SUBROUTINE ZHBTRD( VECT, UPLO, N, KD, AB, LDAB, D, E, Q, LDQ,
$ WORK, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER UPLO, VECT
INTEGER INFO, KD, LDAB, LDQ, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), E( * )
COMPLEX*16 AB( LDAB, * ), Q( LDQ, * ), WORK( * )
* ..
*
* Purpose
* =======
*
* ZHBTRD reduces a complex Hermitian band matrix A to real symmetric
* tridiagonal form T by a unitary similarity transformation:
* Q**H * A * Q = T.
*
* Arguments
* =========
*
* VECT (input) CHARACTER*1
* = 'N': do not form Q;
* = 'V': form Q;
* = 'U': update a matrix X, by forming X*Q.
*
* UPLO (input) CHARACTER*1
* = 'U': Upper triangle of A is stored;
* = 'L': Lower triangle of A is stored.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* KD (input) INTEGER
* The number of superdiagonals of the matrix A if UPLO = 'U',
* or the number of subdiagonals if UPLO = 'L'. KD >= 0.
*
* AB (input/output) COMPLEX*16 array, dimension (LDAB,N)
* On entry, the upper or lower triangle of the Hermitian band
* matrix A, stored in the first KD+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(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
* if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
* On exit, the diagonal elements of AB are overwritten by the
* diagonal elements of the tridiagonal matrix T; if KD > 0, the
* elements on the first superdiagonal (if UPLO = 'U') or the
* first subdiagonal (if UPLO = 'L') are overwritten by the
* off-diagonal elements of T; the rest of AB is overwritten by
* values generated during the reduction.
*
* LDAB (input) INTEGER
* The leading dimension of the array AB. LDAB >= KD+1.
*
* D (output) DOUBLE PRECISION array, dimension (N)
* The diagonal elements of the tridiagonal matrix T.
*
* E (output) DOUBLE PRECISION array, dimension (N-1)
* The off-diagonal elements of the tridiagonal matrix T:
* E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'.
*
* Q (input/output) COMPLEX*16 array, dimension (LDQ,N)
* On entry, if VECT = 'U', then Q must contain an N-by-N
* matrix X; if VECT = 'N' or 'V', then Q need not be set.
*
* On exit:
* if VECT = 'V', Q contains the N-by-N unitary matrix Q;
* if VECT = 'U', Q contains the product X*Q;
* if VECT = 'N', the array Q is not referenced.
*
* LDQ (input) INTEGER
* The leading dimension of the array Q.
* LDQ >= 1, and LDQ >= N if VECT = 'V' or 'U'.
*
* WORK (workspace) COMPLEX*16 array, dimension (N)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* Further Details
* ===============
*
* Modified by Linda Kaufman, Bell Labs.
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
COMPLEX*16 CZERO, CONE
PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ),
$ CONE = ( 1.0D+0, 0.0D+0 ) )
* ..
* .. Local Scalars ..
LOGICAL INITQ, UPPER, WANTQ
INTEGER I, I2, IBL, INCA, INCX, IQAEND, IQB, IQEND, J,
$ J1, J1END, J1INC, J2, JEND, JIN, JINC, K, KD1,
$ KDM1, KDN, L, LAST, LEND, NQ, NR, NRT
DOUBLE PRECISION ABST
COMPLEX*16 T, TEMP
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, ZLACGV, ZLAR2V, ZLARGV, ZLARTG, ZLARTV,
$ ZLASET, ZROT, ZSCAL
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, DCONJG, MAX, MIN
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INITQ = LSAME( VECT, 'V' )
WANTQ = INITQ .OR. LSAME( VECT, 'U' )
UPPER = LSAME( UPLO, 'U' )
KD1 = KD + 1
KDM1 = KD - 1
INCX = LDAB - 1
IQEND = 1
*
INFO = 0
IF( .NOT.WANTQ .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( KD.LT.0 ) THEN
INFO = -4
ELSE IF( LDAB.LT.KD1 ) THEN
INFO = -6
ELSE IF( LDQ.LT.MAX( 1, N ) .AND. WANTQ ) THEN
INFO = -10
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZHBTRD', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Initialize Q to the unit matrix, if needed
*
IF( INITQ )
$ CALL ZLASET( 'Full', N, N, CZERO, CONE, Q, LDQ )
*
* Wherever possible, plane rotations are generated and applied in
* vector operations of length NR over the index set J1:J2:KD1.
*
* The real cosines and complex sines of the plane rotations are
* stored in the arrays D and WORK.
*
INCA = KD1*LDAB
KDN = MIN( N-1, KD )
IF( UPPER ) THEN
*
IF( KD.GT.1 ) THEN
*
* Reduce to complex Hermitian tridiagonal form, working with
* the upper triangle
*
NR = 0
J1 = KDN + 2
J2 = 1
*
AB( KD1, 1 ) = DBLE( AB( KD1, 1 ) )
DO 90 I = 1, N - 2
*
* Reduce i-th row of matrix to tridiagonal form
*
DO 80 K = KDN + 1, 2, -1
J1 = J1 + KDN
J2 = J2 + KDN
*
IF( NR.GT.0 ) THEN
*
* generate plane rotations to annihilate nonzero
* elements which have been created outside the band
*
CALL ZLARGV( NR, AB( 1, J1-1 ), INCA, WORK( J1 ),
$ KD1, D( J1 ), KD1 )
*
* apply rotations from the right
*
*
* Dependent on the the number of diagonals either
* ZLARTV or ZROT is used
*
IF( NR.GE.2*KD-1 ) THEN
DO 10 L = 1, KD - 1
CALL ZLARTV( NR, AB( L+1, J1-1 ), INCA,
$ AB( L, J1 ), INCA, D( J1 ),
$ WORK( J1 ), KD1 )
10 CONTINUE
*
ELSE
JEND = J1 + ( NR-1 )*KD1
DO 20 JINC = J1, JEND, KD1
CALL ZROT( KDM1, AB( 2, JINC-1 ), 1,
$ AB( 1, JINC ), 1, D( JINC ),
$ WORK( JINC ) )
20 CONTINUE
END IF
END IF
*
*
IF( K.GT.2 ) THEN
IF( K.LE.N-I+1 ) THEN
*
* generate plane rotation to annihilate a(i,i+k-1)
* within the band
*
CALL ZLARTG( AB( KD-K+3, I+K-2 ),
$ AB( KD-K+2, I+K-1 ), D( I+K-1 ),
$ WORK( I+K-1 ), TEMP )
AB( KD-K+3, I+K-2 ) = TEMP
*
* apply rotation from the right
*
CALL ZROT( K-3, AB( KD-K+4, I+K-2 ), 1,
$ AB( KD-K+3, I+K-1 ), 1, D( I+K-1 ),
$ WORK( I+K-1 ) )
END IF
NR = NR + 1
J1 = J1 - KDN - 1
END IF
*
* apply plane rotations from both sides to diagonal
* blocks
*
IF( NR.GT.0 )
$ CALL ZLAR2V( NR, AB( KD1, J1-1 ), AB( KD1, J1 ),
$ AB( KD, J1 ), INCA, D( J1 ),
$ WORK( J1 ), KD1 )
*
* apply plane rotations from the left
*
IF( NR.GT.0 ) THEN
CALL ZLACGV( NR, WORK( J1 ), KD1 )
IF( 2*KD-1.LT.NR ) THEN
*
* Dependent on the the number of diagonals either
* ZLARTV or ZROT is used
*
DO 30 L = 1, KD - 1
IF( J2+L.GT.N ) THEN
NRT = NR - 1
ELSE
NRT = NR
END IF
IF( NRT.GT.0 )
$ CALL ZLARTV( NRT, AB( KD-L, J1+L ), INCA,
$ AB( KD-L+1, J1+L ), INCA,
$ D( J1 ), WORK( J1 ), KD1 )
30 CONTINUE
ELSE
J1END = J1 + KD1*( NR-2 )
IF( J1END.GE.J1 ) THEN
DO 40 JIN = J1, J1END, KD1
CALL ZROT( KD-1, AB( KD-1, JIN+1 ), INCX,
$ AB( KD, JIN+1 ), INCX,
$ D( JIN ), WORK( JIN ) )
40 CONTINUE
END IF
LEND = MIN( KDM1, N-J2 )
LAST = J1END + KD1
IF( LEND.GT.0 )
$ CALL ZROT( LEND, AB( KD-1, LAST+1 ), INCX,
$ AB( KD, LAST+1 ), INCX, D( LAST ),
$ WORK( LAST ) )
END IF
END IF
*
IF( WANTQ ) THEN
*
* accumulate product of plane rotations in Q
*
IF( INITQ ) THEN
*
* take advantage of the fact that Q was
* initially the Identity matrix
*
IQEND = MAX( IQEND, J2 )
I2 = MAX( 0, K-3 )
IQAEND = 1 + I*KD
IF( K.EQ.2 )
$ IQAEND = IQAEND + KD
IQAEND = MIN( IQAEND, IQEND )
DO 50 J = J1, J2, KD1
IBL = I - I2 / KDM1
I2 = I2 + 1
IQB = MAX( 1, J-IBL )
NQ = 1 + IQAEND - IQB
IQAEND = MIN( IQAEND+KD, IQEND )
CALL ZROT( NQ, Q( IQB, J-1 ), 1, Q( IQB, J ),
$ 1, D( J ), DCONJG( WORK( J ) ) )
50 CONTINUE
ELSE
*
DO 60 J = J1, J2, KD1
CALL ZROT( N, Q( 1, J-1 ), 1, Q( 1, J ), 1,
$ D( J ), DCONJG( WORK( J ) ) )
60 CONTINUE
END IF
*
END IF
*
IF( J2+KDN.GT.N ) THEN
*
* adjust J2 to keep within the bounds of the matrix
*
NR = NR - 1
J2 = J2 - KDN - 1
END IF
*
DO 70 J = J1, J2, KD1
*
* create nonzero element a(j-1,j+kd) outside the band
* and store it in WORK
*
WORK( J+KD ) = WORK( J )*AB( 1, J+KD )
AB( 1, J+KD ) = D( J )*AB( 1, J+KD )
70 CONTINUE
80 CONTINUE
90 CONTINUE
END IF
*
IF( KD.GT.0 ) THEN
*
* make off-diagonal elements real and copy them to E
*
DO 100 I = 1, N - 1
T = AB( KD, I+1 )
ABST = ABS( T )
AB( KD, I+1 ) = ABST
E( I ) = ABST
IF( ABST.NE.ZERO ) THEN
T = T / ABST
ELSE
T = CONE
END IF
IF( I.LT.N-1 )
$ AB( KD, I+2 ) = AB( KD, I+2 )*T
IF( WANTQ ) THEN
CALL ZSCAL( N, DCONJG( T ), Q( 1, I+1 ), 1 )
END IF
100 CONTINUE
ELSE
*
* set E to zero if original matrix was diagonal
*
DO 110 I = 1, N - 1
E( I ) = ZERO
110 CONTINUE
END IF
*
* copy diagonal elements to D
*
DO 120 I = 1, N
D( I ) = AB( KD1, I )
120 CONTINUE
*
ELSE
*
IF( KD.GT.1 ) THEN
*
* Reduce to complex Hermitian tridiagonal form, working with
* the lower triangle
*
NR = 0
J1 = KDN + 2
J2 = 1
*
AB( 1, 1 ) = DBLE( AB( 1, 1 ) )
DO 210 I = 1, N - 2
*
* Reduce i-th column of matrix to tridiagonal form
*
DO 200 K = KDN + 1, 2, -1
J1 = J1 + KDN
J2 = J2 + KDN
*
IF( NR.GT.0 ) THEN
*
* generate plane rotations to annihilate nonzero
* elements which have been created outside the band
*
CALL ZLARGV( NR, AB( KD1, J1-KD1 ), INCA,
$ WORK( J1 ), KD1, D( J1 ), KD1 )
*
* apply plane rotations from one side
*
*
* Dependent on the the number of diagonals either
* ZLARTV or ZROT is used
*
IF( NR.GT.2*KD-1 ) THEN
DO 130 L = 1, KD - 1
CALL ZLARTV( NR, AB( KD1-L, J1-KD1+L ), INCA,
$ AB( KD1-L+1, J1-KD1+L ), INCA,
$ D( J1 ), WORK( J1 ), KD1 )
130 CONTINUE
ELSE
JEND = J1 + KD1*( NR-1 )
DO 140 JINC = J1, JEND, KD1
CALL ZROT( KDM1, AB( KD, JINC-KD ), INCX,
$ AB( KD1, JINC-KD ), INCX,
$ D( JINC ), WORK( JINC ) )
140 CONTINUE
END IF
*
END IF
*
IF( K.GT.2 ) THEN
IF( K.LE.N-I+1 ) THEN
*
* generate plane rotation to annihilate a(i+k-1,i)
* within the band
*
CALL ZLARTG( AB( K-1, I ), AB( K, I ),
$ D( I+K-1 ), WORK( I+K-1 ), TEMP )
AB( K-1, I ) = TEMP
*
* apply rotation from the left
*
CALL ZROT( K-3, AB( K-2, I+1 ), LDAB-1,
$ AB( K-1, I+1 ), LDAB-1, D( I+K-1 ),
$ WORK( I+K-1 ) )
END IF
NR = NR + 1
J1 = J1 - KDN - 1
END IF
*
* apply plane rotations from both sides to diagonal
* blocks
*
IF( NR.GT.0 )
$ CALL ZLAR2V( NR, AB( 1, J1-1 ), AB( 1, J1 ),
$ AB( 2, J1-1 ), INCA, D( J1 ),
$ WORK( J1 ), KD1 )
*
* apply plane rotations from the right
*
*
* Dependent on the the number of diagonals either
* ZLARTV or ZROT is used
*
IF( NR.GT.0 ) THEN
CALL ZLACGV( NR, WORK( J1 ), KD1 )
IF( NR.GT.2*KD-1 ) THEN
DO 150 L = 1, KD - 1
IF( J2+L.GT.N ) THEN
NRT = NR - 1
ELSE
NRT = NR
END IF
IF( NRT.GT.0 )
$ CALL ZLARTV( NRT, AB( L+2, J1-1 ), INCA,
$ AB( L+1, J1 ), INCA, D( J1 ),
$ WORK( J1 ), KD1 )
150 CONTINUE
ELSE
J1END = J1 + KD1*( NR-2 )
IF( J1END.GE.J1 ) THEN
DO 160 J1INC = J1, J1END, KD1
CALL ZROT( KDM1, AB( 3, J1INC-1 ), 1,
$ AB( 2, J1INC ), 1, D( J1INC ),
$ WORK( J1INC ) )
160 CONTINUE
END IF
LEND = MIN( KDM1, N-J2 )
LAST = J1END + KD1
IF( LEND.GT.0 )
$ CALL ZROT( LEND, AB( 3, LAST-1 ), 1,
$ AB( 2, LAST ), 1, D( LAST ),
$ WORK( LAST ) )
END IF
END IF
*
*
*
IF( WANTQ ) THEN
*
* accumulate product of plane rotations in Q
*
IF( INITQ ) THEN
*
* take advantage of the fact that Q was
* initially the Identity matrix
*
IQEND = MAX( IQEND, J2 )
I2 = MAX( 0, K-3 )
IQAEND = 1 + I*KD
IF( K.EQ.2 )
$ IQAEND = IQAEND + KD
IQAEND = MIN( IQAEND, IQEND )
DO 170 J = J1, J2, KD1
IBL = I - I2 / KDM1
I2 = I2 + 1
IQB = MAX( 1, J-IBL )
NQ = 1 + IQAEND - IQB
IQAEND = MIN( IQAEND+KD, IQEND )
CALL ZROT( NQ, Q( IQB, J-1 ), 1, Q( IQB, J ),
$ 1, D( J ), WORK( J ) )
170 CONTINUE
ELSE
*
DO 180 J = J1, J2, KD1
CALL ZROT( N, Q( 1, J-1 ), 1, Q( 1, J ), 1,
$ D( J ), WORK( J ) )
180 CONTINUE
END IF
END IF
*
IF( J2+KDN.GT.N ) THEN
*
* adjust J2 to keep within the bounds of the matrix
*
NR = NR - 1
J2 = J2 - KDN - 1
END IF
*
DO 190 J = J1, J2, KD1
*
* create nonzero element a(j+kd,j-1) outside the
* band and store it in WORK
*
WORK( J+KD ) = WORK( J )*AB( KD1, J )
AB( KD1, J ) = D( J )*AB( KD1, J )
190 CONTINUE
200 CONTINUE
210 CONTINUE
END IF
*
IF( KD.GT.0 ) THEN
*
* make off-diagonal elements real and copy them to E
*
DO 220 I = 1, N - 1
T = AB( 2, I )
ABST = ABS( T )
AB( 2, I ) = ABST
E( I ) = ABST
IF( ABST.NE.ZERO ) THEN
T = T / ABST
ELSE
T = CONE
END IF
IF( I.LT.N-1 )
$ AB( 2, I+1 ) = AB( 2, I+1 )*T
IF( WANTQ ) THEN
CALL ZSCAL( N, T, Q( 1, I+1 ), 1 )
END IF
220 CONTINUE
ELSE
*
* set E to zero if original matrix was diagonal
*
DO 230 I = 1, N - 1
E( I ) = ZERO
230 CONTINUE
END IF
*
* copy diagonal elements to D
*
DO 240 I = 1, N
D( I ) = AB( 1, I )
240 CONTINUE
END IF
*
RETURN
*
* End of ZHBTRD
*
END