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chbgst.f
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*> \brief \b CHBGST
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download CHBGST + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/chbgst.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/chbgst.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/chbgst.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE CHBGST( VECT, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, X,
* LDX, WORK, RWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO, VECT
* INTEGER INFO, KA, KB, LDAB, LDBB, LDX, N
* ..
* .. Array Arguments ..
* REAL RWORK( * )
* COMPLEX AB( LDAB, * ), BB( LDBB, * ), WORK( * ),
* $ X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> CHBGST reduces a complex Hermitian-definite banded generalized
*> eigenproblem A*x = lambda*B*x to standard form C*y = lambda*y,
*> such that C has the same bandwidth as A.
*>
*> B must have been previously factorized as S**H*S by CPBSTF, using a
*> split Cholesky factorization. A is overwritten by C = X**H*A*X, where
*> X = S**(-1)*Q and Q is a unitary matrix chosen to preserve the
*> bandwidth of A.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] VECT
*> \verbatim
*> VECT is CHARACTER*1
*> = 'N': do not form the transformation matrix X;
*> = 'V': form X.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in] KA
*> \verbatim
*> KA is INTEGER
*> The number of superdiagonals of the matrix A if UPLO = 'U',
*> or the number of subdiagonals if UPLO = 'L'. KA >= 0.
*> \endverbatim
*>
*> \param[in] KB
*> \verbatim
*> KB is INTEGER
*> The number of superdiagonals of the matrix B if UPLO = 'U',
*> or the number of subdiagonals if UPLO = 'L'. KA >= KB >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*> AB is COMPLEX array, dimension (LDAB,N)
*> On entry, the upper or lower triangle of the Hermitian band
*> matrix A, stored in the first ka+1 rows of the array. The
*> j-th column of A is stored in the j-th column of the array AB
*> as follows:
*> if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka).
*>
*> On exit, the transformed matrix X**H*A*X, stored in the same
*> format as A.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= KA+1.
*> \endverbatim
*>
*> \param[in] BB
*> \verbatim
*> BB is COMPLEX array, dimension (LDBB,N)
*> The banded factor S from the split Cholesky factorization of
*> B, as returned by CPBSTF, stored in the first kb+1 rows of
*> the array.
*> \endverbatim
*>
*> \param[in] LDBB
*> \verbatim
*> LDBB is INTEGER
*> The leading dimension of the array BB. LDBB >= KB+1.
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*> X is COMPLEX array, dimension (LDX,N)
*> If VECT = 'V', the n-by-n matrix X.
*> If VECT = 'N', the array X is not referenced.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X.
*> LDX >= max(1,N) if VECT = 'V'; LDX >= 1 otherwise.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is COMPLEX array, dimension (N)
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*> RWORK is REAL array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup complexOTHERcomputational
*
* =====================================================================
SUBROUTINE CHBGST( VECT, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, X,
$ LDX, WORK, RWORK, INFO )
*
* -- LAPACK computational routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. Scalar Arguments ..
CHARACTER UPLO, VECT
INTEGER INFO, KA, KB, LDAB, LDBB, LDX, N
* ..
* .. Array Arguments ..
REAL RWORK( * )
COMPLEX AB( LDAB, * ), BB( LDBB, * ), WORK( * ),
$ X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
COMPLEX CZERO, CONE
REAL ONE
PARAMETER ( CZERO = ( 0.0E+0, 0.0E+0 ),
$ CONE = ( 1.0E+0, 0.0E+0 ), ONE = 1.0E+0 )
* ..
* .. Local Scalars ..
LOGICAL UPDATE, UPPER, WANTX
INTEGER I, I0, I1, I2, INCA, J, J1, J1T, J2, J2T, K,
$ KA1, KB1, KBT, L, M, NR, NRT, NX
REAL BII
COMPLEX RA, RA1, T
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL CGERC, CGERU, CLACGV, CLAR2V, CLARGV, CLARTG,
$ CLARTV, CLASET, CROT, CSSCAL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC CONJG, MAX, MIN, REAL
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
WANTX = LSAME( VECT, 'V' )
UPPER = LSAME( UPLO, 'U' )
KA1 = KA + 1
KB1 = KB + 1
INFO = 0
IF( .NOT.WANTX .AND. .NOT.LSAME( VECT, 'N' ) ) THEN
INFO = -1
ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( KA.LT.0 ) THEN
INFO = -4
ELSE IF( KB.LT.0 .OR. KB.GT.KA ) THEN
INFO = -5
ELSE IF( LDAB.LT.KA+1 ) THEN
INFO = -7
ELSE IF( LDBB.LT.KB+1 ) THEN
INFO = -9
ELSE IF( LDX.LT.1 .OR. WANTX .AND. LDX.LT.MAX( 1, N ) ) THEN
INFO = -11
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CHBGST', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
INCA = LDAB*KA1
*
* Initialize X to the unit matrix, if needed
*
IF( WANTX )
$ CALL CLASET( 'Full', N, N, CZERO, CONE, X, LDX )
*
* Set M to the splitting point m. It must be the same value as is
* used in CPBSTF. The chosen value allows the arrays WORK and RWORK
* to be of dimension (N).
*
M = ( N+KB ) / 2
*
* The routine works in two phases, corresponding to the two halves
* of the split Cholesky factorization of B as S**H*S where
*
* S = ( U )
* ( M L )
*
* with U upper triangular of order m, and L lower triangular of
* order n-m. S has the same bandwidth as B.
*
* S is treated as a product of elementary matrices:
*
* S = S(m)*S(m-1)*...*S(2)*S(1)*S(m+1)*S(m+2)*...*S(n-1)*S(n)
*
* where S(i) is determined by the i-th row of S.
*
* In phase 1, the index i takes the values n, n-1, ... , m+1;
* in phase 2, it takes the values 1, 2, ... , m.
*
* For each value of i, the current matrix A is updated by forming
* inv(S(i))**H*A*inv(S(i)). This creates a triangular bulge outside
* the band of A. The bulge is then pushed down toward the bottom of
* A in phase 1, and up toward the top of A in phase 2, by applying
* plane rotations.
*
* There are kb*(kb+1)/2 elements in the bulge, but at most 2*kb-1
* of them are linearly independent, so annihilating a bulge requires
* only 2*kb-1 plane rotations. The rotations are divided into a 1st
* set of kb-1 rotations, and a 2nd set of kb rotations.
*
* Wherever possible, rotations are generated and applied in vector
* operations of length NR between the indices J1 and J2 (sometimes
* replaced by modified values NRT, J1T or J2T).
*
* The real cosines and complex sines of the rotations are stored in
* the arrays RWORK and WORK, those of the 1st set in elements
* 2:m-kb-1, and those of the 2nd set in elements m-kb+1:n.
*
* The bulges are not formed explicitly; nonzero elements outside the
* band are created only when they are required for generating new
* rotations; they are stored in the array WORK, in positions where
* they are later overwritten by the sines of the rotations which
* annihilate them.
*
* **************************** Phase 1 *****************************
*
* The logical structure of this phase is:
*
* UPDATE = .TRUE.
* DO I = N, M + 1, -1
* use S(i) to update A and create a new bulge
* apply rotations to push all bulges KA positions downward
* END DO
* UPDATE = .FALSE.
* DO I = M + KA + 1, N - 1
* apply rotations to push all bulges KA positions downward
* END DO
*
* To avoid duplicating code, the two loops are merged.
*
UPDATE = .TRUE.
I = N + 1
10 CONTINUE
IF( UPDATE ) THEN
I = I - 1
KBT = MIN( KB, I-1 )
I0 = I - 1
I1 = MIN( N, I+KA )
I2 = I - KBT + KA1
IF( I.LT.M+1 ) THEN
UPDATE = .FALSE.
I = I + 1
I0 = M
IF( KA.EQ.0 )
$ GO TO 480
GO TO 10
END IF
ELSE
I = I + KA
IF( I.GT.N-1 )
$ GO TO 480
END IF
*
IF( UPPER ) THEN
*
* Transform A, working with the upper triangle
*
IF( UPDATE ) THEN
*
* Form inv(S(i))**H * A * inv(S(i))
*
BII = REAL( BB( KB1, I ) )
AB( KA1, I ) = ( REAL( AB( KA1, I ) ) / BII ) / BII
DO 20 J = I + 1, I1
AB( I-J+KA1, J ) = AB( I-J+KA1, J ) / BII
20 CONTINUE
DO 30 J = MAX( 1, I-KA ), I - 1
AB( J-I+KA1, I ) = AB( J-I+KA1, I ) / BII
30 CONTINUE
DO 60 K = I - KBT, I - 1
DO 40 J = I - KBT, K
AB( J-K+KA1, K ) = AB( J-K+KA1, K ) -
$ BB( J-I+KB1, I )*
$ CONJG( AB( K-I+KA1, I ) ) -
$ CONJG( BB( K-I+KB1, I ) )*
$ AB( J-I+KA1, I ) +
$ REAL( AB( KA1, I ) )*
$ BB( J-I+KB1, I )*
$ CONJG( BB( K-I+KB1, I ) )
40 CONTINUE
DO 50 J = MAX( 1, I-KA ), I - KBT - 1
AB( J-K+KA1, K ) = AB( J-K+KA1, K ) -
$ CONJG( BB( K-I+KB1, I ) )*
$ AB( J-I+KA1, I )
50 CONTINUE
60 CONTINUE
DO 80 J = I, I1
DO 70 K = MAX( J-KA, I-KBT ), I - 1
AB( K-J+KA1, J ) = AB( K-J+KA1, J ) -
$ BB( K-I+KB1, I )*AB( I-J+KA1, J )
70 CONTINUE
80 CONTINUE
*
IF( WANTX ) THEN
*
* post-multiply X by inv(S(i))
*
CALL CSSCAL( N-M, ONE / BII, X( M+1, I ), 1 )
IF( KBT.GT.0 )
$ CALL CGERC( N-M, KBT, -CONE, X( M+1, I ), 1,
$ BB( KB1-KBT, I ), 1, X( M+1, I-KBT ),
$ LDX )
END IF
*
* store a(i,i1) in RA1 for use in next loop over K
*
RA1 = AB( I-I1+KA1, I1 )
END IF
*
* Generate and apply vectors of rotations to chase all the
* existing bulges KA positions down toward the bottom of the
* band
*
DO 130 K = 1, KB - 1
IF( UPDATE ) THEN
*
* Determine the rotations which would annihilate the bulge
* which has in theory just been created
*
IF( I-K+KA.LT.N .AND. I-K.GT.1 ) THEN
*
* generate rotation to annihilate a(i,i-k+ka+1)
*
CALL CLARTG( AB( K+1, I-K+KA ), RA1,
$ RWORK( I-K+KA-M ), WORK( I-K+KA-M ), RA )
*
* create nonzero element a(i-k,i-k+ka+1) outside the
* band and store it in WORK(i-k)
*
T = -BB( KB1-K, I )*RA1
WORK( I-K ) = RWORK( I-K+KA-M )*T -
$ CONJG( WORK( I-K+KA-M ) )*
$ AB( 1, I-K+KA )
AB( 1, I-K+KA ) = WORK( I-K+KA-M )*T +
$ RWORK( I-K+KA-M )*AB( 1, I-K+KA )
RA1 = RA
END IF
END IF
J2 = I - K - 1 + MAX( 1, K-I0+2 )*KA1
NR = ( N-J2+KA ) / KA1
J1 = J2 + ( NR-1 )*KA1
IF( UPDATE ) THEN
J2T = MAX( J2, I+2*KA-K+1 )
ELSE
J2T = J2
END IF
NRT = ( N-J2T+KA ) / KA1
DO 90 J = J2T, J1, KA1
*
* create nonzero element a(j-ka,j+1) outside the band
* and store it in WORK(j-m)
*
WORK( J-M ) = WORK( J-M )*AB( 1, J+1 )
AB( 1, J+1 ) = RWORK( J-M )*AB( 1, J+1 )
90 CONTINUE
*
* generate rotations in 1st set to annihilate elements which
* have been created outside the band
*
IF( NRT.GT.0 )
$ CALL CLARGV( NRT, AB( 1, J2T ), INCA, WORK( J2T-M ), KA1,
$ RWORK( J2T-M ), KA1 )
IF( NR.GT.0 ) THEN
*
* apply rotations in 1st set from the right
*
DO 100 L = 1, KA - 1
CALL CLARTV( NR, AB( KA1-L, J2 ), INCA,
$ AB( KA-L, J2+1 ), INCA, RWORK( J2-M ),
$ WORK( J2-M ), KA1 )
100 CONTINUE
*
* apply rotations in 1st set from both sides to diagonal
* blocks
*
CALL CLAR2V( NR, AB( KA1, J2 ), AB( KA1, J2+1 ),
$ AB( KA, J2+1 ), INCA, RWORK( J2-M ),
$ WORK( J2-M ), KA1 )
*
CALL CLACGV( NR, WORK( J2-M ), KA1 )
END IF
*
* start applying rotations in 1st set from the left
*
DO 110 L = KA - 1, KB - K + 1, -1
NRT = ( N-J2+L ) / KA1
IF( NRT.GT.0 )
$ CALL CLARTV( NRT, AB( L, J2+KA1-L ), INCA,
$ AB( L+1, J2+KA1-L ), INCA, RWORK( J2-M ),
$ WORK( J2-M ), KA1 )
110 CONTINUE
*
IF( WANTX ) THEN
*
* post-multiply X by product of rotations in 1st set
*
DO 120 J = J2, J1, KA1
CALL CROT( N-M, X( M+1, J ), 1, X( M+1, J+1 ), 1,
$ RWORK( J-M ), CONJG( WORK( J-M ) ) )
120 CONTINUE
END IF
130 CONTINUE
*
IF( UPDATE ) THEN
IF( I2.LE.N .AND. KBT.GT.0 ) THEN
*
* create nonzero element a(i-kbt,i-kbt+ka+1) outside the
* band and store it in WORK(i-kbt)
*
WORK( I-KBT ) = -BB( KB1-KBT, I )*RA1
END IF
END IF
*
DO 170 K = KB, 1, -1
IF( UPDATE ) THEN
J2 = I - K - 1 + MAX( 2, K-I0+1 )*KA1
ELSE
J2 = I - K - 1 + MAX( 1, K-I0+1 )*KA1
END IF
*
* finish applying rotations in 2nd set from the left
*
DO 140 L = KB - K, 1, -1
NRT = ( N-J2+KA+L ) / KA1
IF( NRT.GT.0 )
$ CALL CLARTV( NRT, AB( L, J2-L+1 ), INCA,
$ AB( L+1, J2-L+1 ), INCA, RWORK( J2-KA ),
$ WORK( J2-KA ), KA1 )
140 CONTINUE
NR = ( N-J2+KA ) / KA1
J1 = J2 + ( NR-1 )*KA1
DO 150 J = J1, J2, -KA1
WORK( J ) = WORK( J-KA )
RWORK( J ) = RWORK( J-KA )
150 CONTINUE
DO 160 J = J2, J1, KA1
*
* create nonzero element a(j-ka,j+1) outside the band
* and store it in WORK(j)
*
WORK( J ) = WORK( J )*AB( 1, J+1 )
AB( 1, J+1 ) = RWORK( J )*AB( 1, J+1 )
160 CONTINUE
IF( UPDATE ) THEN
IF( I-K.LT.N-KA .AND. K.LE.KBT )
$ WORK( I-K+KA ) = WORK( I-K )
END IF
170 CONTINUE
*
DO 210 K = KB, 1, -1
J2 = I - K - 1 + MAX( 1, K-I0+1 )*KA1
NR = ( N-J2+KA ) / KA1
J1 = J2 + ( NR-1 )*KA1
IF( NR.GT.0 ) THEN
*
* generate rotations in 2nd set to annihilate elements
* which have been created outside the band
*
CALL CLARGV( NR, AB( 1, J2 ), INCA, WORK( J2 ), KA1,
$ RWORK( J2 ), KA1 )
*
* apply rotations in 2nd set from the right
*
DO 180 L = 1, KA - 1
CALL CLARTV( NR, AB( KA1-L, J2 ), INCA,
$ AB( KA-L, J2+1 ), INCA, RWORK( J2 ),
$ WORK( J2 ), KA1 )
180 CONTINUE
*
* apply rotations in 2nd set from both sides to diagonal
* blocks
*
CALL CLAR2V( NR, AB( KA1, J2 ), AB( KA1, J2+1 ),
$ AB( KA, J2+1 ), INCA, RWORK( J2 ),
$ WORK( J2 ), KA1 )
*
CALL CLACGV( NR, WORK( J2 ), KA1 )
END IF
*
* start applying rotations in 2nd set from the left
*
DO 190 L = KA - 1, KB - K + 1, -1
NRT = ( N-J2+L ) / KA1
IF( NRT.GT.0 )
$ CALL CLARTV( NRT, AB( L, J2+KA1-L ), INCA,
$ AB( L+1, J2+KA1-L ), INCA, RWORK( J2 ),
$ WORK( J2 ), KA1 )
190 CONTINUE
*
IF( WANTX ) THEN
*
* post-multiply X by product of rotations in 2nd set
*
DO 200 J = J2, J1, KA1
CALL CROT( N-M, X( M+1, J ), 1, X( M+1, J+1 ), 1,
$ RWORK( J ), CONJG( WORK( J ) ) )
200 CONTINUE
END IF
210 CONTINUE
*
DO 230 K = 1, KB - 1
J2 = I - K - 1 + MAX( 1, K-I0+2 )*KA1
*
* finish applying rotations in 1st set from the left
*
DO 220 L = KB - K, 1, -1
NRT = ( N-J2+L ) / KA1
IF( NRT.GT.0 )
$ CALL CLARTV( NRT, AB( L, J2+KA1-L ), INCA,
$ AB( L+1, J2+KA1-L ), INCA, RWORK( J2-M ),
$ WORK( J2-M ), KA1 )
220 CONTINUE
230 CONTINUE
*
IF( KB.GT.1 ) THEN
DO 240 J = N - 1, J2 + KA, -1
RWORK( J-M ) = RWORK( J-KA-M )
WORK( J-M ) = WORK( J-KA-M )
240 CONTINUE
END IF
*
ELSE
*
* Transform A, working with the lower triangle
*
IF( UPDATE ) THEN
*
* Form inv(S(i))**H * A * inv(S(i))
*
BII = REAL( BB( 1, I ) )
AB( 1, I ) = ( REAL( AB( 1, I ) ) / BII ) / BII
DO 250 J = I + 1, I1
AB( J-I+1, I ) = AB( J-I+1, I ) / BII
250 CONTINUE
DO 260 J = MAX( 1, I-KA ), I - 1
AB( I-J+1, J ) = AB( I-J+1, J ) / BII
260 CONTINUE
DO 290 K = I - KBT, I - 1
DO 270 J = I - KBT, K
AB( K-J+1, J ) = AB( K-J+1, J ) -
$ BB( I-J+1, J )*CONJG( AB( I-K+1,
$ K ) ) - CONJG( BB( I-K+1, K ) )*
$ AB( I-J+1, J ) + REAL( AB( 1, I ) )*
$ BB( I-J+1, J )*CONJG( BB( I-K+1,
$ K ) )
270 CONTINUE
DO 280 J = MAX( 1, I-KA ), I - KBT - 1
AB( K-J+1, J ) = AB( K-J+1, J ) -
$ CONJG( BB( I-K+1, K ) )*
$ AB( I-J+1, J )
280 CONTINUE
290 CONTINUE
DO 310 J = I, I1
DO 300 K = MAX( J-KA, I-KBT ), I - 1
AB( J-K+1, K ) = AB( J-K+1, K ) -
$ BB( I-K+1, K )*AB( J-I+1, I )
300 CONTINUE
310 CONTINUE
*
IF( WANTX ) THEN
*
* post-multiply X by inv(S(i))
*
CALL CSSCAL( N-M, ONE / BII, X( M+1, I ), 1 )
IF( KBT.GT.0 )
$ CALL CGERU( N-M, KBT, -CONE, X( M+1, I ), 1,
$ BB( KBT+1, I-KBT ), LDBB-1,
$ X( M+1, I-KBT ), LDX )
END IF
*
* store a(i1,i) in RA1 for use in next loop over K
*
RA1 = AB( I1-I+1, I )
END IF
*
* Generate and apply vectors of rotations to chase all the
* existing bulges KA positions down toward the bottom of the
* band
*
DO 360 K = 1, KB - 1
IF( UPDATE ) THEN
*
* Determine the rotations which would annihilate the bulge
* which has in theory just been created
*
IF( I-K+KA.LT.N .AND. I-K.GT.1 ) THEN
*
* generate rotation to annihilate a(i-k+ka+1,i)
*
CALL CLARTG( AB( KA1-K, I ), RA1, RWORK( I-K+KA-M ),
$ WORK( I-K+KA-M ), RA )
*
* create nonzero element a(i-k+ka+1,i-k) outside the
* band and store it in WORK(i-k)
*
T = -BB( K+1, I-K )*RA1
WORK( I-K ) = RWORK( I-K+KA-M )*T -
$ CONJG( WORK( I-K+KA-M ) )*AB( KA1, I-K )
AB( KA1, I-K ) = WORK( I-K+KA-M )*T +
$ RWORK( I-K+KA-M )*AB( KA1, I-K )
RA1 = RA
END IF
END IF
J2 = I - K - 1 + MAX( 1, K-I0+2 )*KA1
NR = ( N-J2+KA ) / KA1
J1 = J2 + ( NR-1 )*KA1
IF( UPDATE ) THEN
J2T = MAX( J2, I+2*KA-K+1 )
ELSE
J2T = J2
END IF
NRT = ( N-J2T+KA ) / KA1
DO 320 J = J2T, J1, KA1
*
* create nonzero element a(j+1,j-ka) outside the band
* and store it in WORK(j-m)
*
WORK( J-M ) = WORK( J-M )*AB( KA1, J-KA+1 )
AB( KA1, J-KA+1 ) = RWORK( J-M )*AB( KA1, J-KA+1 )
320 CONTINUE
*
* generate rotations in 1st set to annihilate elements which
* have been created outside the band
*
IF( NRT.GT.0 )
$ CALL CLARGV( NRT, AB( KA1, J2T-KA ), INCA, WORK( J2T-M ),
$ KA1, RWORK( J2T-M ), KA1 )
IF( NR.GT.0 ) THEN
*
* apply rotations in 1st set from the left
*
DO 330 L = 1, KA - 1
CALL CLARTV( NR, AB( L+1, J2-L ), INCA,
$ AB( L+2, J2-L ), INCA, RWORK( J2-M ),
$ WORK( J2-M ), KA1 )
330 CONTINUE
*
* apply rotations in 1st set from both sides to diagonal
* blocks
*
CALL CLAR2V( NR, AB( 1, J2 ), AB( 1, J2+1 ), AB( 2, J2 ),
$ INCA, RWORK( J2-M ), WORK( J2-M ), KA1 )
*
CALL CLACGV( NR, WORK( J2-M ), KA1 )
END IF
*
* start applying rotations in 1st set from the right
*
DO 340 L = KA - 1, KB - K + 1, -1
NRT = ( N-J2+L ) / KA1
IF( NRT.GT.0 )
$ CALL CLARTV( NRT, AB( KA1-L+1, J2 ), INCA,
$ AB( KA1-L, J2+1 ), INCA, RWORK( J2-M ),
$ WORK( J2-M ), KA1 )
340 CONTINUE
*
IF( WANTX ) THEN
*
* post-multiply X by product of rotations in 1st set
*
DO 350 J = J2, J1, KA1
CALL CROT( N-M, X( M+1, J ), 1, X( M+1, J+1 ), 1,
$ RWORK( J-M ), WORK( J-M ) )
350 CONTINUE
END IF
360 CONTINUE
*
IF( UPDATE ) THEN
IF( I2.LE.N .AND. KBT.GT.0 ) THEN
*
* create nonzero element a(i-kbt+ka+1,i-kbt) outside the
* band and store it in WORK(i-kbt)
*
WORK( I-KBT ) = -BB( KBT+1, I-KBT )*RA1
END IF
END IF
*
DO 400 K = KB, 1, -1
IF( UPDATE ) THEN
J2 = I - K - 1 + MAX( 2, K-I0+1 )*KA1
ELSE
J2 = I - K - 1 + MAX( 1, K-I0+1 )*KA1
END IF
*
* finish applying rotations in 2nd set from the right
*
DO 370 L = KB - K, 1, -1
NRT = ( N-J2+KA+L ) / KA1
IF( NRT.GT.0 )
$ CALL CLARTV( NRT, AB( KA1-L+1, J2-KA ), INCA,
$ AB( KA1-L, J2-KA+1 ), INCA,
$ RWORK( J2-KA ), WORK( J2-KA ), KA1 )
370 CONTINUE
NR = ( N-J2+KA ) / KA1
J1 = J2 + ( NR-1 )*KA1
DO 380 J = J1, J2, -KA1
WORK( J ) = WORK( J-KA )
RWORK( J ) = RWORK( J-KA )
380 CONTINUE
DO 390 J = J2, J1, KA1
*
* create nonzero element a(j+1,j-ka) outside the band
* and store it in WORK(j)
*
WORK( J ) = WORK( J )*AB( KA1, J-KA+1 )
AB( KA1, J-KA+1 ) = RWORK( J )*AB( KA1, J-KA+1 )
390 CONTINUE
IF( UPDATE ) THEN
IF( I-K.LT.N-KA .AND. K.LE.KBT )
$ WORK( I-K+KA ) = WORK( I-K )
END IF
400 CONTINUE
*
DO 440 K = KB, 1, -1
J2 = I - K - 1 + MAX( 1, K-I0+1 )*KA1
NR = ( N-J2+KA ) / KA1
J1 = J2 + ( NR-1 )*KA1
IF( NR.GT.0 ) THEN
*
* generate rotations in 2nd set to annihilate elements
* which have been created outside the band
*
CALL CLARGV( NR, AB( KA1, J2-KA ), INCA, WORK( J2 ), KA1,
$ RWORK( J2 ), KA1 )
*
* apply rotations in 2nd set from the left
*
DO 410 L = 1, KA - 1
CALL CLARTV( NR, AB( L+1, J2-L ), INCA,
$ AB( L+2, J2-L ), INCA, RWORK( J2 ),
$ WORK( J2 ), KA1 )
410 CONTINUE
*
* apply rotations in 2nd set from both sides to diagonal
* blocks
*
CALL CLAR2V( NR, AB( 1, J2 ), AB( 1, J2+1 ), AB( 2, J2 ),
$ INCA, RWORK( J2 ), WORK( J2 ), KA1 )
*
CALL CLACGV( NR, WORK( J2 ), KA1 )
END IF
*
* start applying rotations in 2nd set from the right
*
DO 420 L = KA - 1, KB - K + 1, -1
NRT = ( N-J2+L ) / KA1
IF( NRT.GT.0 )
$ CALL CLARTV( NRT, AB( KA1-L+1, J2 ), INCA,
$ AB( KA1-L, J2+1 ), INCA, RWORK( J2 ),
$ WORK( J2 ), KA1 )
420 CONTINUE
*
IF( WANTX ) THEN
*
* post-multiply X by product of rotations in 2nd set
*
DO 430 J = J2, J1, KA1
CALL CROT( N-M, X( M+1, J ), 1, X( M+1, J+1 ), 1,
$ RWORK( J ), WORK( J ) )
430 CONTINUE
END IF
440 CONTINUE
*
DO 460 K = 1, KB - 1
J2 = I - K - 1 + MAX( 1, K-I0+2 )*KA1
*
* finish applying rotations in 1st set from the right
*
DO 450 L = KB - K, 1, -1
NRT = ( N-J2+L ) / KA1
IF( NRT.GT.0 )
$ CALL CLARTV( NRT, AB( KA1-L+1, J2 ), INCA,
$ AB( KA1-L, J2+1 ), INCA, RWORK( J2-M ),
$ WORK( J2-M ), KA1 )
450 CONTINUE
460 CONTINUE
*
IF( KB.GT.1 ) THEN
DO 470 J = N - 1, J2 + KA, -1
RWORK( J-M ) = RWORK( J-KA-M )
WORK( J-M ) = WORK( J-KA-M )
470 CONTINUE
END IF
*
END IF
*
GO TO 10
*
480 CONTINUE
*
* **************************** Phase 2 *****************************
*
* The logical structure of this phase is:
*
* UPDATE = .TRUE.
* DO I = 1, M
* use S(i) to update A and create a new bulge
* apply rotations to push all bulges KA positions upward
* END DO
* UPDATE = .FALSE.
* DO I = M - KA - 1, 2, -1
* apply rotations to push all bulges KA positions upward
* END DO
*
* To avoid duplicating code, the two loops are merged.
*
UPDATE = .TRUE.
I = 0
490 CONTINUE
IF( UPDATE ) THEN
I = I + 1
KBT = MIN( KB, M-I )
I0 = I + 1
I1 = MAX( 1, I-KA )
I2 = I + KBT - KA1
IF( I.GT.M ) THEN
UPDATE = .FALSE.
I = I - 1
I0 = M + 1
IF( KA.EQ.0 )
$ RETURN
GO TO 490
END IF
ELSE
I = I - KA
IF( I.LT.2 )
$ RETURN
END IF
*
IF( I.LT.M-KBT ) THEN
NX = M
ELSE
NX = N
END IF
*
IF( UPPER ) THEN
*
* Transform A, working with the upper triangle
*
IF( UPDATE ) THEN
*
* Form inv(S(i))**H * A * inv(S(i))
*
BII = REAL( BB( KB1, I ) )
AB( KA1, I ) = ( REAL( AB( KA1, I ) ) / BII ) / BII
DO 500 J = I1, I - 1
AB( J-I+KA1, I ) = AB( J-I+KA1, I ) / BII
500 CONTINUE
DO 510 J = I + 1, MIN( N, I+KA )
AB( I-J+KA1, J ) = AB( I-J+KA1, J ) / BII
510 CONTINUE
DO 540 K = I + 1, I + KBT
DO 520 J = K, I + KBT
AB( K-J+KA1, J ) = AB( K-J+KA1, J ) -
$ BB( I-J+KB1, J )*
$ CONJG( AB( I-K+KA1, K ) ) -
$ CONJG( BB( I-K+KB1, K ) )*
$ AB( I-J+KA1, J ) +
$ REAL( AB( KA1, I ) )*
$ BB( I-J+KB1, J )*
$ CONJG( BB( I-K+KB1, K ) )
520 CONTINUE
DO 530 J = I + KBT + 1, MIN( N, I+KA )
AB( K-J+KA1, J ) = AB( K-J+KA1, J ) -
$ CONJG( BB( I-K+KB1, K ) )*
$ AB( I-J+KA1, J )
530 CONTINUE
540 CONTINUE
DO 560 J = I1, I
DO 550 K = I + 1, MIN( J+KA, I+KBT )
AB( J-K+KA1, K ) = AB( J-K+KA1, K ) -
$ BB( I-K+KB1, K )*AB( J-I+KA1, I )
550 CONTINUE
560 CONTINUE
*
IF( WANTX ) THEN
*
* post-multiply X by inv(S(i))
*
CALL CSSCAL( NX, ONE / BII, X( 1, I ), 1 )
IF( KBT.GT.0 )
$ CALL CGERU( NX, KBT, -CONE, X( 1, I ), 1,
$ BB( KB, I+1 ), LDBB-1, X( 1, I+1 ), LDX )
END IF
*
* store a(i1,i) in RA1 for use in next loop over K
*
RA1 = AB( I1-I+KA1, I )
END IF
*
* Generate and apply vectors of rotations to chase all the
* existing bulges KA positions up toward the top of the band
*
DO 610 K = 1, KB - 1
IF( UPDATE ) THEN
*
* Determine the rotations which would annihilate the bulge
* which has in theory just been created
*
IF( I+K-KA1.GT.0 .AND. I+K.LT.M ) THEN
*
* generate rotation to annihilate a(i+k-ka-1,i)
*
CALL CLARTG( AB( K+1, I ), RA1, RWORK( I+K-KA ),
$ WORK( I+K-KA ), RA )
*
* create nonzero element a(i+k-ka-1,i+k) outside the
* band and store it in WORK(m-kb+i+k)
*
T = -BB( KB1-K, I+K )*RA1
WORK( M-KB+I+K ) = RWORK( I+K-KA )*T -
$ CONJG( WORK( I+K-KA ) )*
$ AB( 1, I+K )
AB( 1, I+K ) = WORK( I+K-KA )*T +