```      SUBROUTINE SSBTRD( 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 ..
REAL               AB( LDAB, * ), D( * ), E( * ), Q( LDQ, * ),
\$                   WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  SSBTRD reduces a real symmetric band matrix A to symmetric
*  tridiagonal form T by an orthogonal similarity transformation:
*  Q**T * 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) REAL 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.
*
*  LDAB    (input) INTEGER
*          The leading dimension of the array AB.  LDAB >= KD+1.
*
*  D       (output) REAL array, dimension (N)
*          The diagonal elements of the tridiagonal matrix T.
*
*  E       (output) REAL 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) REAL 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.
*
*  LDQ     (input) INTEGER
*          The leading dimension of the array Q.
*          LDQ >= 1, and LDQ >= N if VECT = 'V' or 'U'.
*
*  WORK    (workspace) REAL 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 ..
REAL               ZERO, ONE
PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+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
REAL               TEMP
*     ..
*     .. External Subroutines ..
EXTERNAL           SLAR2V, SLARGV, SLARTG, SLARTV, SLASET, SROT,
\$                   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( 'SSBTRD', -INFO )
RETURN
END IF
*
*     Quick return if possible
*
IF( N.EQ.0 )
\$   RETURN
*
*     Initialize Q to the unit matrix, if needed
*
IF( INITQ )
\$   CALL SLASET( '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 SLARGV( 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
*                    SLARTV or SROT is used
*
IF( NR.GE.2*KD-1 ) THEN
DO 10 L = 1, KD - 1
CALL SLARTV( 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 SROT( 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 SLARTG( 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 SROT( 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 SLAR2V( 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
*                    SLARTV or SROT 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 SLARTV( 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 SROT( 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 SROT( 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 SROT( NQ, Q( IQB, J-1 ), 1, Q( IQB, J ),
\$                                1, D( J ), WORK( J ) )
50                   CONTINUE
ELSE
*
DO 60 J = J1, J2, KD1
CALL SROT( 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 SLARGV( 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
*                    SLARTV or SROT is used
*
IF( NR.GT.2*KD-1 ) THEN
DO 130 L = 1, KD - 1
CALL SLARTV( 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 SROT( 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 SLARTG( 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 SROT( 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 SLAR2V( 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
*                    SLARTV or SROT 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 SLARTV( 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 SROT( 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 SROT( 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 SROT( NQ, Q( IQB, J-1 ), 1, Q( IQB, J ),
\$                                1, D( J ), WORK( J ) )
170                   CONTINUE
ELSE
*
DO 180 J = J1, J2, KD1
CALL SROT( 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 SSBTRD
*
END

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