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*> \brief \b DSBTRD
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSBTRD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsbtrd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsbtrd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsbtrd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSBTRD( VECT, UPLO, N, KD, AB, LDAB, D, E, Q, LDQ,
* WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO, VECT
* INTEGER INFO, KD, LDAB, LDQ, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AB( LDAB, * ), D( * ), E( * ), Q( LDQ, * ),
* $ WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSBTRD reduces a real symmetric band matrix A to symmetric
*> tridiagonal form T by an orthogonal similarity transformation:
*> Q**T * A * Q = T.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] VECT
*> \verbatim
*> VECT is CHARACTER*1
*> = 'N': do not form Q;
*> = 'V': form Q;
*> = 'U': update a matrix X, by forming X*Q.
*> \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 matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*> KD is INTEGER
*> The number of superdiagonals of the matrix A if UPLO = 'U',
*> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
*> On entry, the upper or lower triangle of the symmetric 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.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= KD+1.
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The diagonal elements of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*> E is 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'.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*> Q is DOUBLE PRECISION 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 orthogonal matrix Q;
*> if VECT = 'U', Q contains the product X*Q;
*> if VECT = 'N', the array Q is not referenced.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q.
*> LDQ >= 1, and LDQ >= N if VECT = 'V' or 'U'.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION 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.
*
*> \ingroup doubleOTHERcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Modified by Linda Kaufman, Bell Labs.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DSBTRD( VECT, UPLO, N, KD, AB, LDAB, D, E, Q, LDQ,
$ WORK, 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 ..
CHARACTER UPLO, VECT
INTEGER INFO, KD, LDAB, LDQ, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION AB( LDAB, * ), D( * ), E( * ), Q( LDQ, * ),
$ WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+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 TEMP
* ..
* .. External Subroutines ..
EXTERNAL DLAR2V, DLARGV, DLARTG, DLARTV, DLASET, DROT,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC 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( 'DSBTRD', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Initialize Q to the unit matrix, if needed
*
IF( INITQ )
$ CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
*
* Wherever possible, plane rotations are generated and applied in
* vector operations of length NR over the index set J1:J2:KD1.
*
* The cosines and 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 tridiagonal form, working with upper triangle
*
NR = 0
J1 = KDN + 2
J2 = 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 DLARGV( 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
* DLARTV or DROT is used
*
IF( NR.GE.2*KD-1 ) THEN
DO 10 L = 1, KD - 1
CALL DLARTV( 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 DROT( 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 DLARTG( 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 DROT( 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 DLAR2V( 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
IF( 2*KD-1.LT.NR ) THEN
*
* Dependent on the the number of diagonals either
* DLARTV or DROT 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 DLARTV( 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 DROT( 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 DROT( 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 DROT( NQ, Q( IQB, J-1 ), 1, Q( IQB, J ),
$ 1, D( J ), WORK( J ) )
50 CONTINUE
ELSE
*
DO 60 J = J1, J2, KD1
CALL DROT( N, Q( 1, J-1 ), 1, Q( 1, J ), 1,
$ D( J ), 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
*
* copy off-diagonal elements to E
*
DO 100 I = 1, N - 1
E( I ) = AB( KD, I+1 )
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 tridiagonal form, working with lower triangle
*
NR = 0
J1 = KDN + 2
J2 = 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 DLARGV( 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
* DLARTV or DROT is used
*
IF( NR.GT.2*KD-1 ) THEN
DO 130 L = 1, KD - 1
CALL DLARTV( 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 DROT( 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 DLARTG( 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 DROT( 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 DLAR2V( 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
* DLARTV or DROT is used
*
IF( NR.GT.0 ) THEN
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 DLARTV( 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 DROT( 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 DROT( 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 DROT( NQ, Q( IQB, J-1 ), 1, Q( IQB, J ),
$ 1, D( J ), WORK( J ) )
170 CONTINUE
ELSE
*
DO 180 J = J1, J2, KD1
CALL DROT( 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
*
* copy off-diagonal elements to E
*
DO 220 I = 1, N - 1
E( I ) = AB( 2, I )
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 DSBTRD
*
END