSE
H( K, K-1 ) = -H( K, K-1 )
END IF
V2 = V( 2 )
T2 = T1*V2
IF( NR.EQ.3 ) THEN
V3 = V( 3 )
T3 = T1*V3
DO 60 J = K, N
SUM = H( K, J ) + V2*H( K+1, J ) + V3*H( K+2, J )
H( K, J ) = H( K, J ) - SUM*T1
H( K+1, J ) = H( K+1, J ) - SUM*T2
H( K+2, J ) = H( K+2, J ) - SUM*T3
60 CONTINUE
DO 70 J = 1, MIN( K+3, N )
SUM = H( J, K ) + V2*H( J, K+1 ) + V3*H( J, K+2 )
H( J, K ) = H( J, K ) - SUM*T1
H( J, K+1 ) = H( J, K+1 ) - SUM*T2
H( J, K+2 ) = H( J, K+2 ) - SUM*T3
70 CONTINUE
END IF
120 CONTINUE
10 CONTINUE
*
RETURN
END
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