```      SUBROUTINE DTBSV(UPLO,TRANS,DIAG,N,K,A,LDA,X,INCX)
*     .. Scalar Arguments ..
INTEGER INCX,K,LDA,N
CHARACTER DIAG,TRANS,UPLO
*     ..
*     .. Array Arguments ..
DOUBLE PRECISION A(LDA,*),X(*)
*     ..
*
*  Purpose
*  =======
*
*  DTBSV  solves one of the systems of equations
*
*     A*x = b,   or   A'*x = b,
*
*  where b and x are n element vectors and A is an n by n unit, or
*  non-unit, upper or lower triangular band matrix, with ( k + 1 )
*  diagonals.
*
*  No test for singularity or near-singularity is included in this
*  routine. Such tests must be performed before calling this routine.
*
*  Arguments
*  ==========
*
*  UPLO   - CHARACTER*1.
*           On entry, UPLO specifies whether the matrix is an upper or
*           lower triangular matrix as follows:
*
*              UPLO = 'U' or 'u'   A is an upper triangular matrix.
*
*              UPLO = 'L' or 'l'   A is a lower triangular matrix.
*
*           Unchanged on exit.
*
*  TRANS  - CHARACTER*1.
*           On entry, TRANS specifies the equations to be solved as
*           follows:
*
*              TRANS = 'N' or 'n'   A*x = b.
*
*              TRANS = 'T' or 't'   A'*x = b.
*
*              TRANS = 'C' or 'c'   A'*x = b.
*
*           Unchanged on exit.
*
*  DIAG   - CHARACTER*1.
*           On entry, DIAG specifies whether or not A is unit
*           triangular as follows:
*
*              DIAG = 'U' or 'u'   A is assumed to be unit triangular.
*
*              DIAG = 'N' or 'n'   A is not assumed to be unit
*                                  triangular.
*
*           Unchanged on exit.
*
*  N      - INTEGER.
*           On entry, N specifies the order of the matrix A.
*           N must be at least zero.
*           Unchanged on exit.
*
*  K      - INTEGER.
*           On entry with UPLO = 'U' or 'u', K specifies the number of
*           super-diagonals of the matrix A.
*           On entry with UPLO = 'L' or 'l', K specifies the number of
*           sub-diagonals of the matrix A.
*           K must satisfy  0 .le. K.
*           Unchanged on exit.
*
*  A      - DOUBLE PRECISION array of DIMENSION ( LDA, n ).
*           Before entry with UPLO = 'U' or 'u', the leading ( k + 1 )
*           by n part of the array A must contain the upper triangular
*           band part of the matrix of coefficients, supplied column by
*           column, with the leading diagonal of the matrix in row
*           ( k + 1 ) of the array, the first super-diagonal starting at
*           position 2 in row k, and so on. The top left k by k triangle
*           of the array A is not referenced.
*           The following program segment will transfer an upper
*           triangular band matrix from conventional full matrix storage
*           to band storage:
*
*                 DO 20, J = 1, N
*                    M = K + 1 - J
*                    DO 10, I = MAX( 1, J - K ), J
*                       A( M + I, J ) = matrix( I, J )
*              10    CONTINUE
*              20 CONTINUE
*
*           Before entry with UPLO = 'L' or 'l', the leading ( k + 1 )
*           by n part of the array A must contain the lower triangular
*           band part of the matrix of coefficients, supplied column by
*           column, with the leading diagonal of the matrix in row 1 of
*           the array, the first sub-diagonal starting at position 1 in
*           row 2, and so on. The bottom right k by k triangle of the
*           array A is not referenced.
*           The following program segment will transfer a lower
*           triangular band matrix from conventional full matrix storage
*           to band storage:
*
*                 DO 20, J = 1, N
*                    M = 1 - J
*                    DO 10, I = J, MIN( N, J + K )
*                       A( M + I, J ) = matrix( I, J )
*              10    CONTINUE
*              20 CONTINUE
*
*           Note that when DIAG = 'U' or 'u' the elements of the array A
*           corresponding to the diagonal elements of the matrix are not
*           referenced, but are assumed to be unity.
*           Unchanged on exit.
*
*  LDA    - INTEGER.
*           On entry, LDA specifies the first dimension of A as declared
*           in the calling (sub) program. LDA must be at least
*           ( k + 1 ).
*           Unchanged on exit.
*
*  X      - DOUBLE PRECISION array of dimension at least
*           ( 1 + ( n - 1 )*abs( INCX ) ).
*           Before entry, the incremented array X must contain the n
*           element right-hand side vector b. On exit, X is overwritten
*           with the solution vector x.
*
*  INCX   - INTEGER.
*           On entry, INCX specifies the increment for the elements of
*           X. INCX must not be zero.
*           Unchanged on exit.
*
*
*  Level 2 Blas routine.
*
*  -- Written on 22-October-1986.
*     Jack Dongarra, Argonne National Lab.
*     Jeremy Du Croz, Nag Central Office.
*     Sven Hammarling, Nag Central Office.
*     Richard Hanson, Sandia National Labs.
*
*
*     .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER (ZERO=0.0D+0)
*     ..
*     .. Local Scalars ..
DOUBLE PRECISION TEMP
INTEGER I,INFO,IX,J,JX,KPLUS1,KX,L
LOGICAL NOUNIT
*     ..
*     .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
*     ..
*     .. External Subroutines ..
EXTERNAL XERBLA
*     ..
*     .. Intrinsic Functions ..
INTRINSIC MAX,MIN
*     ..
*
*     Test the input parameters.
*
INFO = 0
IF (.NOT.LSAME(UPLO,'U') .AND. .NOT.LSAME(UPLO,'L')) THEN
INFO = 1
ELSE IF (.NOT.LSAME(TRANS,'N') .AND. .NOT.LSAME(TRANS,'T') .AND.
+         .NOT.LSAME(TRANS,'C')) THEN
INFO = 2
ELSE IF (.NOT.LSAME(DIAG,'U') .AND. .NOT.LSAME(DIAG,'N')) THEN
INFO = 3
ELSE IF (N.LT.0) THEN
INFO = 4
ELSE IF (K.LT.0) THEN
INFO = 5
ELSE IF (LDA.LT. (K+1)) THEN
INFO = 7
ELSE IF (INCX.EQ.0) THEN
INFO = 9
END IF
IF (INFO.NE.0) THEN
CALL XERBLA('DTBSV ',INFO)
RETURN
END IF
*
*     Quick return if possible.
*
IF (N.EQ.0) RETURN
*
NOUNIT = LSAME(DIAG,'N')
*
*     Set up the start point in X if the increment is not unity. This
*     will be  ( N - 1 )*INCX  too small for descending loops.
*
IF (INCX.LE.0) THEN
KX = 1 - (N-1)*INCX
ELSE IF (INCX.NE.1) THEN
KX = 1
END IF
*
*     Start the operations. In this version the elements of A are
*     accessed by sequentially with one pass through A.
*
IF (LSAME(TRANS,'N')) THEN
*
*        Form  x := inv( A )*x.
*
IF (LSAME(UPLO,'U')) THEN
KPLUS1 = K + 1
IF (INCX.EQ.1) THEN
DO 20 J = N,1,-1
IF (X(J).NE.ZERO) THEN
L = KPLUS1 - J
IF (NOUNIT) X(J) = X(J)/A(KPLUS1,J)
TEMP = X(J)
DO 10 I = J - 1,MAX(1,J-K),-1
X(I) = X(I) - TEMP*A(L+I,J)
10                     CONTINUE
END IF
20             CONTINUE
ELSE
KX = KX + (N-1)*INCX
JX = KX
DO 40 J = N,1,-1
KX = KX - INCX
IF (X(JX).NE.ZERO) THEN
IX = KX
L = KPLUS1 - J
IF (NOUNIT) X(JX) = X(JX)/A(KPLUS1,J)
TEMP = X(JX)
DO 30 I = J - 1,MAX(1,J-K),-1
X(IX) = X(IX) - TEMP*A(L+I,J)
IX = IX - INCX
30                     CONTINUE
END IF
JX = JX - INCX
40             CONTINUE
END IF
ELSE
IF (INCX.EQ.1) THEN
DO 60 J = 1,N
IF (X(J).NE.ZERO) THEN
L = 1 - J
IF (NOUNIT) X(J) = X(J)/A(1,J)
TEMP = X(J)
DO 50 I = J + 1,MIN(N,J+K)
X(I) = X(I) - TEMP*A(L+I,J)
50                     CONTINUE
END IF
60             CONTINUE
ELSE
JX = KX
DO 80 J = 1,N
KX = KX + INCX
IF (X(JX).NE.ZERO) THEN
IX = KX
L = 1 - J
IF (NOUNIT) X(JX) = X(JX)/A(1,J)
TEMP = X(JX)
DO 70 I = J + 1,MIN(N,J+K)
X(IX) = X(IX) - TEMP*A(L+I,J)
IX = IX + INCX
70                     CONTINUE
END IF
JX = JX + INCX
80             CONTINUE
END IF
END IF
ELSE
*
*        Form  x := inv( A')*x.
*
IF (LSAME(UPLO,'U')) THEN
KPLUS1 = K + 1
IF (INCX.EQ.1) THEN
DO 100 J = 1,N
TEMP = X(J)
L = KPLUS1 - J
DO 90 I = MAX(1,J-K),J - 1
TEMP = TEMP - A(L+I,J)*X(I)
90                 CONTINUE
IF (NOUNIT) TEMP = TEMP/A(KPLUS1,J)
X(J) = TEMP
100             CONTINUE
ELSE
JX = KX
DO 120 J = 1,N
TEMP = X(JX)
IX = KX
L = KPLUS1 - J
DO 110 I = MAX(1,J-K),J - 1
TEMP = TEMP - A(L+I,J)*X(IX)
IX = IX + INCX
110                 CONTINUE
IF (NOUNIT) TEMP = TEMP/A(KPLUS1,J)
X(JX) = TEMP
JX = JX + INCX
IF (J.GT.K) KX = KX + INCX
120             CONTINUE
END IF
ELSE
IF (INCX.EQ.1) THEN
DO 140 J = N,1,-1
TEMP = X(J)
L = 1 - J
DO 130 I = MIN(N,J+K),J + 1,-1
TEMP = TEMP - A(L+I,J)*X(I)
130                 CONTINUE
IF (NOUNIT) TEMP = TEMP/A(1,J)
X(J) = TEMP
140             CONTINUE
ELSE
KX = KX + (N-1)*INCX
JX = KX
DO 160 J = N,1,-1
TEMP = X(JX)
IX = KX
L = 1 - J
DO 150 I = MIN(N,J+K),J + 1,-1
TEMP = TEMP - A(L+I,J)*X(IX)
IX = IX - INCX
150                 CONTINUE
IF (NOUNIT) TEMP = TEMP/A(1,J)
X(JX) = TEMP
JX = JX - INCX
IF ((N-J).GE.K) KX = KX - INCX
160             CONTINUE
END IF
END IF
END IF
*
RETURN
*
*     End of DTBSV .
*
END

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